a model predictive control approach to aircraft … · variables, a model predictive control (mpc)...

POLITECNICO DI MILANODIPARTIMENTO DI ELETTRONICA INFORMAZIONE E BIOINGEGNERIA

DOCTORAL PROGRAMME IN INFORMATION TECHNOLOGYSYSTEMS AND CONTROL AREA

A MODEL PREDICTIVE CONTROL APPROACH TO

AIRCRAFT MOTION CONTROL

Doctoral Dissertation of:Luca Deori

Supervisor:Prof. Simone GarattiCo-Supervisor:Prof.ssa Maria PrandiniTutor:Prof. Marco LoveraThe Chair of the Doctoral Program:Prof. Carlo Fiorini

2015 – XXVIII November

Abstract

Air traffic is expected to increase over the next few decades, and this growthis likely to lead to route congestions and delays. For this reason new effi-cient methods to manage air space has been developed: they rely on 4-Dtrajectories and Target Windows (TWs). TWs impose time-space specifica-tions on the aircraft trajectory so as to enhance airspace capacity and safety.In this work we aim to develop a control strategy for aircraft motion able toaddress time-space specifications on the aircraft trajectory as those imposedby 4-D trajectories and TWs. Furthermore, the controller has to account forthe aircraft dynamics and physical limitations, and it has to be robust withrespect to disturbances, in particular, with respect to wind. A Model Pre-dictive Control (MPC) approach is adopted given its capability to handleconstraints. MPC involves the formulation of a finite horizon optimizationproblem to be solved on-line, whose solution is applied according to a re-ceding horizon strategy. Application of MPC to aircraft motion control isnot straightforward. Indeed aircraft dynamics is non linear and constraintshave to be accounted for to model its physical limitations. In this work,feedback linearization is exploited to a obtain linear model with respect toa new set of state and input variables which exactly matches the originalaircraft dynamics. However, aircraft motion constraints expressed in thenew variables result non convex. An effort is then put in reformulatingthem by means of convex approximations that allow to recover computa-tional tractability. As the TWs specifications involve large time and spacescales, they require to be addressed focusing on very large time horizons, arequirement that cannot be fulfilled in the finite horizon problem underly-ing the construction of the MPC control law. For this reason, we developa method to design trajectories that comply with TWs specifications andaccount for aircraft motion capability. The designed trajectories are thenused as reference for the MPC controller that is required to keep the air-craft as close as possible to it. The controller should track the referencerobustly with respect to the wind, which is modelled as an unbounded sup-port stochastic disturbance. To address this issue, chance-constraints areenforced on the aircraft position, but due to the complexity of the windmodel, the chance-constrained problem result computationally unafford-able. Therefore, we resort to randomized solution algorithms, specificallythe scenario approach, to find approximate but guaranteed solutions to thechance-constrained finite horizon optimization problems at relatively lowcomputational effort. The effectiveness of the proposed approach is shownby means of simulations.

I

Contents

1 Introduction 11.1 Problem description and sketch of the proposed solution . . 3

1.1.1 Structure of the thesis and contributions . . . . . . . 5

2 Aircraft model 72.1 Aircraft model (simple) . . . . . . . . . . . . . . . . . . . . 82.2 Aircraft model (complex) . . . . . . . . . . . . . . . . . . . 92.3 Aircraft constraints . . . . . . . . . . . . . . . . . . . . . . 112.4 Wind model . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Feedback linearization & Constraint convexification 193.1 Feedback linearization (simple model) . . . . . . . . . . . . 203.2 Feedback linearization (complex model) . . . . . . . . . . . 22

3.2.1 Time discretization . . . . . . . . . . . . . . . . . . 293.3 Reformulation of the constraints in the new variables . . . . 293.4 Feasibility of the constraints . . . . . . . . . . . . . . . . . 38

4 Trajectory generation 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 Sketch of the proposed design approach . . . . . . . 404.2 Path design . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Motion primitives . . . . . . . . . . . . . . . . . . . 424.2.2 Path composition . . . . . . . . . . . . . . . . . . . 484.2.3 Choice of lr1, lu1 and gridding of lr7 and sV II . . . . 554.2.4 System resolution . . . . . . . . . . . . . . . . . . . 57

III

Contents

4.2.5 Bounds for s1 . . . . . . . . . . . . . . . . . . . . . 604.3 Choice of initial and final conditions . . . . . . . . . . . . . 644.4 Time law design . . . . . . . . . . . . . . . . . . . . . . . 67

4.4.1 PWA speed profile . . . . . . . . . . . . . . . . . . . 674.4.2 Choice of initial and final conditions for the speed

profile . . . . . . . . . . . . . . . . . . . . . . . . . 694.4.3 Conversion from s to t . . . . . . . . . . . . . . . . . 70

4.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 71

5 Tracking problem formulation and solution 815.1 Finite horizon optimization problem . . . . . . . . . . . . . 815.2 Modelling wind in the optimization problem . . . . . . . . . 84

5.2.1 Deterministic wind component . . . . . . . . . . . . 855.2.2 Stochastic wind component . . . . . . . . . . . . . . 86

5.3 Solution of the optimization problem via the scenario approach 945.4 Alternative formulations . . . . . . . . . . . . . . . . . . . 99

5.4.1 Disturbance feedback . . . . . . . . . . . . . . . . . 1035.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 105

5.5.1 Simulation 1 . . . . . . . . . . . . . . . . . . . . . . 1065.5.2 Simulation 2 . . . . . . . . . . . . . . . . . . . . . . 1145.5.3 Simulation 3 . . . . . . . . . . . . . . . . . . . . . . 1215.5.4 Validation . . . . . . . . . . . . . . . . . . . . . . . 125

6 Uncertainty on aircraft mass 1296.1 Feedback linearization with mass uncertainty . . . . . . . . 1306.2 Finite horizon optimization problem with mass uncertainty . 132

6.2.1 Multiplicative disturbance . . . . . . . . . . . . . . . 1336.2.2 Additive disturbance . . . . . . . . . . . . . . . . . . 134

6.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 136

7 Conclusions 143

A Randomized methods for stochastic constrained control 147A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 147

A.1.1 Bibliographical remarks . . . . . . . . . . . . . . . . 150A.1.2 Possible extensions . . . . . . . . . . . . . . . . . . 151

A.2 Notational issues . . . . . . . . . . . . . . . . . . . . . . . 151A.3 Trading performance for state constraint feasibility . . . . . 153

A.3.1 Additional penalization term in the control cost . . . 154A.3.2 Two-step approach based on a pre-defined admissible

deterioration of the control cost . . . . . . . . . . . . 154

IV

Contents

A.4 Approximate solution to the optimization problems . . . . . 155A.4.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . 156A.4.2 Chance-constrained feasibility of the obtained approx-

imate solutions . . . . . . . . . . . . . . . . . . . . . 159A.5 Choice of µ and α and trade-off between J and h . . . . . . 161A.6 Numerical example . . . . . . . . . . . . . . . . . . . . . . 164A.7 Relaxation approach focused on constraints . . . . . . . . . 169A.8 Scenario-based resolution scheme . . . . . . . . . . . . . . 171A.9 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . 174A.10Numerical example . . . . . . . . . . . . . . . . . . . . . . 179

B Constraint reformulation with LMI 185

C Alternative choices of the desired path length 191

Bibliography 199

V

CHAPTER1Introduction

Air transportation is nowadays a widespread service that plays an impor-tant role in modern world. It allows people and goods to reach far awaylocations across the whole planet with times and costs were not achievablebefore. Air transportation has, hence, become an essential means to boosteconomic and cultural exchanges among people and societies, [1].In order to ensure the safety and efficiency of aircraft operations Air Traf-fic Management (ATM) systems have been developed, [13, 14, 30, 38, 43,49, 58]. They aim to provide cost effective solutions to manage and in-crease airspace capacity, and to guarantee safety requirements predictingand avoiding possible conflicts between aircraft flying too close each other.Traditional ATM systems are rigidly structured: they are mainly based onpredefined routes that the aircraft are required to follow; these routes areconnected by way-points that also act as crossroads.Air traffic is expected to increase rapidly over the next decades, [65], andthis growth is likely to be unsustainable for rigidly structured ATM systemsleading to the saturation of the routes, congestions, and delays. In orderto face the increasing air traffic, new concepts are needed to exploit moreeffectively the airspace, ensuring more efficiency and safety.A possible solution that has been proposed in the literature rests on the con-

1

Chapter 1. Introduction

cept of 4-D trajectories and Target Windows (TWs), [2–4]. TWs representconstraints on the 4-D trajectories requiring that the aircraft passes througha 2-D region (typically a rectangle) in the 3-D space within a given time in-terval, see Figure 1.1. In the SESAR [2] and in the CATS projects [3], TWsare viewed as a key enabler of new ATM systems involving all different ac-tors (airlines, airports, air navigation service providers) in the managementprocess. TWs should allow for a more efficient use of the airspace, enhanc-ing predictability of the aircraft trajectories, improving safety and airspacecapacity.In perspective, each aircraft will be assigned a sequence of TWs to meet,where the TWs altogether are designed with the twofold objective of bet-ter exploiting the airspace capacity and of avoiding conflicts. Coping withTWs is not straightforward though because TWs impose non trivial con-straints on the aircraft motion in the time-space domain, which have tobe satisfied while explicitly accounting for physical limitations on aircraftspeed and accelerations and for other constraints related to passengers com-fort. Furthermore, the aircraft motion is affected by uncertainty, in partic-ular, the presence of wind, which leads to unpredictability of the aircrafttrajectory, [10, 11, 25, 37, 53, 63].

TW

Figure 1.1: An example of Target Window.

2

1.1. Problem description and sketch of the proposed solution

1.1 Problem description and sketch of the proposed solution

The main goal of this work is to develop a motion control able to steerthe aircraft along a reference trajectory, which is designed by means ofan original approach so as to be compatible with both the aircraft motioncapabilities and the TWs time space requirements. Tracking is performedrobustly with respect to the wind disturbance and accounting for the aircraftphysical limitations. Hence, by keeping the aircraft as close as possible tothe reference trajectory, compatibly with the effect of the wind disturbance,TW specifications are satisfied also by the actual aircraft trajectory. Theintermediate steps to achieve these goals are now discussed in details.Given its capability of handling constraints on the input and on the statevariables, a Model Predictive Control (MPC) approach to control design[21, 44, 46] is adopted. A discrete time model of the aircraft is used. MPCthen involves minimizing at each sampled time instant a suitable finite hori-zon cost subject to suitable constraints so as to enforce trajectory trackingand the satisfaction of the aircraft physical limitations and passenger com-fort requirements. The so obtained control action is applied at the currenttime instant only, and the process is repeated at every sampled time (reced-ing horizon).In formulating the finite horizon constrained optimization problem, a keyissue for computational and solvability reasons is that of achieving con-vexity. Given that the model of the aircraft dynamics is non-linear, weapply feedback linearization so as to find a linear model with new inputand state variables, whose dynamics exactly matches the original one. Theso obtained linear model is then time discretized so as to be embedded inthe MPC controller. The constraints on the aircraft physical limitationsand passenger comfort rewritten with respect to the new state and inputvariables of the feedback-linearized model are non-convex. Hence, a ma-jor effort is put in reformulating these constraints as convex constraints,introducing some relaxation when needed. Such relaxations are appropri-ately designed so as to guarantee that the original constraints are satisfiedat least for the first time instant of the current finite horizon. This way, theactually implemented MPC controller is guaranteed to satisfy the originalconstraints at all time steps, because of the receding horizon.The idea of using feedback linearization followed by a convexification ofthe constraints is not new as it has been adopted in [39, 40, 68], while [64]discusses the problem from a general perspective. The contribution is toapply this methodology to address a control problem that was never tackledin this way before to the best of our knowledge. In this respect, the main

3


results which will be obtained are the solution of a non-trivial global feed-back linearization problem and the development of an ad-hoc method forthe convexification of the constraints. This latter, in particular, is a prob-lem that still lacks a general solution and its resolution will require specificanalysis for each type of constraint. For a different non linear MPC basedapproach that does not resort to feedback linearization see [52].TWs specifications usually involve large time and space scales, and for thisreason they are hard to be handled in the finite horizon problem to be solvedat each time step in MPC. Therefore, in order to address TWs specificationswe resort to generate reference trajectories that comply with TWs require-ments accounting at the same time for aircraft motion capabilities. To thispurpose a method to suitably design such trajectories is developed. Thismethod patches together simple motion primitives so as to eventually ob-tain a smooth trajectory that can be easily followed by the aircraft. Patchingis obtained by solving a highly non linear system of equations by means ofprocedures specifically conceived to this purpose.The obtained trajectory is then used as reference for the MPC controller,which has the critical task to keep the aircraft as close as possible to it,compatibly with the aircraft motion constraints and the disturbances. Notethat the trajectory is composed by a path and a time law, and hence in orderto track it the motion controller is required to steer the aircraft so as to makeit reach the proper position at a proper timing.To pursue the objective of tracking the reference trajectory, constraints onthe aircraft position are included in the optimization problem. These con-straints depend on the wind disturbance which makes the problem challeng-ing because: (i) the wind disturbance depends non linearly on the aircraftposition, and (ii) the wind disturbance has unbounded support, a fact thathampers the feasibility of the problem.The former issue is addressed by replacing the original wind disturbancemodel with a local approximation, around the aircraft current position,which is accurate in the region of the airspace that the aircraft will be po-tentially flying into along the look-ahead time horizon of MPC. As for thelatter issue, probabilistic – instead of robust – constraints are consideredto avoid infeasibility, thus leading to a chance-constraint optimization pro-gram. More specifically, we require the controller to keep the aircraft po-sition error small for all wind realizations except for a set of prescribedprobability ε.To handle the probabilistic constraints, we resort eventually to the scenarioapproach, [17,18,22,24], which allows to find approximate but guaranteedsolutions to chance-constraint optimization problems at low computational

4

1.1. Problem description and sketch of the proposed solution

effort. In particular we rely on the approaches of [28, 29, 54, 61], wherethe scenario approach has been tailored to the MPC framework. This way,we eventually obtain an optimization problem that can be efficiently solvedat every time step and which returns in receding horizon a well perform-ing, robust with respect to wind, control action as revealed by numericalsimulations.

1.1.1 Structure of the thesis and contributions

• In Chapter 2 two aircraft models are presented: in the first model asimpler description of the aircraft dynamics is adopted, whereas in thesecond model additional dynamics and variables are included so asto achieve a more accurate representation of the aerodynamic forces.Constraints on the state and control variables of both models are in-troduced so as to account for the aircraft physical limitations and pas-sengers comfort. Moreover, a model for the wind disturbance thataccounts for its correlation in space and time is described.

• In Chapter 3 feedback linearization is applied to both the consideredaircraft models. Two different non linear feedback laws, one for eachmodel, are computed to obtain a linear model with respect to a new setof state and input variables that corresponds exactly to the original air-craft dynamics. Constraints on aircraft physical limitations expressedin the newly introduced variables are non convex: for each of thema specific convexification procedure is developed. Moreover some ofthe constraints take different expressions accordingly to the consid-ered aircraft model, and, consequently, different convex approxima-tions are needed to deal with them.

• In Chapter 4 a method to design reference trajectories complying withtime space specifications is developed. The method starts definingsimple motions primitives that are constructed so as to satisfy air-craft physical limitations constraints. These motion primitives arethen joined together to define a smooth path connecting several TargetWindows. Properly patching motion primitives involves the solutionof a non linear system of equations which is addressed by a specificprocedure that combines numerical and analytical methods with ad-hoc rules to determine all the path parameters. Eventually, a time lawis applied to the designed path so as to define a trajectory that meetstime and space specifications.

• In Chapter 5 the design of the aircraft control via the MPC approach

5


is addressed. The finite horizon optimization problem involved inMPC is formulated defining a suitable cost function and includingconstraints on the aircraft position in order to track the reference tra-jectory. The constraints developed in Chapter 3 accounting for air-craft physical limitations are included as well. As the constraints onaircraft position are affected by the wind uncertainty they are formu-lated as chance-constraints: they result computationally intractabledue the complex model of the wind disturbance. For this reason, totackle them, we resort to approximate local models of the wind distur-bance and to randomized methods. In particular the scenario approachis adopted to solve the chance-constrained program at low computa-tional effort providing guarantees on the robustness of the obtainedsolution. The effectiveness of the proposed controller is assessed bymeans of a number of numerical simulations.

• In Chapter 6 we focus on uncertainty on aircraft mass. Differentlyfrom the previous chapters where exact knowledge of the mass is as-sumed, here we consider that only an estimation of it is available. Thesteps developed in the previous chapters to design the MPC, in partic-ular the feedback linearization and the finite horizon control problemformulation, are revisited to assess the effects of mass uncertainty onthe achieved performances.

• In Chapter 7 some concluding remarks are drawn.

• In Appendix A the constrained control of stochastic systems by meansof randomized methods is discussed from a general perspective. Inparticular, different approaches to deal with chance-constraints on thestate variables in presence of additive stochastic disturbances with un-bounded support are developed. A tailored version of these methodsis adopted in the aircraft application.

• In Appendix B an alternative convexification procedure of the con-straints on aircraft physical limitations based on LMI is discussed. Asthe obtained constraint approximations turns out to be less effectivethan those in Chapter 3 they have not been adopted.

• In Appendix C a possible extension of the trajectory designed proce-dure of Chapter 4 is discussed.

6

CHAPTER2Aircraft model

In this chapter, models of the aircraft motion dynamics are described. Ad-mittedly, the models here considered are derived based on some simplify-ing assumptions that make them approximation of the actual dynamics ofan aircraft. Nevertheless, we decided to investigate them for the followingreasons:

1. it is customary to use simplified models when addressing ATM appli-cations, see e.g. [45] and [5].

2. more complex and accurate models can be derived as extensions ofthe ones here considered, so that the approaches developed in the fol-lowing retain validity for enhanced models as well.

The considered models are able to describe the aircraft dynamics from anintermediate perspective in which some relevant physical quantities relatedto aircraft motion are accounted for, but, on the other hand, some abstrac-tion is made so that the models result simple enough to be suitable to beused in the design of the controller.In the following Sections 2.1 and 2.2 two aircraft models are described.They share a similar structure, in the first one a simpler description of aero-

7

Chapter 2. Aircraft model

dynamic forces is considered, the second model is more detailed and in-cludes additional variables and dynamics. Note that in order to properlymodel the aircraft physical limitations and to ensure safety of operationsand passenger comfort, constraints on aircraft velocities, accelerations andrelevant physical angles have to be considered, as described in Section 2.3.Moreover, since wind significantly affects the aircraft motion, in Section2.4 a model of the wind is described.

2.1 Aircraft model (simple)

Following [31, 45], we consider a six-state, flat earth, trimmed, point massmodel for the aircraft dynamics. The state variables are given by the posi-tion of the aircraft expressed in Cartesian coordinates x, y, z with respect toa fixed frame, the True Air Speed (TAS) of the aircraft V (which is the rel-ative aircraft speed with respect to the surrounding air), the heading angleψ (which is the angle between the projection of the aircraft velocity on thex-y plane and the x-axis), and the mass m of the aircraft. The inputs of thesystem are the path angle γ (which is the angle between the velocity and thex-y plane), the bank angle φ (which is the angle between the lift force andthe plane containing the aircraft velocity and the z-axis), and the enginethrust T . The presence of the wind is modelled by adding a disturbance(wx, wy, wz) to the aircraft velocity along the x, y, z axes respectively. Thesystem evolves according to the following equations

x

y

z

V

ψ

m

=

V cosψ cos γ + wx

V sinψ cos γ + wy

V sin γ + wz

−CDρSV2

2m− g sin γ + T

mCLρSV

2msinφ

−ηT

, (2.1)

whereCD andCL are the drag and lift coefficients, ρ is the air density whichis a function of the altitude z, S is the surface of the wings, g is gravitationalacceleration, η is a fuel consumption coefficient. According to [5], the liftcoefficient is set equal to

CL =2mg

ρSV 2 cosφ,

so that the weight force is always compensated by the lift force (trimmedcondition), while the drag coefficient is given by CD = CD0 + CD2C

2L,

8

2.2. Aircraft model (complex)

where CD0 and CD2 are suitable coefficients. As a result, the dynamics forthe heading angle becomes

ψ =g

Vtanφ.

Thus, the final model of the aircraft becomes:

x

y

z

V

ψ

m

=

V cosψ cos γ + wx

V sinψ cos γ + wy

V sin γ + wz

−KDV 2

m− g sin γ + T

mgV

tanφ

−ηT

, (2.2)

where we let KD = CDρS2

.

2.2 Aircraft model (complex)

In this section, following [16, 31, 59], a model in which the trimmed con-dition assumption is removed is considered, so as to get a more accuratedescription of the aircraft dynamics. The new model shows a similar struc-ture of the model in (2.1). A new input variable, the Angle of Attack (AoA)α (which is the difference between the pitch and the path angles) is intro-duced, while the path angle γ is regarded as a state variable and no more asan input. The newly introduced input α allows to consider a more detaileddescription of the aerodynamic forces, and of the aircraft dynamics. In par-ticular, the aerodynamic coefficients that determine lift and drag forces areno more fixed but they show a dependence on the angle of attack. Moreoverα slightly influences how the engine thrust affects TAS, heading angle andpath angles dynamics.In this flat earth, point mass model for the aircraft dynamics the state vari-ables are given by the position of the aircraft expressed in Cartesian coor-dinates x, y, z with respect to a fixed frame, the True Air Speed (TAS) ofthe aircraft V , the heading angle ψ, the path angle γ and the mass m of theaircraft. The inputs are the Angle of Attack (AoA) α, the bank angle φ,and the engine thrust T . The presence of the wind is modelled by addinga disturbance (wx, wy, wz) to the aircraft velocity along the x, y, z axes

9


respectively. The system evolves according to the following equations

x

y

z

V

ψ

γ

m

=

V cosψ cos γ + wx

V sinψ cos γ + wy

V sin γ + wz

−Dm− g sin γ + T cosα

mL+T sinαmV cos γ

sinφL+T sinα

mVcosφ− g

Vcos γ

−ηT

, (2.3)

where D(z, V, α) = 12ρ(z)SV 2Cd(1 + b1α + b2α

2) is the drag force thatopposes the aircraft motion in the direction of the TAS and L(z, V, α) =12ρ(z)SV 2Cl(1 + aα) is the lift that provides the force to oppose gravity,Cd, Cl, b1, b2, a are suitable positive aerodynamic coefficients, see [5].Note that the model in (2.1) can be obtained from the one in (2.3) underthe simplifying assumption that the angle of attack is always null, properlysetting the lift force so as to compensate weight force, and consideringthe path angle as an input. Indeed, in usual aircraft operations the angleof attack takes always small values, and it has to be set so that the liftforce almost compensates the weight force in order to keep the verticalacceleration small or to maintain altitude.It is worth noticing that the models (2.1) and (2.3) are well defined andrepresentative of the aircraft dynamics on a specific domain only. Moreprecisely, the states x, y, z can take value in R3, instead the TAS has tobe positive V ∈ (0,+∞), the heading angle can take any value in R, but,since it is an angle, we can limit its domain to ψ ∈ [0, 2π), the path anglecan take values γ ∈ (−π/2, π/2), and m ∈ (0,+∞). Indeed, being theTAS a speed, it cannot be negative, and if it is null the heading and the pathangle become not well defined. Moreover, also when the aircraft velocity isaligned with the z-axis ( γ = ±π/2 ) the heading angle is not well defined.As for the inputs we consider that T must be non negative T ∈ [0,+∞),the bank angle and the angle of attack are limited to φ ∈ (−π/2, π/2)and α ∈ (−π/2, π/2). Besides these limitations that define the domainin which the considered aircraft models are well defined, other constraintshave to be enforced on the aircraft states and inputs as described in thefollowing section.

10

2.3. Aircraft constraints

2.3 Aircraft constraints

In order to account for physical limitations of the aircraft, passenger com-fort and safety requirements, several constraints both on the state variablesand on the input variables have to be considered as follows.

• True Air Speed VVmin ≤ V ≤ Vmax, (2.4)

where Vmin and Vmax depend on the aircraft type, Vmin is also relatedto the stall velocity of the aircraft, [5].

• Longitudinal Acceleration

− aL ≤ V ≤ aL, (2.5)

where aL = 2 ft/s2 = 2 · 0.3048 m/s2.

• Engine Thrust TTmin ≤ T ≤ Tmax, (2.6)

where Tmin and Tmax can be computed according to [5] and dependon atmospheric conditions and aircraft type.

• Bank Angle φ− φ ≤ φ ≤ φ. (2.7)

According to [5] φmax can vary from 25 to 45 depending on the typeof aircraft.

• Path Angle γ

γmin ≤ γ ≤ γmax. (2.8)

• Vertical Acceleration zIn [5] the following constraint is considered:

− aN ≤∆γV

∆t≤ aN , (2.9)

where aN = 5 ft/s2 = 5 · 0.3048 m/s2, and the term ∆γV∆t

seems toapproximate the derivative of the normal velocity V cos γγ. We preferto translate it in a constraint on the vertical acceleration z:

− aZ ≤ z ≤ aZ . (2.10)

11


Indeed, since the path angle is always small, the vertical accelera-tion is close to the derivative of the normal velocity. More precisely,from (2.1) results z = V cos γγ + V sin γ, and, recalling constraint(2.5) on V and constraint (2.8) on γ, the absolute of V sin γ can-not be greater than aL sin γmax aN (e.g. for γmax = 5 resultsaL sin γmax = 0.053 m/s2), so that (2.10) is a very good approximationof (2.9). Moreover, whether we set aZ = aN − aL sin γmax, enforc-ing (2.10) guarantees that the bounds on the derivative of the normalvelocity are met.

The values of γmin and γmax should be chosen so that if the aircraft is ina feasible initial condition, then there exists a feasible choice of constantinputs such that the aircraft keeps fling at constant TAS and constant pathand heading angles, and, hence, it continues to satisfy the constraints duringits motion evolution. To be more precise if the path angle is set to γmin orγmax, the engine thrust should be able to maintain the TAS V within itsadmissible range [Vmin, Vmax].Some of the aircraft and constraint parameters show a dependence on atmo-spheric conditions (pressure, altitude, temperature, geopotential pressure)and on the phase of the flight (climbing, cruise, landing). However in or-der to simplify the model we fix their values indeed, to our purpose, theyshould be only accurate enough to represent usual flight conditions. More-over, they can be considered constant in the single finite horizon controlproblem, and they can be adjusted according to the current aircraft condi-tion, at each time step according to the receding horizon strategy.

2.4 Wind model

The wind velocities wx, wy, wz are modelled as a time varying vector field(wx = wx(x, y, z, t), wy = wy(x, y, z, t), wz = wz(x, y, z, t)) obtainedas the sum of two contributions: a deterministic term that represent theforecast of the wind, and a stochastic term accounting for the uncertaintyaffecting the actual wind encountered by the aircraft, that is:

wx = wxf + wxs

wy = wyf + wys

wz = wzf + wzs.

As for the forecast, we rely on the National Oceanic and Atmospheric Ad-ministration (NOAA) Rapid Refresh (RAP) model [48] which provides thewind velocities wxf , wyf , wzf in correspondence of a grid of points in the

12

2.4. Wind model

x-y-z space. As for the x, y coordinates, the grid has a fixed resolutionof 13 km, while as for the z coordinate it is divided in 37 levels each cor-responding to a decrease of pressure of 2500 Pa. The altitude z and thepressure p can be related by the following equation:

z =T0

L

(1−

(p

p0

)RLgM

); (2.11)

where

• p0 = 101325 Pa is the sea level standard pressure;

• L = 0.006 K/m is a thermal drift coefficient;

• T0 = 288.15 K is the sea level standard temperature;

• M = 0.0289644 kg/mol is the dry air molar mass;

• R = 8.31447 J/(mol K) is the perfect gas constant.

In order to retrieve the value of the wind velocities wxf = wxf (x, y, z, t),wyf = wyf (x, y, z, t), wzf = wzf (x, y, z, t) in the generic position x, y, zat the generic time t, we perform a linear interpolation of the forecast windvelocities in correspondence of the points of the grid at the vertices of thecell containing x, y, z, and, then, we linear interpolate over time.The forecast wind velocities on the x-y plane wxf , wyf are directly pro-vided by the RAP, while the computation of wzf is slightly more involved.The RAP model provides the vertical velocity pressure Ω [Pa/s] that canbe converted into vertical wind velocities wzf by means of the followingequations:

wz = − Ω

ρg

ρ =p

Rdry_airT,

where ρ is the air density, Rdry_air = 287.05 J/(Kg K) is the specific gasconstant for the dry air, and T is the forecast temperature provided by theRAP. Figure 2.1 shows the forecast wind intensity and direction in the x-yplane for different altitude z at a given time instant.

13


z=4.5301

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150z=4.8653

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

z=5.2131

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150z=5.5746

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

z=5.951

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150z=6.3438

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

z=6.7547

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150z=7.1856

x

y

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

Figure 2.1: Forecast wind intensity and direction in the x-y plane for different altitude zat a given time instant.

According to [31, 42], the stochastic wind components wxs, wys, wzs aremodelled as a time varying random field where for every x, y, z, t, the windcomponentswxs(x, y, z, t),wys(x, y, z, t),wzs(x, y, z, t) are zero mean Gaus-

14

2.4. Wind model

sian random variables with the following correlation structure:

E[wxs(x, y, z, t)wxs(x′, y′, z′, t′)] = ρw(x, x′, y, y′, z, z′, t, t′) (2.12)

E[wys(x, y, z, t)wys(x′, y′, z′, t′)] = ρw(x, x′, y, y′, z, z′, t, t′)

E[wzs(x, y, z, t)wzs(x′, y′, z′, t′)] = ρwz(x, x

′, y, y′, z, z′, t, t′)

ρw(x, x′, y, y′, z, z′, t, t′) =

= k(z)k(z′)e−σ1|t−t′|e−σ2‖[x−x′ y−y′]‖e−σ3|z−z′|

ρwz(x, x′, y, y′, z, z′, t, t′) =

= kz(z)kz(z′)e−σ1z |t−t′|e−σ2z‖[x−x′ y−y′]‖e−σ3z |z−z′|

E[wxs(x, y, z, t)wys(x′, y′, z′, t′)] = 0

E[wxs(x, y, z, t)wzs(x′, y′, z′, t′)] = 0

E[wys(x, y, z, t)wzs(x′, y′, z′, t′)] = 0,

where k(z) and kz(z) represent the variance of the wind velocities at a givenaltitude z and the coefficients σ1, σ2, σ3 and σ1z, σ2z, σ3z regulate the expo-nential decrease of the correlation between wind velocities at different posi-tions and time instants as their corresponding spatial and temporal distanceincreases. According to the correlation structure in (2.12), wind velocitiesalong different axes are independent and, moreover, wind is isotropic i.e.the correlation structure is invariant under rotations of the x-y plane. Notethat according to this model the wind disturbance has unbounded support.A possible approach to generate a random field that satisfy the correlationstructure in (2.12) is described in [31, 42]. It requires to consider a grid ofsayNg points in the x-y-z space, e.g. the same grid used in the RAP model,and to gather the values of the stochastic wind velocities corresponding toeach point in the grid at each time instant in the vectors:

Wxs(t) =

wxs(x1, y1, z1, t)

wxs(x2, y2, z2, t)...

wxs(xNg , yNg , zNg , t)

Wys(t) =

wys(x1, y1, z1, t)

wys(x2, y2, z2, t)...

wys(xNg , yNg , zNg , t)

Wzs(t) =

wzs(x1, y1, z1, t)

wzs(x2, y2, z2, t)...

wzs(xNg , yNg , zNg , t)

,

15


where [xi, yi, zi] represent a point in the grid. Then, we compute:

R0 = E[Wxs(t)Wxs(t)T ] = E[Wys(t)Wys(t)

T ] = Q0QT0

R0z = E[Wzs(t)Wzs(t)T ] = Q0zQ

T0z,

where the expected value is computed according to (2.12) considering thespace distance between the points in the grid and setting the time differenceto 0. The matrix Q0 and Q0z can be compute as the Cholesky decomposi-tion ofR0 andR0z respectively. At the initial time instant t = 0 we computea realization of the random field in correspondence of the points in the gridas:

Wxs(0) = Q0ex(0) ex ∼ WGN(0, I)

Wys(0) = Q0ey(0) ey ∼ WGN(0, I)

Wzs(0) = Q0zez(0) ez ∼ WGN(0, I).

As for the following time steps a realization of the random field is computedby means of the multidimensional AR(1) processes:

Wxs(t+ Ts) = aWxs(t) +Qex(t+ Ts)

Wys(t+ Ts) = aWys(t) +Qey(t+ Ts)

Wzs(t+ Ts) = azWzs(t) +Qzez(t+ Ts),

where

a = e−αTs az = e−αzTs (2.13)

QQT = (1− a2)R0 QzQTz = (1− a2

z)R0z.

Setting the parameters of the AR(1) processes as in (2.13) ensures thatthe random field realizations generated according to the described method,match the correlation structure in (2.12), see [31, 42]. The values of thestochastic wind velocities wxs, wys, wzs in space positions other than thepoints considered in the grid can be computed by means of linear interpo-lation, introducing some approximation in the correlation structure. Figure2.2 shows the stochastic wind intensity and direction on the x-y plane forsome time steps at an altitude of 6 Km.

16

2.4. Wind model

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=0

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=5

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=10

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=15

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=20

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=25

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=30

−150 −100 −50 0 50 100 150

−150

−100

−50

0

50

100

150

x

y

t=35

Figure 2.2: Stochastic wind intensity and direction on the x-y plane for subsequent timesteps of length 5s at altitude of 6 Km.

17

CHAPTER3Feedback linearization & Constraint

convexification

In this chapter feedback linearization is applied to the aircraft models in(2.2) and (2.3) to obtain a linear model in a new set of input and statevariables that exactly matches the original aircraft dynamics. The linearmodels achieved by feedback linearizing (2.2) or (2.3) present the sameinput and state variables and the same dynamics. On the contrary, the non-linear relationship between the new input and state of the linear model andthose of the original models are different and specific to each of the aircraftmodels (2.2) and (2.3).After the change of variables, the constraints on aircraft physical limitationsand passenger comfort in Section 2.3 expressed as function of the new stateand input variables, turn out to be not convex. In order to recover efficientlytractable constraints, suitable convex approximations are introduced.Note that, some of the constraints take the same expression both for model(2.2) and (2.3) and, hence, they can be dealt with by means of the sameapproach. Instead, other constraints are different depending on to the con-sidered aircraft model and, hence, specific approximations are developed todeal with them.

19

Chapter 3. Feedback linearization & Constraint convexification

Remark. Performing feedback linearization seems to be a sensible choicefor this kind of application, [51]. An alternative approach could be to lin-earize the aircraft model dynamics around a working point or a workingtrajectory. However this latter approach shows some difficulties. Firstly, itis not clear around which operating point or trajectory the dynamics hasto be linearized: indeed, as the TAS cannot be null, the system is never insteady state, and linearizing around an operating point is not viable. There-fore the most natural choice is to linearize the system dynamics around anominal trajectory, such as the one designed in Chapter 4. However, notethat because of the action of the wind disturbance on the aircraft, the sys-tem may operate in a state quite different from the one corresponding tothe nominal trajectory. Consequently, as the dynamics are strongly non lin-ear, the linearized dynamics may be quite inaccurate. Indeed, for example,when the wind blows against or toward the nominal path direction, in or-der to properly track the trajectory counteracting the disturbance, the TAShas to be modified and, hence, it may significantly differ from the nominalTAS. A similar reasoning holds when a lateral, with respect to the nominalpath, position error has to be compensated and the heading angle has to beadjusted with respect to the nominal one.Furthermore, the linearized dynamics would be time varying, whereas thedynamics obtained by feedback linearization are time invariant. Note thatthe linearization around a given trajectory leads to approximations in theconstraints on aircraft physical limitations and on aircraft position (theselatter will be discussed in Chapter 5), indeed the approximate linearizeddynamics has to be embedded in their formulation.

3.1 Feedback linearization (simple model)

In this section, we perform the feedback linearization of system (2.2) so asto obtain a linear model for the aircraft with new input and state variables.The adopted procedure is inspired by [47, 66], where, however, a simplermodel of an aircraft in a 2-D airspace is considered.First, equations (2.2) are simplified by neglecting the mass dynamics whichis quite slow with respect to other dynamics, so that mass can be consideredconstant along the time horizon which will be considered while developingMPC. Moreover the regulation of m is not an objective of the controller tobe designed. Then, we set:

T = KDV2 +mg sin γ +mτ, (3.1)

ϕ = tanφ, (3.2)

20

3.1. Feedback linearization (simple model)

so obtaining V = τ and ψ = gVϕ, which are linear equations in the new

input variables τ and ϕ. Note that τ represents the part of the accelerationprovided by the engine thrust that is still available once the drag and possi-bly part of the weight force have been compensated.If we define new state variables

x1 = x (3.3)x2 = y

x3 = z

x4 = V cosψ cos γ

x5 = V sinψ cos γ

x6 = V sin γ

then, the equations governing the system evolution become:

x1 = x4 + wx

x2 = x5 + wy

x3 = x6 + wz

x4 = τ cosψ cos γ − V cos γ sinψg

Vϕ− V cosψ sin γγ

x5 = τ sinψ cos γ + V cos γ cosψg

Vϕ− V sinψ sin γγ

x6 = V cos γγ + τ sin γ.

