professor mario di bernardo faculty of engineering, naples

Aeroelastic flutter non linear control

Professor Mario Di Bernardo

Faculty of engineering, Naples

Masters degree in control engineering

Giovanni Pugliese Carratelli M58/30

"Heavier than air flying machines are impossible"

Lord Kelvin

Summary

The purpose of this document is to investigate some non linear control strategies for a 2 degree of freedom

(DOF) wing section subject to aero elastic flutter. In the beginning of the document it will be shown what aero

elastic flutter is with some examples. A mathematical model is then shown, the BACT model, with system some

analysis. After the system analysis results, two control strategies are developed and results and simulations are

shown.

5

Contents

Summary 5

1 Model and system analysis 9

1.1 What is aereoelastic fluttering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 The NASA BACT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 BACT MODEL with two control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 State space representation and system open loop analysis . . . . . . . . . . . . . . . . . . . . . . 16

1.3.1 State space representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Open loop analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Input Output Feedback linearization 25

2.1 Pitch FBL control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.2 Hidden dynamics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.2.1 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Plunge FBL control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2 Hidden dynamics analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2.2.1 Lyapunov stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.2.2 Bifurcation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 MRAC Minimum Controller Synthesis control for flutter 43

3.1 MCS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.1 MCS extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1.1 MCS-LQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.1.2 MCSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1.3 EMCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1.4 NEMCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.1.1.5 LQ-NEMCSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2 MCS control synthesis and simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.2.0.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.0.7 MCS controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2.0.8 MCSI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.0.9 EMCSI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.0.10 NEMCSI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.0.11 LQ-NEMCSI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.0.12 Gain-locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.0.13 Velocity variation rejection and comparison . . . . . . . . . . . . . . . . . . . . . 63

3.2.0.14 Hybrid parameter variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7

3.2.0.15 Chaos recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 72

List of figures 73

List of tables 79

Chapter 1

Model and system analysis

It is possible to fly without motors, but not without knowledge and skill. This I conceive to be fortunate, for man, by

reason of his greater intellect, can more reasonably hope to equal birds in knowledge than to equal nature in perfection

of her machinery.

Wilbur Wright, Letter to the Western society of engineers, 1900

1.1 What is aereoelastic fluttering

Aeroelasticity is the interaction of structural, inertial and aerodynamic forces. It occurs to systems subject to an

airstream (or more generally in a fluid stream), for example to airfoils or even bigger structures such as bridges

or buildings. Aeroelasticity is under certain conditions characterised by what is called flutter. Aeroelastic flutter

is an oscillatory aeroelastic instability characterized by the loss of elastic recall and low damping due to the

presence of aerodynamic loads. In the aerospace industry this is a very well known problem as it happens to

occur to airplanes wings.

The conditions under which flutter can be observed are various and depend mainly on: the speed at which the

structure is moving in the fluid, the elastic recall to which the foil is subject to (as the foil is a structure it has

an elastic behaviour) and the angle between the fluid and the foil (also known as angle of attack AoA). Indeed,

in the case of an airplane, wing structural deformation leads to higher aerodynamic forces making flutter a

self-feeding mechanism that may lead to catastrophe, moving so from an equilibrium point to a Limit Cycle

9


Oscillation (LCO) or to chaos. If the damping is not adequate, the imbalance between energy input and the

structural dissipation will result in very large oscillations or unconstrained increase of amplitude.

A simple example can show qualitatively what happens: consider first a foil in a steady airstream. Such foil in

a fluid stream is subject to lift, air resistance and a pitch moment as can be seen in Fig.1.1 .

Figure 1.1: Lift (L), resistance (R) and pitch moment (P) of a foil in a steady airstream

Lift, resistance and pitch moment depend, among other factors, on the square of the airstream velocity, the

exposed surface to the airstream and the AoA, α.

L = Cl(α)12

ρU2CxA

R = Cr(α)12

ρU2CxA

P = Cm(α)12

ρU2CxA

Functions Cl,Cr and Cm are non linear functions of α and are different for every foil. Some typical behaviours

are depicted in Fig.1.2.

Figure 1.2: Typical Cl,Cm and Cr behaviour

A simple model to show how LCO occur on such system is to consider a 2nd order torsional ODE as follows:

Consider now a foil, that is immerged in an airstream and is subject also to an elastic recall K, for example a

wing section like as Fig.1.3.