Now, let

u1 = τ cosψ cos γ − V cos γ sinψg

Vϕ− V cosψ sin γγ (3.4a)

u2 = τ sinψ cos γ + V cos γ cosψg

Vϕ− V sinψ sin γγ (3.4b)

u3 = V cos γγ + τ sin γ. (3.4c)

Solving (3.4a) and (3.4b) for τ and ϕ gives

τ =1

cos γ(u1 cosψ + u2 sinψ + V sin γγ) (3.5)

ϕ =1

g cos γ(−u1 sinψ + u2 cosψ). (3.6)

By reconstructing γ from (3.4c), we can rewrite (3.5) as

τ = u1 cos γ cosψ + u2 cos γ sinψ + u3 sin γ, (3.7)

21


which, together with (3.6), recalling (3.1) and (3.2), gives:

T = KDV2 +mg sin γ +mτ (3.8)

φ = arctanϕ = arctan

(1

g cos γ(−u1 sinψ + u2 cosψ)

)(3.9)

γ = arcsin(x6

V

)= arcsin

(u3,0 +

∫ t0u3,σdσ

V

). (3.10)

Equations (3.8)-(3.10) define the nonlinear feedback making the dynamicsof x1, x2, x3, x4, x5, x6 linear with respect to the new inputs u1, u2, u3:

x1

x2

x3

x4

x5

x6

=

[03×3 I3

03×3 03×3

]

x1

x2

x3

x4

x5

x6

+

[03×3

I3

]u1

u2

u3

+

[I3

03×3

]wxwywz

. (3.11)

Note that the new state and input variables have a precise physical meaning:the state is composed by the position of the aircraft in Cartesian coordinates(x1, x2, x3) and by the velocity along the Cartesian axes (x4, x5, x6),while the inputs u1, u2, u3 are the accelerations along the x, y, z axesrespectively. Moreover the original state variables can be recovered fromthe new ones as follows

x = x1 (3.12)y = x2

z = x3

V =√x2

4 + x25 + x2

6

ψ = atan2(x5, x4),

where atan2(·, ·) denotes the four quadrant arctangent of its argument.

3.2 Feedback linearization (complex model)

In this section, feedback linearization is applied the model in (2.3) so asto make the model of the aircraft linear with respect to a new set of inputand state variables. The procedure is similar to the one adopted in previous

22

3.2. Feedback linearization (complex model)

section for the model in (2.2), but, in this case the non-linear feedback ismore complicated because of the presence of the path angle dynamics andof the angle of attack α.As before, we start by simplifying (2.3) by neglecting the mass dynamics.As a matter of fact, the considerations made in Section 3.1 still apply here.The engine thrust T is set to:

T = (D +mg sin γ + τm)1

cosα, (3.13)

where τ is a new auxiliary input replacing the thrust T . Substituting in(2.3), the equation (3.14) for the dynamics of V become:

V = τ. (3.14)

Note that, also in this model, τ represents the part of the longitudinal ac-celeration provided by the engine thrust that is still available once the dragforce and possibly part of the weight force have been compensated.

Similarly to (3.3), we define new state variables as follows:

x1 = x (3.15)x2 = y

x3 = z

x4 = V cosψ cos γ

x5 = V sinψ cos γ

x6 = V sin γ.

The new state variables have the same physical meaning of the correspond-ing variables defined in the previous section: x1, x2, x3 give the positionof the aircraft in Cartesian coordinates, while x4, x5, x6 represent the trueair speed Cartesian components. Note that (3.15) is clearly a bijection andthe original state variables can be recovered from the new ones according

23


to the following inverse relation:

x = x1 (3.16)y = x2

z = x3

V =√x2

4 + x25 + x2

6

ψ = atan2 (x5, x4)

γ = arcsin

(x6√

x24 + x2

5 + x26

),

which is always well defined except for x4 = x5 = 0, a situation which isruled out in the considered range of operation of the aircraft.Differentiating (3.15) and substituting from (2.3) and (3.14) the equationsgoverning the new state variables evolution become:

x1 =x4 + wx (3.17)x2 =x5 + wy

x3 =x6 + wz

x4 =τ cos γ cosψ − V sin γ cosψ

(L+ T sinα

mVcosφ− g

Vcos γ

)+

− V cos γ sinψ

(L+ T sinα

mV cos γsinφ

)x5 =τ cos γ sinψ − V sin γ sinψ

(L+ T sinα

mVcosφ− g

Vcos γ

)+

+ V cos γ cosψ

(L+ T sinα

mV cos γsinφ

)x6 =τ sin γ + V cos γ

(L+ T sinα

mVcosφ− g

Vcos γ

).

24


Now define new input variables u1, u2, u3 as follows:

u1 = τ cos γ cosψ − V sin γ cosψ

(L+ T sinα

mVcosφ− g

Vcos γ

)+

− V cos γ sinψ

(L+ T sinα

mV cos γsinφ

)(3.18a)

u2 = τ cos γ sinψ − V sin γ sinψ

(L+ T sinα

mVcosφ− g

Vcos γ

)+

+ V cos γ cosψ

(L+ T sinα

mV cos γsinφ

)(3.18b)

u3 = τ sin γ + V cos γ

(L+ T sinα

mVcosφ− g

Vcos γ

). (3.18c)

Equations (3.18a)-(3.18c) are now solved for τ , φ and α, the original inputof the aircraft model, as function of the new inputs u1, u2, u3 and of thestate variables x, y, z, V , ψ, γ, so obtaining a non-linear feedback lineariz-ing (3.17).

Calculating u1 cosψ + u2 sinψ and −u1 sinψ + u2 cosψ from (3.18a) and(3.18b) gives:

τ =1

cos γ

(u1 cosψ + u2 sinψ + sin γ

(L+ T sinα

mcosφ− g cos γ

))(3.19)

(L+ T sinα) sinφ = m (−u1 sinψ + u2 cosψ) , (3.20)

while from (3.18c) we have that:

L+ T sinα

mVcosφ− g

Vcos γ =

u3 − τ sin γ

V cos γ, (3.21)

which is the same as:

(L+ T sinα) cosφ = mg cos2 γ + u3 − τ sin γ

cos γ. (3.22)

Replacing the left-hand side of (3.21) appearing in (3.19) with the right-hand side of (3.21) and solving for τ yields:

τ = cos γ(u1 cosψ + u2 sinψ + u3 tan γ). (3.23)

Note that equation (3.23) is not affected by the expression taken by the pathangle dynamics, and, indeed τ takes the same expression in (3.7).

25


Define the auxiliary variables:

ν1 = u3 − τ sin γ + g cos2 γ (3.24)ν2 = −u1 sinψ + u2 cosψ.

If ν1 6= 0, then the bank angle φ can be computed as:

φ = arctan

(ν2 cos γ

ν1

)∈(−π

2,π

2

), (3.25)

which is exactly the domain of φ. If ν1 = 0, then ν2 must be 0 as well,because, otherwise, ν1 = 0 and ν2 6= 0 would imply φ = ±π/2 while weimposed that φ ∈ (−π/2, π/2). In the case when ν1 = 0 and ν2 = 0, itturns out that L+T sinα = 0 so that φ can be arbitrarily chosen. Note thatif L+T sinα = 0, then the value of φ does not affect the system dynamics,see (2.3). To fix a value we set φ = 0 when ν1 = 0 and ν2 = 0. Sinceφ ∈ (−π/2, π/2), L+ T sinα must take the sign of ν1, hence, substitutingthe expressions for sinφ and for cosφ from (3.20), (3.22) into the equationsin2 φ+ cos2 φ = 1 one obtains:

L+ T sinα = m

√ν2

1

cos2 γ+ ν2

2 sign (ν1). (3.26)

Note that given the choice in (3.25) by which φ ∈ (−π/2;π/2) and becauseof (3.22), L+T sinα takes always the sign of ν1. Note however that, giventhe bounds on the vertical and longitudinal accelerations and on the pathangle, ν1 will be always strictly positive, see also the constraints formulatedin the following Section 3.3.Substituting in (3.26) the expression of T given in (3.13) and writing ex-plicitly the dependence of L and D on α, it is seen that the AoA α has tosatisfy the following equation:

m

√ν2

1

cos2 γ+ ν2

2 sign (ν1) = 12ρSV 2Cl(1 + aα)+ (3.27)

+(mτ + gm sin γ + 1

2ρSV 2Cd(1 + b1α + b2α

2))

tanα,

which requires slightly more inspection than (3.23) and (3.25). In orderto find a value of the original input α that actually feedback linearizes theaircraft dynamics it is necessary that equation (3.27) admits at least onesolution. This is so if e.g. the right-hand side of (3.27) spans all the val-ues between −∞ and +∞ as α varies between −π/2 and +π/2. To this

26


purpose, note that the right-hand side of (3.27) is the sum of a linear termin α and a quadratic term in α multiplied by tanα. Since the aerodynamiccoefficients Cl, a, Cd, b1, b2 are positive, a sufficient condition for the right-hand side of (3.27) to span all the possible values between −∞ and +∞as α varies between −π/2 and +π/2, is that the roots of the quadratic termare not real, or, if real, are both strictly inside (−π/2, π/2). Indeed, underthis condition, the quadratic term takes positive values both when α is closeto −π/2 and +π/2, and, being this quadratic term multiplied by tanα thattends to−∞ and +∞ respectively, this makes the right-hand side of (3.27)assuming all the values between −∞ and +∞. Requiring that the roots ofthe quadratic term belong to (−π/2, π/2) amounts to letting:

−Cdb1 −√

(Cdb1)2 − 4Cdb2

(Cd +m τ+g sin γ

12ρSV 2

)2Cdb2

> −π2,

which can be rewritten as:

m(τ + g sin γ) >

(−π

2

4b2 +

π

2b1 − 1

)1

2ρSV 2Cd. (3.28)

Note that (3.28) should be enforced for the feedback linearization to bewell-posed. This in principle may limit the domain of validity of the feed-back linearization. Notably this is not the case for standard aircraft operat-ing conditions since it can be verified that (3.28) is automatically satisfiedwhen the constraints of Section 2.3 are enforced (see the example of Sec-tion 5.5 for a specific instance)1.Note that, in principle, while solving (3.27) for αwe may find a solution for(3.27) that, according to (3.13), leads to T < 0, which does not belong tothe admissible domain of the engine thrust. In other words, the solution to(3.27) does not automatically satisfy the condition T > 0. In order to avoidthis situation, a constraint to meet the specification on the engine thrust hasto be enforced, and in Section 3.3 is shown how to constrain u1, u2, u3 soas to obtain the desired specification on T . Notably, if a solution to (3.27)is found such that T > 0, this solution is unique. Indeed, we show that, theright-hand side of (3.27) is monotonically increasing with α when T > 0.The derivative of the right-hand side of (3.27) with respect to α is:

12ρSV 2Cla+

(mτ + gm sin γ + 1

2ρSV 2Cd(1 + b1α + b2α

2))

1cos2 α

+

+(

12ρSV 2Cd(b1 + 2b2α)

)tanα. (3.29)

1Perhaps it is worth noticing that the above argument is for purposes of proving existence only. In no way itis meant that α has to be close to −π/2 or +π/2. On the contrary, in normal aircraft operations, the left-handside of (3.27) is almost completely compensated by the lift L and, correspondingly, the AoA α keeps small.

27


The term mτ +gm sin γ+ 12ρSV 2Cd(1+ b1α+ b2α

2) = T cosα is positivewhen T > 0, so that a sufficient condition for (3.29) to be positive is:

aCl + Cd(b1 + 2b2α) tanα > 0. (3.30)

Note that since the coefficients Cd, b1, b2 are positive, the term Cd(b1 +2b2α) tanα is not positive only for α ∈ [− b1

2b2, 0], and in this interval

Cd(b1 + b2α) tanα ≥ −Cdb1 tan(b12b2

). In view of this, a sufficient condi-

tion to ensure that (3.30) holds is:

aCl − Cd tan

(b1

2b2

)b1 > 0.

This condition is usually satisfied because a and Cl are positive and, more-over, Cl Cd for standard aircraft.

Thus, in conclusion, we have shown that equation (3.27) in the unknownα always admits a solution, and, moreover, in the considered range of air-craft operation, as specified by the constraint is Section 2.3, this solution isunique.To find α as a function of the other variables, equation (3.27) can be easilynumerically solved, e.g. by bisection. This along with equations (3.13),(3.23), (3.24) and (3.25), gives the (non-linear) feedback that makes thedynamics of x1, x2, x3, x4, x5, x6 linear with respect to the new inputs u1,u2, u3:

x1

x2

x3

x4

x5

x6

=

[03×3 I3

03×3 03×3

]

x1

x2

x3

x4

x5

x6

+

[03×3

I3

]u1

u2

u3

+

[I3

03×3

]wxwywz

. (3.31)

Note that, the new inputs u1, u2, u3 have again the physical meaning ofCartesian components of the aircraft acceleration with respect to air. More-over, as said, the linear model (3.31) is the same achieved in (3.11), while,as is clear, the non-linear feedback laws needed to obtain such linear dy-namics starting from model (2.3) or from model (2.2) are different.

28

3.3. Reformulation of the constraints in the new variables

3.2.1 Time discretization

To ease subsequent MPC developments, system (3.31) is eventually dis-cretized by applying a constant input in the intervals [t0 + kTs, t0 + (k +1)Ts), k = 0, 1, . . ., where Ts is the sampling time. This leads to thediscrete-time system

x1,k+1

x2,k+1

x3,k+1

x4,k+1

x5,k+1

x6,k+1

=

[I3 TsI3

03×3 I3

]

x1,k

x2,k

x3,k

x4,k

x5,k

x6,k

+

[T 2s

2I3

TsI3

]u1,k

u2,k

u3,k

+

[TsI3

03×3

]wx,kwy,k

wz,k

,

(3.32)

which can be rewritten as

xk+1 = Axk +Buk +Bwwk,

for short (where xk stands for x(t + kTs) and likewise for the other vari-ables). Note that, in the following we will consider the discrete time dy-namics in (3.32) for both aircraft model (2.2) and (2.3).

3.3 Reformulation of the constraints in the new variables

Feedback linearization performed in the previous sections permitted us torewrite the aircraft models, either (2.2) or (2.3), as a linear system whichhas the undoubtedly advantage of simplifying MPC design. On the otherhand, state and input constraints turn out to be non-convex when rewrit-ten in terms of new state and input variables x1, x2, x3, x4, x5, x6, u1, u2,u3. In this section we proceed by reformulating the constraints over a finitehorizon in view of subsequent MPC application, and suitable relaxation areintroduced when necessary in order to convexify them (actually all con-straints except one will be linear). The introduced relaxation, however, aresuch that the original constraint is satisfied at least at the first time instant,so that, by receding horizon, the original constraints are met by the MPCsolution.The constraints on vertical acceleration, TAS, longitudinal acceleration andpath angle take the same expression irrespectively from the choice of theaircraft model (2.2) or (2.3) and, hence, they are dealt with in the same wayfor both models. On the contrary, the constraints on bank angle and engine

29


thrust take a different expressions depending on the choice of the aircraftmodel. They are treated separately and different convex approximations areadopted accordingly.

3.3.0.1 Vertical Acceleration

Since in both aircraft models, the input u3 is equal to the vertical accelera-tion z, constraint (2.10) can be straightforwardly written as:

− aZ ≤ u3,k+i ≤ aZ i = 0, . . . ,M − 1. (3.33)

3.3.0.2 True Air Speed V

Being V 2 = x24 + x2

5 + x26, see (3.12) and (3.16), constraints (2.4) on TAS

is rewritten as:

x24,k+i + x2

5,k+i + x26,k+i ≤ V 2

max i = 1, . . . ,M (3.34)

x24,k+i + x2

5,k+i + x26,k+i ≥ V 2

min i = 1, . . . ,M. (3.35)

Constraints (3.34) are already convex, and their interpretation is that theaircraft velocity2 lies inside a sphere of radius Vmax. Constraints (3.35)are instead concave as they require that the aircraft velocity lies outside asphere of radius Vmin. To attain convexity, constraints (3.35) are modifiedby requiring that the aircraft velocity stays beyond a plane tangent to sphereof radius Vmin and oriented according to the current, i.e. at time k, headingangle ψk and path angle γk, which are available since the state is measur-able. The new constraints can be expressed by means of the following linearinequalities:

[1 0 0

]RyRz

x4,k+i

x5,k+i

x6,k+i

≥ Vmin i = 1, . . . ,M, (3.36)

where the rotation matrices Rz(ψk) and Ry(γk) are defined as:

Rz =

cosψk sinψk 0

− sinψk cosψk 0

0 0 1

Ry =

cos γk 0 sin γk

0 1 0

− sin γk 0 cos γk

.Note that (3.34) and (3.36) together pose a stricter condition than (3.34) and(3.35), so that the original constraints are always satisfied. The drawback

2Note that the aircraft velocity with respect to the surrounding air is a vector with modulus equal to V andorientation given by the heading angle ψ and by the path angle γ. Its Cartesian components are x4, x5, x6.

30


with (3.36) is that, having the plane tangent to the sphere of radius Vminfixed orientation, for values of TAS close to Vmin, the aircraft is forced toaccelerate in order to turn in the finite horizon problem. However this side-effect is negligible for higher values of TAS, which correspond to the usualflight conditions, and, moreover, thanks to the receding horizon strategy,the plane orientation is recomputed at each time step according the actualvalues of ψk and γk, and this drawback is further mitigated in the actualtrajectory of the aircraft.

3.3.0.3 Longitudinal Acceleration

From V = τ and equations (3.7) and (3.23), in both the considered air-craft models (2.2) or (2.3), the constraint (2.5) on longitudinal accelerationrewrites as:

−aL ≤ cos γk+i(u1,k+i cosψk+i + u2,k+i sinψk+i+ (3.37)+ u3,k+i tan γk+i) ≤ aL i = 0, . . . ,M − 1.

Further substituting the expressions for γk+i and ψk+i as functions of x4,k+i,x5,k+i, x6,k+i as in (3.12) and (3.10) or (3.16) for model (2.2) or model (2.3)respectively, it is easily seen that the constraints would be not convex. Wedecide then to approximate them by replacing the values of the headingangle ψk+i and of the path angle γk+i with their initial values ψk and γk,which are available. This approximation seems to be acceptable for theheading angle cannot vary too much in the considered finite time horizonwhile, given the limitations on it, the path angle always keeps close to 0.This yields:

−aL ≤ cos γk(u1,k+i cosψk + u2,k+i sinψk+ (3.38)+ u3,k+i tan γk) ≤ aL i = 0, . . . ,M − 1,

which are linear constraints on uk+i, i = 0, . . . ,M − 1. Notably, for i = 0,the constraint on uk is exactly equivalent to the original one, without anyapproximation. Thus, if u is designed to satisfy (3.38), and a receding hori-zon is considered, then the original constraint (2.5) is not violated by theactual aircraft motion because each time only the first u is indeed actuated.

3.3.0.4 Path Angle γ

Recalling that x6 = V sin(γ) both from equation (3.3) for model (2.2) orfrom equation (3.15) for model (2.3), the constraint (2.8) writes as

Vk+i sin γmin ≤ x6,k+i ≤ Vk+i sin γmax i = 1, . . . ,M, (3.39)

31


which is however non-convex because Vk+i =√x2

4,k+i + x25,k+i + x2

6,k+i.Constraint (3.39) can be approximated by keeping V fixed to its initial valueVk for all i = 1, . . . ,M , leading to

Vk sin γmin ≤ x6,k+i ≤ Vk sin γmax i = 1, . . . ,M. (3.40)

Although the approximation is mild as V cannot vary too much along theprediction horizon, note, however that (3.40) does not guarantee the satis-faction of the original constraint (2.8), not even for i = 1 (note that con-straint (3.39) is a state constraint and, it has to be posed for i = 1, . . . ,M ).Hence, we prefer to tackle (3.39) by replacing Vk+i with the worst-caseprediction of the values assumed by it along the considered prediction hori-zon. Given the constraints (2.4) and (2.5) on TAS and on the longitudinalacceleration, the maximal decrease of TAS over the prediction horizon isapproximately computed as:

V wc−

k+i = maxVk − iTsaL, Vmin i = 0, . . . ,M, (3.41)

while the maximal increase as:

V wc+

k+i = minVk + iTsaL, Vmax i = 0, . . . ,M. (3.42)

Turning back to (3.39), since sin γmin is negative and sin γmax is positivethe worst-case is achieved when Vk+i takes the smallest values, which leadsto enforce the constraints

V wc−

k+i sin γmin ≤ x6,k+i ≤ V wc−

k+i sin γmax i = 1, . . . ,M, (3.43)

in place of (3.39). The constraints in (3.43) pose a stricter condition than(3.39), so that when (3.43) is applied the satisfaction of the original con-straint is guaranteed along the whole prediction horizon. A compromisebetween (3.40) and (3.43) consist in fixing V to the first step worst caseprediction V wc−

k+1 , that is:

V wc−

k+1 sin γmin ≤ x6,k+i ≤ V wc−

k+1 sin γmax i = 1, . . . ,M. (3.44)

This way the original constraint is guaranteed for i = 1 only, but its satisfac-tion along all the aircraft operation is recovered thanks to receding horizon.On the other hand, the conservatism introduced by worst case safeguardingis mitigated and a better performance is achieved.

3.3.0.5 Bank Angle φ

The constraint on the bank angle needs to be treated differently for model(2.2) and model (2.3). Indeed despite the fact that the constraint share a

32


similar structure in both models, the feedback linearization of model (2.3)establishes a dependence between the bank angle and lift force (see equa-tion (3.20)), which in turn is related to the angle of attack α. Hence, wedevelop a first constraint approximation to be used with model (2.2) andanother approximation that accounts for the presence of α to be used withmodel (2.3).

• Bank angle constraints for model (2.2)The constraint (2.7) on the bank angle presents a structure similar tothat of the constraints on the longitudinal acceleration, that is, on τ .Indeed, by (3.2) and (3.6), (2.7) can be rewritten as:

− tan φ ≤ 1

g cos γk+i

(−u1,k+i sinψk+i + u2,k+i cosψk+i) ≤ (3.45)

≤ tan φ i = 0, . . . ,M − 1.

Similarly to (3.38), the values of γk+i and ψk+i are fixed to their initialvalue γk and ψk in order to recover linear constraints with respect tothe variables u. We have

− tan φ ≤ 1

g cos γk(−u1,k+i sinψk + u2,k+i cosψk) ≤ (3.46)

≤ tan φ i = 0, . . . ,M − 1.

Note that the constraint on uk at i = 0 is exactly equivalent to the orig-inal constraints on the bank angle, so that the actual aircraft operationas given by MPC satisfies (2.7) thanks to the receding horizon.

• Bank angle constraints for model (2.3)In view of (3.20), the constraint (2.7) on the bank angle can be rewrit-ten as:

| − u1,k+i sinψk+i + u2,k+i cosψk+i| ≤ (3.47)

≤ sin φLk+i + Tk+i sinαk+i

mi = 0, . . . ,M − 1.

Taking the squares and recalling (3.26), (3.24):

ν22,k+i ≤ sin2 φ(

ν21,t+i

cos2 γk+i

+ ν22,k+i) i = 0, . . . ,M − 1,

which in turn rewrites as

ν22,k+i

cos2 φ

sin2 φcos2 γk+i ≤ ν2

1,k+i i = 0, . . . ,M − 1.

33


Recalling the definition of ν1,k+i and because of the limitations onvertical and longitudinal accelerations, and on the path angle, it canbe easily seen than ν1, cos γ are always positive. Hence, taking thesquare root we have that

|ν2,k+i|cos φ

sin φcos γk+i ≤ ν1,k+i i = 0, . . . ,M − 1,

so that replacing the expression of ν1,k+i, ν2,k+i in (3.24), the con-straints (3.47) are eventually rewritten as:

| − u1,k+i sinψk+i + u2,k+i cosψk+i|cos φ

sin φcos γk+i ≤

≤ u3,k+i + g cos2 γk+i − sin γk+i cos γk+i(u1,k+i cosψk+i+

+ u2,k+i sinψk+i + u3,k+i tan γk+i) i = 0, . . . ,M − 1.

Similarly to previous cases, convexity is recovered replacing ψk+i andγk+i with their initial value γk and ψk:

| − sinψku1,k+i + cosψku2,k+i|cos φ

sin φcos γk ≤ (3.48)

≤ u3,k+i + g cos2 γk − sin γk cos γk(u1,k+i cosψk+

+ u2,k+i sinψk + tan γku3,k+i) i = 0, . . . ,M − 1.

Note that, for each i the constraint in (3.48) is linear and for i = 0is exactly equivalent to the original constraint on the bank angle, sothat, thanks to the receding horizon, the actual aircraft operation canbe enforced to satisfies (2.7) for all time steps through (3.48).

3.3.0.6 Engine Thrust T

The constraint on the engine thrust expressed as a function of the new stateand input variables rewrites differently depending on the adopted aircraftmodels (2.2) or (2.3) and on their corresponding feedback linearizations.Note that in models (2.2) and (2.3) T has a similar, but different expression,namely T = D + mg sin γ + τm (see (3.1)) and T = (D + mg sin γ +τm) 1

cosα(see (3.13)), where the presence of the angle of attack α in the

latter expression reflects the more detailed description of the aerodynamicforces in model (2.3). Hence, it is necessary to develop a first approachto deal with the engine thrust constraint when model (2.2) is considered,while a different approach is needed to explicitly account for the presenceof the angle of attack α when model (2.3) is used.

34


• Engine thrust constraint for model (2.2)Recalling (2.6) and (3.1), the constraint on the engine thrust can bewritten as

Tminm≤ τ + g sin γ +

KD

mV 2 ≤ Tmax

m,

which in turn, recalling the expression for τ in (3.7) and sin γ = x6

V,

becomes:

Tmin ≤ m cos γk+i(u1,k+i cosψk+i + u2,k+i sinψk+i + u3,k+i tan γk+i)

+mg

Vk+i

x6,k+i +KD(x3,k+i)V2k+i ≤ Tmax i = 0, . . . ,M − 1.

(3.49)

Constraint (3.49) is non-convex because of its dependence on Vk+i,ψk+i, γk+i and x3,k+i.We, hence, proceed as follows. As in (3.38) the value of the head-ing angle ψk+i and of the path angle γk+i are replaced by their initialvalue ψk and γk which introduces a mild approximation as discussedabove. Note to this purpose that, though γk+i keeps close to 0, re-placing sin γk+i with sin γk may lead to poor approximation becauseit is multiplied by a big factor like mg. Hence, it is of paramountimportance to first write sin γ = x6

Vso that the constraint is actually

imposed on x6 before substituting γk+i with γk. Unfortunately, theconstraints are still non-convex because of the presence of Vk+i =√x2

4,k+i + x25,k+i + x2

6,k+i. We replace x3,k+i and Vk+i with their ini-tial values x3,k and Vk. Again this approximation seems to be settablebecause the limitation on the vertical acceleration, on TAS and on thepath angle implies that V and x3 cannot vary too much in the con-sidered finite horizon (note that KD depends on x3 only through thefunction ρ(x3) which describes how air density varies with altitude,and this further reduces the impact of variation of x3). Eventually, theconstraint on engine is approximated with the following linear con-straints:

Tmin ≤ m cos γk(u1,k+i cosψk + u2,k+i sinψk + u3,k+i tan γk)+

+mg

Vkx6,k+i +KDV

2k ≤ Tmax i = 0, . . . ,M − 1. (3.50)

A second approach, instead, consists in replacing Vk+i with worst-casepredictions along the considered horizon, computed as in (3.41) and

35


(3.42). In this case the approximate constraints on the engine thrustwould write:

Tmin −KD(V wc+

k+i )V wc−2k+i ≤ (3.51)

m cos γt(u1,k+i cosψt + u2,k+i sinψt + u3,k+i tan γt) +mg

Vkx6,k+i

≤ Tmax −KD(V wc−

k+i )V wc+2k+i , i = 0, . . . ,M − 1.

We do not use worst case predictions for the TAS in the term gVx6 be-

cause the sign of x6 is unknown, and because the constraints would beexcessively conservative, due the significant contribution of mg.Remarkably, both (3.50) and (3.51) are linear constraints, and are suchthat the first constraint corresponding to i = 0 is exactly equivalent tothe original constraint on the engine thrust. Thus, again, thanks to thereceding horizon strategy, the actual operation of the aircraft satisfies(2.6).The constraints (3.51) are closer to the original constraints than (3.50),and using (3.51) it is more likely that the original constraint on the en-gine thrust is satisfied over the whole finite horizon. Yet, in manycases (3.51) are too conservative and simulations reveal that the con-straints (3.50) lead to better performance.

• Engine thrust constraint for model (2.3)Recalling (3.13), the constraint on the engine thrust can be written as:

Tmin ≤(mτk+i +D(x3,k+i, Vk+i, αk+i)+

+mg sin γk+i)1

cosαk+i

≤ Tmax i = 0 . . .M − 1,

which in turn, by rewriting τ as in (3.23) and sin γ = x6

V, becomes:

Tmin ≤(m cos γk+i(u1,k+i cosψk+i + u2,k+i sinψk+i+ (3.52)

+ u3,k+i tan γk+i) +D(x3,k+i, Vk+i, αk+i)+

+mgx6,k+i

Vk+i

) 1

cosαk+i

≤ Tmax i = 0 . . .M − 1.

Similarly to constraint (3.49), constraint (3.52) is highly non-convexbecause of its dependence on Vk+i, ψk+i, γk+i, x3,k+i and αk+i.As for Vk+i, ψk+i, γk+i, x3,k+i we proceed as in the previous case re-placing them with their initial value Vk, ψk, γk, x3,k which introduces

36


a mild approximation as discussed above. Hence the constraint onengine thrust rewrites:

Tmin ≤(m cos γk(u1,k+i cosψk + u2,k+i sinψk+ (3.53)

+ u3,k+i tan γk) +D(x3,k, Vk, αk+i)+

+mgx6,k+i

Vk

) 1

cosαk+i

≤ Tmax i = 0 . . .M − 1.

which is still non-convex because of the presence of the angle of at-tack. Note that the previous approximation obtained replacing αk+i

with αk does not work anymore because αk is a control input whichdepends on the actual choice of u1,k, u2,k, u3,k. We, hence, decide toenforce the constraint robustly with respect to all the possible valuestaken by α as u1,k, u2,k, u3,k vary in accordance to the other con-straints. More precisely, we first compute the admissible range ofvalues for α, [αmin(xk), αmax(xk)], where the bounds αmin(xk) andαmax(xk) are the minimum and the maximum value for α achievedby solving equation (3.27) when the state is kept fixed to the currentvalue xk and the inputs u1,k, u2,k, u3,k take all the feasible values inaccordance to the constraints on the vertical acceleration (3.33), thelongitudinal acceleration (3.38), the path angle (3.44) and the bankangle (3.48). Then, we compute the worst case bounds for constraint(3.53):

Tmin = maxα∈[αmin(xk), αmax(xk)]

Tmin cosα−D(Vk, α)

Tmax = minα∈[αmin(xk), αmax(xk)]

Tmax cosα−D(Vk, α) ,

and the robust constraint with respect to the values that can be possiblytaken by α on the engine thrust is eventually enforced:

Tmin ≤mgx6,k+i

Vk+m cos γk(u1,k+i cosψk + u2,k+i sinψk+ (3.54)

+ u3,k+i tan γk) ≤ Tmax i = 0, . . . ,M − 1.

Remarkably constraints (3.54) are linear constraints. Moreover the in-troduced approximations are such that the original constraints (3.52)is satisfied for i = 0. As a matter of fact, the replacement of ψk+i,γk+i, Vk+i, x3,k+i with ψk, γk, Vk, x3,k, introduces no error for i = 0,while being robust with respect to the values taken by α in correspon-dence of xk implies that (3.54) for i = 0 is a stricter condition than

37


(3.52) for i = 0. This is of most importance because it implies that acontrol action designed so as to satisfies (3.54) and implemented alonga receding horizon actually satisfies (3.52) at all time instants.Note that while computing the bounds for (3.54), it is of paramountimportance that the admissible range [αmin(xk), αmax(xk)] is adaptedto the current state xk. This introduces a possible error at the subse-quent time instants for i ≥ 1. Being robust with respect to a largerrange of admissible values for α, e.g. for those α corresponding toall possible feasible states, leads to more guaranteed constraints at alltime instants k+ i, i = 0, . . . ,M − 1. On the other hand, however, anextreme conservatism is introduced, which leads to poor performanceor even to infeasibility. Constraint (3.54) seems to achieve a goodlevel of approximation and allows to obtain good performance.

Remark. A possible alternative approach to the convexification of the con-straints on longitudinal acceleration, engine thrust and bank angle so as toobtain constraints which are closer to the original ones along the predictionhorizon is discussed in Appendix B. However, simulation results reveal thatthe linear approximations described in this section are more effective andlead to better performance and, hence, they are preferred in the following.

3.4 Feasibility of the constraints

It is perhaps worth noticing that the limitations on path angle posed by γminand γmax keep the projection of the weight force on the longitudinal direc-tion limited. As it can be verified, this ensures that for every admissibleoperating condition the engine thrust is always (that is without violatingthe limitations on it) able to counteract the drag and possibly the longitudi-nal component of the weight force without enforcing the aircraft to increaseor decrease the TAS. This means that keeping TAS, heading angle and pathangle constant is always feasible, that is the constraints of this section aresatisfied by the solution uk+i = 0, i = 0, . . . ,M − 1. This guarantees thatthe feasibility of the finite horizon optimization problem to be solved ateach time instant k is not compromised by the constraints on aircraft phys-ical limitations, safety and passenger comfort discussed in this section.

38

CHAPTER4Trajectory generation

4.1 Introduction

In this chapter, we focus on the design of trajectories that comply with TWstime and space requirements and that, at the same time, account for aircraftphysical limitations and passenger comfort constraints. In this way, the de-signed trajectory is feasible to be followed by the aircraft, and, if followed,it makes the aircraft reaching the TWs at the proper time. This trajec-tory will be used next as reference in the aircraft motion control. In orderwords, the TW requirements on large time and space scale are dealt within the design of the reference trajectory only, while the MPC controller, tobe developed in the next Chapter 5 will address the task of keeping the air-craft as close as possible to the reference trajectory, while counteracting theaction of the wind disturbance and ensuring the satisfaction of the aircraftmotion constraints. As the reference trajectory is suitably tracked, the TWsspecifications are met also by the actual aircraft motion, leading to an effec-tive solution to the problem of TWs constraint satisfaction. This problemsplitting between reference design and reference tracking is key to obtain asolution. As a matter of fact, if, instead, only a target position was given,it would not be easy to enforce in MPC control time specifications so as

39

Chapter 4. Trajectory generation

to guarantee reaching the target position in the right time interval, becauseTW specifications involve a time scale much larger than the one that canbe considered in the finite horizon optimization problem. In addition therewould be computational difficulties related to modelling TW requirementsby means of convex constraints, and to considering very long time intervalin the optimization problem.

4.1.1 Sketch of the proposed design approach

The reference trajectory is composed by a path and by a time law withwhich the path has to be covered. Hence, the trajectory provides at everytime instant a reference position and a velocity. As previously explained,having a reference position for each time instant is fundamental to keep theproper timing as required by TW specifications.Our aim is to design trajectories that:

• accounts for aircraft dynamics and physical and comfort constraintsso that they are feasible to be followed by aircraft;

• accounts for the space and time specifications imposed by TWs so thataircraft can meet TW requirements by tracking the designed trajecto-ries.

In doing so, a path is designed first by ensuring that the spatial centres ofthe TWs belong to the trajectory, and then a time law is attached to it. Tothis purpose we first try to adjust the path length so as to meet the TWstime specifications by means of a constant speed time law, namely keepingthe aircraft flying at constant TAS. If this would be infeasible, we try todesign a time law based on a piecewise affine (PWA) speed profile that stillensures that the spatial centres of the TWs are reached with proper timing.

Path design relies on three simple motion primitives corresponding to: fly-ing along a straight line, to adjusting the heading angle, and to adjusting thepath angle. These motion primitives are selected in such a way that whencovered by the aircraft at constant speed within the admissible range, allthe constraints on aircraft motion in Section 2.3 are satisfied.

Then, the motion primitives are suitably joined together to design a pathbetween an initial and a final point consisting of the centres of two TWs insuccession. More precisely, the initial and the final point are connected by 7motion primitives consisting in a first heading adjustment, a path angle ad-justment, a straight segment, a second heading adjustment, another straight

40

4.1. Introduction

segment, a second path angle adjustment and a final heading adjustment.Depending on the TWs displacement, it may be that some of these motionshave zero length (i.e. they are not exploited in the path definition), whilein more complex situations all of them are used. Altogether the 7 motionprimitives allow enough degrees of freedom to design a smooth path andto properly approach the final point. Continuity of position and velocity atthe junction points is imposed to ensure that the designed path is connectedand smooth. Additional conditions enforce the path to have a length suchthat when the path is covered at constant speed, the final point is reachedat the proper time instant. In order to find the parameters that define thepath according to the specifications above, a system of equations has to besolved. Its resolution is carried over by means of a method in which fewparameters are found relying on ad-hoc rules, resorting to gridding, or bymeans of mono-dimensional numerical search, while all the other parame-ters can be analytically determined.

The procedure to design a path between two points is repeated to find apath between the spatial centres of each couple of consecutive TWs, so that,eventually, a path that connects multiple TWs is defined. As said, the pathlength between a TW and the following one is designed so as to reach thislatter with proper timing covering the path at constant speed. If this is notfeasible, the time law is changed and a PWA speed profile is designed so asto still met time specification. The velocity direction with which each TWcentre is approached, is chosen accounting for the time and space specifi-cations of the previous and the following TWs. This provides a link amongthe several path designs: indeed, in each path design, the velocity directionat the final point (which will be the initial point in the next design) is chosenaccounting for the tightness of the next TW requirements, so as to foster thedesign of a shorter or longer path according to the distance between TWsand the allowed time to cover it.

The resulting overall path passes through all the TW centres at the propertime instant so as to met TW specification. This procedure may sometimesdo not exploit the full maneuverability of the aircraft. However, limitationsor even infeasibility arise only when the TW specifications are excessivelytight (e.g. when time space requirements are incompatible with aircraftmotion constraints), while specifications corresponding to normal aircraftoperations are successfully addressed. The main advantage of the proposedapproach is that it can easily dealt with a very large number of TWs thatcan span large time and space scales. Moreover, it allows to design smooth

41


paths that are easy to be followed and that leave enough degrees of freedomfor compensating disturbances acting on the aircraft. These characteristicsmake this approach suitable for applications in which smoothness of thetrajectory and scalability to a large number of TWs have a critical role.

In the following sections the trajectory design procedure is described in fulldetails. We warn the reader that, despite the design approach is guided bythe simple ideas described above, some parts of its implementation are quitetechnical. Nonetheless they may be skipped without affecting the overallunderstanding.

4.2 Path design

4.2.1 Motion primitives

The path is designed starting from simple motion primitives joined to-gether. We choose 3 basic motion primitives corresponding to flying alonga straight line, an adjustment of the heading angle and an adjustment of thepath angle.

4.2.1.1 Straight Line R

The first motion primitive, corresponding to straight line flight in the x-y-zspace, is parametrized through a parameter s ∈ [0; sf ], sf ≥ 0, as follows:

x = mxs+ qx

y = mys+ qy

z = mzs+ qz

, (4.1)

wheremx,my,mz, qx, qy, qz, along with sf are tunable coefficients throughwhich the displacement and the length of the straight line segment can bevaried. Assigning a time law consists in specifying the time dependence ofs on time t, that is s = s(t). The speed1 with which the path is covered isthen given by:

V (t) = ‖[x y z

]‖2 =

√(x′s)2 + (y′s)2 + (z′s)2 =

√m2x +m2

y +m2z s,

(4.2)

where we used ‘ ′ ’ to denote the derivative with respect to s, while ‘ ˙ ’ todenote the time derivative.

1Note that the velocity denoted with V corresponds to the aircraft TAS V .

42

4.2. Path design

To keep the speed constant V (t) = V , the time law for s results:

s(t) =V√

m2x +m2

y +m2z

t+ const. (4.3)

and, under such flight conditions imposed by (4.1) considering a constantspeed V (t) = V , the following considerations about the aircraft motionconstraints in Section 2.3 hold.

• TAS The constraint is satisfied by taking any admissible value for theconstant TAS V = V ∈ [Vmin, Vmax].

• Longitudinal Acceleration At constant TAS, V = V , the longitudi-nal acceleration is V = 0, automatically satisfying the constraint onthe longitudinal acceleration.

• Bank Angle During a flight along a straight line the bank angle φremains always equal to 0, so that the bank angle constraint is alwayssatisfied.

• Path Angle Note that sin γ = zV

= mz√m2x+m2

y+m2z

. This implies that

the path angle γ is constant during straight line flight and keeps equalto its initial value as specified by mx, my, mz.

• Vertical Acceleration The vertical acceleration is z = z′′s2 +z′s = 0,which always satisfies the constraint on the vertical acceleration.

• Engine Thrust Being the longitudinal acceleration null, the enginethrust has to compensate the drag and eventually part of the weightforce when γ 6= 0. The engine thrust constraint then is satisfied aslong as V and γ are within their admissible values.

Summarizing, in order to obtain a feasible trajectory the only requirement isto properly choose the value for the speed V and to start with an admissiblepath angle γ.

4.2.1.2 Heading adjustment Cxy

A second motion primitive is defined to allow adjustments of the headingangle ψ while keeping a constant path angle γ. The parametrization in thevariable s is as follows:

x = R cos(s) + xi

y = lrR sin(s) + yi

z = a(s− sinit) + b

, (4.4)

43


where s ∈ [si; sf ], with si ≤ sf (note that si and sf are not necessarilypositive and they can take also negative values). The parametrization in(4.4) represents a circular arc in the x-y plane and a straight line in the zdirection. si, sf , R, lr, xi, yi, a, and b are tunable parameters through whichthe displacement of the path can be adjusted. In particular,R corresponds tothe radius of the circular arc, while the parameter lr can take only the values1 and −1 and determines whether the circumference is covered clockwise(CW) or counter clockwise (CCW) as s increases. We can compute thespeed with which the path is covered as:

V (t) = ‖[x y z

]‖2 =

√(x′s)2 + (y′s)2 + (z′s)2 = (4.5)

=√R2 sin2(s) + l2rR

2 cos2(s) + a2s =√R2 + a2 s.

If the speed V (t) is constant, V (t) = V , the time law for s is:

s(t) =V√

R2 + a2t+ const. (4.6)

Similarly to the previous case, the following considerations for the aircraftconstraints in correspondence of the flight condition imposed by (4.5) con-sidering a constant speed V (t) = V hold.



• Bank Angle We start considering the aircraft model in (2.3). In orderto make the aircraft move along a circular arc in the x-y plane, thebank angle as to be set as:

sinφ = −lrmV 2 cos γ

(L+ T sinα)√R2 + a2

,

Indeed, accounting for the fact that cos γ = R√R2+a2 ,

ψ = L+T sinαmV cos γ

sinφ = −lr V√R2+a2

→ ψ = −lr(

V√R2+a2

t+ const.)

x = V cos γ cos

(V√R2+a2

t+ const.)

→ x = R sin

(V√R2+a2

t+ const.)

+ xi

y = −lrV cos γ sin

(V√R2+a2

t+ const.)→ y = lrR cos

(V√R2+a2

t+ const.)

+ yi,

44

4.2. Path design

where the resulting expressions for x and y clearly describe a circum-ference. In order to satisfy the constraint on the bank angle we mustenforce:

mV 2 cos γ

(L+ T sinα)√R2 + a2

≤ sin φ. (4.7)

Condition (4.7) can be satisfied by taking R large enough, so that forall the admissible values of m, V , γ, and L+ T sinα:

R ≥ mV 2 cos γ

(L+ T sinα)

1

sin φ. (4.8)

This way we can avoid modifying the radius depending on the aircraftoperating conditions.

A similar reasoning can be done considering the aircraft model in(2.2). In order to make the aircraft fly along a circumference in thex-y plane we have to set tanφ = −lr V

2

gRcos γ so that results:

ψ = gV

tanφ = −lr VR cos γ → ψ = −lr(VR

cos γt+ const.)

x = V cos γ cos(VR

cos γt+ const.)

→ x = R sin(VR

cos γt+ const.)

y = −lrV cos γ sin(VR

cos γt+ const.)→ y = lrR cos

(VR

cos γt+ const.).

Again the expressions for x and y describe a circumference. Hence,choosing the parameter R that defines the turn radius so as to satisfy

R ≥ V 2

g tan φ, (4.9)

ensures that the designed path is feasible for the bank angle constraint.

Note that for both the considered aircraft models it is sufficient to takethe turning radius large enough to keep the bank angle within properranges.


= a√R2+a2 . This implies that the path

angle γ is constant and keeps equal to its initial value.

• Vertical Acceleration The vertical acceleration is z = z′′s2 +z′s = 0,which always satisfies the constraint on the vertical acceleration.

• Engine Thrust Being the longitudinal acceleration null, the enginethrust has to compensate the drag and eventually part of the weight

45


force when γ 6= 0. The engine thrust constraint then is satisfied aslong as V and γ are within their admissible values.

Hence, summarizing, properly setting the values for the radius R and forthe speed V a feasible paths is achieved as long as the initial path angle γis admissible at the beginning.

4.2.1.3 Path Angle adjustment Cz

A third motion primitive is defined to allow adjustments of the path angleγ while keeping the heading angle ψ constant. The parametrization in thevariable s is:

x = lukRγ sin(s) + xi

y = lukRγ sin(s) + yi

z = Rγ cos(s) + zi

, (4.10)

where s ∈ [si; sf ], si ≤ sf . The path corresponds to an arc of circum-ference in the plane determined by the z-axis and the current heading di-rection, and a straight line in the x-y plane. As usual, lu, k, k, Rγ , xi, yi,zi, si and sf are parameters through which the path displacement can beadjusted. In particular, the parameter lu can take only the values 1 and −1and determines whether the path angle is increased (lu = −1) or decreased(lu = +1). The parameters k and k accounts for properly splitting the pro-jection of the velocity into the x-y plane between the x and the y direction;they are chosen so as to satisfy k2 + k2 = 1. The speed with which the pathis covered is computed as:

V (t) = ‖[x y z

]‖2 =

√(x′s)2 + (y′s)2 + (z′s)2 = (4.11)

=√

(lukRγ)2 cos2(s) + (lukRγ)2 cos2(s) +R2γ sin2(s) s = Rγ s

so that the time law for s corresponding to a constant speed V (t) = V is:

s(t) =V

Rγ

t+ const. (4.12)

The consideration about the aircraft constraints in correspondence of theflight condition imposed by (4.1) considering a constant speed V (t) = Vare as follows:


46

4.2. Path design


• Bank Angle During a flight along a straight line in the x-y plane thebank angle φ remains always equal to 0, so that the bank angle con-straint is always satisfied.


= −Rγ sin(s)

VVRγ

= sin(−s). Hence,in order to satisfy the constraints on the path angle we have to enforcethe following conditions:

s ∈ [−γmax,−γmin] if lu = +1 (descreasing path angle) (4.13)

(i.e. si ≥ −γmax, sf ≤ γmin)

s ∈ [π + γmin, π + γmax] if lu = −1 (increasing path angle)

(i.e. si ≥ π + γmin, sf ≤ π + γmax)

• Vertical Acceleration Being z = z′′s2 + z′s = − V 2

Rγcos(s), the con-

straints on the vertical acceleration write:∣∣∣∣ V 2

Rγ

cos(s)

∣∣∣∣ ≤ aZ . (4.14)

Condition (4.14) is satisfied taking the radiusRγ large enough, so that:

Rγ ≥V 2

aZ,

for all admissible value of V .

• Engine Thrust Being the longitudinal acceleration null, the enginethrust has to compensate the drag and eventually part of the weightforce when γ 6= 0. The engine thrust constraint then is satisfied aslong as V and γ are within their admissible values.

Summarizing, in order to attain a feasible path when covered at constantspeed it is enough to properly chose the values of V and of the radius Rγ ,and to limit the range of the admissible values for the variable s according to(4.13) so as to bound the path angle. Note that keeping the path angle withinthe admissible range in this phase of the path, which is the only one thatallows for path angle modification, implies that γ keeps in its admissiblerange also in the other phases of the path.

47


4.2.2 Path composition

In this section we address the problem of design a path between an initialand a final given points. This can be then used to design the trajectory be-tween a couple of consecutive TWs by attaching a suitable time law. Weassume that the initial and final point coordinates xI , yI , zI and xF , yF , zFrespectively and the initial and final velocities vxI , vyI , vzI , vxF , vyF , vzFare given, together with a desired path length Dref . We take as initial andfinal points the centres of two consecutive TWs, while the initial and finalvelocity directions can be chosen so as to properly head for the successiveTW, according to the method proposed in Section 4.3. As said, in the pathdesign we consider a constant speed profile V (t) = V , and hence we setDref = V∆T , where ∆T is the time interval with which the path has to becovered. ∆T can be taken as the time interval between the time centres ofthe consecutive TWs.Exploiting the motion primitives described in Section 4.2.1, the path be-tween two points is composed by

1. an heading adjustment Cxy1x = R cos(s) + x1

y = lr1R sin(s) + y1

z = a1(s− sI) + b1

s ∈ [sI , s1] (4.15)

2. a path angle adjustment Cz2x = lu1k1Rγ sin(s) + x2

y = lu1k2Rγ sin(s) + y2

z = Rγ cos(s) + z2

s ∈ [sII , s2] (4.16)

3. a straight line segment R3x = mxs+ qx

y = mys+ qy

z = mzs+ qz

s ∈ [0, s3] (4.17)



z = a4(s− sIV ) + b4

s ∈ [sIV , s4] (4.18)

48

4.2. Path design

5. a straight line segment R5x = Mxs+Qx

y = Mys+Qy

z = Mzs+Qz

s ∈ [0, s5] (4.19)

6. a path angle adjustment Cz6x = lu2K1Rγ sin(s) + x6

y = lu2K2Rγ sin(s) + y6

z = Rγ cos(s) + z6

s ∈ [sV I , s6] (4.20)



z = a7(s− sV II) + b7

s ∈ [sV II , s7] (4.21)

This composition should guarantee enough degrees of freedom to make thepath reach the final point and to cope with the TW time space requirements.The motion primitive composition is chosen so that immediately after theinitial point the aircraft manoeuvres to head for the final point modifyingits initial heading and path angles, then it proceeds in straight line. The lasttwo path sections Cz6 and Cxy7 are needed to foster a proper approach tothe final point accordingly to the characteristics of next couple of TWs, seealso Section 4.3. The central heading adjustment Cxy5 is needed for a betterregulation of the path length.It remains clear that the initial an final point should be not too much badlyplaced: they should be at such a distance large enough that the turn radii Rand Rγ are sufficiently small to not hamper the reaching of the final point.Moreover the allowed time interval to reach the final point must be com-patible with the distance between the points and the range of admissibleaircraft speed.

The different path sections specified by equations (4.15)-(4.21) have to bejoined together enforcing the continuity of positions and velocities. This,together with the fact that each motion primitive is specified so as to satisfythe aircraft motion constraints, guarantees that the resulting path is feasibleto be followed by the aircraft, at least when covered at constant speed.

49


The position and velocity continuity conditions give the following systemof equations. Note that the continuity of velocities is imposed only on theratio among its different components. Indeed, the parametrization in thevariable s describes the path only, and hence it only specifies the ratio withwhich, the velocity V is split among the x, y, z directions. Note that in thisway the continuity of velocity for the resulting trajectory is always achievedby simply imposing a continuous speed V (t) by means of a proper time laws(t).

1. Initial condition - Cxy1

xI = R cos sI + x1 (4.22)

yI = lr1R sin(sI) + y1

zI = a1(sI − sI) + b1vzI√

v2xI + v2

yI

=a1

R(4.23)

vxIvyI

=−R sin(sI)

lr1R cos(sI)

2. Cxy1 - Cz2

R cos(s1) + x1 = k1lu1Rγ sin(sII) + x2 (4.24)

lr1R sin(s1) + y1 = k1lu1Rγ sin(sII) + y2

a1(s1 − sI) + b1 = Rγ cos(sII) + z2

−R sin(s1)

lr1R cos(s1)=− sin(s1)

lr1 cos(s1)=k1lu1Rγ cos(sII)

k1lu1Rγ cos(sII)=k1

k1

(4.25)

a1√(−R sin(s1))2 + (lr1R cos(s1))2

=a1

R=

=−Rγ sin(sII)√

(k1lu1Rγ cos(sII))2 + (k1lu1Rγ sin(sII))2=− sin(sII)

| cos(sII)|

Recall that lr1 ∈ −1, 1, lu1 ∈ −1, 1 and k21 + k2

1 = 1.

3. Cz2 - R3

k1lu1Rγ sin(s2) + x2 = qx (4.26)

k1lu1Rγ sin(s2) + y2 = qy

Rγ cos(s2) + z2 = qz

− sin(s2)

| cos(s2)| =mz√

m2x +m2

y

(4.27)

k1

k1

=mx

my

50

4.2. Path design

4. R3 - Cxy4

mxs3 + qx = R cos(sIV ) + x4 (4.28)

mys3 + qy = lr4R sin(sIV ) + y4

mzs3 + qz = a4(sIV − sIV ) + b4mz√

m2x +m2

y

=a4

R(4.29)

mx

my=− sin(sIV )

lr4 cos(sIV )

5. Cxy4 - R5

R cos(s4) + x4 = Qx (4.30)

lr4R sin(s4) + y4 = Qy

a4(s4 − sIV ) + b4 = Qz (4.31)a4

R=

Mz√M2x +M2

y

(4.32)

− sin(s4)

lr4 cos(s4)=Mx

My

6. R5 - Cz6Mxs5 +Qx = k2lu2Rγ sin(sV I) + x6 (4.33)

Mys5 +Qy = k2lu2Rγ sin(sV I) + y6

Mzs5 +Qz = Rγ cos(sV I) + z6

Mz√M2x +M2

y

=− sin(sV I)

| cos(sV I)|(4.34)

Mx

My=k2

k2

7. Cz6 - Cxy7

k2lu2Rγ sin(s6) + x6 = R cos(sV II) + x7 (4.35)

k2lu2Rγ sin(s6) + y6 = lr7R sin(sV II) + y7

Rγ cos(s6) + z6 = a7(sV II − sV II) + b7

− sin(s6)

| cos(s6)| =a7

R(4.36)

k2

k2

=− sin(sV II)

lr7 cos(sV II)

51


8. Cxy7 - Final Condition

R cos(s7) + x7 = xF (4.37)

lr7R sin(s7) + y7 = yF

a7(s7 − sV II) + b7 = zFa7

R=

vzF√v2xF + v2

yf

(4.38)

− sin(s7)

lr7 cos(s7)=vxFvyF

An additional condition must be eventually imposed so that the length D ofthe path is equal to desired one Dref :

D =√R2 + a2

1(s1 − sI) +Rγ(s2 − sII) +√m2x +m2

y +m2zs3+

+√R2 + a2

4(s4 − sIV ) +√M2

x +M2y +M2

z s5+

+Rγ(s6 − sV I) +√R2 + a2

7(s7 − sV II) = Dref . (4.39)

In order to find the path parameters we have to solve the quite intricate sys-tem of equations given by (4.22)-(4.39). We propose an effective approachin which only few parameters are set according to ad-hoc rules or retrievedby means of mono-dimensional search (which requires low computationaleffort), while the values of all the other parameters are analytically deter-mined.The idea is as follows. Since the initial and final conditions xI , yI , zI , xF ,yF , zF and vxI , vyI , vzI , vxF , vyF , vzF are given, if the direction param-eters lr1, lr7, lu1 and the parameters s1, s2, sV II were known, a solutionof a reduced system of equations obtained by discarding (4.31) and (4.39)from (4.22)-(4.39) could be analytically found as explained in full detailsin Section 4.2.4. In order to find the solution of the overall system, we havethen to find a method to satisfy also the two discarded equations (4.31) and(4.39) and to find the value of the parameters that we assumed to be known.To this purpose, the direction parameters lr1, lu1 are set according to logicrules as described in Section 4.2.3, we resort to gridding lr7 and sV II whilenumerical mono-dimensional search is used to find the values of s1 and s2

so as to satisfy equations (4.31) and (4.39). Note that gridding is not criti-cal being limited to two non critical variables, one of which takes only thevalues ±1. Clearly we could resort to gridding all the parameters but thiswould be computationally unaffordable; in particular the gridding of s1 ands2 is quite critical. Indeed s1 and s2 regulate the orientation of the straight

52

4.2. Path design

line segment R3 and a small error propagates along all the segment lengthinducing arbitrarily large errors. As for the mono-dimensional search werely on standard MATLAB search tools. However is necessary to providesuitable bounds for the range of the free variable to achieve effective solu-tions. This issue is discussed in Section 4.2.5.The overall algorithm to design a path between an initial and a final pointis specified by the following pseudo code.

53


Algorithm 1Input: initial and final position xI , yI , zI , xF , yF , zF and velocity

vxI , vyI , vzI , vxF , vyF , vzF , a desired length for the path Dref = V∆T

1: Pbest is empty εD,best = +∞ [Denotes the optimal path to be found]

2: for (sV II , lr7) ∈ SV II × −1, 1 [See Section 4.2.3]

3: Compute lr1, lu1 and define the s2 range S2 [See Section 4.2.3]

4: Perform a mono-dimensional search:

mins2∈S2

εZ

s.t. (P, εZ , εD) = F2(s2)

Denote with (P ?, ε?Z , ε?D) the solution

5: if ε?D = 0 ∧ ε?Z = 0 then6: Pbest = P ? εD,best = ε?D7: return Pbest εD,best [Solution found]

8: end if9: if ε?D > 0 ∧ ε?Z = 0 then

10: if Pbest is empty ∨ εD,best > ε?D then11: Pbest = P ? εD,best = ε?D12: end if13: end if14: end for15: if Pbest is not empty then16: return Pbest, εD,best [Path with lengthD 6= Dref (try a design a proper speed profile)]

17: else18: return [No feasible path found]

19: end if

(P, εZ , εD) = F2(s2)

1: Compute the s1 range S1(s2) [See Section 4.2.5]


mins1∈S1

εD

s.t. (P, εD) = F1(s1)

Denote with (P ?, ε?D) the solution3: P = P ? εZ = |(a4(s4 − sIV ) + b4)−Qz| εD = ε?D4: return the path P , the error on equation (4.31) εZ , and the error on equation (4.39) εD

(P, εD) = F1(s1)

1: Solve the system (4.22)-(4.39) without (4.31), (4.39) to determine P [See Section 4.2.4]

2: Compute the path length D3: εD = |Dref −D|4: return the current path P , and the error on the distance equation (4.39) εD

54

4.2. Path design

The most critical steps of the algorithm are explained in details in the fol-lowing sections.The algorithm can terminate in three possible states: i) a solution satisfy-ing (4.31) and (4.39) is found (line 7); ii) a solution satisfying (4.31) butnot (4.39) is found (line 16); iii) no solution satisfying (4.31) is found (line18). Note that if a solution satisfies (4.31) but not the distance equation(4.39) (ii) is still acceptable, however it requires to design a different speedprofile than V (t) = V to satisfy the TW time specification, Section 4.4will discuss this issue. On the contrary, a solution that do not satisfy (4.31)(iii) is infeasible because it shows a position discontinuity in the z directionbetween Cxy4 and R5. Note that there are 2 mono-dimensional searchesone nested in the other. This is due to the fact that the bound on s1 can becomputed only in correspondence of fixed values of s2.

4.2.3 Choice of lr1, lu1 and gridding of lr7 and sV IIThe parameter lr1 regulates whether the circumference in Cxy1 is coveredclockwise (CW) or counter-clockwise(CCW). Its choice is determined ac-cording to the following logic: lr1 is chosen so that projection of the pathsection Cxy1 onto x-y domain lies in the half plane determined by straightline xI + αvxI , yI + αvyI and that contains the final point xF , yF . Informulas:

lr1 = − sign(arctan 2(vyI , vyI)− arctan 2(yF − yI , xF − xI)).

This choice corresponds to make the path going toward the final point sincethe beginning. If the final point is on the boundary, namely it results lr1 = 0,then lr1 is chosen so that the path sectionCxy1 lies in the half plane that doesnot contain the vector [vxF , vyF ]:

lr1 = sign(arctan 2(vyI , vxI)− arctan 2(vyF , vxF )).

This choice should foster a proper approach to the final point. If again re-sults lr1 = 0, i.e. both the final point and the final velocity are aligned tothe initial velocity, then lr1 can be arbitrarily chosen.

The parameter lu1 regulates whether the path angle is increased or de-creased in Cz2.The choice of lu1 is based on the following reasoning: suppose to keep theinitial path angle as it is without making any adjustment in Cz2 and solvethe reduced system as described in Section 4.2.4 setting s2 = sII so that thevalue of lu1 is irrelevant. Then we can recover the right value of lu1 based on

55


the obtained error εZ on equation (4.31): if εZ is positive it means that thealtitude reached at the end of Cxy4 is higher with respect to the one reachedat the beginning of R5 (determined so as to reach the final position and thefinal velocity), and hence a decrease of the path angle in Cz2 is needed. Onthe contrary, if εZ is negative it means that the altitude reached at the endof Cxy4 is lower with respect to the one reached at the beginning of R5 and,hence, an increase of the path angle in Cz2 is needed. More precisely,

1: a1 = RvzI√v2xI+v2xI

2: sII = s2 = arctan 2(−a1, R)3: compute the s1 range S1(s2) [See Section 4.2.5]


mins1∈S1

εD

s.t. (P, εD) = F1(s1)

Denote with (P ?, ε?D) the solution5: P = P ? εZ = |(a4(s4 − sIV ) + b4)−Qz| εD = ε?D6: lu1 = sign((a4(s4 − sIV ) + b4)−Qz)7: if lu1 = 1 then8: S2 = [sII , −γmin]9: else

10: sII = π − sII11: S2 = [sII , π + γmax]12: end if

The function F1(s1) is defined as in algorithm 1. The proper range S2 fors2 is determined according to the sign of lu1 so as to keep the path anglein the feasible range. Note that if it results lu1 = 0 it means that the pathobtained skipping Cz2 (sII = s2) is already feasible, being equation (4.31)already satisfied.

As for the choice of lr7 and sV II we resort to gridding. The first point inthe grid SV II corresponds to sV II = s7 (see (4.50)), namely we start tryingto skip Cxy7: the value of lr7 is, in this case, irrelevant. If, by means of thischoice a feasible path is not found, we proceed by taking decreasing valuesfor sV II which corresponds to increasing length of Cxy7. For each choice ofthe length of Cxy7 we try both CW and CCW way, setting lr7 = 1 and lr7 =−1 accordingly. The feasible path with the shortest Cxy7 should be found,otherwise we stop when the length of Cxy7 covers a whole circumference.

56

4.2. Path design

4.2.4 System resolution

Assuming that the initial and final conditions xI , yI , zI , xF , yF , zF andvxI , vyI , vzI , vxF , vyF , vzF are given, and that the direction parameterslr1, lr7, lu1, lu2 and s1, s2, sV II are known, it is possible to analyticallycompute the value of all the other path parameters by means of the solutionof system (4.22)-(4.39), where (4.31) and (4.39) are discarded2.

The equations are solved starting from the initial condition and proceeding forward in the path.

sI = arcsin(−vxI√v2xI + v2

yI

) (4.40)

if cos(sI)−vyI√

v2xI + v2

yI

6= 0 then sI = π − sI

if lr1 = −1 then sI = −(sI + π)

a1 = RvzI√

v2xI + v2

yI

(4.41)

x1 = xI −R cos(sI) (4.42)

y1 = yI − lr1R sin(sI)

b1 = zI

sII = arctan 2(−a1, R) (4.43)

if lu1 = −1 then sII = π − sII

if lr1 = +1 then k1 = − sin(s1) k1 = cos(s1) (4.44)

if lr1 = −1 then k1 = − sin(π − s1) k1 = cos(π − s1)

x2 = R cos(s1) + x1 − k1lu1Rγ sin(sII) (4.45)

y2 = lr1R sin(s1) + y1 − k1lu1Rγ sin(sII)

z2 = a1(s1 − sI) + b1 −Rγ cos(sII)

mx = k1 (4.46)

my = k1

mz =− sin(s2)

| cos(s2)|2Some of the system equations admits multiple solutions, logic rules are needed to choose the right one.

E.g. equations whose solution is an angle may admit multiple solutions but only one of them corresponds to aconvenient path.

57


qx = k1lu1Rγ sin(s2) + x2 (4.47)

qy = k1lu1Rγ sin(s2) + y2

qz = Rγ cos(s2) + z2

a4 = mzR (4.48)

From this point on the equations are solved from the final condition going backward in the path.

if vzF is defined then a7 =RvzF√v2xF + v2

xF

(4.49)

if vzF is not defined then vzF = a7 = a4

Whether vzF is defined or not is explained in Section 4.3.

s7 = arcsin(−vxF√v2xF + v2

yF

) (4.50)

if cos(s7)− vyF√v2xF + v2

yF

6= 0 then s7 = π − s7

if lr7 = −1 then s7 = −(s7 + π)

x7 = xF −R cos(s7) (4.51)

y7 = yF − lr7R sin(s7)

b7 = zF − a7(s7 − sV II)

s6 = arctan(−a7, R) (4.52)

if − sin(s2) ≤ − sin(s6) then lu2 = 1

else lu2 = −1 s6 = π − s6

Note that lu2 is defined according to whether the path angle resulting from s2 should be increased ordecreased to reach the path angle imposed by the final condition on the vertical velocity representedthrough s6.

if lr7 = +1 then k2 = − sin(sV II) k2 = cos(sV II) (4.53)

if lr7 = −1 then k2 = − sin(π − sV II) k2 = cos(π − sV II)

Mx = k2 (4.54)

My = k2

58

4.2. Path design

lr4 = − sign(arctan(my,mx)− arctan(My,Mx)) (4.55)

if lr1 = lr4 then sIV = s1

else if lr1 = 1 then sIV = −(s1 + π)

else sIV = π − s1

Note that lr4 is chosen accordingly to the shortest feasible path given the orientation of the segmentsR3 and R5 in the x-y plane provided by the parameters mx, my, Mx, My . In general othersolutions may exist.

if lu1 = lu2 then sV I = s2 (4.56)

else then sV I = π − s2

Mz =− sin(sV I)

| cos(sV I)|(4.57)

if lr4 = lr7 then s4 = sV II (4.58)

else if lr7 = 1 then s4 = −(sV II + π)

else s4 = π − sV IIwhile s4 < sIV s4 = s4 + 2π

while s4 > sIV + 2π s4 = s4 − 2π

If the last line is ignored we still get a feasible solution corresponding to a path in which the circum-ference in Cxy4 is completely covered at least once. However we stick to the solution in which theshortest arc is covered.

x6 = R cos(sV II) + x7 − k2lu2Rγ sin(s6) (4.59)

y6 = lr7R sin(sV II) + y7 − k2lu2Rγ sin(s6)

z6 = b7 −Rγ cos(s6)

The following linear system joins together the initial forward computed part of path with the finalbackward computed part of the path[mx my

Mx My

][s3

s5

]=

[k2lu2Rγ sin(sV I) + x6 − qx −R(cos(s4)− cos(sIV ))

k2lu2Rγ sin(sV I) + y6 − qy − lr4R(sin(s4)− sin(sIV ))

]. (4.60)

Note that the matrix is not invertible when the segment R3 and R5 belong to the same straight line:in this case s3 and s5 can be arbitrarily chosen within a certain range. Furthermore note that s3 ands5 must be always non-negative: this issue is addressed in Section 4.2.5.

x4 = mxs3 + qx −R cos(sIV ) (4.61)

y4 = mys3 + qy − lr4R sin(sIV )

b4 = mzs3 + qz

59


Qx = k2lu2Rγ sin(sV I) + x6 −Mxs5 (4.62)

Qy = k2lu2Rγ sin(sV I) + y6 −Mys5

Qz = Rγ cos(sV I) + z6 −Mzs5

The discarded equation (4.31) that must be satisfied to get a feasible path,rewrites:

a4(s4 − sIV ) +mzs3 + qz = Rγ cos(sV I) + z6 −Mzs5. (4.63)

If (4.31) is not satisfied the path shows a position discontinuity in the z-axis between Cxy4 and R5 that are not properly joined together. Equation(4.39) that has also been discarded in the system resolution, has not to benecessarily satisfied to get a feasible path since it enforces a condition onthe path length only. The satisfaction of both equation (4.31) and (4.39)is addressed in algorithm (1) by means of two nested mono-dimensionalsearches.The solution of the system, together with the logic rules of Section 4.2.3 andthe mono-dimensional searches, determines all the path parameters. Notethat all the parameters but si i = 1, 2, . . . , 7 are needed to make each ofthe motion primitives start at the right position with proper derivatives. Theparameters si i = 1, 2, . . . , 7, instead, determine the length of each motionprimitive and are needed to satisfy conditions on the whole path such as thepath length and the reaching of the final point with proper derivatives.The directions parameters may give origin to multiple solutions and theyshould be set so as to obtain convenient path (e.g. so as to avoid to turn 270

right in order to turn 90 left). Only sV II acts as a partial degree of freedomthat can be chosen within a certain admissible range, note, however, that insome situations the presence of Cxy7 is strictly necessary: e.g. when initialand final points and velocities are aligned.

4.2.5 Bounds for s1

In this section we compute suitable bounds for the values of s1 so as toachieve affective solutions to the system in (4.22)-(4.39), more precisely toobtain that the resulting s3 and s5, corresponding to the length of R3 andR5 are non-negative. Hence we focus on:[

mx my

Mx My

][s3

s5

]=

[k2lu2Rγ sin(sV I) + x6 − qx −R(cos(s4)− cos(sIV ))

k2lu2Rγ sin(sV I) + y6 − qy − lr4R(sin(s4)− sin(sIV ))

].

60

4.2. Path design

Assuming that the matrix is invertible and replacing the terms according toequations (4.22)-(4.39) we obtain:mx = − sin(s1) my = lr1 cos(s1) Mx = sin(sV II) My = lr7 cos(sV II)

k2 = − sin(sV II) k2 = lr7 cos(sV II)

sin(sV I) = sin(s2) sin(s4) = sin(sV II) sin(sIV ) = sin(s1)

cos(s4) = lr4lr7 cos(sV II) cos(sIV ) = lr1lr4 cos(s1)

qx = − sin s1lu1Rγ(sin(s2)− sin(sII)) + x1 + R cos(s1)

qy = lr1 cos(s1)lu1Rγ(sin(s2)− sin(sII)) + y1 + lr1R sin(s1)

x6 = x7 + R cos(sV II) + sin(sV II)lu2Rγ sin(s6)

y6 = y7 + lr7R sin(sV II)− lr7 cos(sV II)lu2Rγ sin(s6)

[s3

s5

]=

1

lr1 cos(s1) sin(sV II)− lr7 cos(sV II) sin(s1)

[lr7 cos(sV II) sin(sV II)

−lr1 cos(s1) − sin(s1)

]

− sin sV II lu2Rγ sin(s2 − s6) + x7 + R cos sV II + sin s1lu1Rγ(sin s2 − sin sII)+

−x1 − R cos s1 − R(lr4lr7 cos sV II − lr1lr4 cos s1)

lr7 cos sV IIu2Rγ(sin(s2 − s6) + y7 + lr7R sin sV II − lr1 cos s1lu1Rγ(sin s2 − sin sII)+

−y1 − lr1R sin s1 − lr4R(sin sV II − sin s1)

Note that the determinant ∆ = lr1 cos(s1) sin(sV II)− lr7 cos(sV II) sin(s1)is null when the segment R3 and R5 in the x-y plane are parallel or anti-parallel. Let:

c1 = − sin sV II lu2Rγ sin(s2 − s6) + x7 + R cos(sV II)− x1 − Rlr4lr7 cos(sV II)

c2 = lr7 cos(sV II)u2Rγ(sin(s2 − s6) + y7 + lr7R sin(sV II)− y1 − lr4R sin(sV II).

Performing some computations it results:

s3 =

(lr7 cos(sV II)c1 + sin(sV II)c2 + sin(s1)

(lr7 cos(sV II)lu1Rγ(sin(s2)− sin(sII))

− sin(sV II)R(lr1 − lr4))

+ cos(s1)(− lr7 cos(sV II)R(1− lr1lr4)

− sin(sV II)lr1lu1Rγ(sin(s2)− sin(sII))))

/∆

s3 =K1 + α1 sin(s1) + β1 cos(s1)

∆≥ 0 (4.64)

s5 =(− lr1 cos(s1)c1 − lr1 cos(s1) sin(s1)lu1Rγ(sin(s2)− sin(sII)) + lr1 cos

2(s1)R(1− lr1lr4)

− sin(s1)c2 + lr1 cos(s1)c1 − lr1 cos(s1) sin(s1)lu1Rγ(sin(s2)

− sin(sII)) + sin2(s1)R(lr1 − lr4)

)/∆

=(− lr1 cos(s1)c1 − sin(s1)c2 + R(lr1 − lr4)

)/∆

s5 =K2 + α2 sin(s1) + β2 cos(s1)

∆≥ 0. (4.65)

61


We can study the sign of s3 and s5 as function of s1 exploiting inequalities(4.64) and (4.65). Inequalities of the form a cos(x) + b sin(x) ≥ c, such asthe numerator and the denominator of (4.64) and (4.65) are, can be solvedby means of the normalization method. Hence we can found the range ofvalues for s1 that ensure s3 and s5 to be non-negative.A geometrical interpretation of the bounds for s1 is provided in Figure 4.1:s1 regulates the length of the arc of circumference in Cxy1 and thus theorientation of the straight line corresponding to R3; s1 should be such thatthe intersection between the straight lines in which R3 and R5 lie, is inthe half-lines corresponding to positive s3 and s5. The orientation of R5

depends on the choice of sV II that has to be fixed in advance. Note thatwhen s1 takes non-admissible values, the intersection point moves on thenegative half-line of R3, R5 or both, making the resulting path infeasible.This simplified geometric interpretation is not entirely accurate due to thefact that the presence of Cz2, Cz6 and of the arc of circumference Cxy4

between R3 and R5 should be accounted for, as it is done in the boundcomputation.

62

4.2. Path design

(a) Infeasible solution: s1 = 0, s3 > 0, s5 < 0. (b) Infeasible solution: s1 > 0, s3 > 0, s5 < 0.Infeasibility is due to the presence of Cxy4.

(c) Feasible solution: s1 > 0, s3 > 0, s5 = 0. (d) Feasible solution: s1 > 0, s3 > 0, s5 > 0.

(e) Infeasible solution: R3 and R5 become anti-parallel.

(f) Infeasible solution: s1 > 0, s3 < 0, s5 > 0.

Figure 4.1: Choice of s1.