10

1.1. What is aereoelastic fluttering

Figure 1.3: Foil subject to elastic recall K

A simple model to show how LCO occur on such system is to consider a 2nd order torsional ODE as follows:

Iαα + cαα + Kα(α) = M(α, α) (1.1)

M(α, α) is the pitch moment, cα(α) is the damping term and Kα(α) is the torsional elastic recall. Assuming

that M is only function of α, possible for low velocities, the model can be recast to the following form:

Iαα + cαα + (Kα(α) − M(α)) = 0 (1.2)

When hit by an airstream at a critical speed U0 the foil will be subject to a pitch moment increase, that will

lead to higher values of the AoA. Pitch angle will increase if Kα(α) − M(α) < 0; so the structure elastic recall

force plays a crucial role. Indeed when for a certain value α0 of AoA, after which Kα(α)−M(α) > 0, the elastic

recall will be greater than pitching moment and bring the foil towards α = 0. As the system damping is low,

which is typical for structures, overshooting from α = 0 will occur. This will result in a negative AoA and an

increasing pitch moment in opposite direction as shown in Fig.1.4

Figure 1.4: Foil subject to elastic recall K in opposite direction after overshoot

After overshooting, very much as what happens for positive AoA, the structure’s recall is lower than pitching

moment and so there will be a negative increase of AoA. Since the airstream is steady, pitching moment will grow

until it is greater than the elastic recall for negative values of α, increasing the AoA until Kα(α) − M(α) > 0.

The major elastic recall over the pith moment will bring the system towards α = 0, and an overshoot will occur

again for positive values of the AoA. A a self feeding mechanism will so begin, this is called flutter.

This example shows how flutter is in this simple dynamical system a LCO. More complex models are of course

possible with more than one degree of freedom where chaotic behaviour is possible.

1.1.1 Examples

Flutter is not only limited to the aerospace industry, it can indeed occur to other structures in a fluid, as for

instance chimneys and bridges, or even simply sign posts. Although the focus of this report is towards airplane

(or gliders) wing active fluttering control, in the past many other ways to avoid the phenomena have been

successfully applied in aeronautics (i.e. passive or structural fluttering control). The main strategy along this

line is to dimension the structure in such a way that the energy introduced in the system is well damped in

11


normal use (mass distribution is the main parameter on which to work).

Some examples of what flutter can lead to are reported in the following. As for the aerospace industry an

example of wing flutter was shown in a test flight by NASA in 1966. The test flight pilots brings the Piper

PA-38 to speed that causes flutter and slows down just before any structural failure.

Figure 1.5: The tail of the Piper during an LCO

Some accidents have happened due to fluttering, the first example is the Tackoma bridge in 1940. The bridge

was so that it would oscillate at its natural mode when subject to wind at approximately 67mph. When this

occurred the bridge structure failed as shown by these impressive images.

In recent times some studies have qualitatively related how flutter and dry Coulomb friction are a close

phenomena as shown in [2]. In this paper a simple 2 Degree of Freedom (DoF) mechanical arm is built with

two elastic hinges on which a load is applicable. The arm is shown in Fig., and known as Ziegler Column.

12

1.1. What is aereoelastic fluttering

Figure 1.6: The Ziegler Column 2 DoF arm

This is a two-degree-of-freedom structure made up of two rigid bars, internally jointed with an elastic

rotational spring, externally fixed at one end through another elastic rotational spring, while at the other end

subject to a tangential follower load P coaxial to the rod to which it is applied as schematically show in 1.7.

One side of the arm is fixed to a and on the other a wheel is mounted so to create a follower force P with a

movable metal plate that allows friction between the wheel and the plate.

Figure 1.7: Schematic representation of the 2 DoF mechanical arm. vp is the plate velocity

It is shown that the structure becomes dynamically unstable at a certain load level , so that it evidences

flutter and, at higher load, divergence instability. In Fig.1.8 it is quite clear how the system moves towards a

LCO both from the model predictions and the experimental data.

Figure 1.8: Experimental data and model prediction for the Ziegler Column

13


1.2 The NASA BACT model

In the past many approaches have been used to model and actively control flutter. The major modeling results

are by Theodorsen who developed the classical unsteady aerodynamic theory which accounts for the aerodynamic

damping at different conditions, and showed how flutter depends on it. After Theodorsen, Mukhopadhyay and

Gangsaas created 20th and 50th order models,and strategies for order reduction. They used state feedback as

the control method, and implemented estimators to describe unmeasured states. They employed proportional

gain feedback methods developed from root locus plots. A quasi-steady aerodynamic model coupled with a two

degree-of-freedom structural model was used to develop several types of feedback. The development directly

feeds one of four variables to the control surface through a proportional gain.