Note that in the computation of the bounds on s1 we assume that the ini-tial and final conditions, and the values for lr1, lr4, lr7, lu1 and s2 areknown. All of them but lr4 can be fixed according to the methods discussedin Sections 4.2.3 and 4.3. As for lr4 we have to guess its value and verify aposteriori that the guess lr4 is coherent with the value of lr4 obtained from(4.55). In view of this the function F2(s2) in algorithm (1) should be moreproperly written as:

63


(P, εZ , εD) = F2(s2)

1: Set the guess lr4 = 12: Compute the s1 range S1(s2, lr4)3: Perform a mono-dimensional search:

mins1∈S1

εD

s.t. (P, εD) = F1(s1)

Denote with (P ?, ε?D) the solution4: if (lr4 in P ?) 6= lr4 then5: Set the guess lr4 = −16: Compute the s1 range S1(s2, lr4)7: Perform a mono-dimensional search:

mins1∈S1

εD

s.t. (P, εD) = F1(s1)

Denote with (P ?, ε?D) the solution8: end if9: P = P ? εZ = |(a4(s4 − sIV ) + b4)−Qz| εD = ε?D

10: Return current path P , and error on equation (4.31) εZ

4.3 Choice of initial and final conditions

We can generate a path that covers multiples TWs repeatedly applying theapproach described above for each couple of consecutive TWs. For eachcouple of consecutive TWs we set the path initial and the final positionsequal to the space centres of the considered TWs, and the desired lengthof the path Dref = V∆T , where ∆T is the time interval between the timecentres of the TWs. Alternative approaches to set the value of Dref may beconsidered, their discussion is postponed in Appendix C for sake of clarity.The initial and final velocities direction in the x-y plane, are chosen as aweighted average of the directions given by the straight lines that join eachTW centre with the previous and following ones, see Figure 4.2.

64

4.3. Choice of initial and final conditions

1

2

3

𝑣𝑠𝑙,1

𝑣𝑠𝑙,2

𝑣𝐹 = 𝛼𝑣𝑠𝑙,1 + (1 − 𝛼)𝑣𝑠𝑙,2

Figure 4.2: Choice of vxF and vyF .

The weight α is set so that the more a TW centre is critical to be reachedwith respect to the previous and the following one, the more the boundaryvelocity for the current TW centre is aligned with the straight line direc-tion toward the next TW centre. The idea is that when the initial velocityheads directly toward the final point a shorter path should be designed. Theevaluation of the criticism of couples of consecutive TWs is based on theratio between their distance and the time allowed to cover it. The mathe-matical computation of the velocity direction and of the weight is given inalgorithm 2.The vertical velocity vzF is assigned so that when two consecutive TWs areplaced at the same altitude the path between them lies at that altitude too,and no path angle adjustments are made. Hence when TWs are placed atthe same altitude the vertical velocity vzF is set to 0. On the contrary whenTWs are not at the same altitude the final vertical velocity is not assignedand the second path angle adjustment in Cz6 is skipped (s6 = sV I): thevertical velocity direction remains defined as the one imposed at the end ofCz2 by s2, as needed to reach the final point.The procedure to set the boundary conditions, to be repeated for each cou-ple of consecutive TWs, is reported in algorithm 2.

65


Algorithm 2Input: a set of TWs Tw.Input: initial velocities vxI , vyI , vzI

1: for j = 1, . . . , |Tw| − 12: [ xI yI zI tI ] = centre of TWj

3: [ xF yF zF tF ] = centre of TWj+1

4: if j < |Tw| − 1 then5: [ xN yN zN tN ] = centre of TWj+2

6: else7: [ xN yN zN tN ] = 2 · ( centre of TWj+1)− centre of TWj

8: end if9: Vsl = ‖[xI ,yI ,zI ]−[xF ,yF ,zF ]‖2

tF−tI − V10: Vsl+ = ‖[xF ,yF ,zF ]−[xN ,yN ,zN ]‖2

tN−tF − V11: if VslVsl+ ≥ 0 then12: α = Vsl+

Vsl+Vsl+13: else

14: α =

0 if Vsl < Vsl+

1 if Vsl > Vsl+15: end if

16:

[vxF

vyF

]= α [xF , yF ]−[xI , yI ]

‖[[xF , yF ]−[xI , yI ]]‖2 + (1− α) [xN , yN ]−[xF , yF ]‖[[xN , yN ]−[xF , yF ]]‖2

17: if zI = zF ∨ zF = zN then18: vzF = 019: else20: vzF is not defined.21: end if22: Dref = V (tF − tI)23: Generate the path as in algorithm (1)24: if vzF is not defined then25: vzF = a7

R ‖[vxF , vyF ]‖226: end if27: [ vxI vyI vzI ] = [ vxF vyF vzF ]28: end for

Note that the method with which the boundary velocity is assigned pro-vides a link between the different path designs for each couple of consec-utive TWs. This allows for a design that accounts also for the next sectionrequirements: indeed a not critical couple of TWs is exploited to approachits final point with a velocity direction that fosters the design of a good pathfor the following more critical section and, vice versa, in a critical sectionthe final point is approached accounting for the current needs relying on theless criticism of the next couple of TWs. This becomes clear e.g. when thecurrent time specification enforce to cover the distance between two points

66

4.4. Time law design

in straight line: in this case it is necessary in the previous section to ap-proach the final point with a velocity already headed toward the final pointof the current section. Following the same reasoning the initial velocity ofthe next section is determined accordingly to the needs of the current criti-cal section. It is clear that in most cases a proper boundary velocity shouldbe a compromise between the straight line directions of the previous andthe next sections.

4.4 Time law design

In this section we design a time law to be applied to the path to obtain atrajectory.In the previous sections we try to design for each couple of consecutiveTWs a path with a length D = V∆T such that it steers the aircraft fromthe initial to the final position with the right timing ∆T when covered atconstant speed V . If this is the case it is enough to use a constant speedprofile setting V (t) = V . The corresponding time law s(t) is obtainedfrom equations (4.3), (4.6) or (4.12) for the different motion primitives.On the contrary, when the designed path has a length D different from thedesired one Dref = V∆T , the path cannot be covered at constant speed Vin order to satisfy the time specification. Hence we try to design a PWAspeed profile that allows to meet the time specification.

4.4.1 PWA speed profile

In order to design a PWA speed profile to be applied to a path between twoconsecutive TWs, with length D to be covered in a time interval ∆T , weadopt the following parametrization for V (t):

V (t) =

V1 t ∈ [0, t1)

α1(t− t1) + β1 t ∈ [t1, t1 + t2)

Vc t ∈ [t1 + t2, t1 + t2 + t3)

α2(t− t3 − t2 − t1) + β2 t ∈ [t1 + t2 + t3, t1 + t2 + t3 + t4)

V2 t ∈ [t1 + t2 + t3 + t4, t1 + t2 + t3 + t4 + t5)

.

(4.66)

The speed profile is composed by three intervals in which the speed is keptconstant and by two intervals covered at constant acceleration. We considerthat the initial and final velocities V1 and V2 are set as explained in Section4.4.2, and the absolute values of accelerations |α1| and |α2| are fixed. Weenforce conditions to obtain continuity of velocity:

V1 = β1 α1t2 + β1 = Vc Vc = β2 α2t4 + β2 = V2. (4.67)

67


We ensure the satisfaction of the time specification enforcing that the speedprofile covers a distance equal to the path length D in the allowed timeinterval ∆T :

∆T = t1 + t2 + t3 + t4 + t5 (4.68)

D = V1t1 + β1t2 + α1t222

+ Vct3 + β2t4 + α2t242

+ V2t5. (4.69)

Moreover we enforce a symmetric condition:

t1 = t5. (4.70)

We can find the parameters that define the speed profiles as function of Vcby solving system in (4.66)-(4.70):

β1 = V1

β2 = Vc

α1 = sign(Vc − V1)|α1|α2 = sign(−Vc + V2)|α2|

t2 =Vc − V1

α1

t4 =−Vc + V2

α2

t1 =D − V1t2 + α

t222

+ vc(∆T − t2 − t4) + Vct4 + α2t242

V1 + V2 − 2Vct5 = t1

t3 = ∆T − 2t1 − t2 − t4.

The remaining degree of freedom Vc is dealt with so as to minimize (2t1 −t3)2, this should foster the balance between the lengths of the intervals cor-responding to constant speed and the smoothness of the designed speed pro-file. The optimization is performed by means of mono-dimensional search:the variable Vc is bounded so as to take values only in the admissible range[Vmin, Vmax], and to find effective solutions. To this purpose, consider thedistance that it is covered in the time interval ∆T at average speed betweenV1 and V2:

Davg =V1 + V2

2∆T ,

68

4.4. Time law design

then assign bounds on Vc as follow:

if Davg > D then Vc ∈ [Vmin,V1 + V2

2]

if Davg < D then Vc ∈ [V1 + V2

2, Vmax].

The idea is that when the distance Davg covered at average speed is shorterthan path length D the speed should be increased, on the contrary when thedistance Davg is longer than path length D the speed should be decreased.It is worth noticing that by means of this approach we can ensure the sat-isfaction of the constraints on TAS (properly choosing V1, V2 and Vc) andthe constraint on the longitudinal acceleration (properly choosing |α1| and|α2|). The constraint on the path angle is automatically satisfied becauseit depends on the path only, the constraint on the bank angle can be satis-fied choosing R large enough. On the other hand, the constraints on thevertical acceleration and on the engine thrust are not explicitly accountedfor in the design of the PWA speed profile because they depends both onthe speed profile phase (constant speed or constant acceleration) and on thecorresponding motion primitive in the path. This fact make enforcing con-straints on them quite difficult, however it remains clear that no constraintviolations may occur at constant speed.

4.4.2 Choice of initial and final conditions for the speed profile

The choice of the boundary conditions for the speed profile between eachcouple of consecutive TWs is described in algorithm (3). Whether the de-signed path can be covered at constant speed V , its boundary velocities areset to V as well, so that the constant speed profile is preserved. If this is notthe case, the boundary speed is set as the mean of the average speed of theprevious and the following path sections.

69


Algorithm 3Input: a set of path PInput: a reference velocity V

1: V1 = V2: for j = 1, 2, . . . |P|3: Vact =

D(j)

∆T,(j)

4: if j < |P| then5: Vnext =

D(j+1)

∆T,(j+1)

6: else7: Vnext = V8: end if9: if Vact = V ∨ Vnext = V then

10: V2 = V11: else12: V2 = Φ[Vmin,Vmax](

Vact+Vnext2 ) [Φ[a,b](·) denotes the saturation function between a and b.]

13: end if14: if V1 = V2 = V then15: Use a constant speed profile V (t) = V16: else17: Use a PWA speed profile18: end if19: V1 = V2

20: end for

4.4.3 Conversion from s to t

In this section it is discussed how to recover the position on a trajectory(composed by a path and a time law) corresponding to a certain time instantt.We can compute the covered distance in two ways: integrating the pathwith respect to the variable s or integrating the velocity profile with respectto the time t.∫

V (τ)dτ =

V t+ const. when constant speed

βt+ α t2

2+ const. when constant acceleration

= D(t)

(4.71)∫ s

sI

‖[x′(σ) y′(σ) z′(σ)]‖2dσ = (4.72)

=

√R2 + a2(s− sI) + const. for Cxy

Rγ(s− sI) + const. for Cz√m2x +m2

y +m2z(s− sI) + const. for R

= D(s).

70

4.5. Numerical results

Hence we can find the s that corresponds to a given time instant t from:

D(s) = D(t).

Once s has been found one can simply evaluate the path at s to find theposition corresponding to time t on the trajectory.x(t)

y(t)

z(t)

=

x(s)

y(s)

z(s)

.Note that in order to generate the reference for the MPC controller we haveto find the position in correspondence of every sample time instants kTs.Note that, thanks to the fact that the integrals can be analytically solvedand takes easy expressions in s and t, see (4.71) and (4.72), we can exactlyfound the s corresponding to a t: some care has to be taken to manage thechanges of section (and hence of the expression taken by the integral) in thepath and in the speed profiles. Once the distance covered by the velocityprofile at a certain time t is computed we can integrate the path sectionone by one and until the same distance is covered by the path and find thecorresponding value for s. The integration can be done as follow:

• if at the current path section the distanceD is not reached, we integratethe current path section and proceed to the following section

• if at the current path section the distance D is exceeded, we can findthe value of s corresponding the distance D solving for s the properexplicit expression of the integral in (4.72) (s of course will belong tothe current path section).

Then we have just to evaluate the path in correspondence of the so foundvalue of s to get x(kTs), y(kTs) and z(kTs).

4.5 Numerical results

In this section, we report some trajectories designed by means of the ap-proach described in Sections 4.2 and 4.4. The considered sets of TWs re-quire to pass through points that are quite close and bad placed with respectto the usual aircraft operations but they are chosen so as to test the proposeddesign approach. Note that, however, the points cannot be closer than thedistance required by the aircraft to turn and to change altitude representedthrough the parameters R = 8km and Rγ = 41km. We set V = 850km/h.

71


The sets of TWs for which we want to design a reference trajectory arereported in Table 4.1.

Table 4.1: TW sets.

(a)

TW xc [km] yc [km] zc [km] tc [s]

1 -40 0 3 02 0 0 4 1803 40 60 5 5004 60 -20 5 9505 100 0 4 11506 140 0 4 13407 140 40 3 15208 140 60 3 16209 100 80 4 1820

(b)


1 -40 0 3 02 0 0 4 1703 40 60 5 5004 60 -20 5 9305 100 0 4 11156 140 0 4 13507 140 40 3 15158 140 60 3 16009 100 80 4 1820

(c)


1 -60 -60 6 02 -40 -30 5 1603 0 -30 4 3504 -15 0 5 5205 20 0 5 7206 40 20 4 8407 60 40 4 9608 30 70 4 11709 40 100 5 1340

10 60 80 4 160011 70 20 3 178012 60 -20 2 1960

The trajectory designed for the fisrt TW set in Table 4.1(a) is depicted inFigure 4.3, in this case a constant speed profile is always feasible.

72


−40 −20 0 20 40 60 80 100 120 140 160−40

−20

0

20

40

60

80

x [km]

y [k

m]

(a) x− y view

−50

0

50

100

150

−40−20

020

4060

80

3

3.5

4

4.5

5

y [km]x [km]

z [k

m]

(b) 3D view

Figure 4.3: Designed path. Cxy blue, Cz red, R green, TW centre magenta.

73


The designed path is quite smooth, as it is clear, even though the points 6,7, 8 are aligned in the x-y plane the path among them is not a straight line:this is due to the fact that the path is designed so as to satisfy the time spec-ification when covered at constant speed. Note that being the points 3 and 4at the same altitude, the path between them lies at that altitude, as allowedby the presence of path angle adjustments in the previous and followingpaths.

A second design is done considering the same set of TWs of the first ex-ample, but with a different time schedule, as reported in Table 4.1(b): inthis case a PWA speed profile is required to met time specifications. Thedesigned path and the speed profile are reported in Figure 4.4 and 4.5 re-spectively.

74


−40 −20 0 20 40 60 80 100 120 140 160−40

−20

0

20

40

60

80

x [km]

y [k

m]

(a) x− y view

−50

0

50

100

150

−40

−20

0

20

40

60

80

3

3.5

4

4.5

5

y [km]

x [km]

z [k

m]

(b) 3D view

Figure 4.4: Designed path. Cxy blue, Cz red, R green, TW centres magenta.

75


0 200 400 600 800 1000 1200 1400 1600 1800 20000.22

0.225

0.23

0.235

0.24

0.245

0.25

t [s]

V [k

m/s

]

Figure 4.5: Designed velocity profile in blue, V red, tc green.

Note that according to the new time specification the path among the points6, 7, 8 becomes almost a straight line so as to reduce the path length asmuch as possible. This is not enough to reach point 7 from point 6 in time,and, hence, it is necessary to increase the speed as seen in the velocity pro-file, Figure 4.5. The same happens between point 4 and 5, which are joinedby a straight line path. Note that also between points 1 and 2 the path is theshortest possible given the initial velocity. On the contrary, accordingly tothe new time specification, the path between point 8 and 9 is longer thanthe one designed in Figure 4.3, and also a decrease of speed is needed tosatisfy the new time requirement. Note that the design of shorter (or short-est) paths between the most critical couples of TWs is allowed by a suitablechoice of the boundary velocity direction, which accounts for the tightnessof the time space requirements of the previous and of the following coupleof TWs.

A third design is performed considering the TW set reported in Table 4.1(c).The achieved path is depicted in Figure 4.6, the corresponding velocityprofile is constant.

76


−60 −40 −20 0 20 40 60 80 100−60

−40

−20

0

20

40

60

80

100

120

x [km]

y [k

m]

(a) x− y view

−60 −40 −20 0 20 40 60 80 100−100

−50

0

50

100

150

2

2.5

3

3.5

4

4.5

5

5.5

6

y [km]

x [km]

z [k

m]

(b) 3D view


77


The designed path, also in this case, is quite smooth. As one can see, whenconsecutive points are at the same altitude the path does not present anypath angle adjustments, instead, when some consecutive points are not atthe same altitude the second path angle adjustment Cz6 is skipped. Notethat both path angle adjustments are exploited between points 5 and 6 soas to avoid path angles modifications in the previous and the following sec-tions.

The same set of TWs is considered for the design of a trajectory in whichthe objective is to achieve the shortest path: this can be done simply settingDref = 0 in the design procedure. Of course, in this case, it is needed toheavily rely on PWA speed profiles to satisfy the given time schedule.The resulting path and speed profile are depicted in Figure 4.7 and 4.8 re-spectively.

78


−60 −40 −20 0 20 40 60 80 100−60

−40

−20

0

20

40

60

80

100

120

x [km]

y [k

m]

(a) x− y view

−60 −40 −20 0 20 40 60 80 100−100

−50

0

50

100

150

2

2.5

3

3.5

4

4.5

5

5.5

6

y [km]

x [km]

z [k

m]

(b) 3D view


79


0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

t [s]

V [k

m/s

]

Figure 4.8: Designed velocity profile in blue, V red, tc green.

The designed path is even smoother than the ones in the previous exam-ples, however, this comes at the price that the required speed profile ismuch more irregular: the avoided manoeuvres in the path are replaced bythe need of many speed adjustments which make the overall trajectory notconvenient. Note that, as expected, the path is mainly composed by straightlines joined with arcs of circumference in correspondence of the TWs cen-tres. Also path angles adjustments are needed: in the section between points3 and 4 the path has to be slightly lengthened to allow for the needed pathangle adjustments. Indeed the initial negative path angle has to be increasedso as to climb to reach the proper altitude, and, then, the path angle is takenback to 0 as required to avoid adjustments in the following path section.Note that this effect is due to the fact that the points are quite close eachother with respect to the turning radius R.

80

CHAPTER5Tracking problem formulation and solution

5.1 Finite horizon optimization problem

Let (xR1 , xR2 , x

R3 ), and (xR1 , x

R2 , x

R3 ) denote the reference trajectory and ve-

locity that the aircraft is required to track. We assume that they have beensuitably designed as described in Chapter 4 so as to be compatible withthe aircraft motion capabilities and to meet TW specifications. Given thatthe aircraft dynamics is affected by wind, the aircraft may deviate from thereference trajectory, which motivates the introduction of an MPC controllerthat steers it back and makes it following the reference trajectory robustlywith respect to the wind disturbance, compatibly with the aircraft physicallimitations and passenger comfort constraints that are explicitly accountedfor.To this purpose, we need to formulate a finite horizon optimization problemwhere a suitable cost function is chosen and additional constraints on theaircraft position are included besides those discussed in Section 3.3 on theaircraft motion capability.Given the current time k, we define at each time k + i, i = 0, . . . ,M theposition error ξk+i as the difference between aircraft position x1,k+i, x2,k+i,x3,k+i and reference position xR1,k+i, x

R2,k+i, x

R3,k+i expressed in longitudinal,

81

Chapter 5. Tracking problem formulation and solution

lateral and vertical components with respect to the reference trajectory:

ξk+i = Rz(ψR,k+i)

x1,k+i

x2,k+i

x3,k+i

−x

R1,k+i

xR2,k+i

xR3,k+i

with

Rz(ψR,k+i) =

cos(ψR,k+i) sin(ψR,k+i) 0

− sin(ψR,k+i) cos(ψR,k+i) 0

0 0 1

,where the reference heading angle ψR,k+i can be easily recovered from thereference trajectory velocity as ψR,k+i = arctan(

xR,k+i

yR,k+i).

The position error ξk+i should be kept below some suitably chosen thresh-old. Note, however, that the error ξ depends additively on the wind dis-turbance w through x1, x2, and x3 (see equation (3.32)) and that w hasunbounded support. Hence, it is not possible to enforce a robust constraintwhich holds for every and each disturbance realization of w. We, instead,resort to a probabilistic constraint, namely a constraint that has to be satis-fied only with a given (high) probability 1− ε. That is:

P‖ξk+i(wk, . . . ,wk+i−1)‖22 ≤ hi, i = 1, . . . ,M ≥ 1− ε, (5.1)

where the squared 2-norm of the error is considered so as to bound the dis-tance of the aircraft from the reference position. A delicate issue here arisesin the choice of thresholds hi, i = 1, . . . ,M . Constraint (5.1), indeed, maybe infeasible if thresholds are not compatible with the disturbance char-acteristics, the other constraints on the input and on the state, the systeminitial condition and the allowed violation ε. Moreover a priori verifyingcompatibility is quite difficult and would be quite sensitive to model errorof wind disturbance. Possible approaches to address this issue are discussedin Appendix A. Here, we follow the approach in which rather than seeingthem as fixed thresholds, hi, i = 1, . . . ,M , are regarded as variables to beminimized along with other possible objectives. More precisely, the costfunction J of the finite horizon optimization problem is defined as the sumof two terms: one that depends on the input acceleration u only and ac-counts for fuel consumption and passenger comfort, and a second term thataccounts for the position error thresholds hi, i = 1, . . . ,M :

J =M−1∑i=0

µiuduTk+iRuk+i + µhc

M∑i=1

µi−1hd hi, (5.2)

82

5.1. Finite horizon optimization problem

where R, µhc, µud, µhd are weights that are now discussed. The weightingmatrix R is chosen as follows:

R = RTrotR

TnorRcRnorRrot,

where

Rrot =

cosψk sinψk 0

− sinψk cosψk 0

0 0 1

and Rnor =

1aL

0 0

0 1g tan φ

0

0 0 1aN

.In other words, Rrot is a rotation matrix that transforms u1 and u2 (namely,the accelerations along the x and y axes) into the longitudinal and lateral ac-celerations with respect to the initial value of the heading angle ψk, whereasRnor is a normalization matrix, chosen according to the limits on acceler-ations. Eventually, matrix Rc allows one to weight the longitudinal andlateral accelerations, as well as the vertical acceleration, which are directlyrelated to fuel consumption and comfort. The weight matrix Rc, togetherwith weight µhc regulates the relative importance given to input and posi-tion error, so as to achieve a proper trade-off between saving the control in-put and keeping the position error small, while the weights µud, µhd ∈ (0 1]2

can be used to give greater importance to the first time steps, which are themost important for the actual aircraft response because of the adopted re-ceding horizon strategy.Thus, given the discrete-time model of Section 3.2.1, the convex constraintsdiscussed in Section 3.3, the probabilistic constraint in (5.1) and the costfunction (5.2) described above, the overall finite horizon optimization prob-lem is as follows:

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (5.3)

s.t.

dynamics (3.32)constraint (3.33) (3.34) (3.36) (3.38) (3.44) constraint (3.46) (3.50) for aircraft model (2.2)

orconstraint (3.48) (3.54) for aircraft model (2.3)

P‖ξk+i(wk, . . . ,wk+i−1)‖2

2 ≤ hi i = 1, . . . ,M ≥ 1− ε.

83


Note that the same discrete time linear dynamics are considered for bothaircraft models (2.2) and (2.3), but slightly different constraints for the en-gine thrust and the bank angle have to considered accordingly to the chosenmodel, as discussed in Chapter 3.The MPC control is then obtained by applying the first element uk of thecomputed control actions and repeating the optimization (5.3) at each timestep (receding horizon). Note that problem (5.3) is very hard to solve be-cause of the presence of the probabilistic constraint and of the complexprobabilistic model of the wind, which altogether make it non convex. As amatter of fact, w is not a simple additive disturbance, but it depends on theaircraft position which in turn is a function of the input to be optimized, asdescribed in Section 2.4. In Section 5.2 we hence revisit the wind model sothat in (5.3) w can be regarded as an additive disturbance, that can be han-dled in the optimization problem. Then, in Section 5.3 problem (5.3) withthe revisited wind model is solved by means of a randomized approachfor computational reasons. In Section 5.4 possible alternative formulationsof the finite horizon optimization problem are discussed along with somenumerical implementation issues.

5.2 Modelling wind in the optimization problem

Both the forecast that provides the wind deterministic component, and therandom field that models the wind stochastic component introduce a non-linear dependence of the wind velocity on the aircraft position (x1, x2, x3),which compromises the convexity of the optimization problem (5.3). In-deed, the wind forecast is a look-up table that maps the aircraft positioninto the wind velocities, and the covariance matrices that define the windrandom field in (2.12) depend on the position of the aircraft as well. Notehowever that, being performed over a finite horizon, optimization at eachtime step requires the model of the wind over a neighbourhood of the cur-rent aircraft position only. Since both the deterministic and stochastic com-ponents typically show a limited variability in space, the idea is then tobuild an approximated local model of the wind that does not depend on x1,x2 and x3. This model is updated at each time step so as to track the aircraftchange of position in accordance to the receding horizon implementationof the MPC controller.

84

5.2. Modelling wind in the optimization problem

5.2.1 Deterministic wind component

As for the wind deterministic component wf,k+i(x1,k+i, x2,k+i, x3,k+i), i =0, . . . ,M we simply approximate it with wf,k+i which is, i by i, the averageof the forecast wind over the hyper-rectangle:

[−VkTs, VkMTs]× [−VkTsM

2, VkTs

M

2]× [VkTsM sin γmin, VkTsM sin γmax],

where the origin is centred in the current aircraft position x1,k+i, x2,k+i,x3,k+i and axes are oriented according to the current aircraft velocity orien-tation given by ψk+i, γk+i according to the following rotation matrices:

Rz(ψk) =

cos(ψk) − sin(ψk) 0

sin(ψk) cos(ψk) 0

0 0 1

Ry(γk) =

cos(γk) 0 − sin(γk)

0 1 0

sin(γk) 0 cos(γk)

.Note that the size of the hyper-rectangle is determined by the amount of

space that the aircraft can cover in the finite horizon M according to thecurrent TAS Vk. wf,k+i is computed by taking a grid of points P over thehyper-rectangle, calculating the wind deterministic components in corre-spondence of these grid points by linear interpolating the data provided bywind forecast and, eventually, by averaging these values:

wxf,k+i =1

|P|∑

(xp,yp,zp)∈P

wxf,k+i(xp, yp, zp) (5.4)

wyf,k+i =1

|P|∑

(xp,yp,zp)∈P

wyf,k+i(xp, yp, zp) i = 0, . . . ,M.

wzf,k+i =1

|P|∑

(xp,yp,zp)∈P

wzf,k+i(xp, yp, zp)

Possibly a weighted average may be considered, e.g. based on the distanceof the points from the current aircraft position, so as to give greater im-portance to the value of the wind in correspondence of points closer to thecurrent aircraft position.This way wf,k+i is a function of time only and it is straightforward to ac-count for it in the optimization problem. Despite its simplicity, this ap-proach works quite well thanks to the limited variability of the wind fore-cast over the distances travelled in the considered finite prediction horizon.

A possible alternative is to consider models for the deterministic wind com-ponent that depend linearly on the aircraft position. This way the depen-dence of the wind on the spatial position is partially preserved, the model

85


has a local validity only as the wind velocity is unlikely to show a lineardependence on spatial position over large space regions. The consideredmodel for the deterministic wind components in this case is:wxf,k+i

wyf,k+i

wzf,k+i

=

pTwx,k+i

pTwy,k+i

pTwz,k+i

x1,k+i

x2,k+i

x3,k+i

1

= i = 0, . . . ,M. (5.5)

= Pw,k+i

x1,k+i

x2,k+i

x3,k

+ Pw,k+i

where the parameters of the model pTwx, pTwy, p

Twz are identified based on the

wind data computed in correspondence of the grid points P in the hyper-rectangle as described above. The identification is performed by means ofthe least squares algorithm independently for each component of the windvelocity, and it is repeated at each time step following the receding horizonstrategy of the controller.The obtained model for the deterministic wind component is included inthe aircraft dynamics (3.32), so that the overall dynamics become linear,though time-varying because at each time step a different dynamics matrixhas to be considered. That is

xk+1 = (A+BwPw,k)xk +Buk +Bw(Pw,k + ws,k).

It seems that the linear model for the deterministic wind is not significantlymore effective or accurate than the constant model obtained by simply av-eraging, possibly because of the fact that the wind is slowly changing in thetime and space scales considered in the finite horizon optimization prob-lem. On the other hand, it is rather more difficult to deal with the linearmodel, because of the time-varying dynamics. For this reason, in the fol-lowing we decide to stick to the constant model (5.4) for the deterministicwind component.

5.2.2 Stochastic wind component

As for the wind stochastic component, various approaches are developed.Some approaches are based on the covariance structure of the wind ran-dom field in (2.12), in which an approximation of the dependence of thewind on aircraft position is made so as to decouple the wind model fromthe current optimization variables. Other approaches, instead, see the wind

86


disturbance directly as a time series and model it by means of stochasticprocesses, whose parameters are identified based on the data collected bythe aircraft along its trajectory. Note that the space dependence that is ne-glected in these latter approximated wind models, is somehow indirectlyrecovered thanks to the fact that the identification is repeated at each timestep considering wind data closer to current aircraft position.

5.2.2.1 Modelling wind as a time series

In a first approach, ws,k+i(x1,k+i, x2,k+i, x3,k+i) is approximately modelledby means of three discrete time stochastic Auto-Regressive (AR) processes,whose parameters are identified based on the past wind values experiencedalong the aircraft trajectory up to time instant k − 1 preceding the currenttime instant k. As a matter of fact, past wind velocities along the aircrafttrajectory are easily recovered from the aircraft dynamics in (3.32) as:wx,lwy,l

wz,l

=1

Ts

x1,l+1

x2,l+1

x3,l+1

−x1,l

x2,l

x3,l

− Tsx4,l

x5,l

x6,l

− T 2s

2

u1,l

u2,l

u3,l

(5.6)

l = 0, 1, . . . , k − 1,

from which the past stochastic wind components can be computed by sim-ply subtracting the deterministic ones. These computed values wxs,l, wys,l,wzs,l, l = 0, 1, . . . , k− 1, are seen as time series and are used to recursivelyidentify the following AR models, one for each wind stochastic component:

wxs,l = ϕTxs,lθx + exs,l exs ∼ WGN(0, λ2x)

wys,l = ϕTys,lθy + eys,l eys ∼ WGN(0, λ2y) (5.7)

wzs,l = ϕTzs,lθz + ezs,l ezs ∼ WGN(0, λ2z),

where WGN stands for White Gaussian Noise and

ϕxs,l = [wxs,l−1, . . . , wxs,l−ma , 1]T

ϕys,l = [wys,l−1, . . . , wys,l−ma , 1]T

ϕzs,l = [wzs,l−1, . . . , wzs,l−ma , 1]T

are the regressors, ma is the model order, and θx, θy, θz are the model pa-rameter vectors. Note that, given the strong wind correlation with respectto both time and space it may be that non zero mean models have to be

87


preferred to best fit the available data records: the ones in the regressors areintroduced to this purpose. The identified AR models are used as modelsfor the stochastic wind components over the finite horizon [k, k +M ].Eventually, at each time step, a new data point, computed via (5.6), be-comes available and the AR models have to be updated. To this purpose weresort to the Recursive Least Square (RLS) algorithm with forgetting factorµ: θj,k = θj,k−1 + Sj,kϕjs,k−1(wjs,k−1 − ϕTjs,k−1θj,k−1)

Sj,k = 1µ(Sj,k−1 −

Sj,k−1ϕjs,k−1ϕTjs,k−1Si,k−1

µ+ϕTjs,k−1Sj,kϕjs,k−1)

j = x, y, z.

(5.8)

The white noise variances are also estimated as:

λ2j =

∑k−1i=1 µ

k−i(wjs,i − ϕTjs,iθj,k)2∑k−1i=1 µ

k−ij = x, y, z.

Note that in the AR models the dependence of the wind with respect tospace position is neglected. However, thanks to the fact that the identifica-tion is repeated at each time step and that a forgetting factor is introducedso as to discard past data that are no more representative of the current sit-uation, the model is tuned to the wind characteristics in the region of spaceclose to the current aircraft position. The strong correlation of the wind intime and space should further foster the identification of a good model forthe wind. Note also that the identified wind model has to be initialized withthe last available data, so as to make it consistent with the actual seen wind.The proposed approach can be used irrespective of the availability of thestochastic model of the wind field in (2.12). Also it can accounts for alldisturbances other than wind acting on the aircraft model as due to modelerrors, noisy measurements or noisy reconstructions of the state variables,etc. This way its usage can enforce additional robustness to the design ofthe controller.

Other approaches

The framework described above for the stochastic wind identification maybe slightly modified so as to consider different wind models or identifi-cation algorithms. More precisely, one can consider the Extended LeastSquare (ELS) algorithm that allow to identify Auto Regressive Moving Av-erage (ARMA), instead of AR only, models for the wind components andthe Recursive Instrumental Variable (RIV) algorithm that can be exploited

88


for a better identification of the AR part of the data irrespectively of thecolor of the noise. Both are derived from the RLS algorithm.We may consider the model in (5.7) for the wind stochastic components inwhich the regressors ϕ are extended to contain also the past mm values ofthe white noises:

ϕjs,l = [wjs,l−1, . . . , wjs,l−ma , ejs,l−1, . . . , ejs,l−mm , 1]T j = x, y, z,

so as to obtain ARMA processes of order ma, mm. Note that, in theidentification, the past values of the white noises are not available. Theidea of the ELS algorithm is to replace them with the prediction errorsεj,l+1 = wjs,l+1 − ϕTjs,l+1θj,l, j = x, y, z, so that the same update equationsin (5.8) of the RLS algorithm can be exploited to recursively estimate themodel parameters.The RIV algorithm is similar to RLS but it make use of two regressors: theusual one ϕ and the instrumental variable ς . This latter gathers the datashifted backward of ms time steps with respect to the data in ϕ:

ςjs,l = [wjs,l−1−ms , . . . , wjs,l−ma−ms , 1]T j = x, y, z.

This algorithm allows to better identify the AR part of the process that gen-erates the data in presence of an unmodelled MA part, as far as ms is suit-ably chosen. More precisely the instrumental variable ς and the regressionϕmust be not uncorrelated in order to keep the regression matrix invertible,and the instrumental variable ς and the unmodelled MA part of the processmust be uncorrelated so as to not introduce a bias in the identification. Theupdate equations to recursively estimate the model parameters are: θj,k = θj,k−1 + Sj,kςjs,k−1(wjs,k−1 − ϕTjs,k−1θj,k−1)

Sj,k = 1µ(Sj,k−1 −

Sj,k−1ςjs,k−1ϕTjs,k−1Si,k−1

µ+ϕTjs,k−1Sj,kςjs,k−1)

j = x, y, z.

(5.9)

The RIV algorithm should foster an identification of models more suitedto simulation purposes, but, on the other hand, the reduced correlation be-tween ϕ and ς may lead to a more difficult identification and hence to lessaccurate models.

5.2.2.2 Wind model based on random field

Another approach to model the stochastic wind component in the finitehorizon optimization problem is based directly on the structure of the stochas-tic wind random field described in Section 2.4.

89


Following the approach in [31], we consider the following model for thestochastic wind components:

Wx,l = QxVx Vx ∼ N(0, I) (5.10)Wy,l = QyVy Vy ∼ N(0, I)

Wz,l = QzVz Vz ∼ N(0, I),

where the vectorsWx,l,Wy,l,Wz,l are formed from np values of the stochas-tic wind velocities

Wx,l = [wxs,l−1, . . . , wxs,l−np ]T

Wy,l = [wys,l−1, . . . , wys,l−np ]T

Wz,l = [wzs,l−1, . . . , wzs,l−np ]T ,

andQx,Qy,Qz are lower triangular matrices, whose not null coefficient hasto be properly set, depending on the time instant and space position corre-sponding to the wind velocities in Wx,l, Wy,l, Wz,l, so that model (5.10)satisfies the covariance structure of the wind random field in (2.12). Moreprecisely, we rewrite the wind model as1:

wxs,l−1−np+i =i∑

j=1

qi,jvj i = 1, . . . , np v ∼ N(0, 1).

Then the not null coefficients qi,j of matricesQx, Qy, Qz can be determinedas described in [31], by computing the covariances between each couple ofwind velocities in Wx,l, Wy,l, Wz,l and, then, enforcing them to be equalto the covariances of the random field in (2.12). The variance of wxs,l−npresults:

E[w2xs,l−np ] = q2

1,1 = ρ(xl−np ,xl−np , l − np, l − np).

The covariances of wxs,l−1−np+i i = 2, . . . , np with respect to the first pointwxs,l−np result:

E[wxs,l−1−np+iwxs,l−np ] = qi,1q1,1 i = 2, . . . , np

= ρ(xl−1−np+i,xl−np , l − 1− np + i, l − np)

qi,1 =ρ(xl−1−np+i,xl−np , l − 1− np + i, l − np)

q1,1i = 2, . . . , np.

1In other to lighten the notation, formulas are provided with reference to the x component of the wind only,the same have to be applied also for the y and z components.

90


The covariances of each wind velocities in Wx,l, Wy,l, Wz,l with respect tothe previous ones 2 result:

E[wxs,l−1−np+iwxs,l−1−np+j ] =

j∑k=1

qi,kqj,k i = 2, . . . , np j = 2, . . . , i− 1

= ρ(xl−1−np+i,xl−1−np+j , l − 1− np + i, l − 1− np + j)

qi,j =ρ(xl−1−np+i,xl−1−np+j , l − 1− np + i, l − 1− np + j)−

∑j−1k=1 qi,kqj,k

qj,j

i = 2, . . . , np j = 2, . . . , i− 1.