Aeroelastic systems typically contain nonlinearities which are either neglected or simplified to a linear form

for analysis. Nonlinearities which occur in aeroelastic systems include control saturation, free play, hysteresis,

piece-wise linear, and continuous nonlinearities.

Later on it was shown that a poor agreement between theory and experiment in flutter is most likely due to

nonlinear structural elastic models. So detailed examination of many types of nonlinearities that may affect

aeroelastic systems is presented in various articles. Tang and Dowell introduced a free play nonlinearity in the

torsional stiffness and examined the nonlinear aeroelastic response. For various initial condition they show that

LCO is dependent upon free stream velocity, initial pitch condition, magnitude of the free play nonlinearity and

initial conditions.

One the major developments was given by NASA in 1997, by building the Benchmark Active Control Technology

(BACT). It is a two degree of freedom model where the pitching movement and the plunging one, are respectively

restrained by a pair of springs attached to the elastic axis(EA)of the airfoil. A prototype was also built as shown

in Fig.1.9 where one or two trailing-edge control surfaces are used to control the air flow, thereby providing

maneuverability to suppress instability. The BACT model is accurate for airfoils at low velocity and has been

confirmed by wind tunnel experiments.

Figure 1.9: Configuration of the nonlinear 2-D prototypical aeroelastic wing

In this report the BACT model will be shown, thus with respect to the example in equation 1.2, a second

degree of freedom known as plunge is taken into account. Plunge is introduced so to consider also flapping of the

considered wing section. The model is a simple representation of an aeroelastic system for low speed, where all

non linear terms from experimental data are taken into account as shown in [9]. Hence the equations governing

the motion of the aereoelastic system are:

[

m mxαb

mxαb Iα

] {

h

α

}

+

[

ch 0

0 cα

] {

h

α

}

+

[

Kh 0

0 Kα(α)

] {

h

α

}

=

{

−L

M

}

(1.3)

where h is the plunge motion and α is the pitch or AoA. In equation 1.3, m is the mass of the considered section

of the wing, and Iα is the mass moment of inertia about the elastic axis. The position of the elastic axis with

respect to the center of mass of the considered wing section can be varied and is referred as xα. Constants ch

and cα are linear damping constants of the system. Kh is the spring constant for the plunge motion and Kα(α)

is the non linear stiffness of the pitch motion. In this report the non linear stiffness Kα(α) is assumed to be a

14

1.2. The NASA BACT model

4th order polynomial function:

Kα(α) =4

∑

i=0

Kαiαi = Kα0

+ Kα1α + Kα2

α2 + Kα3α3 + Kα4

α4 (1.4)

Figure 1.10: Wing cross-section representation

M and L, in case of a single control surface as in Fig. 1.10, are respectively the input moment and the

quasi-steady aerodynamic lift and are modeled as [3]:

L = ρU2bclα(α +

h

U+ (

12

− a)bα

U) + ρU2bclβ

β (1.5)

M = ρU2b2cmα(α +

h

U+ (

12

− a)bα

U) + ρU2b2cmβ

β (1.6)

where ρ is air density, U is the air stream velocity, β is the angle between the foil and the trailing edge control

surface. clα and cmα are the lift and momentum coefficients for the AoA and clβ and cmβ are respectively the

lift and moment coefficients for the control surface. a is the distance between mid-chord1 and the elastic axis

(EA) as shown in Fig.1.11.

Figure 1.11: Wing cross-section schematic representation showing a and mid-chord b

After substituting the quasi-stead forces from equations 1.5 and 1.6 into equation 1.3:[

m mxαb

mxαb Iα

] {

h

α

}

+

[

ch + ρUbclαρUb2clα

(12 − a)

ρUb2cmαcα − ρUb3cmα

(12 − a)

] {

h

α

}

+ (1.7)

[

Kh ρU2bclα

0 −ρU2b2cmα+ Kα(α)

] {

h

α

}

=

{

−ρbclββ

ρb2cmββ

}

U2

1Chord is the imaginary line joining the trailing edge and the center of curvature of the leading edge of the cross-section of an

airfoil

15


It is important to note that the only source of non linearity is given by the stiffness and an extension for higher

velocity of the model is possible by considering the pith moment as a quadric function of α.

1.2.1 BACT MODEL with two control surfaces

A extension of the model presented in equation 1.7 can be derived for when two control surfaces are available

on the trailing edge as depicted in Fig. 1.12 The model can be derived considering the following lift and pitch:

Figure 1.12: Two trailing edge control surfaces

L = ρU2bclα(α +

h

U+ (

12

− a)bα

U) + ρU2bclβ2

β2 + ρU2bclβ2β2 (1.8)

M = ρU2b2cmα(α +

h

U+ (

12

− a)bα

U) + ρU2b2cmβ1

β1 + ρU2b2cmβ2β2 (1.9)

That substituted in equation 1.3 leads to:

[

m mxαb

mxαb Iα

] {

h

α

}

+

[

ch + ρUbclαρUb2clα

(12 − a)

ρUbcmαcα − ρUb3cmα

(12 − a)

] {

h

α

}

+ (1.10)

+

[

Kh ρU2bclα

0 −ρU2cmαKα(α)

] {

h

α

}

=

[

−ρbclβ1−ρbclβ2

ρb2cmβ1ρb2cmβ2

] {

β1

β2

}

U2

1.3 State space representation and system open loop analysis

1.3.1 State space representation

It is convenient for the analysis that follows, and for control design to have a state space formulation (SS) of

the system in equations 1.7 and 1.10. For this purpose the following state variable vector is used:

x =

x1

x2

x3

x4

=

h

α

h

α

, x ∈ ℜ4 (1.11)

and β is the control input.

The SS formulation expresses the system in the following affine form:

x = fµ,xα(x) + g(x)µβ

where µ = U2 and it is to note the equations are dependant on the airstream velocity and the elastic axis

location xα. The notation with the two subscripts µ and xα emphasizes the system dependance on the two

16

1.3. State space representation and system open loop analysis

parameters and the solutions are in fact a two-parameter family of solutions.

It is possible to obtain a SS representation of an equation in the following form:

Mx + Dx + Kx = Fu(t)

by using the well know formula:[

x1

x2

]

=

[

0 I

−M−1K −M−1D

] [

x1

x2

]

+

[

0

−M−1F

]

In the considered case matrix M−1 will be:

M−1 =

[

Iα

d−mxαb

d−mxαb

dmd

]

so a synthetic SS formulation with a single trailing edge control surface is the following:

fµ,xα(x) =

x3

x4

−k1x1 − (k2µ + p(x2))x2 − c1x3 − c2x4

−k3x1 − (k4µ + q(x2))x2 − c3x3 − c4x4

, g(x) =

0

0

g3

g4

(1.12)

In the following table the system term of equation 1.12 are explicated:

d = m(Iα − mx2αb2)

c1 = Iα(ch+ρUbclα )d

c2 = IαρUb2clα ( 1

2−a)−mxαbcα+mxαρUb4cmα ( 1

2−a)

d

c3 = −mxαb(ch+ρUbclα )+mρUb2cmα

dc4 = −mxαρUb3clα ( 1

2−a)+mclα −mρUb3cmα ( 1

2−a)

d

p(x2) = −mxαbd

kα(x2) q(x2) = md

kα(x2)

k1 = Iαkh

dk2 = Iαρclα +mxαb3ρcmα

d

k3 = mxαbkh

dk4 = −mxαb2ρclα −mρb2cmα

d

g3 =−Iαρclβ

−mxαb3ρcmβ

dg4 =

mxαb2ρclβ+mxαb2ρcmβ

d

Table 1.1: System SS variables

The parameters for the following investigation are obtained from [8]. The non linear elastic recall, defined

in 1.4, that will be used is defined as follows:

Kα(α) = 2.82(1 − 22.1α + 1315.5α2 + 8580α3 + 17289.7α4)

The other parameters used are shown in the following table: Most of the calculations and integrations for the

m = 12.3870Kg Iα = 0.065Kgm2

rcg = 0.0873 − (b + ab) rcg

b

b = 0.135m ρ = 1.118 Kgm3

kh = 2844.4 Nm

ch = 27.43 Nsm

clα= 6.28 clβ

= 3.358

cmα= (1

2 + a)clαcmβ

= −0.635

Table 1.2: System SS parameters

system have been carried out using the MATLAB/SIMULINK environment. At this purpose in Fig.1.13 the

SIMULINK block diagram is shown; it is possible to any shape to the parameters signals of air stream (U),

density of air (ρ) and Elastic Axis location (a).