Finally their variances result:

E[wxs,l−1−np+iwxs,l−1−np+i] =

i∑j=1

q2i,j = i = 2, . . . , np

= ρ(xl−1−np+i,xl−1−np+i, l − 1− np + i, l − 1− np + i)

q2i,i = ρ(xl−1−np+i,xl−1−np+i, l − 1− np + i, l − 1− np + i)−

i−1∑j=1

q2i,j i = 2, . . . , np.

This way all the entries of Qx, Qy, Qz are determined so as to satisfy thecovariance structure in (2.12).In order to include this model of the stochastic wind component in the finitehorizon optimization problem we proceed in two steps: first we exploit thepast wind data recovered via (5.6) to initialize the model, then, the modelis exploited to characterize the wind along the finite horizon considered inthe optimization problem.At each time step k we collect the last np wind data in the vectors Wx, Wy,Wz and we compute the corresponding matrices Qx, Qy, Qz as describedabove (note that the position corresponding to each past data is known) andwe recover the value of Vx, Vy, Vz by inversion ofQx, Qy, Qz. These matri-ces are lower triangular and the elements on the diagonals are variances sothat they are invertible. The so obtained Vx, Vy, Vz are used as initializationof the model.In order to model ws,k the matrices Qx, Qy, Qz are extended with a rightcolumn of zeros and a bottom row, qx,lr, qy,lr, qz,lr respectively, such thatthe covariance between the past wind valuesWx, Wy, Wz and the new windvelocities ws,k in the current aircraft position xt matches the random field

2Here the wind velocities are ordered according to time index, but any order can be used.

91


structure in (2.12). Hence the current wind velocities are modelled as:

wxs,k = qx,lr

[Vx

vx,k

]wys,k = qy,lr

[Vy

vy,k

]wzs,k = qz,lr

[Vz

vz,k

], (5.11)

where vx,k, vy,k, vz,k are Guassian random variables N(0, 1).We can iterate this procedure to model the stochastic wind along the fi-nite prediction horizon ws,k+i, i = 1, . . . ,M , but unfortunately the corre-sponding spatial positions to be reached at time steps k + 1, . . . , k + Mare unknown as they will be determined by the current input uk+i, i =0, . . . ,M−1 to be designed. In order to decouple the stochastic wind modelfrom the current optimization variables we approximate the future aircraftpositions with the positions attained by steering the aircraft according tothe finite horizon solution computed at the previous time step. Eventually,i by i, we can model the stochastic wind component iteratively enlargingthe matrices Qx, Qy, Qz whose new coefficients are computed based on theaircraft positions determined by the input provided by the previous step (atk − 1) finite horizon solution.By means of this approach we can obtain a quite accurate model for thewind which approximates the random field structure in (2.12), and is inde-pendent from the current optimization variables. Hence, the wind becomesa purely additive disturbance which can be handled in the finite horizon op-timization problem.Note that, also in this approach, the initialization, namely the recovery ofthe past noise values corresponding to the wind data, is of paramount im-portance to obtain a model that is conditioned to the last data and that isconsistent with the seen actual wind.This approach heavily relies on the model of the wind described in Sec-tion 2.4 and, hence, it may suffer from inaccurate modelling of the truewind by the random field defined in (2.12). On the contrary, the recursiveidentification of AR models may better adapt to different conditions.

5.2.2.3 AR(1) approximation of the random field

A rough but extremely simple approach consists in modelling the stochasticwind components by means of three AR(1) processes, whose parameter areset so as to approximate the correlation structure of the random field de-scribed in Section 2.4. Consider the following model for wxs,l, wys,l, wzs,l:

wxs,l+1 = awxs,l + λex,l+1 ex,l ∼ WGN(0, 1)

wys,l+1 = awys,l + λey,l+1 ey,l ∼ WGN(0, 1)

wzs,l+1 = azwzs,l + λzez,l+1 ez,l ∼ WGN(0, 1).

92


In order to properly set the values of the model parameters a, az and λ, λzwe compute:

E[wxs,0wxs,0] = E[wys,0wys,0] =λ2

1− a2

E[wzs,0wzs,0] =λ2z

1− a2z

E[wxs,1wxs,0] = E[wys,1wys,0] =λ2a

1− a2

E[wzs,1wzs,0] =λ2zaz

1− a2z

,

and we enforce them to be equal to the covariances of wxs,l, wys,l, wzs,lcomputed according to (2.12):

E[wxs,0wxs,0] = E[wys,0wys,0] = k(z0)2

E[wzs,0wzs,0] = kz(z0)2

E[wxs,1wxs,0] = E[wys,1wys,0] = k(z1)k(z0)e−αTse−β‖[x1−x0 y1−y0]‖e−γ|z1−z0|

E[wzs,1wzs,0] = kz(z1)k(z0)e−αzTse−βz‖[x1−x0 y1−y0]‖e−γz |z1−z0|.

Hence, the AR(1) parameters result:

a = k(z1)k(z0)−1e−αTse−β‖[x1−x0 y1−y0]‖e−γ|z1−z0| (5.12)

az = kz(z1)k(z0)−1e−αzTse−βz‖[x1−x0 y1−y0]‖e−γz |z1−z0|

λ = k(z0)2(1− a2)

λz = kz(z0)2(1− a2z).

Note that the parameter values are fixed considering that we are interestedin modelling the wind stochastic component only at time steps of lengthTs. The AR(1) models are a good local approximation of the wind randomfield only as far as the difference of the spatial positions between couplesof consecutive time steps does not change too much, so that accounting forthe spatial difference corresponding to the first two time steps only whilesetting the model parameters is still accurate enough to model the windvelocities also at other time steps.In the finite horizon optimization problem, this kind of model may be accu-rate enough thanks to the facts that: i) usually the difference of the aircraftpositions between consecutive time steps is not rapidly changing, and ii) theAR(1) models are used only over the considered finite horizon M . Indeed,following the receding horizon strategy of the controller, at each time stepk the model parameters are reset according to (5.12), considering as initial

93


positions xk−1 and xk, or, alternatively xk and an approximation of xk+1

computed accounting for aircraft dynamics and assuming that the aircraftis steered according the previous step finite horizon solution and constantwind speed. This latter approximation is needed to decouple the currentdecision variables from the wind model, so as to make the finite horizonoptimization problem solvable.Also in this case, the initialization of the AR models with the last availablewind data is important to achieve a model which is consistent with the seenactual wind. This is especially true when the AR(1) shows a strong corre-lation (i.e. when a and az are close to 1).Simulations have revealed that even this extremely simple model for thestochastic wind component returns suitable approximations, thanks to thestrong correlation of the wind, and to the fact that the wind model is localand it is updated at each time step.

5.3 Solution of the optimization problem via the scenario ap-proach

The finite horizon optimization problem (5.3) can be reformulated as fol-lows by simply replacing w with w = wf + ws as given by one of theapproximate wind models proposed in Sections 5.2.1 and 5.2.2.1, 5.2.2.2,5.2.2.3:

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (5.13)

s.t.

dynamics (3.32) with w in place of wconstraint (3.33) (3.34) (3.36) (3.38) (3.44) constraint (3.46) (3.50) for aircraft model (2.2)


P‖ξk+i(wk, . . . , wk+i−1)‖2

2 ≤ hi i = 1, . . . ,M ≥ 1− ε.

In this new formulation the wind w is an additive disturbance that is in-dependent from the state x, and, hence, from the decision variable uk+i,i = 1, . . . ,M . This was a first fundamental step to make the optimizationproblem tractable. The resolution of (5.13), however, may still be prob-lematic because of the presence of a chance-constraint, which is in general

94

5.3. Solution of the optimization problem via the scenario approach

a non-convex constraint, making the problem hard to be dealt with. Thisissue has to be carefully addressed in the present aircraft motion control ap-plication, because problem (5.13) is meant to be solved on board in a smallfraction of the sampling time with limited computational resources.We decide to resort to the scenario approach, a randomized method to ap-proximately solve chance-constrained problems such as (5.13), which hasbeen recently introduced and discussed in [17,18] and applied to stochasticconstrained control and MPC in [20,24,28,54,61]. This approach has beenchosen because it allows to find an approximate but guaranteed solution tothe chance-constrained optimization problem by solving a standard convexquadratic program. Furthermore it has the advantage to be applicable to awide range of situations: indeed it can be applied considering any of thewind models described in Section 5.2 and possibly considering other addi-tive or multiplicative disturbances affecting the aircraft dynamics providedthat sample realizations are available either from data or by simulation.Moreover it can be easily applied also when different formulations of theposition error constraint, such as the ones discussed in the next Section 5.4,are considered.The idea of the scenario approach is very simple: a bunch of N realizationsof the disturbance, say

[w(j)k w

(j)k+1 . . . w

(j)k+M−1]T j = 1, . . . , N,

are generated according to the last available wind model. We can con-sider any of the wind models proposed in Section 5.2, update its parametersbased on the wind data available at time k, and generate the wind samplesby simulation. Note that the last wind observations wk−1, wk−2, . . . have tobe used as initialization of the wind model.Then, the probabilistic constraint is replaced with the N constraints ob-

95


tained in correspondence of the extracted disturbance realizations. Namely:

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (5.14)

s.t.



‖ξk+i(w(j)k , . . . , w

(j)k+i−1)‖2

2 ≤ hii = 1, . . . ,M

j = 1, . . . , N.

Note that the newN constraints replacing the probabilistic one are quadraticconstraints, and, overall, (5.14) is a quadratically constrained quadratic pro-gram which can be efficiently solved by means of standard solver like e.g.CPLEX, [6]. Moreover, despite its apparent naivety the scenario approachis grounded on a solid theory that provides precise guarantees about thefeasibility of the solution obtained solving problem (5.14) for the originalchance-constrained problem (5.13). More precisely, according to the re-sults of [22] if N is chosen so as to satisfy

β ≥d∑i=0

(N

i

)εi(1− ε)N−i, (5.15)

where d is the number of optimization variables (in our setup d = 4M ),then the solution of problem (5.14) is feasible for problem (5.13) with con-fidence 1 − β. In [7, 8] an explicit expression for N that guarantees thesatisfaction of (5.15) is provided:

N ≥ 1

ε

(d+ 1 + ln(1/β) +

√2(d+ 1) ln(1/β)

).

Thanks to the logarithmic dependence, very small values of β such as 10−6

or even 10−9 can be enforced without affecting N too much, and, with suchsmall values for β, the solution achieved solving (5.14) can be deemedfeasible for (5.13) beyond any reasonable doubts.The guarantee inherited from the results of [22] described above, regardsthe solution of each finite horizon optimization problem (5.14). However

96

5.3. Solution of the optimization problem via the scenario approach

in MPC, the solution is recomputed at each time step by letting the hori-zon recedes, and one is also interested in evaluating the behavior of the soobtained closed loop control. More precisely, we let (u?k, h

?k+1) be the first-

time-instant part of the solution of problem (5.14) which is, k by k, actuallyapplied to the aircraft system, and we let x?k+1 and ξ?k+1 be the correspond-ing closed loop aircraft state and position error respectively. Because theway constraints (3.33), (3.34), (3.36), (3.38), (3.44), (3.46), (3.48), (3.50),(3.54) are constructed it is easily seen that u?k, x?k+1 are guaranteed to sat-isfy the original physical and comfort limitations posed in (2.4)-(2.10) forevery k. Moreover by letting

vk =

1 if ‖ξ?k‖2

2 ≤ h?k0 otherwise

,

by application of the strong law of large number it can be proved that, if Nsatisfies (5.15), then

limk→∞

inf1

k

k∑j=0

vj ≥ 1− ε (5.16)

with probability 1. In other words, the ratio of the number of times inwhich the actual position error ‖ξ?k‖2

2 is within the bound h?k is almost surelyasymptotically bigger or equal to 1− ε.Actually a more specific analysis, tailored to the problem of evaluating thisasymptotic ratio, permits one to show that

limk→∞

inf1

k

k∑j=0

vj ≥ 1− d

N + 1, (5.17)

see [62]. This shows that if the initial objective was that of attain (5.16)and not to guarantee the satisfaction at each time instant of the probabilisticconstraint in (5.13), it is enough to take N such that 1− d

N+1≥ 1− ε that

is

N ≥ d

ε− 1, (5.18)

with a reduction of the disturbance extractions required at each time k.Condition (5.15) leads to a more guaranteed solution, and, accordingly, re-quires an higher number of extracted scenarios, on the other hand, condition(5.18) provides less guaranteed finite horizon solutions, but it is specificallytailored to the MPC application because it focuses on the behavior of thereceding horizon solution.

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Eventually, note that only the satisfaction of threshold h?k at the currenttime k enters the definition of vk, because only the first-time-instant partof the solution of (5.14) is actually applied in accordance to the recedinghorizon strategy. In view of this, it may be more convenient to reformulatethe finite horizon optimization problem as a multiple chance-constrainedprogram where the first time instant position error constraint is accountedfor separately from the constraint corresponding to the other time steps:

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (5.19)

s.t.



P‖ξk+1(wk)‖2

2 ≤ h1 ≥ 1− εP‖ξk+i(wk, . . . , wk+i−1)‖2

2 ≤ hi i = 2, . . . ,M ≥ 1− ε′.

In this case the scenario program becomes:

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (5.20)

s.t.



‖ξk+1(w

(j)k )‖2

2 ≤ h1 j = 1, . . . , N

‖ξk+i(w(j)k , . . . , w

(j)k+i−1)‖2

2 ≤ hii = 2, . . . ,M

j = N + 1, . . . , N +N ′,

where N + N ′ extractions of w are considered. In this case, extractionsN + 1, . . . , N +N ′ play no role as for guaranteeing (5.16), and, hence, N ′

98

5.4. Alternative formulations

can be heuristically decided. Instead, according to [61,62], (5.17) becomes

limk→∞

inf1

k

k∑j=0

vj ≥ 1− ζ

N + 1,

where ζ is is the number of optimization variables appearing in the first-time-instant constraint only (in the case at hand ζ = 4). Accordingly con-dition (5.18) can be rewritten as:

N ≥ ζ

ε− 1, (5.21)

which further reduces the number of required disturbance realizations, andhence the computational effort.

5.4 Alternative formulations

In this section we propose some alternative formulations of the finite hori-zon optimization problem that can be exploited to achieve slightly differentperformance and guarantees on the position error ξk. All the optimizationproblems consider the wind disturbance w modelled as described in Sec-tion 5.2, and they are solved resorting to the scenario approach.A first straightforward extension is that of replacing the 2-norm of the posi-tion error ξk+i in the chance-constraint (5.1) with other norms. One can optfor weighted norms by means of which it is possible to regulate the relativeimportance of the position error components. It is also worth noticing thatwhen the infinite-norm or the 1-norm is considered the sampled version ofthe position error constraint is a linear constraint so that the required com-putational effort to solve the finite horizon optimization problem is furtherreduced.Another interesting reformulation of the position error constraint is the fol-lowing:

P|ξk+i(wk, . . . , wk+i−1)| ≤[hL,i hl,i hv,i

]T, i = 1, . . . ,M ≥ 1− ε,

(5.22)

where inequalities are understood component wise, i.e. different boundshL,i, hl,i, hv,i are imposed on the absolute value of longitudinal, lateral andvertical components of the position error ξk+i. Note that the bound on thenorm of the position error does not provide informations on how the error issplit among the different components. Instead by means of the constraint in(5.22) a more detailed description of the position error is provided because

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the bounds are adapted to each component of the position error and give amore accurate information on it. The drawback is that an higher numberof decision variables have to be considered, and, consequently in order toapply the theory of the scenario approach and provide guarantees on thesolution, an higher number of disturbance sample realizations and of corre-sponding constraints are required. Indeed the choice of N is dependent onthe number of decision variables d or ζ . However, the sampled version ofconstraint (5.22) is a linear constraint which can be dealt with at very lowcomputational effort3.In order to account for the newly introduced optimization variables the costfunction is modified as follow:

J =M−1∑i=0

µiuduTk+iRuk+i + µLc

M∑i=1

µi−1Ld hL,i+ (5.23)

+ µlc

M∑i=1

µi−1ld hl,i + µvc

M∑i=1

µi−1vd hv,i,

where the weights R, µLc, µlc, µvc regulate the relative importance of thebounds and of the input, whereas the weights µud, µLd, µld regulate therelative importance of the time steps along the finite horizon.Hence, we formulate the finite horizon optimization problem as:

minuk+i i = 0 . . .M − 1

hL,i, hl,i, hv,i i = 1 . . .M

J (5.24)

s.t.



P|ξk+i(wk, . . . , wk+i−1)| ≤ [hL,i hl,i hv,i]

T , i = 1, . . . ,M ≥ 1− ε.

It is worth noticing that being wl a Guassian process (at least when the windmodels proposed in Section 5.2 are considered) that enters additively the

3We can achieve an even more detailed description of the position error considering 6 different bounds oneach component of ξk+i and −ξk+i, so as to provide bounds on the maximum and minimum value of eachcomponent of the position error. However, as this extension is straightforward in the following we stick to theformulation in (5.22).

100


state equation and that is independent from the input uk+i, i = 1, . . . ,M , itcan be proved that the probabilistic constraint in (5.24) is indeed a convexconstraint, see [27, 55]. However, for computational reasons we solve itagain resorting to the scenario approach:

minuk+i i = 0 . . .M − 1

hL,i, hl,i, hv,i i = 1 . . .M

J (5.25)

s.t.



|ξk+i(w(j)k , . . . , w

(j)k+i−1)| ≤ [hL,i hl,i hv,i]

T ,i = 1, . . . ,M

j = 1, . . . , N.

We can also consider the formulation with multiple chance-constraints,where a probabilistic constraint account for the first-time-step part of thesolution that will be actually applied to the system, while the other timesteps in the finite horizon are considered in a separate constraint:

minuk+i i = 0 . . .M − 1

hL,i, hl,i, hv,i i = 1 . . .M

J (5.26)

s.t.



P|ξk+1(wk)| ≤ [hL,1 hl,1 hv,1]T ≥ 1− εP|ξk+i(wk, . . . , wk+i−1)| ≤ [hL,i hl,i hv,i]

T , i = 2, . . . ,M ≥ 1− ε′.

101


In this case the scenario program becomes:

minuk+i i = 0 . . .M − 1

hL,i, hl,i, hv,i i = 1 . . .M

J (5.27)

s.t.



|ξk+1(w

(j)k )| ≤ [hL,1 hl,1 hv,1]T , j = 1 . . . N

|ξk+i(w(j′)k , . . . , w

(j′)k+i−1)| ≤ [hL,i hl,i hv,i]

T ,i = 2, . . . ,M

j′ = N + 1, . . . , N +N ′

where N +N ′ extractions of w are considered.Problems (5.25) and (5.27) can be very efficiently solved thanks to the par-ticular structure and to convexity. Indeed, accounting for the system dy-namics (3.32), the position constraints in problem (5.25) and (5.27) rewriteas:∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣Rz(ψR,k+i)

x1,k + Ts

i−1∑l=0

x4,k+l(u1) +T 2s

2

i−1∑l=0

u1,k+l + Ts

i−1∑l=0

w(j)x,k+l

x2,k + Ts

i−1∑l=0

x5,k+l(u2) +T 2s

2

i−1∑l=0

u2,k+l + Ts

i−1∑l=0

w(j)y,k+l

x3,k + Ts

i−1∑l=0

x6,k+l(u3) +T 2s

2

i−1∑l=0

u3,k+l + Ts

i−1∑l=0

w(j)z,k+l

−

xR,k+i

yR,k+i

zR,k+i

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣≤

hL,k+i

hl,k+i

hv,k+i

i = 1, . . . ,M j = 1, . . . , N. (5.28)

We can then isolate in each row the term that depends on the wind distur-bance, and consider the constraints that correspond to the worst case amongthe extracted wind realizations, in place of all the constraints. That is the

102


position constraints (5.28) are equivalent to:

−

hL,k+i

hl,k+i

hv,k+i

− minj=1,...,N

Rz(ψR,k+i)Ts

∑i−1l=0 w

(j)x,k+l∑i−1

l=0 w(j)y,k+l∑i−1

l=0 w(j)z,k+l

(5.29)

≤ Rz(ψR,k+i)

x1,k + Ts

∑i−1l=0 x4,k+l(u1) +

T2s2

∑i−1l=0 u1,k+l

x2,k + Ts∑i−1l=0 x5,k+l(u2) +

T2s2

∑i−1l=0 u2,k+l

x3,k + Ts∑i−1l=0 x6,k+l(u3) +

T2s2

∑i−1l=0 u3,k+l

−xR,k+i

yR,k+i

zR,k+i

≤

hL,k+i

hl,k+i

hv,k+i

− maxj=1,...,N

Rz(ψR,k+i)Ts

Ts∑i−1l=0 w

(j)x,k+l

Ts∑i−1l=0 w

(j)y,k+l

Ts∑i−1l=0 w

(j)z,k+l

, i = 1, . . . ,M,

where min and max have to be computed row-wise. The computationof these minimum and maximum values is extremely straightforward andthe resulting convex optimization problem obtained replacing the positionconstraints in (5.25) or (5.27) with the formulation in (5.29) can be solvedat very low computational effort.The control algorithm based on this latter formulation of the finite horizonoptimization problem has been implemented in MATLAB equipped withan IBM ILOG CPLEX solver, [6], running on a desktop pc with two IntelXeon E5-2630 2.30 GHz processors and 64Gb of RAM: the resolution ofthe optimization problem with a finite horizon M = 20 requires about 0.3s. The computational time can be further reduced by means of an embed-ded implementation and problem-specific solvers achieving performancescompatible with an on-line implementation.

5.4.1 Disturbance feedback

In all the optimization problems above we design an open-loop finite hori-zon control law, and we rely on the receding horizon strategy to providestate feedback to the controller. However, one may want to introduce statefeedback also in the design of the finite horizon solution. To this purpose, aconvenient parametrization of the control input uk+i, originally introducedin [35, 36], which has the advantage to make the input and the state affinefunction of the design parameters is adopted:

uk+i = ci +i−1∑l=0

θi,lϕ(wk+l), (5.30)

where the open loop terms ci and the disturbance feedback gains θi,l aredegrees of freedom to be optimized, and ϕ(·) is a proper scalar function

103


applied to each component of w, that can be chosen e.g. as the identitymap or as the saturation function. The disturbance feedback policy (5.30)is indeed a state feedback policy because the past disturbance value arerecovered as a function of the state via (5.6), see [32]. Note that the con-vex constraints in the state and input variables remain convex in the newoptimization variables ci and θi,l. Further details on this input parametriza-tion are given in appendix A. However, in this specific application, a majordrawback of this approach is that the constraints on the aircraft physicallimitations and on the passenger comfort, which were independent fromw, become affected by the wind disturbance because of the disturbancefeedback parametrization of the input. In order to address this issue weformulate a chance-constraint which accounts for both the position errorconstraint and the constraints on aircraft physical limitations and passengercomfort. In this case the finite horizon optimization problem results:

minci i = 0 . . .M − 1

θi,l i = 0 . . .M − 1, l = 0 . . . i− 1

hi i = 1 . . .M

J (5.31)

s.t.

dynamics (3.32) with w in place of winput parametrization (5.30)

P

constraint (3.33) (3.34) (3.36) (3.38) (3.44) constraint (3.46) (3.50) for aircraft model (2.2)or

constraint (3.48) (3.54) for aircraft model (2.3)

‖ξk+i(wk, . . . , wk+i−1)‖2

2 ≤ hi i = 1, . . . ,M

≥ 1− ε.

Likewise all the alternative formulations described before can be applied:we can e.g. consider multiple chance-constraints, and different formula-tions for the position error constraint, etc. For simplicity all the commentshere will refer to the ‘basic’ problem (5.31).Problem (5.31) can also be approximately solved resorting to the scenarioapproach, which can be applied replacing the chance-constraint in (5.31)with the constraints on position error and on aircraft physical limitationsobtained in correspondence of a bunch of extracted wind realizations only.It is worth noticing that since the constraints on aircraft physical limitationsat the first time step are not affected by the wind disturbance, indeed theydepend on c1 only, the receding horizon solution is still guaranteed to satisfythe aircraft motion constraints.

104


Note that an alternative approach to deal with the uncertainty in the con-straints on the aircraft physical limitations (3.33), (3.34), (3.36), (3.38),(3.44), (3.46), (3.50), (3.48), (3.54) can be considered. As a matter of fact,the constraints on aircraft physical limitations depend on the wind distur-bance only through the input. Hence, if we choose the function ϕ(·) to bebounded for every value of w, then, following [35, 36], we can enforce theaircraft physical limitations constraints as robust constraints, a procedurethat is computationally affordable thanks to convexity.In spite of all these considerations, however, it turns out that by means ofthe disturbance feedback parametrization in (5.30) little or no performanceimprovement is achieved in the actual receding horizon solution with re-spect to the case in which only the open loop term is considered. Thisis probably due to the receding horizon that already provides state feed-back, and to the characteristics of the wind disturbance. On the other hand,the solution of the finite horizon problem with the disturbance feedbackparametrization of the input is much more computationally demanding asnew variables and new probabilistic constraints have to be introduced, sothat it has been deemed not very well suited for the application at hand.


In this section, we report some numerical results obtained from simulationsin which the developed MPC controller is applied to the aircraft model.

In the simulations we consider the linear aircraft model and the convex con-straints in Chapter 3, the reference trajectory is designed as described inChapter 4, and we apply the MPC controller developed in this chapter. Ouraim is to evaluate the capability of the controller to steer the aircraft so as tokeep it close to the reference trajectory accounting for both space positionand correct timing, while counteracting the action of the wind. Moreoverwe want to verify that the introduced approximations of the constraints donot too much adversely affect the achieved performance.

In all the following simulations the stochastic wind disturbance realizationsthat act on the aircraft are generated according to the model in Section 2.4,

105


in which the wind random field parameters are set as follow:

σ1 = 6 10−4 σ1z = 6 10−4

σ2 = 1.6 10−5 σ2z = 1.5 10−4

σ3 = 1.5 10−4 σ3z = 1.6 10−5

k(z) =z

3+ 5 kz(z) = 0.5(

z

3+ 5).

5.5.1 Simulation 1

A simulation is performed considering the aircraft model in Section 2.1and the controller developed in Sections 5.1- 5.3, where probabilistic con-straints on the 2-norm of the position error are enforced. The controller isrequired to steer the aircraft along a given reference trajectory computedas described in Chapter 4, compatibly with the physical limitations andcomfort constraints, and counteracting the action of the wind, which is ex-plicitly accounted for in the design of the control action. Note that, in orderto test the performance of the controller, we choose a reference trajectorywith many turns and altitude changes quite close one each other, that doesnot correspond to usual aircraft operations. Moreover the reference speedprofile is not constant.The model parameters and constraint bounds are set as follow:

Vmax = 900 km/h Vmin = 600 km/hγmax = 5 γmin = −3

Tmax = 2 · 276 kN Tmin = Tmax/200

φ = 40

m = 150 103 kg S = 0.28 103 m2,

the sampling time is Ts = 5s, the initial position and velocity coincidewith the starting point and the initial velocity of the reference trajectory:x0 = [−40 0 3 601 601 0]T . The finite prediction horizon is M = 12and the weights in the cost function of the optimization problem are set asfollows:

Rc = 0.02I3 µud = 1

µhc = 10 µhd = 0.64.

106


Note that long prediction horizons and large weights on the farthest timesteps position error should be avoided because the approximations intro-duced in the constraints may lead to inaccurate predictions. Moreover weare actually more interested in achieving a good performance in the timesteps closer to the current time instant. On the other hand, however, tooshort prediction horizons lead to poor performance of the controller, so thata proper compromise as to be reached by a suitable choice of the weightparameters.The allowed violation level for the probabilistic constraint on the 2-norm ofthe position error is ε = 0.1, the confidence parameter for the scenario ap-proach is β = 10−6, so that according to (5.15) the correspondent numberof required wind realizations is N = 876, instead if we consider condi-tion (5.18) to provide guarantees on the receding horizon solution only thenumber of required disturbance realizations reduces to N = 39.The wind realizations needed in the scenario optimization problem are gen-erated relying on the approach presented in Section 5.2. More precisely, thewind forecast is accounted for by means of the constant average model inSection 5.2.1, while the stochastic wind component is modelled by meansof AR(3) processes whose parameters are recursively identified at each timestep by means of the RLS algorithm with a forgetting factor µ = 0.992, seeSection 5.2.2.1. We perform the wind identification and compute somewind realizations also by means of the approaches in Sections 5.2.2.2 and5.2.2.3. The wind realizations obtained by means of these latter two meth-ods are not used in the controller design but are compared with the actualwind so as to evaluate the performance of the different approaches.The obtained aircraft trajectory and the other relevant physical quantitiesare reported in Figures 5.1, 5.2, 5.3.

107


−40 −20 0 20 40 60 80 100 120 140 160−40

−20

0

20

40

60

80

x [Km]

y[K

m]

(a) x-y view.

0

50

100−20

0

20

40

60

80

3

3.5

4

4.5

5

y [Km]

x [Km]

z[K

m]

(b) 3D view.

Figure 5.1: Aircraft trajectory, receding horizon solution in blue stars, finite horizon so-lution computed at each time step in colored squares, and reference trajectory in redcircles.

108


0 200 400 600 800 1000 1200 1400 1600 1800−40

−20

0

20

40

60

80

100

120

140

160

Time [s]

xyz[K

m]

(a) Positions x (blue), y (green), z (red).

0 200 400 600 800 1000 1200 1400 1600 1800−200

−150

−100

−50

0

50

100

150

200

Time [s]

ψ[deg

rees]

(b) Heading angle ψ.

0 200 400 600 800 1000 1200 1400 1600 1800−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

x4x5x6[K

m/s]

(c) Velocities x4 (blue), x5 (green), x6 (red).

0 200 400 600 800 1000 1200 1400 1600 18000.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0.26

Time [s]

V[K

m/s]

(d) TAS V (blue), reference speed (cyan).

0 200 400 600 800 1000 1200 1400 1600 1800−10

−8

−6

−4

−2

0

2

4

6

8x 10−3

Time [s]

u1u2u3[K

m/s2]

(e) Acceleration u1 (blue), u2 (green), u3 (red).

0 200 400 600 800 1000 1200 1400 1600 1800−50

−40

−30

−20

−10

0

10

20

30

40

50

Time [s]

γφ[deg

rees]

(f) Path angle γ (blue), bank angle φ (green).

Figure 5.2

109


0 200 400 600 800 1000 1200 1400 1600 1800−8

−6

−4

−2

0

2

4

6

8x 10−4

Time [s]

τ[K

m/s2]

(a) Longitudinal acceleration τ .

0 200 400 600 800 1000 1200 1400 1600 18000

100

200

300

400

500

600

Time [s]

T[K

N]

(b) Engine thrust T .

Figure 5.3

As one can see from Figure 5.1 the controller is able to steer the aircraftso as to well follow the reference trajectory. Again the finite horizon so-lution are slightly affected by the constraint approximations that preventthe aircraft to make rapid turn, however the effect of the approximations ismuch attenuated in the actual aircraft trajectory thanks to receding horizon.The constraints on the aircraft physical limitations are satisfied as shownin Figures 5.2, 5.3. Note that the TAS is not equal to the designed speedprofile: this is due to the fact that the TAS is controlled so as to counteractthe action of the wind. In Figure 5.4 is depicted the 2-norm of the actualposition error ξ?k , the corresponding bound h?k and the bounds on the 2 and3 steps ahead position error h?k+2, h?k+3.

110


0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [s]

√

h[K

m]

Figure 5.4: Actual position error ‖ξ?k‖2 (black), and the corresponding probabilisticbounds h?1 (red), h?2 (blue), h?3 (green).

The position error keeps often below 100m, so that the aircraft is close tothe reference trajectory. Around time 1100s, there is a strong deviationfrom the reference of about 600m: note however that the TAS is set to itsmaximum admissible value (see Figure 5.2(d)), that the speed profile re-quires an increase of TAS and that the wind disturbance acts against thecurrent aircraft heading, so that the achieved performance is acceptable. Asshown in Figure 5.4 the bound h?k is quite close to ξ?k and it is not excessivelyconservative with respect to the actual position error. As expected from theprobabilistic guarantees on the solution obtained by means of the scenarioapproach, the bound is violated by the actual position error only 1 timeover the 350 steps performed in the simulation. Indeed, we solve the finitehorizon optimization problems with N = 876 as determined by condition(5.15), and if we plug such value for N in (5.18) the obtained correspond-ing bound for the asymptotic constraint violation ratio of the closed-loopsolution is 0.0046, which is compatible with the empirical value achievedin the simulation. Note that the bounds for the 2 and 3 steps ahead positionerror are slightly larger with respect to h?k, indeed the prediction of the statebecomes less accurate and more affected by the presence of the stochastic

111


wind disturbance as farther time steps in the future are considered.In Figures 5.5, 5.6, 5.7 are reported some of the extracted wind realizationsof length M obtained at each time step by simulation of the approximatewind models described in Sections 5.2.2.1, 5.2.2.2, 5.2.2.3 respectively,along with the actual wind realization.

0 200 400 600 800 1000 1200 1400 1600 1800−100

−50

0

50

100

time [s]

wx[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800−100

−50

0

50

100

time [s]

wy[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800−40

−20

0

20

40

time [s]

wz[K

m/h]

Figure 5.5: Actual wind (red), and 10 wind realizations extracted at each time step ac-cording to the AR(3) models identified by RLS algorithm (blue).

112


0 200 400 600 800 1000 1200 1400 1600 1800−100

−50

0

50

100

time [s]

wx[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800−100

−50

0

50

100

time [s]

wy[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800−40

−20

0

20

40

time [s]

wz[K

m/h]

Figure 5.6: Actual wind (red), and 10 wind realizations extracted at each time step ac-cording to the AR(1) models based on the wind random field correlation structure(blue).

0 200 400 600 800 1000 1200 1400 1600 1800−100

−50

0

50

100

time [s]

wx[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800−100

−50

0

50

100

time [s]

wy[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800−40

−20

0

20

40

time [s]

wz[K

m/h]

Figure 5.7: Actual wind (red), and 10 wind realizations extracted at each time step bymeans of the approach based on random field reconstruction (blue).

113


The extracted wind realizations obtained by means of the AR(3) models re-cursively identified at each time step by the RLS algorithm, well track theactual wind and are slightly more accurate with respect to the wind realiza-tions obtained by means of the other two approaches of Section 5.2.2.2 and5.2.2.3, especially on the vertical wind component. Despite the fact thatsometimes the RLS algorithm may require some steps to adapt the modelsto the wind changes (see e.g. the initial drift on the wx component, or therapid change in wy around time 1000s), the identified models close matchthe actual wind local characteristics. The other two approaches based onapproximation of the wind random field correlation structure show slightlymore spread wind realizations and seem to not as tightly fit the local windcharacteristics as the RLS-identified models. This may be due to the factthat they have a fixed structure and only their initialization is based on thelast available data.

5.5.2 Simulation 2

We perform a simulation in which the more accurate aircraft model (2.3)is considered, and the finite horizon optimization problem is formulatedas in Section 5.4. The objective of the controller is to steer the aircraftso as to track a reference trajectory designed by means of the approach inChapter 4. As in the previous section, many turns and altitude changes areintentionally introduced in the reference trajectory, so as to better evaluatethe controller capability.The system parameters and the bounds for the constraints on physical lim-itations and passenger comfort are set as follow:

Vmax = 910 km/h Vmin = 650 km/hγmax = 5 γmin = −3

Tmax = 2 · 276 kN Tmin = Tmax/200

φ = 40

m = 150 103 kg S = 0.28 103 m2

Cd = 0.026 Cl = 0.24

b1 = 12.6 a = 59

b2 = 377.

The model is time-discretized with a sample time Ts = 2s. We sup-pose that the aircraft initial position and velocity coincide with the start-ing position and velocity of the reference trajectory, namely x0 = [−60 −60 6 600 600 0]T .

114


We set the prediction horizonM = 20, and the weights of the finite horizonoptimization problem as follow:

Rc =

0.125 0 0

0 1 0

0 0 0.25

µud = 0.72

µLc = 8 µlc = 6 µvc = 6

µLd = 0.72 µld = 0.72 µvd = 0.72.

In this case we do not explicitly provide guarantees on the finite horizonsolutions, instead the number of extracted wind realizations N is chosen soas to have guarantees tailored directly on the receding horizon solution: theasymptotic bound on the closed loop constraint violation ratio is ε = 0.1,and according to (5.18) the corresponding number of sample wind realiza-tions is N = 59. The wind disturbance realizations required for the appli-cation of the scenario approach are generated according to AR(3) modelsrecursively identified at each time step by means of RLS algorithm as de-scribed in Section 5.2.

The results achieved by implementing the MPC approach to the presentsetup are reported in Figures 5.8, 5.9, where an x-y view, a 3-D view and azoomed particular of the actual trajectory of the aircraft are depicted alongwith the reference trajectory.

115


−60 −40 −20 0 20 40 60 80 100−60

−40

−20

0

20

40

60

80

100

120

x [Km]

y[K

m]

(a) x-y view

−100

−50

0

50

100 −60 −40 −20 0 20 40 60 80 100 120

2.5

3

3.5

4

4.5

5

5.5

6

6.5

y [Km]x [Km]

z[K

m]

(b) 3-D view

Figure 5.8: Aircraft trajectory, receding horizon solution in blue stars, finite horizon so-lution computed at each time step in colored squares, and reference trajectory in redcircles.

116


−15

−10

−5

0 −35

−30

−25

−20

−15

−103.5

4

4.5

5

y [Km]

x [Km]

z[K

m]

Figure 5.9: A zoomed particular of the aircraft trajectory, receding horizon solution inblue stars, finite horizon solution computed at each time step in colored squares, andreference trajectory in red circles.