17


x1

xa

a xa

k4

d

rho

xa

cma

k4

k3

d

xa

k3

k2

d

rho

xa

cma

k2

k1

d k1

g4

d

rho

xa

g4

g3

d

rho

xa

g3

d

xa d

cma

a cma

c4

d

rho

xa

U

a

cma

c4

c3

d

rho

xa

cma

U

c3

c2

d

rho

xa

cma

U

a

c2

c1

d

rho

xa

cma

U

c1

a9

−m

a8

[c3]

a7

m

a6

[c1]

a5

[U]

a4

U

a3

[rho]

a2

rho

a10

b

a1

a

a

[a]

Scope9

Scope8

Scope7

Scope6

Scope5

Scope4

Scope3

Scope2

Scope10

Scope

Product9

Product8

Product7

Product6

Product5

Product4

Product3

Product2

Product14

Product13

Product12

Product11Product10 Product1

Product

MathFunction5

1

u

MathFunction4

u2

MathFunction3

u2

MathFunction2

u2

MathFunction1

u2

MathFunction

1

u

Ka1

P(u)O(P) = 4

Ka

P(u)O(P) = 4

Integrator

1s

Goto9

[g3]

Goto8

[c2]

Goto7

[k1]

Goto6

[cma]

Goto5

[d]

Goto4

[xa]

Goto3

[x4]

Goto2

[x3]

Goto14

[k4]Goto13

[k2]

Goto12

[k3]

Goto11

[c4]

Goto10

[g4]

Goto1

[x2]

Goto

[x1]

From9

[x2]From8

[x2]

From70

[xa] From7

[x4]

From69

[d]

From68

[U]From67

[k2]

From66

[U]From65

[k4]

From64

[x4]

From63

[x3]

From62

[c4]

From61

[c3]

From60

[U]

From6

[x3]

From59

[g4]

From58

[U]From57

[g3]From56

[x4]

From55

[x3]

From54

[k1]

From53

[xa]

From52

[rho]

From51

[d]

From50

[xa]

From5

[c2]

From49

[rho]

From48

[d]

From47

[a]From46

[U]

From45

[cma]

From44

[xa]From43

[rho]From42

[d]

From41

[U]From40

[cma]

From4

[c1]

From39

[xa]From38

[rho]From37

[d]

From36

[a]From35

[U] From34

[cma]From33

[xa] From32

[rho]From31

[d]

From30

[U]

From3

[k3]

From29

[cma]From28

[xa]From27

[rho]From26

[d]

From25

[cma]From24

[xa]From23

[rho]From22

[d]

From21

[xa]

From20

[d]

From2

[x1]

From19

[rho]

From18

[cma]From17

[xa]

From16

[d]

From15

[d]

From14

[a]

From13

[a]

From12

[xa]

From11

[x2]

From10

[x2]

From1

[d]

From

[x1]

Add3

Add2

Add1

Add

beta1

x_dot

x3_dot

x4_dot

Figure 1.13: Simulink block diagram for the system’s open loop dynamics

18


1.3.2 Open loop analysis

A first simulation of an open loop response of the system is carried out considering the air flow velocity U = 15 ms

and a = −0.4 with the following initial conditions α = 0.1[rad], h = 0.1[m] and α = h = 0 as depicted in Fig.

1.14a.

−6 −4 −2 0 2 4 6 8−150

−100

−50

0

50

100

150

α (x2) [deg]

x 4 [deg

/s]

(a) AoA Phase portrait with α = 0.1[rad], h = 0.1[m] and

α = h = 0,U = 15 ms

,a = −0.4

−0.015 −0.01 −0.005 0 0.005 0.01−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

α (x2) [deg]

x 4 [deg

/s]

(b) Plunge phase portrait with α = 0.1[rad], h = 0.1[m]

and α = h = 0,U = 15 ms

,a = −0.4

The system clearly shows a LCOs due to the structural non linearity in equation 1.4 and due to the low

damping. The suppression of the LCO and of a possible chaotic behaviour will be in the following chapters, one

of the main goals for the controller.

Other simulation are for different initial conditions. It is interesting to notice that the system other than LCO

behaviour also shows a chaotic behaviour 2 for some initial conditions. Indeed for low values of h and h, the

system exhibits chaos as shown in the following figures.