As it appears, the reference trajectory is well followed by the aircraft,though the approximation introduced to attain convexity hampers the fi-nite horizon solutions to make fast turns (see Figure 5.9), it seems that thisapproximation does not adversely affect the actual behavior of the aircraft,thanks to the beneficial effect of the receding horizon. All the physicallimitations and comfort constraints are satisfied by the receding horizon so-lution as seen in Figures 5.10, 5.11

117


0 200 400 600 800 1000 1200 1400 1600 1800−60

−40

−20

0

20

40

60

80

100

120

Time [s]

xyz[K

m]

(a) Positions x (blue), y (green), z (red).

0 200 400 600 800 1000 1200 1400 1600 1800−200

−150

−100

−50

0

50

100

150

200

Time [s]

ψ[deg

rees]

(b) Heading angle ψ.

0 200 400 600 800 1000 1200 1400 1600 1800−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

Time [s]

x4x5x6[K

m/s]

(c) Velocities x4 (blue), x5 (green), x6 (red).

0 200 400 600 800 1000 1200 1400 1600 18000.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

0.26

Time [s]

V[K

m/s]

(d) TAS V , reference speed (cyan).

0 200 400 600 800 1000 1200 1400 1600 1800−8

−6

−4

−2

0

2

4

6

8x 10−3

Time [s]

u1u2u3[K

m/s2]

(e) Acceleration u1 (blue), u2 (green), u3 (red).

0 200 400 600 800 1000 1200 1400 1600 1800−50

−40

−30

−20

−10

0

10

20

30

40

50

Time [s]

γφ[deg

rees]

(f) Path angle γ (blue), bank angle φ (green).

Figure 5.10

118


0 200 400 600 800 1000 1200 1400 1600 1800−8

−6

−4

−2

0

2

4

6

8x 10−4

Time [s]

τ[K

m/s2]

(a) Longitudinal acceleration τ .

0 200 400 600 800 1000 1200 1400 1600 18000

100

200

300

400

500

600

Time [s]

T[K

N]

(b) Engine thrust T .

0 200 400 600 800 1000 1200 1400 1600 1800−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time [s]

α[deg

rees]

(c) Angle of attack α.

Figure 5.11

Note that despite the fact that the reference speed profile is constant theactual TAS is not, since the controller adapts the TAS so as to counteractthe action of the wind. We can also see that the bank and the path anglesare exploited so as to steer the aircraft along the turns and the changes ofaltitude of the reference trajectory.In Figure 5.12 are depicted the absolute values of the components of theactual position error ξ?k , and the corresponding bound h?L,k, h?l,k h

?v,k and the

bounds on the 2 and 3 steps ahead components on the position error.

119


0 200 400 600 800 1000 1200 1400 1600 18000

0.05

0.1

0.15

0.2

Time [s]

hL[K

m]

0 200 400 600 800 1000 1200 1400 1600 18000

0.02

0.04

0.06

Time [s]

hl[K

m]

0 200 400 600 800 1000 1200 1400 1600 18000

0.01

0.02

0.03

0.04

Time [s]

hv[K

m]

Figure 5.12: Absolute value of the aircraft position error ξ?k in black, h?L,k+i, h?l,k+i,

h?v,k+i for i = 0 (red), i = 1 (blue) and i = 2 (green).

As one can see the position error keeps almost always below 100m and it isusually even smaller especially on the lateral and vertical components. Thecomputed bounds h?L,k, h?l,k, h

?v,k are quite close to the actual position error:

the first step position constraint is violated, namely |ξ?k| is greater than thecorresponding bound [h?L,k h

?l,k h

?v,k] computed at time k − 1, only in the

5.2% of the steps, a performance compatible with the maximum allowedasymptotic bound on the closed-loop constraint violation ratio ε = 0.1.Note that the bounds on the position error become larger as farther stepsahead in the future are considered because the wind disturbance affects theaircraft position making the predicted evolutions of the position error morespread and uncertain.In Figure 5.13 is reported the actual wind realization along the aircraft tra-jectory, together with some of the extracted wind realizations used at eachstep in the solution of the scenario optimization problem. The wind sam-ples are obtained by simulation of 3 recursively identified AR(3) models.

120


0 200 400 600 800 1000 1200 1400 1600 1800 2000−100

−50

0

50

100

Time [s]wx[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800 2000−100

−50

0

50

100

Time [s]

wy[K

m/h]

0 200 400 600 800 1000 1200 1400 1600 1800 2000−50

0

50

Time [s]

wz[K

m/h]

Figure 5.13: Actual wind along the aircraft trajectory (red) and 10 wind samples for eachtime step obtained by simulation of the last identified AR models (blue).

As one can see the identified AR(3) models are step by step adapted to thelocal characteristics of wind faced by the aircraft, and the generated windrealizations well track the actual wind disturbance.

5.5.3 Simulation 3

Considering the same setup described in the previous section, we perform asimulation in which the wind disturbance acts on the aircraft, but the windpresence is not accounted for in the MPC controller design, namely we setw = 0 in the finite horizon optimization problem. The same referencetrajectory and actual wind disturbance realization of the previous simula-tion is used. Our purpose is to compare the performances achieved by theprobabilistic robust solution obtained in the previous section, and by thesolution obtained neglecting the wind disturbance in the controller design.Note however that, also when the wind presence is neglected, the recedinghorizon strategy introduces a feedback in the closed-loop solution, hencewe do not expect that the aircraft position error takes large drifts as it mayhappen if a purely open loop solution were applied.The achieved results for the current setup are shown in Figure 5.14 wherethe receding horizon aircraft trajectory is depicted, together with the finitehorizon solution computed at each time step.

121


−60 −40 −20 0 20 40 60 80 100−60

−40

−20

0

20

40

60

80

100

120

x [Km]

y[K

m]

(a) x-y view.

−100

−50

0

50

100 −60 −40 −20 0 20 40 60 80 100 120

2.5

3

3.5

4

4.5

5

5.5

6

6.5

y [Km]x [Km]

z[K

m]

(b) 3-D view.

Figure 5.14: Aircraft trajectory, receding horizon solution in blue stars, finite horizonsolution computed at each time step in colored squares, and reference trajectory in redcircles.

122


The reference trajectory is quite well followed also in this case, but if welook at the actual position error and at the computed bounds on the positionerror components in Figure 5.15 we can see that the probabilistic robustsolution achieves better performances. More precisely, as shown in Figure5.16, the actual position error obtained by means of the scenario based solu-tion computed in the previous section is almost always significantly smallerthan the actual position error achieved neglecting the wind disturbance inthe controller design: we get that neglecting the wind presence

∑900k=1 |ξ?k|

equals [60.47 27.40 7.80], while accounting for the wind presence we ob-tain [8.74 2.98 0.93]. Furthermore we can see that the bounds on the 2 and3 steps ahead predicted position error components in Figure 5.15 have anopposite trend with respect to the previous case in Figure 5.12: indeed thebounds on the predicted position error become smaller as time step fartherin the future are considered. This is due to the fact that the controller donot account for the presence of the wind, and, hence, it predicts to be ableto reduce the initial position error along the finite horizon. This, however,does not happen because of the actual wind disturbance acting on the air-craft, and the obtained bounds for the position error components is violatedby the actual position error in the 99,8% of the simulation steps.

123


0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

Time [s]

hL[K

m]

0 200 400 600 800 1000 1200 1400 1600 18000

0.05

0.1

0.15

0.2

Time [s]

hl[K

m]

0 200 400 600 800 1000 1200 1400 1600 18000

0.02

0.04

0.06

Time [s]

hv[K

m]

Figure 5.15: Absolute value of the aircraft position error ξ?k in black, h?L,k+i, h?l,k+i,

h?v,k+i for i = 0 (red), i = 1 (blue) and i = 2 (green).

124


0 200 400 600 800 1000 1200 1400 1600 18000

0.2

0.4

Time [s]

|ξ∗ L|[K

m]

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

Time [s]

|ξ∗ l|[K

m]

0 200 400 600 800 1000 1200 1400 1600 18000

0.02

0.04

0.06

Time [s]

|ξ∗ v|[K

m]

Figure 5.16: Absolute value of position error ξ obtained accounting for the wind presence(red) and neglecting the presence of the wind (green) in the finite horizon optimizationproblem.

5.5.4 Validation

By means of the proposed approach, we find bounds hL, hl, hv on the posi-tion error ξ that change at every time step, and that depend on the particularwind realization used in the simulation. Our aim is to validate the per-formance of the controller finding a bound hp on the position error that isguaranteed to hold along the whole considered trajectory with high prob-ability with respect to the wind realizations. The computation of such abound hp can be addressed by means of a chance-constrained optimizationproblem:

minhp

hp (5.32)

s.t. Pmaxk‖ξk(w)?‖2 ≤ hp ≥ 1− εp.

The value of hp achieved solving (5.32) represents a probabilistic boundon the maximal distance between aircraft position and reference trajectory.Problem (5.32) can be solved again exploiting the scenario approach the-ory: the bound hp is computed simply performing Np simulations in which

125


the aircraft is controlled by means of the developed MPC controller for agiven reference trajectory, considering different wind realizations. Then wetake

hp = maxj=1...Np

maxk‖ξk(w)‖(j)

2 .

The number Np of simulations should be chosen according to (5.15) toachieve the desired probabilistic guarantees on the bound hp.A bound on the maximal deviation along the reference trajectory that isrobust, in a probabilistic sense, with respect to the wind realizations canbe exploited in the context of air traffic management to properly designreference trajectories for multiple aircraft.In order to compute the probabilistic bound hp on the maximal aircraft po-sition error along the whole trajectory the same simulation setup as in Sec-tions 5.5.2 and 5.5.3 is considered. We set εp = 0.05, β = 10−6 and,correspondingly, we obtain Np = 270.In Figure 5.17 are reported for 270 simulations of 500 steps the 2-norm ofthe position error ‖ξk(w)?‖2 obtained accounting for the presence of thewind or neglecting it in the controller design. As one can see, in most ofthe simulations, the position error keeps quite small, while only in few ofthem it takes larger values. Moreover, it can be clearly seen that in most ofthe simulations the position errors obtained by the solution that explicitlyaccounts for the wind disturbance take smaller values with respect to theposition errors achieved neglecting the wind disturbance in the controllerdesign. This fact can be further appreciated looking at the sampled distri-butions of maxk ‖ξ?k‖2 and of 1

K

∑Kk=1 ‖ξ?k‖2 reported in Figure 5.18. This

figure shows how the sampled distributions of the position error are signif-icantly shifted toward smaller values when the wind presence is explicitlyaccounted for and a probabilistic robust solution is designed.

126


0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

Time [s]

‖ξk‖ 2

[km]

(a) Aircraft controlled with the scenario based MPC.

0 100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

Time [s]

‖ξk‖ 2

[km]

(b) Aircraft controlled without accounting for wind in MPC design.

Figure 5.17: Obtained ‖ξk‖2 for simulations with different wind realizations.

127


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

maxk ‖ξk‖2 [km]

(a) Sampled distribution of maxk ‖ξk‖2 for simulations with dif-ferent wind realizations obtained accounting for the wind dis-turbance (blue) or neglecting it (red) in the MPC design.

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1K

∑K

k=1 ‖ξk‖2 [km]

(b) Sampled distribution of 1K

∑Kk=1 ‖ξk‖2 for simulations with

different wind realizations obtained accounting for the winddisturbance (blue) or neglecting it (red) in the MPC design.

Figure 5.18

128

CHAPTER6Uncertainty on aircraft mass

In the previous chapters we assume that the state and the parameters of theaircraft model were perfectly known. Here, instead, we consider that thestate is not exactly known but only an estimation of it, affected by uncer-tainty, is available. In particular we focus on the uncertainty affecting theaircraft mass m which is assumed to be not exactly known, but only esti-mated by m. The results presented in the following can be easily extendedto cases where other sources of uncertainty are present due to noisy statemeasurements or model errors. Note that we do not address the problem ofmass estimation, instead we focus on the analysis of the sensitivity of theperformance and on the robustness of the designed solution with respect tothe mass uncertainty.To this purpose we revisit the steps needed to develop the MPC controlscheme taking into account that only m is available, so as to explicitly high-light the effects of uncertainty, and possibly to ensure some level of robust-ness with respect to it. The mass uncertainty hampers the exact feedbacklinarization of the aircraft model introducing an error proportional to themismatch between m and m. Some simplifying approximations are madeso as to regard the committed error as an additional disturbance acting,alongside with the wind disturbance, on the discrete time aircraft dynam-

129

Chapter 6. Uncertainty on aircraft mass

ics. This allows to account for it in the finite horizon optimization problemin a computationally tractable way.

6.1 Feedback linearization with mass uncertainty

Suppose that the actual mass and its estimation are related by:

m = δmm.

We consider again the aircraft model in (2.3) and we perform the feedbacklinearization following the same steps in Section 3.2, accounting for the factthat the mass m is not exactly known but only its estimation m is available.We set the engine thrust T as in (3.13), but for the fact that m is replacedwith m:

T = (mτ + gm sin γ +D)1

cosα, (6.1)

so that the auxiliary variable τ is defined:

τ = −g sin γ +−D + T cosα

m.

The resulting TAS dynamics shows an additional term due to mass estima-tion error:

V = −g sin γ +−D + T cosα

m+−D + T cosα

m− −D + T cosα

m=

= τ +−D + T cosα

m(δm − 1). (6.2)

We define the new state variables as in (3.15), (3.16). Accounting for theaircraft dynamics in (2.3) and for (6.2) the new state dynamics becomes:

x1 =x4 + wx (6.3)

x2 =x5 + wy

x3 =x6 + wz

x4 =τ cos γ cosψ − V sin γ cosψγ − V cos γ sinψL+ T sinα

mV cos γsinφ+

+−D + T cosα

m(δm − 1) cos γ cosψ

x5 =τ cos γ sinψ − V sin γ sinψγ + V cos γ cosψL+ T sinα

mV cos γsinφ+

+−D + T cosα

m(δm − 1) cos γ sinψ

x6 =τ sin γ + V cos γγ +−D + T cosα

m(δm − 1) sin γ.

130

6.1. Feedback linearization with mass uncertainty

We define the new input variables u1, u2, u3 as in (3.18) but replacing mwith m so that the new state dynamics in (6.3) rewrites:

x1 = x4 + wx (6.4)

x2 = x5 + wy

x3 = x6 + wz

x4 = u1 + (δm − 1)(−D+T cosα

mcos γ cosψ − L+T sinα

m(sin γ cosψ cosφ+ sinψ sinφ)

)= u1 + (δm − 1)

((τ + g sin γ) cos γ cosψ −

√ν21

cos2 γ+ ν2

2 (sin γ cosψ cosφ+ sinψ sinφ))

= u1 + dm1

x5 = u2 + (δm − 1)(−D+T cosα

mcos γ sinψ − L+T sinα

m(sin γ sinψ cosφ− cosψ sinφ)

)= u2 + (δm − 1)

((τ + g sin γ) cos γ sinψ −

√ν21

cos2 γ+ ν2

2 (sin γ sinψ cosφ− cosψ sinφ))

= u2 + dm2

x6 = u3 + (δm − 1)(−D+T cosα

msin γ + L+T sinα

mcos γ cosψ

)= u3 + (δm − 1)

((τ + g sin γ) sin γ +

√ν21

cos2 γ+ ν2

2 cos γ cosψ)

= u3 + dm3.

where ν1 and ν2 are defined as in (3.24). Note that the dynamics are com-posed by the linear terms which mirror the dynamics (3.31) obtained as-suming exact knowledge of the aircraft mass, and by the error terms dueto the presence of the estimation m, which make the overall dynamics nonlinear because of their non linear dependence on the state and the input. Itis clear that if m = m then δm = 1 and all the error terms vanish. Thenon linear terms, roughly speaking, represent the error committed on theaccelerations and on the compensation of the drag and of the weight forcedue to the not exact knowledge of the aircraft mass, projected by means ofthe heading, path and bank angles along the Cartesian axes. Note that theerrors committed on the x and y components of the acceleration is usuallysmall, instead the error committed on the z component is relatively largerbecause of the not exact compensation of the weight force. Indeed the erroron the z-axis can be roughly represented by (δm − 1)(u3 + g).Given the new input u1, u2, u3, the value of the original input T , φ, α canbe recovered by means of the relationships (3.23), (3.25), (3.27) in Section3.2.

131


6.2 Finite horizon optimization problem with mass uncertainty

We discretize the dynamics (6.4):

x1,k+1

x2,k+1

x3,k+1

x4,k+1

x5,k+1

x6,k+1

=

[I TsI

0 I

]

x1,k

x2,k

x3,k

x4,k

x5,k

x6,k

+

[T 2s

2I

TsI

]u1,k + dm1,k

u2,k + dm2,k

u3,k + dm3,k

+

[TsI

0

]wx,kwy,k

wz,k

xk+1 = Axk +B(uk + dm,k) +Bwwk. (6.5)

Note that the term dm,k(x,u) that represents the error due to the mass esti-mation m, depends on the state x and, hence, the time discretization is notexact. However, as far as the sample time Ts is small enough, the discretetime dynamics should well approximate the original continuous time ones,so that model (6.5) can be exploited in the development of the MPC controlscheme.The constraints on the aircraft physical limitations and passenger comfortcan be dealt with by means of the formulation in Section 3.31. Note how-ever that, given the dynamics (6.5), the constraints are affected by dm1, dm2,dm3 through the states x4, x5, x6.In order to account for the uncertainty affecting the mass estimation weassume that the mass error δm is a random variable with some probabil-ity distribution that describes the accuracy of the mass estimation. Hencethe term dm,k(x,u, δm) can be seen as an additional stochastic disturbanceacting on the aircraft dynamics.This in turn imply that i) the constraints on aircraft physical limitationsare no more deterministic but they must be formulated as probabilistic (orrobust) constraints so as to account for the presence of the stochastic dis-turbance; ii) as dm1, dm2, dm3 are not simple additive disturbances becausethey depend non-linearly on the state and on the input variables, the con-straints both on aircraft physical limitations and on aircraft position becomenon-convex even for a fixed (not null) value of the mass estimation error δm,making them very hard to handle. In order to address these issues some ap-proximations are introduced to enhance computational tractability. At firstwe try to preserve as much as possible the structure of the disturbances dm1,dm2, dm3: a tractable optimization problem is formulated, but a significant

1Note that in the constraint on the engine thrust the mass m has to be replaced with m, accordingly with thenew definition of T in (6.1)

132

6.2. Finite horizon optimization problem with mass uncertainty

computational effort is required to solve it. Then we propose a less accu-rate approximation of the disturbance that allows to significantly reduce thecomputational effort.

6.2.1 Multiplicative disturbance

A first idea is to replace dm with an approximation of it that shows anaffine dependence on the input u. This way the dynamics becomes linearand the uncertainty affects the coefficients with which the input enters thedynamics (multiplicative disturbance). Moreover, thanks to the linearity ofthe dynamics, the constraints both on aircraft physical limitations and onaircraft position are convex for every fixed value of the uncertainty, allow-ing to recover computational tractability by means of randomized methods.

To this purpose, we start dealing with the term√

ν21

cos2 γ+ ν2

2 : recalling that

|ν2| ≤ ν1tan φcos γ

(see (3.48)) we can bound it by√ν21

cos2 γ(1 + tan2 φ) = ν1

cos γ cos φ.

Then, following the approach in Section 3.3, we fix the values of the head-ing angle ψk+i and of the path angle γk+i to their current value ψk and γk,moreover we fix the value of the bank angle φk taking its last value φk−1.Eventually, we can approximate dm with dm:

dm1,k+i =(δm − 1)(

(τ + g sin γk) cos γk cosψk+

+u3,k+i + τ sin γk + g cos2 γk

cos γk cos φ(− sin γk cosψk cosφk−1 − sinψk sinφk−1)

)dm2,k+i =(δm − 1)

((τ + g sin γk) cos γk sinψk+

+u3,k+i + τ sin γk + g cos2 γk

cos γk cos φ(− sin γk sinψk cosφk−1 + cosψk sinφk−1)

)dm3,k+i =(δm − 1)

((τ + g sin γk) sin γk +

u3,k+i + τ sin γk + g cos2 γk

cos γk cos φcos γk cosψk

)τ = cos γk(u1,k+i cosψk + u2,k+i sinψk + u3,k+i tan γk) i = 0, . . . ,M.

The new term dm is affine in the input u and is meant to replace dm inthe system dynamics (6.5). Accordingly the constraints result convex withrespect to the optimization variables, even though still affected by multi-plicative uncertainty δm. Note that the constraint on the position error isaffected also by the wind uncertainty, which can be handled as in Section5.22. In order to deal with the presence of uncertainty in the constraints

2Note that the wind data can be approximately recovered, indeed, similarly to (5.6), we can first recoverthe disturbance due to mass uncertainty from the dynamics of x4, x5, x6 and then recover the wind velocitiesexploiting the dynamics of x1, x2, x3.

133


we formulate them by means of a chance-constraint that accounts for bothaircraft physical limitations and position error. The resulting finite horizonoptimization problem is:

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (6.6)

s.t.dynamics (6.5) with w in place of w, and d in place of d

P

constraint (3.33) (3.34) (3.36) (3.38) (3.44)

(3.54) (3.48)

‖ξk+i(w, dm)‖22 ≤ hi i = 1, . . . ,M

≥ 1− ε.

As it is clear the position error constraint can be alternatively formulatedas proposed in Section 5.4. Problem (6.6) can be approximately solvedresorting to the scenario approach, achieving the same guarantees describedin Section 5.3 on the obtained solution. Note that in this case every sampledscenario is composed by a wind realization and a mass error realization, sothat we have to extract both w, and δm that appears in dm. One of the mainadvantages of the scenario approach is that it can be applied regardlessof the structure of the uncertainty affecting the constraints, being the onlyrequirement convexity with respect to the optimization variables for everyfixed valued of the uncertainty. Indeed it can be straightforwardly applied toproblem (6.6) where multiplicative and additive uncertainties from differentsources has to be accounted for.By means of the scenario approach we are able to find an approximatebut guaranteed solution to problem (6.6) which because of the presence ofthe chance-constraint and of the multiplicative uncertainty is very hard totackle. On the other hand, despite its the convexity, the scenario programcorresponding to (6.6) is quite computational demanding because all theconstraints depend on the uncertainty. Hence additional approximationsaimed at further reducing the computational effort may be introduced.

6.2.2 Additive disturbance

A way to greatly reduce the required computational effort is to neglect thepresence of the disturbance on the dynamics of x4, x5, x6 so that the con-straints representing aircraft physical limitations and passenger comfort be-come deterministic. Furthermore we disregard the structure of dm1, dm2,dm3 and we see them as purely additive disturbances which sum up with

134

6.2. Finite horizon optimization problem with mass uncertainty

the wind disturbance in the dynamics of x1, x2, x3. In this case the dis-crete time aircraft dynamics considered in the finite horizon optimizationproblem become:

x1,k+1

x2,k+1

x3,k+1

x4,k+1

x5,k+1

x6,k+1

=

[I TsI

0 I

]

x1,k

x2,k

x3,k

x4,k

x5,k

x6,k

+

[T 2s

2I

TsI

]u1,k

u2,k

u3,k

+

[I

0

]wdx,kwdy,k

wdz,k

, (6.7)

where the additive disturbance wd = [wdx, wdy, wdz]T accounts for all

the disturbance contributions acting on the position dynamics. In order tocharacterize wd so as to properly represent, at least locally, the uncertaintydue to both wind and mass estimation, we resort to the approach in Section5.2.2.1. The disturbances wdx, wdy, wdz are modelled by means of time se-ries whose parameters are identified by an RLS algorithm based on the pastdata recovered via (5.6) along the aircraft trajectory. The model parame-ters are updated at each time step following the receding horizon strategyof the controller. Although we neglect the structure of the disturbance, theRLS algorithm should be able to properly set the model parameters so as tomatch the local disturbance characteristics, as seen from data. Indeed thisapproach has the advantage that it can be exploited to identify a concisemodel of all the disturbances acting on the aircraft position dynamics.Following this approach the finite horizon optimization problem can be for-mulated as in Section 5.1, being the disturbance wd a purely additive dis-turbance acting on the dynamics of x1, x2, x3 only.

minuk+i i = 0 . . .M − 1

hi i = 1 . . .M

J (6.8)

s.t.dynamics (6.7)constraint (3.33) (3.34) (3.36) (3.38) (3.44) (3.54) (3.48)P ‖ξk+i(wd,k, . . . ,wd,k+i−1)‖2

2 ≤ hi i = 1, . . . ,M ≥ 1− ε.

Problem (6.8) can be approximately solved resorting to the the scenarioapproach, where the samples for wd,k+i, i = 1, . . . ,M have to be extractedaccording to last available model identified by the RLS algorithm at eachtime step. All the considerations in Sections 5.3, 5.4 apply.

135


By means of this approach the error introduced by the not exact knowl-edge of the aircraft mass is quite roughly approximated, however it maybe enough to achieve good performances. On the other hand, the requiredcomputational effort is much reduced with respect to the case in whichmultiplicative uncertainty affecting all the state dynamics is considered.As discussed in Section 5.4.1, one may want to consider a disturbance feed-back parametrization for the input, as in (5.30), so that a state feedbackterm is introduced in the finite horizon optimization problem. As the dis-turbance introduced by the aircraft mass uncertainty has different charac-teristics with respect to the wind disturbance, it may be that, in this case, thedisturbance feedback parametrization for the input could slightly improveperformance. On the other hand, the disturbance feedback parametriza-tion of the input significantly increases the computational effort required tosolve the finite horizon optimization problem. A possible compromise is toconsider the disturbance feedback parametrization in the formulation of theposition constraints, while in the constraints on aircraft physical limitationswd is replaced by E[wd] so as to keep them deterministic. By means of thisheuristic approach we can keep the computational effort low while partiallyintroducing disturbance feedback in the finite horizon solution design.


In this section we report simulation results obtained neglecting the winddisturbance and considering the aircraft mass available to the controlleronly through the uncertain estimation m. Our purpose is to evaluate theeffects of the mass uncertainty, and to assess the performance of the lowcomputational demanding approach of Section 6.2.2.In the finite horizon optimization problem the disturbance is considered toact only on the position dynamics as in (6.7), and it is modelled by meansof three AR(3) processes, whose parameters are identified by the RLS al-gorithm and updated at each time step as discussed in Section 5.2.2.1. Themodel parameters and constraints bound are the same as in Section 5.5.2,the position error constraint is formulated as in Section 5.4. As for themass estimation m we consider a Gaussian distribution centred in m, withan artificially large variance so as to emphasize the uncertainty effects.The achieved results are shown in Figure 6.1-6.2.

136


−60−40

−200

2040

60 −60

−40

−20

0

20

40

4

4.5

5

5.5

6

y [Km]

x [Km]

z[K

m]

Figure 6.1: Aircraft trajectory, receding horizon solution in blue stars, reference trajec-tory in red circles.

137


0 100 200 300 400 500 600 700 800 900 10000

0.02

0.04

Time [s]

hL[K

m]

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

Time [s]

hl[K

m]

0 100 200 300 400 500 600 700 800 900 10000

0.02

0.04

0.06

0.08

Time [s]

hv[K

m]

Figure 6.2: Absolute value of the aircraft position error ξ?k in black, h?L,k+i, h?l,k+i, h

?v,k+i

for i = 0 (red) and i = 1 (blue).

Despite the mass estimation we adopt in the simulation is not very accu-rate, the reference trajectory is quite well followed by the aircraft as shownin Figure 6.1, especially on the longitudinal and lateral components. Aslightly worst performance is achieved on the vertical component of theposition error which seems more affected by the mass uncertainty. Fig-ure 6.3 shows the true mass m and its uncertain estimation m available tothe controller, while Figure 6.4 shows the disturbance acting on the posi-tion dynamics recovered via (5.6), and some of the disturbance realizationsused in the scenario program obtained by simulation of the AR(3) modelsrecursively identified at each time step.

138


0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 105

Time [s]

m[K

g]

Figure 6.3: Aircraft mass m (red), mass estimation m (blue).

0 100 200 300 400 500 600 700 800 900 1000−0.04

−0.02

0

0.02

0.04

time [s]

dm1

0 100 200 300 400 500 600 700 800 900 1000−0.01

−0.005

0

0.005

0.01

time [s]

dm2

0 100 200 300 400 500 600 700 800 900 1000−0.01

−0.005

0

0.005

0.01

time [s]

dm3

Figure 6.4: Disturbance acting on aircraft position dynamics (red) and 10 disturbancesamples for each time step obtained by simulation of the last identified AR models(blue).

As one can see, the RLS algorithm needs some time steps to properly ad-just the model parameters whose initialization was not tailored to the new

139


disturbance characteristics. Indeed, after the initial steps, the extracted dis-turbance realizations become closer to the actual disturbance. It is worthnoticing that, as expected, the actual disturbance affects the z componentslightly more with respect to the x and y components.Another simulation is performed considering a disturbance feedback parametriza-tion for the input. The wind disturbance is neglected and the same massestimation m as above is used. In order to reduce the computational effortwe adopt a reduced parametrization with respect to (5.30), in which onlythe input u3 is in feedback over the past values of wd3, namely:

u3,k+i = g3,i +l=i−1∑l=0

θi,lwd3,k+l

The idea is that, in view of the results of the previous simulation, the distur-bance feedback is introduced only on the z component that seems to be themost sensible to mass uncertainty. The physical limitation constraints arekept deterministic replacing wd3 with E[wd3] in them. The actual positionerror achieved in this case are compared with those achieved in the previoussimulation in Figure 6.5.

0 100 200 300 400 500 600 700 800 900 10000

0.005

0.01

0.015

0.02

Time [s]

ξ1[K

m]

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

Time [s]

ξ2[K

m]

0 100 200 300 400 500 600 700 800 900 10000

0.02

0.04

0.06

Time [s]

ξ3[K

m]

Figure 6.5: Absolute value of the aircraft position error ξ?k with disturbance feedback onu3 (green) or without it (red).

140


As one can see the performance are almost equal on the x and y positionerror components, instead the error on the z is slightly smaller when thedisturbance feedback term is considered in the control input design.

141

CHAPTER7Conclusions

In this thesis, an MPC controller for aircraft motion has been developed inorder to satisfy time and space requirements imposed on the aircraft tra-jectory by Target Windows. Moreover the designed controller accounts forthe aircraft physical limitations and passenger comfort constraints, and it isrobust with respect to the wind disturbance.A first effort has been put in modelling the aircraft dynamics and its physi-cal limitations by means of linear dynamics and convex constraints: this hasbeen addressed by feedback linearization and approximate reformulationof the constraints. The achieved linear dynamics and the introduced con-straint approximations are needed to allow the usage of MPC and to obtaina convex, computationally tractable, finite horizon optimization problem.Moreover the approximations of the constraints are such that the recedinghorizon application of the solution is guaranteed to satisfy the original con-straints on aircraft physical limitations.The problem of handling TWs specifications has been proven to be chal-lenging, and, in order to tackle it, first a specifically tailored algorithm hasbeen developed to generate reference trajectories that comply with TWsrequirements, accounting for aircraft physical limitations. The design ap-proach aims to satisfy TWs specifications by means of smooth trajectories,

143

Chapter 7. Conclusions

composed by simple motion primitives, that can be easily followed by air-craft. Then, the MPC controller has been designed with the aim of trackingthese trajectories thus allowing the aircraft to meet the TWs specificationsin its actual operations. As far as the tracking MPC is concerned, due to thepresence of the wind, it is not possible to enforce hard constraints on theaircraft position; therefore, we have chosen to resort to chance-constraintsaimed at keeping the error between aircraft position and the reference tra-jectory as small as possible. Different approaches to represent the wind dis-turbance were developed, they rely on models that can be easily includedin the finite horizon optimization problem, and identification methods wereused to adapt the wind model parameters so as to best describe the lo-cal wind characteristics. Randomized methods were exploited to handlethe chance-constraints at low computational effort, achieving probabilisticguarantees on the obtained solution. Overall, as it is also proven by nu-merical simulations, the developed controller seems to be able to steer theaircraft along the reference trajectory with good accuracy, counteracting theaction of the wind, compatibly with the aircraft motion capability.Further developments may focus on a computationally efficient implemen-tation of the proposed controller, compatible with aircraft on board hard-ware. To this purpose also the interactions with the low level controllersthat actually operates the aircraft should be to be accounted for.On the other side, interactions of the developed controller with higher con-trol layers, such as ATM systems that manage airspace and provide timeand space trajectory specifications, may be investigated. In particular, in amulti aircraft scenario, TWs design may benefit of the presence of an air-craft motion controller accounting for both space and time specifications,and that, providing probabilistic guarantees on the robustness of the solu-tion, makes aircraft trajectories more predictable.Many of the proposed approaches are developed with reference to the appli-cation at hand, but they may be exploited in other frameworks. In particularthe trajectory design approach in Chapter 4 may be useful in applicationswhere time space constraints have to be satisfied and the smoothness of thetrajectory is important to account for limited motion capability. In a multiagent, e.g. multi aircraft, framework further developments may be focusedon extending the proposed trajectory design method so as to provide con-flict free trajectories. This may be achieved suitably exploiting some ofthe degrees of freedom allowed by the path design, properly modifying thetime law design, or, eventually, considering extra TWs to embed additionaltime space specifications on the trajectory.The approaches to dealt with the wind disturbance in Chapter 5 may be

144

exploited in applications where the wind plays an important role such asenergy generation by means of wind turbines, and building climate control.The approaches to model the wind in the optimization problem, to adaptmodel parameters to match local wind characteristics based on measureddata, and to account for the uncertainty by means of randomized methodscan be applied and extended to these frameworks. Moreover, more gener-ally, the approaches, described in Appendix A and applied in Chapter 5,to deal with unbounded uncertainty guaranteeing recursive feasibility andachieving a proper trade-off between robustness of the solution and perfor-mance may be exploited in many control applications where the system isaffected by stochastic uncertainty and probabilistic constraints have to beaddressed.

145

APPENDIXARandomized methods for stochastic

constrained control

A.1 Introduction

In this appendix we address from a general perspective the finite horizoncontrol of a stochastic linear system focusing on the issue of ensuring fea-sibility of the finite horizon optimization problem in presence of state con-straints and additive unbounded disturbances. Different methods to man-age the trade off between the two contrasting objectives of state constraintsfeasibility and performance are discussed. In all the proposed methods suit-able chance-constrained optimization problems are formulated, and, then,approximately solved resorting to randomized methods.

We consider a stochastic linear system whose state xt ∈ Rn evolves ac-cording to the equation

xt+1 = Axt +But +Bwwt, (A.1)

where ut ∈ Rm and wt ∈ Rnw represent the control input and a stochasticdisturbance, respectively. Matrices A, B, Bw have appropriate dimensions,

147

Appendix A. Randomized methods for stochastic constrained control

and Bw is assumed to be full rank. The disturbance wt has known probabil-ity distribution over a possibly unbounded support and its actual value canbe reconstructed from the state measurements through

wt = B†w(xt+1 − Axt −But), (A.2)

where B†w denotes the pseudo-inverse of Bw.Our goal is to design a state-feedback control policy so as to minimize somefinite horizon cost, while satisfying saturation constraints on the controlinput and safety constraints on the state. The proposed control policy canbe applied in a receding horizon fashion. However, we shall focus here onthe finite horizon problem only.A convenient parametrization of the control input ut, which makes both thecontrol and state variables affine in the design parameters γt ∈ Rm andθt,j ∈ Rm×nw , is given by

ut = γt +t−1∑j=0

θt,jϕ(wj), (A.3)

where ϕ : R → R is a given scalar function and the notation ϕ(wj) standsfor ϕ(·) applied to each component of the nw-dimensional vector wj . Pol-icy (A.3) is indeed a state-feedback control policy since the disturbance isrecovered from the state according to (A.2). Notably, if the scalar func-tion ϕ(·) is the identity map (i.e., ϕ(α) = α, ∀α ∈ R) and Bw = In,then, we obtain a policy that is equivalent to a feedback policy affine in thestate, [32]. If ϕ(·) is given by the saturation function

ϕ(α) =

−ϕ, α < −ϕα, |α| ≤ ϕ

ϕ, α > ϕ,

(A.4)

(or alternatively by a sigmoidal function), then, the resulting policy is anonlinear function of the state and provides a bounded input even if thedisturbance is unbounded, [35, 36].The system performance is expressed in terms of the average quadratic cost

J = E

[M∑t=1

xTt Qtxt +M−1∑t=0

uTt Rtut

], (A.5)

where Qt ∈ Rn×n and Rt ∈ Rm×m are symmetric positive semidefinitematrices. We consider saturation and safety constraints of the following

148

A.1. Introduction

form:

supt=0,...,M−1

‖ut‖∞ ≤ u,

(A.6)sup

t=1,...,M‖Cxt‖∞ ≤ y.

Note that the value taken by the input and state variables along the refer-ence finte-horizon [0,M ] is uncertain, since it depends on the noise processwt affecting the system evolution. To account for this when formulating theconstraints, one can either enforce the constraints (A.6) to hold for everyand each disturbance realization (hard constraints), even for those realiza-tions that are quite unlikely to occur, or require them to hold only on a setof disturbance realizations of probability at least 1 − ε, with ε ∈ (0, 1) setby the user (soft constraints).As for the control input, hard constraints are typically motivated by thepresence of saturation limits of the actuators. However, they do not take intoaccount the statistical properties of the noise and may lead to conservativesolutions, which motivates the introduction of soft constraints on the input.Note that, whilst both hard and soft constraints on the input are alwaysfeasible (a feasible solution is always obtained by setting all the designparameters equal to zero), hard constraints on the state are not feasible whenthe noise distribution has unbounded support, because wt enters additivelythe state equation and this contribution cannot be canceled through anycontrol action. Hence, if the noise is not bounded, one can only head forsoft state constraints, leading to the following two formulations for the inputand state constraints:

- hard & softsup

t=0,...,M−1‖ut‖∞ ≤ u, ∀(w0, w1, . . . , wM−1)

P

supt=1,...,M

‖Cxt‖∞ ≤ y≥ 1− ε;

(A.7)

- soft & soft

P

supt=0,...,M−1

‖ut‖∞ ≤ u ∧ supt=1,...,M

‖Cxt‖∞ ≤ y≥ 1− ε. (A.8)

When formulating the constraints, the value of y in the state constraints isquite critical, because of the following two reasons:

149


• the feasibility of the soft constraint on the state is not always guaran-teed since y may be not compatible with the disturbance characteris-tics, the system dynamics and initialization, and the saturation limitsimposed on the control input. This is crucial, e.g., in receding hori-zon implementations, since it hampers the achievement of recursivefeasibility;

• even when the soft constraint is feasible, the performance of the ob-tained solution can be too much adversely affected by the presence ofthe state constraints if y takes a conservative value.

As a remedy to prevent the critical issues above, rather than seeing y asa fixed value, one should try to modulate it so as to guarantee feasibility,while achieving the appropriate compromise between safety and perfor-mance.We pursue this approach and, to address the feasibility issue, y is replacedby a decision variable, say h, so that it can be automatically set to a valuecompatible with the system dynamics and initialization, input constraints,and noise characteristics. Appropriate chance-constrained optimization prob-lems depending on some parameter to be tuned are then introduced. In theseoptimization problems both safety and performance are accounted for, andthe value for the parameter determines the trade-off between the objectiveof minimizing the control cost J in (A.5) (performance) and that of mini-mizing h (safety). By tuning this parameter, one can explore the differentpossible compromises between safety and performance – while preservingfeasibility –, and choose the solution that is more satisfactory in terms ofvalues achieved for J and h.

A.1.1 Bibliographical remarks

Alternative approaches to tackle problems with both input and state con-straints and where the disturbance has unbounded support have been pro-posed in [9, 12, 15, 26, 27, 34, 50, 56, 57]. In [9, 56, 57], state constraints arereplaced by a penalization term accounting for the state constraint viola-tion so as to avoid infeasibility. In [15, 26, 27, 34, 50], an analytic convexrelaxation of chance-constraints is adopted, while in [12] the problem isreduced to one with bounded disturbance by suitably cutting the tails ofthe disturbance distribution. In all cases, the disturbance is assumed to bea sequence of i.i.d. (independent and identically distributed) random vari-ables. Many approaches also assume that the disturbance has a Gaussiandistribution, [9, 12, 15, 27, 34, 50].

150

A.2. Notational issues

The proposed methods differ from these approaches in that a randomized-based solution is considered, which allows us to drop the independence andGaussianity assumptions. Indeed, randomized methods have been recentlydeveloped to address design in the presence of uncertainty, making tractableproblems that were otherwise deemed computationally intractable, [67].Other approaches based on randomized techniques have been proposedin [19, 20, 60] under the assumption, however, that the noise has boundedsupport, or in [54] considering input constraints only.

A.1.2 Possible extensions

To avoid cumbersome notations, all the results are given for the constraintsin (A.6). Nonetheless, the presented results are still valid in the more gen-eral setting of multiple affine constraints on u and x,

supt=0,...,M−1

‖Lkut + gk‖∞ ≤ uk, k = 1, . . . , pu,

supt=1,...,M

‖Ckxt + dk‖∞ ≤ yk, k = 1, . . . , px,

through which one can e.g. pose distinct limits on the various input/statecomponents. Also, as for the probabilistic constraints, extensions to thecase when norms other than ‖ · ‖∞ are adopted can be easily worked out,the only requirement being that the norm argument is affine in the optimiza-tion variables.The proposed approach can also be adapted to the case when the distur-bance is directly measurable and a feed-forward disturbance compensatoris adopted.

A.2 Notational issues

Let

x =

x1

x2

...xM

u =

u0

u1

...uM−1

w =

w0

w1

...wM−1

.Then, it is easy to show that

x = Fx0 + Gu + Hw

u = Γ + Θϕ(w),

151


where matrices F, G and H are given by

F =

A

A2

...AM

G =

B 0n×m · · · 0n×m

AB B. . . ...

... . . . . . . 0n×m

AM−1B · · · AB B

H =

Bw 0n×nw · · · 0n×nw

ABw Bw. . . ...

... . . . . . . 0n×nwAM−1Bw · · · ABw Bw

,whereas Γ and Θ are given by

Γ =

γ0

γ1

...γM−1

Θ =

0m×nw 0m×nw · · · 0m×nw

θ1,0 0m×nw. . . ...

... . . . . . . 0m×nwθM−1,0 · · · θM−1,M−2 0m×nw

Both u and x depend linearly on the parameters Γ and Θ.If we set

Q =

Q1 · · · 0n×n... . . . ...

0n×n · · · QM

R =

R0 · · · 0m×m... . . . ...

0m×m · · · RM−1

,mw = E[w], mϕ = E[ϕ(w)] and

V =

[Vϕϕ Vϕw

V Tϕw Vww

],

where Vww and Vϕϕ are the covariance matrices of w and ϕ(w) and Vϕwis the cross covariance matrix of ϕ(w) and w; then, the control cost (A.5)

152

A.3. Trading performance for state constraint feasibility

can be expressed as the following convex function of (Γ,Θ):

J(Γ,Θ) = E[xTQx + uTRu

](A.9)

= (Fx0 + GΓ + GΘmϕ + Hmw)T Q (Fx0 + GΓ + GΘmϕ + Hmw) +

+ tr(Q

12GΘVϕϕΘTGTQ

12

)+ tr

(Q

12HVwwH

TQ12

)+

+ 2tr(Q

12GΘVϕwH

TQ12

)+ (Γ + Θmϕ)T R (Γ + Θmϕ) +

+ tr(R

12 ΘVϕϕΘTR

12

)= (Fx0 + GΓ + GΘmϕ + Hmw)T Q (Fx0+ GΓ + GΘmϕ + Hmw) +

+ tr(Q

12 [GΘ H]V [GΘ H]T Q

12

)+ (Γ + Θmϕ)T R (Γ + Θmϕ) +

+ tr(R

12 ΘVϕϕΘTR

12

).

As for the constraints (A.6), if we set

C =

C 0 . . . 0

0 C...

... . . . 0

0 . . . 0 C

(A.10)

then, they can be expressed in compact form as follows:

‖u‖∞ ≤ u

‖Cx‖∞ ≤ h,

where y has been replaced with the optimization variable h.Thanks to the linear dependence of u and x on Γ and Θ, these constraintsturn out to be convex with respect to the optimization variables, h, Γ, Θ.

A.3 Trading performance for state constraint feasibility

In this section, two parametric optimization problems that account for themodulation of performance in favour of safety are introduced. As antic-ipated in the introduction, in both problems the bound h on the norm ofthe state is regarded as an optimization variable so that feasibility is alwaysrecovered.

153


A.3.1 Additional penalization term in the control cost

In this first method, a penalization term is added to the average quadraticcost J in (A.5) in order to penalize too high values for h:

J ′ = J + µh.

The coefficient µ ≥ 0 is the relative weight between J and h and deter-mines the trade-off between the objective of having a small J (performance)and that of having a small h (safety). Depending on the kind of constraintadopted for the control input, two chance-constrained problems can be for-mulated:

minΓ,Θ,h

J(Γ,Θ) + µh (A.11)

s.t.‖u‖∞ ≤ u, ∀wP ‖Cx‖∞ ≤ h ≥ 1− ε

minΓ,Θ,h

J(Γ,Θ) + µh (A.12)

s.t.P ‖u‖∞ ≤ u ∧ ‖Cx‖∞ ≤ h ≥ 1− ε.

A.3.2 Two-step approach based on a pre-defined admissible deterio-ration of the control cost

In this second approach, the two objectives of minimizing the control costas well as the bound h on the state are handled by solving two optimiza-tion problems in cascade. In the first one, the smallest admissible controlcost is found by minimizing it subject to the control input constraints only,whereas in the second one, h is minimized subject to the constraints on bothstate and control input and a further constraint on the maximum admissibledegradation of the control cost with respect to the value J? computed in thefirst problem: J ≤ J?+α, with α ≥ 0. Again, the coefficient α determinesthe trade-off between performance and safety.When hard constraints are imposed on the control input, the first optimiza-tion problem is given by

minΓ,Θ

J(Γ,Θ) (A.13a)

s.t.‖u‖∞ ≤ u ∀w,

154

A.4. Approximate solution to the optimization problems

while, letting J? be the optimal cost obtained by solving (A.13a), the sec-ond optimization problem is:

minΓ,Θ,h

h (A.13b)

s.t.‖u‖∞ ≤ u ∀wP ‖Cx‖∞ ≤ h ≥ 1− εJ(Γ,Θ) ≤ J? + α

.

If the control input is subject to a probabilistic constraint as well, then, thefirst optimization problem writes

minΓ,Θ

J(Γ,Θ) (A.14a)

s.t.P ‖u‖∞ ≤ u ≥ 1− ε,

while, letting J? denote the optimal cost obtained by solving (A.14a), thesecond optimization problem is:

minΓ,Θ,h

h (A.14b)

s.t.P ‖u‖∞ ≤ u ∧ ‖Cx‖∞ ≤ h ≥ 1− εJ(Γ,Θ) ≤ J? + α

.

As is clear, also a two-step approach where the role of J and h is inverted(namely in the first step the smallest possible value for h subject to inputand state constraints is found while in the second step the control cost isminimized subject to an additional constraint on the maximum admissibledegradation for h) can be considered. However, this approach suffers frommajor drawback in its approximate solution. For this reason it will not betaken into account here, instead an approach focused on the managementof the state bound h will be discussed separately in Section A.7.

A.4 Approximate solution to the optimization problems

The resolution of problems (A.11)–(A.14) posed in Sections A.3.1 andA.3.2 demands for algorithmic methods to tackle the robust and probabilis-tic constraints.

155


As for the robust constraint ‖u‖∞ ≤ u, ∀w, being the support of w un-bounded, function ϕ(·) has to be chosen as a saturation function as in (A.4)because otherwise, if e.g. ϕ(·) is the identity map, the robust constraintwould always lead to solutions with Θ = 0, i.e., to a control policy withoutthe feedback term. Assuming that |ϕ(·)| ≤ ϕ and following [35, 36], therobust constraint can be then replaced by the following finite set of convexconstraints

|Γi|+ ‖Θi‖1ϕ ≤ u, i = 1, . . . ,mM, (A.15)

where Γi denotes the i-th element of vector Γ and Θi the i-th row of Θ. Theidea behind (A.15) is that u cannot be worse than when the components ofϕ(w) have all absolute value equal to ϕ and signs such that the elementsof each row Θiϕ(w) + Γi positively sum up. Plainly, any feasible pointfor (A.15) is also feasible for the original robust constraint, and, moreover,they are equivalent as long as |ϕ(w)| = ϕ for some w. The number ofconstraints in (A.15) is finite and usually small, and (A.15) can be dealtwith by means of standard solvers.The probabilistic constraints

P ‖Cx‖∞ ≤ h ≥ 1− ε (A.16)

and

P ‖u‖∞ ≤ u ∧ ‖Cx‖∞ ≤ h ≥ 1− ε, (A.17)

instead, are much harder to solve because they may be even non-convex,though ‖u‖∞ ≤ u and ‖Cx‖∞ ≤ h are convex for any fixed realizationof w. An exact resolution of the problems (A.11)–(A.14) is therefore notpossible, except for few special cases, and some level of approximationmust be accepted.In the remainder of this section, suitable relaxations of the probabilisticconstraints (A.16) and (A.17) are introduced and discussed so as to refor-mulate problems (A.11)–(A.14) in a way that is amenable of resolution bymeans of standard convex optimization techniques.

A.4.1 Algorithms

Probabilistic constraints are tacked by resorting to the scenario approach,a recently developed randomized method to approximately solve chance-constrained problems, [17, 18, 22–24].

156


The idea behind the scenario approach is very simple. A bunch of N real-izations of the disturbance w, say

w(1) =[w

(1)0 w

(1)1 . . . w

(1)M−1

]w(2) =

[w

(2)0 w

(2)1 . . . w

(2)M−1

]...

w(N) =[w

(N)0 w

(N)1 . . . w

(N)M−1

];

is generated according to the underlying probability distribution of w. Then,the probabilistic constraints are replaced with a finite number N of con-straints of the type ‖Cx‖∞ ≤ h and/or ‖u‖∞ ≤ u, those obtained in cor-respondence of the generated instances of the disturbance. More precisely,writing explicitly the dependence of x and u on w, the constraint

P ‖Cx(w)‖∞ ≤ h ≥ 1− ε

is replaced by‖Cx(w(j))‖∞ ≤ h, j = 1, . . . , N,

whileP ‖u‖∞ ≤ u ∧ ‖Cx‖∞ ≤ h ≥ 1− ε

is replaced by ‖u(w(j))‖∞ ≤ u

‖Cx(w(j))‖∞ ≤ hj = 1, . . . , N.

Summarizing, depending on the chosen type of constraint (robust or inprobability) for the input, and on the chosen method to take into accountthe presence of the optimization variable h, the possible reformulations of(A.11)–(A.14) are the following:

- Additional penalization term and hard constraint on input:

minΓ,Θ,h

J(Γ,Θ) + µh (A.18)

s.t. |Γi|+ ‖Θi‖1ϕ ≤ u, i = 1, . . . ,mM

‖Cx(w(j))‖∞ ≤ h, j = 1, . . . , N

157


- Additional penalization term and soft constraint on input:

minΓ,Θ,h

J(Γ,Θ) + µh (A.19)

s.t.‖u(w(j))‖∞ ≤ u

‖Cx(w(j))‖∞ ≤ hj = 1, . . . , N

- Two-step approach and hard constraint on input:

minΓ,Θ

J(Γ,Θ) (A.20a)

s.t.|Γi|+ ‖Θi‖1ϕ ≤ u, i = 1, . . . ,mM

Let J? be the optimal cost value of (A.20a).

minΓ,Θ,h

h (A.20b)

s.t.|Γi|+ ‖Θi‖1ϕ ≤ u, i = 1, . . . ,mM,

‖Cx(w(j))‖∞ ≤ h, j = 1, . . . , N

J(Γ,Θ) ≤ J? + α

- Two-step approach and soft constraint on input:

minΓ,Θ

J(Γ,Θ) (A.21a)

s.t.

‖u(w(i))‖∞ ≤ u i = 1, . . . , N

Let J? be the optimal cost value of (A.21a).

minΓ,Θ,h

h (A.21b)

s.t.‖u(w(j))‖∞ ≤ u

‖Cx(w(j))‖∞ ≤ hj = 1, . . . , N

J(Γ,Θ) ≤ J? + α

158


Note that all the relaxed optimization problems above are always feasible(just take Γ = 0, Θ = 0 and h = maxj ‖Cx(w(j))‖∞). To this purpose,note that in the two-step approach with soft constraints on input the same re-alizations w(1),w(2), . . .w(N) must be used both in (A.21a) and in (A.21b),because, otherwise, the feasibility of the optimization problem (A.21b),whose constraints depend on the solution of (A.21a), may be compromised.The resolution of (A.18)–(A.21) amounts to solving convex programs witha finite number of constraints and requires no machinery other than standardconvex optimization solvers like those used by CVX, [33], and YALMIP,[41].Despite the apparent naivety of the scenario approach, the obtained solu-tions come with precise guarantees about their feasibility with respect theoriginal probabilistic constraints as long as N is suitably chosen. This isdiscussed in the next section.

A.4.2 Chance-constrained feasibility of the obtained approximate so-lutions

The problems (A.18)–(A.21) are obtained as relaxations of the originalproblems (A.11)–(A.14). The sub-optimality of the obtained solutions isthe price we must pay to enhance computational tractability. However, amain issue is whether the solutions to problems (A.18), (A.19), (A.20),and (A.21) are feasible for the original constraints on u and x in problems(A.11), (A.12), (A.13), and (A.14), respectively.As already discussed, the relaxation introduced for the robust constraint,see (A.15), is such that feasibility with respect to the original hard con-straint is preserved. It is a fact that a similar result holds for the relax-ation of the constraints in probability introduced by the scenario approach,though this is much less evident than the previous case. For simplicity, wewill discuss this point by assuming that the solutions to problems (A.18),(A.19), (A.20), and (A.21) are unique for every fixed value of µ and α. Thefollowing theorem1 provides the fundamental result in this respect.

Theorem 1. Let f(ξ) : Rd → R be a convex function and g(ξ, δ) : Rd ×∆→ R be a parametric family of convex functions (i.e. g(ξ, δ) is convex inξ for any fixed value of δ ∈ ∆). Moreover, let Ξ be any given convex subset

1Theorem 1 is proven in [22]. Theorem 1 is amenable of extensions, like e.g. when some scenario constraintsare removed, [23], which are not considered here.

159


of Rd. For a given positive integer N , consider the optimization problem

minξ∈Ξ⊆Rd

f(ξ)

s.t.

g(ξ, δ(j)) ≤ 0, j = 1, . . . , N,

where δ(1), δ(2), . . . , δ(N) are samples independently extracted according toa given probability P over ∆, and let ξ∗ be the solution, which is assumedto always exists and be unique.For any ε ∈ (0, 1) and β ∈ (0, 1), if

β ≥d∑i=0

(N

i

)εi(1− ε)N−i, (A.22)

then ξ∗ is feasible for the constraint in probability

P g(ξ, δ) ≤ 0 ≥ 1− ε,

with confidence greater than or equal to 1− β.

In [7, 8] is proved that (A.22) can be satisfied enforcing the following ex-plicit condition on N :

N ≥d+ 1 + ln(1/β) +

√2(d+ 1) ln(1/β)

ε.

Note that the feasibility of ξ∗ for the probabilistic constraint can be guar-anteed with high confidence 1 − β only. This is intrinsically so becauseξ∗ is random as it depends on the extracted δ(1), δ(2), . . . , δ(N), and β keepsinto account the possibility of seeing an anomalous sample w(1), . . . ,w(N)

which is not representative enough. However, N depends on β logarithmi-cally so that small values of β, like β = 10−5 or β = 10−7, can be forced inwithout affecting N too much. With such small values for β, the result inTheorem 1 reads as “ξ∗ is feasible for the constraint in probability beyondany reasonable doubt”.

160

A.5. Choice of µ and α and trade-off between J and h

By letting ξ = (Γ,Θ, h), δ = w,

f(ξ) =

J(Γ,Θ) + µh for (A.18),(A.19)h for (A.20b)

,

g(ξ, δ(j)) =

‖Cx(w(j))‖∞ − h for (A.18),(A.20b)max ‖u(w(j))‖∞ − u,

‖Cx(w(j))‖∞ − h for (A.19)

,

and Ξ be the intersection of the other constraints that do not depend on w(j),a direct application of Theorem 1 to (A.18), (A.19), (A.20b) shows that, ifN satisfies (A.22), then the solutions of Problems (A.18), (A.19), (A.20)are feasible with high confidence 1 − β for the constraints on u and x in(A.11), (A.12), (A.13), respectively.As for Problem (A.21), Theorem 1 does not apply in this case. As a matterof fact, J? in (A.21b) should be more properly written as J?(w(1),w(2),. . .,w(N)), being obtained as the optimal value of (A.21a), a program whereconstraints depend on w(1),w(2),. . .,w(N). This means that Ξ = Ξ(w(1),w(2),. . .,w(N)), a setup which is not covered by Theorem 1. Although weexperimentally verified that, for N large enough, the solution of Problem(A.21) is usually feasible for the constraint in probability

P ‖u‖∞ ≤ u ∧ ‖Cx‖∞ ≤ h ≥ 1− ε

(see e.g. the numerical example in Section A.6), we were not able to provethat feasibility holds with high confidence 1 − β for N satisfying (A.22).This latter property for the solution to Problem (A.21) remains a conjecture.

A.5 Choice of µ and α and trade-off between J and h

The parameters µ and α in Problems (A.18)–(A.21) play the role of tuningparameters through which the user can modulate the trade-off between theobjective of minimizing the control cost J and that of having a small h tostrengthen the safety of the system operation.To this purpose, the two extreme cases, where all effort is put in the mini-mization of the sole J or, viceversa, of the sole h, are given by the following

161


two problems:

minΓ,Θ,h

J(Γ,Θ) (A.23)

s.t.

F (Γ,Θ, h,w(1), . . . ,w(N)) ≤ 0,

and

minΓ,Θ,h

h (A.24)

s.t.

F (Γ,Θ, h,w(1), . . . ,w(N)) ≤ 0,

where, the notation F (Γ,Θ, h,w(1), . . . ,w(N)) ≤ 0 is used as a shorthandto refer to

|Γi|+ ‖Θi‖1ϕ ≤ u, i = 1, . . . ,mM

‖Cx(w(j))‖∞ ≤ h, j = 1, . . . , N,

or‖u(w(j))‖∞ ≤ u

‖Cx(w(j))‖∞ ≤ hj = 1, . . . , N,

depending on the chosen approach (hard or soft) to treat input constraints.The properties below aim at showing that µ and α are indeed sensible tuningparameters, since by progressively increasing them over a suitable rangeone can explore all possible trade-off combinations between (A.23) and(A.24).In the following, we denote by (Γµ,Θµ, hµ) a solution to problem

minΓ,Θ,h

J(Γ,Θ) + µh (A.25)

s.t.

F (Γ,Θ, h,w(1), . . . ,w(N)) ≤ 0,

for a given µ and by Jµ = J(Γµ,Θµ) the corresponding cost value. Simi-larly, we denote by (Γα,Θα, hα) a solution to problem

minΓ,Θ,h

h (A.26)

s.t.F (Γ,Θ, h,w(1), . . . ,w(N)) ≤ 0,

J(Γ,Θ) ≤ J? + α,

162

A.5. Choice of µ and α and trade-off between J and h

for a given α, where J? is the optimal cost for problem (A.23). MoreoverJα = J(Γα,Θα). Note that (Γµ,Θµ, hµ) [(Γα,Θα, hα)] can be a solutionto either problem (A.18) or (A.19) [(A.20) or (A.21)], depending on whichset of constraints F (Γ,Θ, h,w(1), . . . ,w(N)) ≤ 0 is representing.Properties are derived under the following assumption.

Assumption 1. The matrices R and

Vϕϕ = E[(ϕ(w)− E[ϕ(w)])(ϕ(w)− E[ϕ(w))T

]are positive definite.

Assumption 1 guarantees that J(Γ,Θ) is a strictly convex function.The proof of the following Properties 1-5 is stated in [29].

Property 1 (well-definiteness of Jµ, hµ, Jα, and hα). For every positivevalue of µ, the solution to problem (A.25) is unique and, hence, both Jµ andhµ takes on a unique value. Moreover, let α be equal to J(Γ,Θ)−J?, where(Γ,Θ, h) denotes any solution to problem (A.24) chosen among those withthe smallest value for J . Then, for every α ∈ (0, α), Jα and hα take on aunique value.

Property 1 ensures that Jµ and hµ [Jα and hα] are indeed single-valuedfunctions of µ [α].

Property 2 (continuity and monotonicity). Let α be as in Property 1. Forµ ∈ (0,+∞) [α ∈ (0, α)], Jµ [Jα] is a monotonically increasing con-tinuous function of µ [α], while hµ [hα] is a monotonically decreasingcontinuous function of µ [α].

Property 3 (initial value). Let Γ,Θ, h be any solution to problem (A.23)chosen among those with the smallest value for h. When µ → 0 [α → 0],Jµ [Jα] tends to J(Γ,Θ), while hµ [hα] tends to h.

Property 4 (final value). Let α and (Γ,Θ, h) be as in Property 1. Whenµ→∞ [α→ α], Jµ [Jα] tends to J(Γ,Θ), while hµ [hα] tends to h.

Property 5 (equivalence). Let α be as in Property 1. For any µ ∈ (0,+∞)there is an α ∈ (0, α) such that Jα = Jµ and hα = hµ. Conversely, for anyα ∈ (0, α) there is a µ ∈ (0,+∞) such that Jµ = Jα and hµ = hα.

The first four properties show that all possible trade-offs between the twoextremes represented by problem (A.23) (whose objective is the minimiza-tion of the sole J) and problem (A.24) (whose objective is the minimizationof the sole h) can be achieved by increasing µ [α] from 0 to∞ [from 0 to

163


α], the behavior of both J and h being monotone and continuous. The lastproperty, instead, establishes a substantial equivalence between the addi-tional penalization term approach and the two-step approach.Heuristically, one can proceed by solving the chosen program among (A.18),(A.19), (A.20), and (A.21) for a grid of values of µ or α, say µ1, µ2, . . . , µkor α1, α2, . . . , αk, each time using the same realizations w(1), w(2),. . .,w(N) of the noise. This way, various solutions are obtained, each returninga different trade-off between J and h. By inspecting the achieved valuesfor J and h, the best solution according to the problem at hand can be even-tually selected.Note that in the two-step approach it is easier to make a suitable griddingfor the tuning parameter α since α represents the maximum degradation ofthe control cost. Yet, the two-step approach has the drawback that, whenconsidering probabilistic constraints on the input, no a-priori guarantees onthe feasibility of the scenario solution are given. In this case, one may resortto the additional penalization term approach.When Theorem 1 holds true, for each value of the grid, the obtained ap-proximate solution is guaranteed to be feasible for the original optimizationproblem except for a set of bad extractions whose probability is at most β.When considering and comparing all the approximate solutions obtainedfor the k gridding points, the guarantees to be feasible for the original op-timization problems hold jointly except for a set whose probability can beupper bounded by kβ.

A.6 Numerical example

The approaches described in Sections A.3.1 and A.3.2 are here applied to anumerical example inspired by [27].

Figure A.1: Scheme of the mechanical system.

We consider the mechanical system reported in Figure A.1, which is com-posed of four masses and four springs. The state of the system is given

164

A.6. Numerical example

by the mass displacements from the equilibrium position when all inputsare zero and their derivatives: x = [d1, d2, d3, d4, d1, d2, d3, d4]T . The con-trol input is u = [u1, u2, u3]T where u1, u2 and u3 are forces acting on themasses as in Figure A.1.We set all masses and stiffness constants equal to 1, i.e., m1 = m2 = m3 =m4 = 1 and k1 = k2 = k3 = k4 = 1, and consider the discrete time modelof the system

xt+1 = Axt +But + wt,

obtained by time discretization under the assumption that the control actionis piecewise constant over the intervals [t, t+ 1) and where the state of thesystem is supposed to be affected by a white Gaussian noise w with zeromean and covariance matrix I8×8.Our goal is to design a state feedback control policy that is able to coun-teract the disturbance w, maintaining the third and the fourth masses closeto their equilibrium positions and keeping the springs within their linearoperating domain. The latter requirement is explicitly accounted for byimposing a constraint on the spring deformations.To the purpose of regulating the third and the fourth masses around theirequilibrium positions, we consider the average control cost (A.5) with aprediction horizon of length M = 5 and constant weight matrices

Q =

I2×210−3 02×2 04×4

02×2 I2×2

04×4 04×4

R = 10−6I3×3.

Let

C =

1 0 0 0

−1 1 0 0 04×4

0 −1 1 0

0 0 −1 1

so that

Cxi =

d1,i

d2,i − d1,i

d3,i − d2,i

d4,i − d3,i

provides the springs deformation at time i. Then, the state constraints, in-troduced to limit the springs deformation, can be expressed as

‖Cx‖∞ ≤ h,

165


where C is defined as in (A.10). Eventually, we suppose that the controlinput is subject to the saturation limit

‖u‖∞ ≤ u,

where u = 4.We shall now illustrate the performance of the approaches in Sections A.3.1and A.3.2. The control policy is parameterized according to (A.3) whereϕ(·) is the saturation function in (A.4) with ϕ set equal to 2. The initialstate is zero, i.e., the system starts at the equilibrium point. We shall focuson the case when constraints on both the control input and the state are ex-pressed in probability with an admissible violation ε = 0.1. In the scenariosolution to the resulting chance-constrained optimization problems, we setβ = 10−5. Correspondingly, the number of disturbance realizations to ex-tract is N = 3455. All scenario problems were solved by running YALMIPover SeDuMi [41].Table A.1 reports the optimal values of J and h obtained by the approachin Section A.3.1 where the penalized control cost J ′ = J + µh is adoptedfor five different values of µ. The last column of Table A.1 also reportsthe estimate ε of the actual probability of constraint violation calculatedthrough the Monte Carlo method over 5000 runs of the controlled system.It is worth noticing that the estimate ε is always smaller than ε = 0.1, as itis guaranteed by the scenario theory with confidence 1− 5 · 10−5.From Table A.1, it is apparent that parameter µ affects the trade-off betweenthe two objectives, i.e., control cost and state constraints: for small valuesof µ, the state constraints are ineffective in practice, whereas for large µ’s,h decreases at the price of a significant increase of J . Results reported in

Table A.1: Results of the approach with additional penalization term in the control cost.

µ J h ε

0 20.41 12.60 0.03660.1 20.50 10.40 0.0394

1 21.69 7.58 0.045210 28.02 5.84 0.0546

100 36.78 5.40 0.0724

Table A.2 refer to the two-step approach in Section A.3.2. The same com-ments as for Table A.1 apply. In particular, the actual constraint violation εis small and lower than ε = 0.1. This is a quite interesting fact, since thereare no a-priori guarantees on the feasibility of the scenario solution in thiscase as discussed in Section A.4.2.

166


Table A.2: Results of the two-step approach.

α/J? J h ε

0 20.41 12.60 0.03660.05 21.43 7.86 0.0460

0.1 22.45 6.99 0.05000.2 24.49 6.35 0.05360.5 30.62 5.64 0.0654

Note that, since we are imposing a probabilistic bound on the control inputu, there may be disturbance realizations such that the bound on u is vio-lated. In the actual operation of the controller, the components of u whoseabsolute value exceeds u are saturated to±u (clipping of the control input).It is then interesting to assess the performance of the clipped version of theobtained controllers. To this purpose, the average control cost (A.5) whenclipping holds, say J , is estimated via Monte Carlo simulations over 5000runs and is reported in Table A.3. As it appears, the values of J are quite

Table A.3: Average control cost for the clipped controllers.

µ J J α/J? J J

0 20.41 20.66 0 20.41 20.660.1 20.50 20.75 0.05 21.43 21.71

1 21.69 21.98 0.1 22.45 22.7410 28.02 28.40 0.2 24.49 24.84

100 36.78 37.18 0.5 30.62 31.84

close to the corresponding values of J in Tables A.1 and A.2. This is notsurprising given that the violation of the constraint on the input has smallprobability to occur, which makes the impact of clipping on performancenegligible. This is not the case for the optimal LQG control policy whereconstraints on both state and input are ignored. Clipped LQG control has acost J = 36.09, with a significant degradation with respect to the optimalLQG cost J = 13.81.As for the constraint on the state, Figures A.2 and A.3 depict the proba-bility distribution function of ‖Cx‖∞ for the designed clipped controllersin correspondence of the different values of µ and α, respectively, togetherwith the probability distribution of ‖Cx‖∞ for the clipped LQG control.These figures reveal that all designed clipped controllers outperform theclipped LQG policy in terms of state constraint guarantees. Moreover, Fig-ures A.2 and A.3 together with Table A.3 show that as µ and α vary, a

167


trade-off between performance and state constraint guarantees similar tothat revealed by Tables A.1 and A.2 is achieved when clipping is active.

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

h

F(h)

LQG

µ= 0

µ= 0.1

µ= 1

µ= 10

µ= 100

Figure A.2: Probability distribution function of ‖Cx‖∞ for the clipped controllers corre-sponding to the different values of µ and for the clipped LQG controller.

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

h

F(h)

LQG

α/J? = 0

α/J? = 0.05

α/J? = 0.1

α/J? = 0.2

α/J? = 0.5

Figure A.3: Probability distribution function of ‖Cx‖∞ for the clipped controllers corre-sponding to the different values of α and for the clipped LQG controller.

168

A.7. Relaxation approach focused on constraints

A.7 Relaxation approach focused on constraints

By means of the approach in Section A.3.2, we can effectively control thedegradation of the cost function J , but the value of the bound h on the stateconstraint cannot be directly chosen because it is determined through thesolution of an optimization problem. In this section we aim to develop anapproach that allows to better control the achieved bound on the state con-straints, this come at the prize that the value of the achieved performance Jis indirectly determined.We consider again the discrete time stochastic linear system (A.1), the inputparametrization in (A.3) and the performance cost function in (A.5), which,in the following, is assumed to be strictly convex with respect to the opti-mization variables Γ,Θ. Given that the input ut and state xt are uncertainsince they both depend on the value taken by the stochastic disturbance wt,constraints are formulated in probabilistic terms:

P f(u0, . . . , uM−1) ≤ u ∧ g(x1, . . . , xM) ≤ y ≥ 1− ε, (A.27)

where f(·) : RmM → Rpu and g(·) : RnM → Rpy are convex and continu-ous vector-valued functions, u ∈ Rpu and y ∈ Rpy , and the inequalities areto be interpreted component-wise.Typically, constraints on the input and state variables are represented bybounds on their norm, e.g.,

f(u0, . . . , uM−1) =

‖u0‖∞

...‖uM−1‖∞

≤ u,

g(x1, . . . , xM) =

‖Cx1‖∞

...‖CxM‖∞

≤ y.

Note that the constraint (A.27) is similar to the one in (A.8), but by meansof this new formulation we want to emphasize the presence of differentgroups of constraints that may correspond to the different time steps alongthe finite horizon, or to different requirements on the state and the input.The proposed approach can be straightforwardly applied also consideringrobust constraints on the input as in (A.7), and the extensions in SectionA.1.2.The possibly unbounded disturbance wt enters additively on the state (see(A.1)). If g(·) grows unbounded when the state increases, as it is the case

169


for norms, then the feasibility domain as given by (A.27) may be void, de-pending on whether or not the value of y is compatible with the systemdynamics, the disturbance characteristics, the input constraint, and the al-lowed violation probability ε. This issue is addressed, also in this case, bymeans of a suitable relaxation of the problem

minΓ,Θ

J(Γ,Θ) (A.28)

s.t.P f(u0, . . . , uM−1) ≤ u ∧ g(x1, . . . , xM) ≤ y ≥ 1− ε

that allows to recover feasibility. The relaxation is conceived so as to en-force the original state bound y whenever is possible, while, otherwise, thesmallest feasible state bound is first determined and then imposed so asto keep the state as close as possible to the desired domain. This translatesinto the cascade of two chance-constrained optimization problems, which ishard to solve. A randomized resolution scheme to enhance computationaltractability is proposed and a theoretical analysis of its properties is pro-vided. The obtained results have immediate implications on the recursivefeasibility of the implementation of the proposed approach over a recedinghorizon as done in model predictive control.The idea to guarantee feasibility is, similarly to the previous approaches, toreplace y in (A.27) with an optimization variable h ∈ Rpy to be minimizedcomponent-wise. This way, the state constraint is always feasible since bytaking h large enough it becomes ineffective. However, in no way taking atoo large h is a good choice and we see that the presence of the new opti-mization variable h requires to handle two typically conflicting objectives:the minimization of the bound on the state so as to satisfy the original stateconstraint (A.27), and the minimization of the cost function J(Γ,Θ) thatrepresents the performance of the system.We deal with these different objectives by means of a two-step approachwhose goal, differently from the two-step approach in Section A.3.2, is tostay as close as possible to the original problem (A.28). In the first stepwe focus on the bound on the state constraint: the optimization variable his minimized and an additional constraint is enforced so as to ensure thath gets not smaller than the original bound y. Then, in the second step, wefocus on the minimization of the cost function J(Γ,Θ), where the optimalh obtained in the first step is taken as bound for the state constraint. That

170

A.8. Scenario-based resolution scheme

is,

minΓ,Θ,h

hTTh (A.29a)

s.t.P f(u(w,Γ,Θ)) ≤ u ∧ g(x(w,Γ,Θ)) ≤ h ≥ 1− εh ≥ y

;

let ho the optimal value for h obtained in (A.29a);

minΓ,Θ

J(Γ,Θ) (A.29b)

s.t.P f(u(w,Γ,Θ)) ≤ u ∧ g(x(w,Γ,Θ)) ≤ ho ≥ 1− ε.

In (A.29), T is a positive definite matrix that can assign a different impor-tance to the different components of h.The idea behind the cascade of problems in (A.29) is as follows. When theoriginal constraint (A.27) is not feasible, the minimization of h in (A.29a)in the first step allows to find the smallest bound on the state that preservesfeasibility; instead, when (A.27) is feasible, ho = y. As a matter of fact,the constraint h ≥ y is introduced in (A.29a) to avoid excessively conser-vative solutions where the bound on the state constraint is smaller than y.Note that the objective function in (A.29a) does not depend on Γ and Θ,and it may be that the same ho is attained for different values of Γ and Θ,each one possibly achieving a different value of the cost function J(Γ,Θ).This extra degree of freedom is exploited in the second step to optimizethe performance of the system. Note that in (A.29b) feasibility is not anissue anymore, since the bound ho computed in the first step is adopted inthe probabilistic constraint. The overall solution returned by the cascadeof problems (A.29) is (Γo,Θo, ho), where (Γo,Θo) determine the controlaction to be implemented and ho is the probabilistically guaranteed boundfor the state constraint. This ho, once computed, can be inspected for com-parison with the original y.