−0.1 −0.05 0 0.05 0.1 0.15

−0.6

−0.4

−0.2

0

0.2

0.4

x2

x 4

(a) AoA chaotic Phase portrait with α = 0.1[rad], h = 0[m]

and α = h = 0,U = 25 ms

a = −0.4

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

−0.4

−0.2

0

0.2

0.4

0.6

X1

X3

(b) Plunge chaotic phase portrait with α = 0.1[rad], h[m]

and α = h = 0,U = 25 ms

a = −0.4

The behaviour shown in Fig.1.14a and 1.14b given by a chaotic vector field for the values of parameters

indicated, and it will be shown in chapter 3 how a MCS controller can recover from chaos.

Setting the initial conditions so to have as solution to the system an LCO behaviour, and because of it’s

dependance on the parameters as shown in equation 1.12, it is possible to do a bifurcation analysis . Before

showing the results of the bifurcation analysis we qualitatively show on a 3d plot what happens for different

values of U and a as depicted in the following figures because it gives a quick glance of how the LCO grow with

respect to velocity:2notice that chaos is possible only for systems of order n > 3, this is a consequence of the Poincaré-Bendixson theorem

19


15

20

25

30

35

40

−15

−10

−5

0

5

10

15

20

25

−800

−600

−400

−200

0

200

400

600

U [m/s]x2 = α[degrees]

x4

=α[d

egre

es/s

]

Figure 1.14: Open loop responses on AoA for the system with α = 0.1[rad], h = 0.1[m] and α = h = 0 and U = 15, 17, 20, 19, 40,

a = −0.4

15

20

25

30

35

40

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

−1.5

−1

−0.5

0

0.5

1

1.5

U [m/s]x1 = h[m]

x3

=h[m

/s]

Figure 1.15: Open loop responses on plunge for the system with α = 0.1[rad], h = 0.1[m] and α = h = 0 and U = 15, 17, 20, 19, 40,

a = −0.4

20


Similarly as what has been done for U , is done for various values of a at a settled velocity of U = 25.

−0.8−0.7

−0.6−0.5

−0.4−0.3

−0.2−0.1

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

−8

−6

−4

−2

0

2

4

6

a

x1

x 3

Figure 1.16: Open loop responses on AoA for the system with α = 0.1[rad], h = 0.1[m] and α = h = 0 and a = −0.1, −0.3, −0.4, −0.6

−0.8−0.7

−0.6−0.5

−0.4−0.3

−0.2−0.1

−0.15−0.1

−0.050

0.050.1

0.150.2

0.25

−8

−6

−4

−2

0

2

4

6

ax

1

x 3

Figure 1.17: Open loop responses on AoA for the system with α = 0.1[rad], h = 0.1[m] and α = h = 0 and a = −0.1, −0.3, −0.4, −0.6

As stated earlier a bifurcation analysis is performed for the LCOs of the system. The bifurcation for the

LCOs is a Hopf bifurcation starting for a = −0.9; no bifurcations for airstream velocity are found. The result

of the continuation calculations is a family of LCOs shown in the following figures.

21


−0.1 −0.05 0 0.05 0.1 0.15 0.2

−5

−4

−3

−2

−1

0

1

2

3

4

x2

x 4

(a) Family of LCOs for AoA, U = 25 ms

−0.2 −0.1 0 0.1 0.2 0.3

−8

−6

−4

−2

0

2

4

6

x2

x 4

(b) Bigger family of LCOs for AoA, U = 25 ms

−20 −15 −10 −5 0 5

x 10−3

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x1

x 3

(c) Family of LCOs for plunge, U = 25 ms

−0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

X1

X3

(d) Bigger family of LCOs for plunge, U = 25 ms

22


−8−6

−4−2

02

46

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

a

x4

x2

(a) 3d visualization of a family of LCOs for AoA, U = 25 ms

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−1

−0.5

0

0.5

1

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

a

x1

x3

(b) 3d visualization of a family of LCOs for plunge, U =

25 ms

The analysis shown for the system is only numerical and qualitative. It is although possible even if highly

complex, to show the behaviour of the system use a Lyapunov approach. Being the system mechanical, as a

Lyapunov function it is possible to use the energy of the system; being the system of the 4th order the candidate

Lyapunov function is the following:

V =12

((x3, x4)M(x3, x4)T + K1,1x21 + x1x2K1,2) +

∫ x2

0

K2,2(ζ)dζ > 0

Where M is the mass matrix, Ki,j are the elements of the elastic matrix as in equation 1.7. The calculations of

the time derivative can be carried out, but from what has been shown in the previous figures it is not possible

to show the V < 0; what can be shown is the presence of an invariant region, so to then use LaSalle’s invariance

principle and thus show analytically the presence of LCO and chaos. Although this is a possible approach the

calculations are particulary difficult, especially to show the invariant regions.