A.8 Scenario-based resolution scheme

Problems (A.29a) and (A.29b) are, in general, hard to solve because of thepresence of a probabilistic constraint, which can be non-convex in spite ofthe convexity of f and g. In order to enhance computational tractability,some approximation has to be accepted. Here, we resort to a randomized

171


scheme that is in the line of the scenario approach, [17, 18, 22, 24]. In thisscheme, the returned solution comes accompanied by precise guaranteesabout feasibility for the original probabilistic constraint in problem (A.29a).Following the scenario approach framework,N disturbance realizations areextracted according to the underlying probability distribution, say

w(1) =[w

(1)0 w

(1)1 . . . w

(1)M−1

]w(2) =

[w

(2)0 w

(2)1 . . . w

(2)M−1

]...

w(N) =[w

(N)0 w

(N)1 . . . w

(N)M−1

];

then, in each optimization problem, the probabilistic constraint is replacedby N deterministic constraints, those obtained in correspondence of theseen disturbance realizations. Precisely, the scenario version of (A.29a)and (A.29b) consists in the following cascade of optimization problems:

minΓ,Θ,h

hTTh (A.30a)

s.t.f(u(w(j),Γ,Θ)) ≤ u

g(x(w(j),Γ,Θ)) ≤ h j = 1, . . . , N,

h ≥ y

;

let h? be optimal value of h obtained in (A.30a);

minΓ,Θ

J(Γ,Θ) (A.30b)

s.t.f(u(w(j),Γ,Θ)) ≤ u

g(x(w(j),Γ,Θ)) ≤ h? j = 1, . . . , N.

Note that (A.30a) and (A.30b) are convex problems with a finite numberof constraints, so that they can be solved by resorting to standard solversfor convex optimization, that is, randomization has led us back to computa-tional tractability. Moreover, in this case, by convexity of constraints, sincehTTh is strictly convex with respect to h the value h? is uniquely deter-mined by (A.30a), and, similarly, the solution to (A.30b), say (Γ?,Θ?) isunique.

172

A.8. Scenario-based resolution scheme

The overall solution to the cascade of problems (A.30) is defined as (Γ?,Θ?,h?). Note that (A.30) can be seen as an empirical counterpart of (A.29),and that all comments made for (A.29) (feasibility of the two problems,uniqueness of the solutions, etc.) applies mutatis mutandis to (A.30). Noteeventually that the solution (Γ?,Θ?, h?) is feasible and optimal for (A.30a),so that the second optimization problem (A.30b) can be thought of as a tiebreak rule to choose among the multiple solutions of the first optimizationproblem (A.30a) the one that minimizes J(Γ,Θ).

Using (Γ?,Θ?, h?) in place of (Γo,Θo, ho) is the price we have to pay inorder to enhance computational tractability. However, one main questionarises about the feasibility of (Γ?,Θ?, h?) for the probabilistic constraint

P f(u(w,Γ,Θ)) ≤ u ∧ g(x(w,Γ,Θ)) ≤ h ≥ 1− ε, (A.31)

so as to establish a link between the obtained scenario-based solution andthe original problem (A.29). The theory of the scenario approach has beenmainly dealt with this question, showing that the answer is indeed affir-mative with high confidence in a number of contexts, provided that N isbig enough, [17, 18, 22, 23]. However, the best available result of [22]has been proven only for scenario optimization programs whose solutionis determined by a specific tie break rule which may differ from the onecorresponding to the cascade of problems in (A.30) (see point 5 in Section2.1 of [22]). The following theorem provides the extension of the resultsobtained in [22] to the present setup where a cascade of problems is con-sidered.

Theorem 2. Let β ∈ (0, 1) be a user-chosen confidence parameter. If thenumber of extracted disturbance realizations N is chosen so as to satisfy

d−1∑i=0

(N

i

)εi(1− ε)N−i ≤ β, (A.32)

where d is the dimensionality of (λ, h), then it holds with confidence at least1− β that

P f(u(w,Γ?,Θ?)) ≤ u ∧ g(x(w,Γ?,Θ?)) ≤ h? ≥ 1− ε.

Proof: see next Section A.9.

In words, the theorem says that the scenario-based solution (Γ?,Θ?, h?) isfeasible for the probabilistic constraint (A.31) with confidence at least 1−β.

173


A.9 Proof of Theorem 2

This proof is inspired by the proof of point 5 in Section 2.1 of [22], wherean extension of the scenario approach to the case of non-unique solutionsis discussed.

In order to lighten the notation we pile up in λ the nonzero components ofΓ and Θ, so that we can adopt the following notations

u = u(w, λ), x = x(w, λ), J = J(λ),

which point out the dependence of input, state, and cost on the optimizationvector λ and the disturbance realization w.For a given (λ, h), define the violation probability of (λ, h) as

V (λ, h) := Pf(u(w, λ)) > u ∨ g(x(w, λ)) > h

= 1− P

f(u(w, λ)) ≤ u ∧ g(x(w, λ)) ≤ h

.

Then, Theorem 2 amounts to showing that

PNV (λ?, h?) > ε ≤ β, (A.33)

where PN is the product probability underlying the independent extractionof the sample w(1), . . . ,w(N) based on which the solution (λ?, h?) is com-puted.

Consider the following auxiliary scenario program

minλ,h

hTTh+1

nJ(λ) (A.34)

s.t.f(u(w(j), λ)) ≤ u

g(x(w(j), λ)) ≤ h j = 1, . . . , N,

h ≥ y

for n = 1, 2, . . ., and denote by (λ?n, h?n) its optimal solution, which exists

and is unique, since the cost function hTTh + 1nJ(λ) has, by the assump-

tions on T and on J , compact level sets for every n ≥ 1 and the optimiza-tion feasibility domain defined by the constraints in (A.34) is close andnonempty. The following two properties hold.

174

A.9. Proof of Theorem 2

1. For every n ≥ 1, it holds that

PNV (λ?n, h?n) > ε ≤ β (A.35)

2. For every multisample w(1), . . . ,w(N), the solution to (A.34) con-verges to the solution to (A.30), namely,

(λ?n, h?n)→ (λ?, h?) as n→∞. (A.36)

Proof of Property 1

By adding a slack variable v ∈ R, problem (A.34) can be rewritten inepigraphic form as:

minλ,h,v

v (A.37)

s.t.f(u(w(j), λ)) ≤ u

g(x(w(j), λ)) ≤ h j = 1, . . . , N,

h ≥ y

hTTh+ 1nJ(λ) ≤ v.

The solution to problem (A.37) is still unique, and the assumptions of The-orem 2.4 in [22] are satisfied. An application of this theorem gives

PNV (λ?n, h?n) > ε ≤

d∑i=0

(N

i

)εi(1− ε)N−i,

where we have d in place of d − 1 because in (A.37) the number of opti-mization variables has been augmented by 1 and is equal to d + 1. On theother hand, since the slack variable v does not enter the expression definingit, the constraint

λ, h, v : f(u(w, λ)) ≤ u ∧ g(x(w, λ)) ≤ h

is, irrespective of w, a cylindroid infinitely extended along the v direc-tion. This entails that the family (with respect to the variability of w) ofconstraints above has a so-called support rank equal to d, according to Def-inition 3.6 of [62]. The conclusion that

PNV (λ?n, h?n) > ε ≤

d−1∑i=0

(N

i

)εi(1− ε)N−i

175


can be eventually drawn, by invoking the observation made in [62] thatTheorem 2.4 of [22] still applies by replacing the optimization domain di-mensionality with the support rank (see Lemma 3.8).

Proof of Property 2

To show that (λ?n, h?n)→ (λ?, h?) as n→∞, consider the sets

Hn =

(λ, h) : (λ, h) is feasible for (A.34) and

hTTh+ 1nJ(λ) ≤ h?TTh? + 1

nJ(λ?)

,

for n = 1, 2, . . .. In words, n by n, Hn is the set of all feasible pointsfor (A.34) that also belong to the smallest level set of the cost function of(A.34) containing the solution (λ?, h?) of (A.30). Note that, while the levelset changes with n, the feasibility domain of (A.34) remains the same forall n and it coincides with the feasibility domain of (A.30a). This entailsthat (λ?, h?) belongs to Hn for all n, showing also that Hn is nonempty.Moreover, n by n, we have that

(λ?n, h?n) ∈ Hn, (A.38)

because (λ?n, h?n) is feasible for (A.34), and, being also optimal, its cost

value must be better than that of (λ?, h?) as required by the first conditiondefiningHn.

A fundamental property of the family of setsHn is that

H1 ⊇ H2 ⊇ · · · ⊇ Hn ⊇ Hn+1 ⊇ · · · , (A.39)

see Figure A.4 for a graphical illustration.To show this, suppose that a (λ, h) belongs toHn+1. From

hTT h+1

n+ 1J(λ) ≤ h?TTh? +

1

n+ 1J(λ?)

it follows that J(λ) ≤ (n+ 1)(h?TTh? − hTTh) + J(λ?). Whence,

hTT h+1

nJ(λ)

≤ hTT h+n+ 1

n(h?TTh? − hTT h) +

1

nJ(λ?)

= h?TTh? +1

n(h?TTh? − hTT h) +

1

nJ(λ?)

≤ h?TTh? +1

nJ(λ?),

176

A.9. Proof of Theorem 2

-1.5 -1 -0.5 0 0.5

0.8

0.9

1

1.1

1.2

1.3

1.4 Feasibility Region

H1

H2

H3

n

6?; h?

Figure A.4: The setsHn’s in a simple case (h, λ ∈ R, T = 1, J(λ) = 3λ2).

where the last inequality follows because h?TTh? − hTT h ≤ 0 beingh?TTh? lowest among feasible points. This shows that (λ, h) ∈ Hn too,that is, (A.39) holds.

From (A.38) and (A.39), it follows that (λ?n, h?n) ∈ H1, ∀n. SetH1 is com-

pact, being the intersection of the feasibility domain of (A.30a), which isclose, with a level set of hTTh + 1

nJ(λ), which is compact thanks to the

assumptions on T and J . It follows that the sequence (λ?n, h?n) is convergent

to some point, say (λ?∞, h?∞), which is feasible for (A.30a) too.

From (A.38) and the definition ofHn, we have that

h?nTTh?n ≤ h?TTh? +

1

n[J(λ?)− J(λ?n)] ,

which in turn implies that

h?∞TTh?∞ = lim

n→∞h?n

TTh?n

≤ h?TTh? + limn→∞

1

n[J(λ?)− J(λ?n)]

= h?TTh?.

Yet, being h?TTh? minimal, it cannot be that a strict inequality holds, sothat eventually h?∞

TTh?∞ = h?TTh?. If h?∞ 6= h?, then (12λ? + 1

2λ?∞,

12h? +

177


12h?∞) would be feasible for (A.30a) thanks to the convexity of the con-

straints, while the positive definiteness of T would give[1

2h? +

1

2h?∞

]TT

[1

2h? +

1

2h?∞

]<

1

2h?TTh? +

1

2h?∞

TTh?∞

= h?TTh?,

so contradicting the minimality of h?TTh?. Hence, h?∞ = h?.

From (λ?n, h?n) ∈ H1, we have that J(λ?n) ≤ h?TTh? − h?nTTh?n + J(λ?)

which, taking the limit, gives

J(λ?∞) ≤ limn→∞

h?TTh? − h?nTTh?n + J(λ?)

= J(λ?).

Plainly, it must be that J(λ?∞) = J(λ?), for, otherwise, being λ?∞ feasiblefor (A.30b), J(λ?∞) < J(λ?) would contradict the minimality of J(λ?).Moreover, if λ?∞ 6= λ?, then 1

2λ? + 1

2λ?∞ would be feasible for (A.30b), and,

because of the strict convexity of J(λ) we would have

J(1

2λ? +

1

2λ?∞) <

1

2J(λ?) +

1

2J(λ?∞) = J(λ?),

contradicting again the minimality of J(λ?). Hence, λ?∞ = λ?, and thisconcludes the proof of Property 2.

We want now to capitalize on (A.35) and (A.36) to show that (A.33) holds.To this purpose, start by fixing a sample w(1), . . . ,w(N) such that V (λ?, h?) >ε, which, we recall, means that

Pf(u(w, λ?)) > u ∨ g(x(w, λ?)) > h?

> ε.

By continuity of f and g, this implies that

Pf(u(w, λ)) > u ∨ g(x(w, λ)) > h?

> ε,

for all (λ, h) : ‖(λ, h) − (λ?, h?)‖ ≤ r for a radius r small enough, and,since (λ?n, h

?n) → (λ?, h?) so that ‖(λ, h) − (λ?, h?)‖ ≤ r for all n bigger

than a suitable n, we can conclude that

V (λ?n, h?n) = P

f(u(w;λ?n)) > u ∨ g(x(w;λ?n)) > h?n

> ε, (A.40)

178


for n > n.

If we now let w(1), . . . ,w(N) vary, and we consider the indicator functionIw(1),...,w(N): V (λ?n,h

?n)>ε, (A.40) yields

IV (λ?,h?)>ε · IV (λ?n,h?n)>ε −−−→

n→∞IV (λ?,h?)>ε,

for all possible realizations of w(1), . . . ,w(N). Applying the Lebesguedominated convergence theorem gives

limn→∞

PNV (λ?n, h?n) > ε

= limn→∞

∫IV (λ?n,h

?n)>εPNdw(1), . . . , dw(N)

≥ limn→∞

∫IV (λ?,h?)>ε · IV (λ?n,h

?n)>εPNdw(1), . . . , dw(N)

=

∫IV (λ?,h?)>εPNdw(1), . . . , dw(N)

= PNV (λ?, h?) > ε.

Hence, PNV (λ?, h?) > ε ≤ limn→∞ PNV (λ?n, h?n) > ε, and since

PNV (λ?n, h?n) > ε ≤ β, ∀n, (A.33) remains proven.

A.10 Numerical example

In this section the proposed approach is applied to a numerical exampleinspired by [27]. We consider again the mechanical system composed by 4masses and 4 springs as depicted in Figure A.1.The state of the system is given by the displacement of masses with re-spect to their nominal positions, and by their derivatives, i.e. x = [d1,d2,d3,d4,d1,d2,d3,d4]T . The control inputs are the forces u1, u2, u3 acting onthe masses shown in Figure A.1. All the masses and the stiffness coeffi-cients are set to 1, and the system dynamics are written as in (A.1) by timediscretization under the assumption that the input is kept constant in theinterval [t, t + Ts), with Ts = 1 s. We also suppose that the masses dis-placements and velocities are affected by a stochastic additive disturbance,which after discretization leads tow ∼ WGN(0, I4) andBw = [0.5I4 I4]T .The initial condition is x0 = [10,−10, 10,−10, 0, 0, 0, 0]T .The objective is to keep the masses positions as close as possible to thenominal ones, while, at the same time, a requirement on the maximumspeed of the masses has to be satisfied. To this purpose, the weight matrices

179


in the cost function J are set so as to penalize deviations from the nominalpositions:

Q =

[I4 04×4

04×4 04×4

]R = 10−6I3,

and the following constraints are enforced on speeds of masses:

‖Cxi‖∞ ≤ yi, i = 1, . . . ,M,

where C = [04×4 I4], yi = 10, i = 1, . . . ,M and M = 8 is the consideredtime horizon. The constraints are enforced in probability with ε = 0.1. Noinputs constraints are imposed.

Following the approach in Section A.7, the bound y was replaced by theoptimization variables hi, i = 1, . . . ,M . We set β = 10−6, returningN = 4614 according to (A.32), and the cascade of problem (A.30) wasthen solved to obtain the scenario-based solution.It turned out that the original yi, i = 1, . . . ,M was not feasible as for thefirst 2 time instants. Indeed, the smallest feasible bounds obtained fromthe solution of (A.30a) were h?1 = 11.62, h?2 = 11.08 and h?i = yi,i = 3, . . . ,M . The optimal scenario-based control policy obtained fromproblem (A.30b) achieved a cost J(Γ?,Θ?) = 2305.55. A Monte Carloverification revealed that the probabilistic constraint (A.31) was satisfiedby (Γ?,Θ?, h?) as guaranteed by Theorem 2.

For the sake of comparison, the obtained scenario control policy was testedagainst a finite horizon LQ controller. In order to somehow account for thespeed requirement, in the design of the LQ controller, the cost function wasmodified so as to assign a penalization on the speed of the masses:

JLQ = E[xTQLQx + uTRLQu]

QLQ =

[qJI4 04×4

04×4 qyI4

]RLQ = 10−6I3,

where the weights qJ and qy permits one to tune the relative importancebetween positions and velocities in the cost function.The performances of the obtained scenario-based control policy and of theLQ controller for various choices of qJ and qy are compared in Table A.4 –where the achieved cost J and the actual probability of violation εy of theoriginal constraint with y, as computed via Monte Carlo simulations, arereported for all the approaches – and in Figure A.5 – where the probability

180


distributions of ‖Cxi‖∞, i = 1, . . . , 8, again computed via Monte Carlosimulations, are depicted for all the approaches.

Table A.4

qJ qy Approach J εy

- - Scenario based 2305.55 0.1248

1 0 LQ 126.44 10 1 LQ 4347.20 0.9724

0.2 9 LQ 2318.50 0.9960

As it appears, the proposed scenario-based approach achieves a good trade-off between J and εy and, in particular, this latter, though not equal to therequired 0.1 value (which was eventually infeasible), is very close to 0.1since in the first step the h?i have been pushed towards y as much as possi-ble.

When the LQ controller is designed accounting for the mass displacementsonly (qJ = 1 qy = 0), the cost function J turns out to be much improved,but the speed limit y is violated by a huge extent, see in particular FigureA.5(b). When on the contrary the LQ controller is designed accounting forspeeds only while displacements are neglected, the cost function J is worsethan the one obtained by the scenario-based solution, while εy is still veryhigh as compared to 0.1. In particular, as shown in Figure A.5(c), the con-straint is significantly violated in the first time step, while, in the other timesteps, the velocity is excessively reduced with respect to the allowed limity. Also in the third case, where qJ = 0.2 and qy = 9 are chosen so as tomake the LQ controller achieving a cost J close to the one obtained by thescenario-based solution, the same kind of violation of the constraint as inthe previous case is obtained, see Figure A.5(d).

The behaviors of the different controllers can be appreciated also analysingthe state trajectories reported in Figure A.6 (displacements) and Figure A.7(velocities) for 100 new disturbance realizations. The scenario-based con-troller, by exploiting the allowed speed, is able to steer the masses closeto their nominal positions. Instead, although the LQ controller violates theconstraint in the first time step, in the other steps it conservatively keeps thespeeds too small, so that the masses are not steered to the nominal position.

181


0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

‖Cxi‖∞

(a) Scenario based solution, h?1 blue dash-dotted line,h?2 red dash-dotted line.

0 5 10 15 20 25 30 35 40 45 50 550

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

‖Cxi‖∞

(b) LQ controller qJ = 1, qy = 0

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

‖Cxi‖∞

(c) LQ controller qJ = 0, qy = 1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

i=1

i=2

i=3

i=4

i=5

i=6

i=7

i=8

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

‖Cxi‖∞

(d) LQ controller qJ = 0.2, qy = 9

Figure A.5: Probability distributions of ‖Cxi‖∞, i = 1, . . . , 8 for the scenario basedsolution and for the LQ controllers; y is represented by the marked vertical solid line.

182


0 1 2 3 4 5 6 7 8−30

−20

−10

0

10

20

Time

Positions

0 1 2 3 4 5 6 7 8−30

−20

−10

0

10

20

Time

Positions

(a) Scenario-based controller0 1 2 3 4 5 6 7 8−30

−20

−10

0

10

20

Time

Positions

0 1 2 3 4 5 6 7 8−30

−20

−10

0

10

20

Time

Positions

(b) LQ controller qJ = 0.2, qy = 9

Figure A.6: Displacements of the masses: d1 (blue diamonds), d2 (green circles), d3 (redsquares), d4 (cyan triangles).

0 1 2 3 4 5 6 7 8−15

−10

−5

0

5

10

15

Time

Velocities

0 1 2 3 4 5 6 7 8−15

−10

−5

0

5

10

15

Time

Velocities

(a) Scenario-based controller0 1 2 3 4 5 6 7 8−15

−10

−5

0

5

10

15

Time

Velocities

0 1 2 3 4 5 6 7 8−15

−10

−5

0

5

10

15

Time

Velocities

(b) LQ controller qJ = 0.2, qy = 9

Figure A.7: Velocities of the masses: d1 (blue diamonds), d2 (green circles), d3 (redsquares), d4 (cyan triangles).

183

APPENDIXBConstraint reformulation with LMI

In this appendix an alternative convex approximation for the constraints onthe bank angle φ and on longitudinal acceleration τ or on the engine thrustT is proposed. The constraints on longitudinal acceleration (i.e. on τ orT ) and on lateral acceleration (i.e. on the bank angle φ) are regarded as aunique constraint which is dealt with by means of LMIs. This approach toobtain convex approximations of the constraints is developed consideringthe simpler aircraft model in (2.2), moreover, in order to ease the notationwe neglect to explicitly write the dependence of input and state variableson time; as it is clear the constraints have to be enforced for every timeinstant along the finite prediction horizon. The LMI approximation of theconstraints on bank angle, longitudinal acceleration and engine thrust maybe used in place of the corresponding linear constraint in Section 3.3, and itshould be enforced along with the constraints on vertical acceleration, pathangle and TAS not discussed here.

Consider constraints of the form:

b ≤ u1 cosψ + u2 sinψ + ¯b ≤ b (B.1)c ≤ −u1 sinψ + u2 cosψ ≤ c. (B.2)

185

Appendix B. Constraint reformulation with LMI

Note that setting:

b =−aLcos γ

b =aL

cos γ¯b = u3 tan γ

(B.1) becomes equivalent to (3.37). Instead if we set:

b =Tmin −KDV

2

m cos γb =

Tmax −KDV2

m cos γ¯b = u3 tan γ + g

x6

V cos γ

(B.1) becomes equivalent to (3.49). Moreover if we set:

c = −g cos γ tan φ c = g cos γ tan φ

(B.2) becomes equivalent to (3.45).Now let:

bm =b+ b

2− ¯b bn =

b− b2

cm =c+ c

2cn =

c− c2

.

If we subtract bm and cm to all members of (B.1) and (B.2) respectively,they rewrite:

−bn ≤ u1 cosψ + u2 sinψ − bm ≤ bn

−cn ≤ −u1 sinψ + u2 cosψ − cm ≤ cn.

These constraints are normalized by bn and cn respectively, and reformu-lated by means of an infinity-norm constraint:∥∥∥∥∥

[1bn

(u1 cosψ + u2 sinψ − bm)1cn

(−u1 sinψ + u2 cosψ − cm)

]∥∥∥∥∥∞

≤ 1. (B.3)

Recalling that the infinity-norm is bounded by the 2-norm, (B.3) can beapproximated by the tighter constraint:

1−

u1

u2

−bm−cm

T

cosψ − sinψ

sinψ cosψ

1 0

0 1

[

1b2n

0

0 1c2n

][cosψ sinψ 1 0

− sinψ cosψ 0 1

]u1

u2

−bm−cm

≥ 0 (B.4)

Q− SRST ≥ 0, S =[u1 u2 −bm −cm

].

The constraint above is in a form that allows for the application of theSchur Lemma:

R 0

Q− SRST 0⇐⇒

[Q S

ST R−1

] 0,

186

but for the fact that in (B.4) matrix R is not invertible because it always has2 null eigenvalues. In order to overcome this issue, we add some new termsthat prevent the matrix to be singular, and we reformulate (B.4) as:

1−

u1

u2

−bm−cm

T

cosψ − sinψ 0 0

sinψ cosψ 0 0

1 0 1 0

0 1 0 1

1b2n

1c2n

1ε2b

1ε2c

cosψ sinψ 1 0

− sinψ cosψ 0 1

0 0 1 0

0 0 0 1

u1

u2

−bm−cm

≥ 0

(B.5)

Q− SRST ≥ 0, S =[u1 u2 −bm −cm

].

The newly added terms imply that the following constraints are enforced:

− 1 ≤ −bmεb≤ 1 − 1 ≤ −cm

εc≤ 1. (B.6)

Note that if the original constraints (B.1) and (B.2) are symmetric, as theconstraints on the bank angle, then bm and cm are null and the artificiallyadded constraints are always ineffective. On the contrary if bm and cm arenot null, as in the constraint on longitudinal acceleration and engine thrust,the choice of εb and εc becomes quite critical: a proper trade off betweenmaking matrix R well conditioned and avoiding (B.6) become excessivelytight or infeasible has to be found.However, as the matrix R in (B.5) is positive definite, we can apply SchurLemma:

1 u1 u2 −bm −cmu1 (b2n + ε2b) cos2 ψ + (c2n + ε2c) sin2 ψ sinψ cosψ(b2n + ε2b − c

2n − ε

2c) −ε2b cosψ ε2c sinψ

u2 sinψ cosψ(b2n + ε2b − c2n − ε

2c) (c2n + ε2c) cos2 ψ + (b2n + ε2b) sin2 ψ −ε2b sinψ −ε2c cosψ

−bm −ε2b cosψ −ε2b sinψ ε2b 0

−cm ε2c sinψ −ε2c cosψ 0 ε2c

0

(B.7)

We can split (B.7) in the sum of two terms:

1 u1 u2 −bm −cmu1 b2n + ε2b + c2n + ε2c 0 −ε2b cosψ ε2c sinψ

u2 0 b2n + ε2b + c2n + ε2c −ε2b sinψ −ε2c cosψ

−bm −ε2b cosψ −ε2b sinψ ε2b 0

−cm ε2c sinψ −ε2c cosψ 0 ε2c

+ (B.8)

+

0 0

cosψ − sinψ

sinψ cosψ

0 0

0 0

[−(c2n + ε2c) 0

0 −(bn + ε2b)

][0 cosψ sinψ 0 0

0 − sinψ cosψ 0 0

] 0

187

Appendix B. Constraint reformulation with LMI

and apply Schur Lemma again:

1 u1 u2 −bm −cm 0 0

u1 b2n + ε2b + c2n + ε2c 0 −ε2b cosψ ε2c sinψ cosψ − sinψ

u2 0 b2n + ε2b + c2n + ε2c −ε2b sinψ −ε2c cosψ sinψ cosψ

−bm −ε2b cosψ −ε2b sinψ ε2b 0 0 0

−cm ε2c sinψ −ε2c cosψ 0 ε2c 0 0

0 cosψ sinψ 0 0 1c2n+ε2c

0

0 − sinψ cosψ 0 0 0 1b2n+ε2

b

0.

Recalling that cosψ = x4

V cos γand sinψ = x5

V cos γwe obtain:

1 u1 u2 −bm −cm 0 0

u1 b2n + ε2b + c2n + ε2c 0 −ε2b x4V cos γ

ε2cx5

V cos γx4

V cos γ− x5V cos γ

u2 0 b2n + ε2b + c2n + ε2c −ε2b x5V cos γ

−ε2c x4V cos γ

x5V cos γ

x4V cos γ

−bm −ε2b x4V cos γ

−ε2b x5V cos γ

ε2b 0 0 0

−cm ε2cx5

V cos γ−ε2c x4

V cos γ0 ε2c 0 0

0 x4V cos γ

x5V cos γ

0 0 1c2n+ε2c

0

0 − x5V cos γ

x4V cos γ

0 0 0 1b2n+ε2

b

0.

(B.9)

If we fix the TAS V and the path value γ taking their initial value Vk andγk, as done in Section 3.3, all entries of the matrix in (B.9) are affine in theinput u1, u2, u3, and, hence, the constraint is convex.We may also consider an alternative formulation of the LMI constraint. Werewrite (B.8) as:

1 u1 u2 −bm −cmu1 b2n + ε2b + c2n + ε2c 0 −ε2b x4

V cos γε2c

x5V cos γ

u2 0 b2n + ε2b + c2n + ε2c −ε2b x5V cos γ

−ε2c x4V cos γ

−bm −ε2b x4V cos γ

−ε2b x5V cos γ

ε2b 0

−cm ε2cx5

V cos γ−ε2c x4

V cos γ0 ε2c

+ (B.10)

+

0 0

x4 −x5

x5 x4

0 0

0 0

− c2n+ε2c

(V cos γ)20

0 − b2n+ε2b(V cos γ)2

[0 x4 x5 0 0

0 −x5 x4 0 0

] 0.

Recalling that (V cos γ)2 = V 2−x26, we apply the Schur Lemma to (B.10)

188

two times, and we obtain the following constraint:

1 u1 u2 −bm −cm 0 0 0 0

u1 b2n + ε2b + c2n + ε2c 0 −ε2bx4

V cos γ ε2cx5

V cos γ x4 −x5 0 0

u2 0 b2n + ε2b + c2n + ε2c −ε2b

x5V cos γ −ε

2c

x4V cos γ x5 x4 0 0

−bm −ε2bx4

V cos γ −ε2bx5

V cos γ ε2b 0 0 0 0 0

−cm ε2cx5

V cos γ −ε2cx4

V cos γ 0 ε2c 0 0 0 0

0 x4 x5 0 0 V 2

c2n+ε2c0 x6 0

0 −x5 x4 0 0 0 V 2

b2n+ε2b

0 x6

0 0 0 0 0 x6 0 c2n + ε2c 0

0 0 0 0 0 0 x6 0 b2n + ε2b

0

(B.11)

Similarly to (B.9), if we replace V , γ with their initial values Vk, γk (B.11)become a convex constraint as all entries in the matrix are affine in theinput.The approximate LMI constraints formulated above show some issues: theycan handle the constraints on the lateral and longitudinal acceleration, butthe constraint on the engine thrust formulated as LMI usually is infeasi-ble because of the introduced approximation. The LMI constraints are ex-tremely sensitive to the choice of the parameters εb and εc which is quitecritical, especially when the original constraints are not symmetric and bmand cm take large values. For these reasons we prefer not to use these LMIreformulations of the constraints.

189

APPENDIXCAlternative choices of the desired path

length

In this appendix it is discussed a possible approach to modify the choice ofthe desired path length Dref so as to foster the design of trajectories withdifferent characteristics. For example, setting Dref = 0 leads to design atrajectory with the shortest possible path, instead, setting Dref = V∆Tleads to a trajectory that should be covered at constant TAS in absence ofthe wind.Here a method to suitably modify the choice of Dref so as to partially com-pensate for the wind disturbance is proposed. In particular we focus on thelongitudinal, with respect to the path, component of the deterministic windvelocity, that, in actual aircraft operation, can be counteracted mainly byadjusting the aircraft TAS. In order to reduce the TAS adjustments, the ideais to set the desired path length so as to reach the final point at the righttiming covering the path with a speed that is given by the sum of the con-stant TAS V of the aircraft and of the longitudinal component of the windvelocity. Namely, if the wind blows against the path direction, instead ofrelying on the MPC controller to increase the TAS so as to compensate forthe wind disturbance, a shorter path is designed so that an increase of the

191

Appendix C. Alternative choices of the desired path length

TAS is no longer required. On the other hand, if the wind pushes the air-craft along the path, instead of decreasing the TAS, a slightly longer path isdesigned.A possible way to achieve this result is to set the desired length for thedesigned path as:

Dref =

∫ tF

tI

V + Vw,L([x y z](s(τ)), τ)dτ, (C.1)

where V is the desired constant TAS of the aircraft and Vw,L is the lon-gitudinal (with respect to the path) component of the deterministic windvelocity evaluated along the path. The fact that the presence of the windis accounted for in the computation of the reference distance as in (C.1),requires the path and time law design algorithms in Sections 4.2 and 4.4 tobe slightly modified as described point by point in the following.

• Computation of Dref .The computation of the distance Dref as the integral of the speedprofile accounting for the deterministic wind as in (C.1) is no morestraightforward: indeed the wind velocity depends on the positionwhich in turn depends on the designed path. Indeed at every timeinstant it is necessary to know the corresponding position along thepath.Recalling that the value of the spatial coordinate s corresponding to agiven time instant t is recovered exploiting the fact that the distancecomputed as integral of speed must equal the distance computed asintegral of the path (as described in Section 4.4.3), we can computeDref as:

Dref = d(tF ) with d(t) solution of:

d = V (t) + Vw,L([x y z](s(d)), t) with i.c. d(tI) = 0. (C.2)

The longitudinal deterministic wind velocity Vw,L is computed asVw,LVw,l

Vw,v

= RyRzwf ([x y z](s(d)), t),

where wf ([x y z], t) is the wind velocity, expressed in Cartesian com-ponents, in the position x, y, z at time t (more precisely at the valueof s corresponding to t, i.e. at s such that D(s) = D(t)), that can

192

be computed from the wind forecast as explained in Section 2.4. Ifit results that D(t) > D(s) t ∈ [tI , tF ], i.e. the designed path endsbefore the final time tF is achieved, the position is evaluated consider-ing that the path continues in straight line in the direction determinedby s7 at the end of Cxy7. The rotation matrices Rz and Ry transformthe wind velocities so as to be expressed in longitudinal, lateral andvertical components with respect to the path:

Rz =

cosψR sinψR 0

− sinψR cosψR 0

0 0 1

Ry =

cos γR 0 sin γR

0 1 0

− sin γR 0 cos γR

ψR = arctan

( y(s)

x(s)

)γR = arcsin

( z(s)

‖[x(s) y(s)]‖2

).

In view of this, in order to solve (C.2) we have to:i) given a distance d compute the corresponding value of s by inver-sion of (4.72);ii) given s evaluate the path position x(s), y(s), z(s) and velocity x(s),y(s), z(s);iii) given the position evaluate the wind velocity wf (x, y, z, t);iv) given wf and x(s), y(s), z(s), compute the longitudinal path com-ponent of wind velocity Vw,L by means of Rz and Ry;v) evaluate the aircraft speed profile V at time t (the velocity can bethe nominal TAS V or the PWA speed in (4.66));vi) integrate d = V + Vw,L.

• Path design.Note that since Dref defined as in (C.1) depends on the path, it cannotbe fixed in advance but it must be evaluated in correspondence of everycomputed path during the design procedure. Hence, the calculation ofDref by means of (C.2) setting V (t) = V has to be embedded infunction F1 (line 3) of algorithm 1.

• Speed profile design.In the design of the PWA speed profile Vpwa defined in (4.66), thewind velocity Vw,L has to be taken into account: condition (4.68) onthe covered distance has to be modified as:

D =

∫ tF

tI

Vpwa(t) + Vw,L([x y z](s(τ)), τ)dτ, (C.3)

where D can be computed resorting again to the differential equationin (C.2). In order to determine the value of the parameters defining

193


the PWA speed profile, the solution of the system of equations (4.66)-(4.70) in which (4.68) is replaced by (C.3) is carried over relying ontwo nested mono-dimensional searches to find the proper values oft1 and Vc satisfying (C.3). Note that the second mono-dimensionalsearch introduced here is needed to determine t1 because, differentlyfrom (4.68), (C.3) cannot be analytically solved. This is described inalgorithm 4.

194

Algorithm 4Input: a path P , initial and final speed V1 and V2, absolute values of the accelerations|α1| = |α2| = α

1: Compute the t1 range T1 [See below for details]


mint1∈T1

εV

s.t. (Vpwa, εV ) = F3(t1)

Denote with (V ?pwa, ε?V ) the solution

3: if ε?V = 0 then4: return the current speed profile V ?pwa [Solution found]

5: else6: return [Solution does not reach the final point with proper timing]

7: end if

(Vpwa, εV ) = F3(t1)

1: Compute the Vc range Vc(t1) [See below for details]


minVc∈Vc

εV

s.t. (Vpwa, εV ) = F4(Vc)

Denote with (V ?pwa, ε?V ) the solution

3: Vpwa = V ?pwa εV = |ε?V |4: return the current speed profile Vpwa, and the error on equation (C.3) εV

(Vpwa, εV ) = F4(Vc)

1: Compute the parameters of the speed profile Vpwa:

β1 = V1

β2 = Vc

α1 = sign(Vc − V1)|α1|α2 = sign(V2 − Vc)|α2|

t2 =Vc − V1

α1

t4 =V2 − Vcα2

t5 = t1

t3 = ∆T − 2t1 − t2 − t4

2: Compute the covered distance by the current speed profile DVpwa as in (C.2).3: εV = |D −DVpwa |4: return the current speed profile Vpwa, and the error on equation (C.3) εV

195


In order to achieve admissible solutions with ti ≥ 0 i = 1, . . . , 5 andVc ∈ [Vmin, Vmax] proper bounds T1 and Vc are imposed on t1 and Vcrespectively. More precisely, since t2 and t4 result automatically non-negative and V1 and V2 are properly bounded in algorithm 3, we needto enforce only the following condition:

t3 = ∆T − 2t1 − t2 − t4 ≥ 0

∆T − 2t1 −|Vc − V1|

α− |V2 − Vc|

α≥ 0.

Hence, when V1 ≤ Vc ≤ V2 or V2 ≤ Vc ≤ V1 we have a condition ont1:

t1 ∈ T1 = [0, (∆T −|V1 − V2|

α)/2],

instead when Vc ≤ V1 ∧ Vc ≤ V2 or Vc ≥ V1 ∧ Vc ≥ V2, we obtain thefollowing bound for Vc:

Vc ∈ Vc =

[maxVmin,

V1 + V2 − α(∆T − 2t1)

2,

minVmax,V1 + V2 + α(∆T − 2t1)

2].

The computation of the boundary velocity V1 and V2 can be done ac-cording to algorithm 3.

• Generation of the MPC reference.As described in Section 4.4.3, in order to define the reference trajec-tory to be used in the MPC controller we have to evaluate the positionalong the path in correspondence of t = kTs k = 1, 2, . . .. This canbe done computing the distance covered at each time instant d(kTs)as in (C.2) so as to account for the presence of the deterministic windvelocity.

The approach described in this appendix significantly increases the com-putational effort required. Indeed, in the path design, equation (C.2) hasto be integrated for every value of s1 and s2 tried in the two nested mono-dimensional searches. Moreover, in the speed profile design, two nestedmono-dimensional searches, instead of a single one, are required, and againequation (C.2) has to be integrated at each evaluation.

196

Note also that this approach do not account for the stochastic wind compo-nent and, hence, in the actual aircraft operation, TAS adjustments may bestill needed to compensate for it. Furthermore the MPC controller shouldcounteract the effects of the lateral and vertical (with respect to the path)wind disturbances.

197

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