A more viable strategy, given the presence of the LCO behaviour is the use a Poincaré Map and Poincaré

sections, which is a typical approach in the case of non planar systems.

23

Chapter 2

Input Output Feedback linearization

In this chapter a non-linear controller based on Feedback linearization (FBL) is developed for the systems

introduced in equation 1.12. The main idea of this control technique is to build a non-linear controller so to

cancel the non-linear dynamic terms of the system. After the linearizing law is built a linear controller on a

second control loop is synthesized with classic methods as in [10] or, for instance, an optimal LQ controller can

be developed.

The control objective is to build a possibly global FBL controller, either with partial FBL or total FBL, to

suppress the LCO shown in the previous chapter. Indeed a FBL presents with respect to linearization the

advantage of being global.

There are some downsides in FBL, the main one is that it is not always possible and trivial to fully linearize a

system, indeed sometimes only partial FBL is possible and in this case hidden dynamics are to be investigated

for the stability of the entire system. The second downside, is that the linearizing control law may (and often

does) use parameters that are not always known; if the system parameters change the FBL law may not yield

the mismatch between the assigned parameter in the law and the actual value of the parameter thus may lead

the system to instability. The last downside concerns structural robustness, indeed non modeled dynamics can

affect the system’s performance.

Input output FBL (IO FBL) strategy is derived by using partial linearization and thus analysing hidden

dynamics for the system with one control surface. Partial FBL will be applied first to the AoA control and then

to the plunge dynamics.

2.1 Pitch FBL control

To derive the control law with IO FBL it is necessary to define an output function y = h(x). In the case of

pitch control the most reasonable variable to considered as output is the AoA since it is also is the easiest to

measure. So another equation is to be added to the system defined in equation 1.12:

y = h(x) = x2

In order to accomplish partial FBL the first step is to calculate1 the relative degree r of the system. The relative

degree of the system is calculated as in [1]:

Lg(h) =∂h

∂xg(x) = 0

Lf (h) =∂h

∂xf(x) = x4

Lg(Lf (h)) = g4 6= 0

1notice the following calculations the Lie derivative Lf (h) is calculated because it will be used as both for finding the relative

degree r, and for the state transformation as shown in [5]

25


Showing that the relative degree of the system is r = 2 allowing to linearize two equations leaving the other

n − r as hidden dynamics. In order to accomplish partial FBL a state transformation is carried out as follows:

x 7→ z

z1 = h(x) = x2

z2 = Lf (h) = x4

z3 = x1

z4 = −g3x4 + g4x3 (2.1)

The criterion in choosing the state transforation is shown in [5]. It is to note that z3 and z4 could have been

chosen differently but a different choice would not have had the benefit of having Lg(zi) = 0, 3 ≤ i ≤ 4(the

n − r hidden dynamics equations ) thus not allowing any effect of the input towards the variables z3 and z4 and

simplifying the hidden dynamics stability analysis. The picked choice for the SS transformation guarantees the

transformation to be invertible.

The new SS representation is the following, again in affine form:

z = fµ,xα(z) + g(z)µβ

where:

fµ,xα(z) =

z2

−(k4µ + q(z1))z1 + (c3g3

g4− c2)z2 − k3z3 − c3

g4z4

1g4

(z4 + g3z2)

((g3k4 − g4k2)µ + g3q(z1) − g4p(z1))z1 + (c3g2

3

g4+ c4g3 − c1g3 − c2g4)z2 + (g3k3 − g4k1)z3 + ( c1g3

g4− c1)z4

,

g(z) =

0

g4

0

0

Partial FBL can be achieved by selecting a control law where the non linear dynamics are compensated2. Thus

by choosing:

β =−(k4µ + q(z1))z1 + (c3

g3

g4

− c2)z2 − k3z3 − c3

g4

z4 + v

µg4

the closed loop system is defined as in equation 2.2:

z1

z2

z3

z4

=

z2

v1g4

(g3z2 + z4)

((g3k4 − g4k2)µ + g3q(z1) − g4p(z1))z1 + (c3g2

3

g4

+ c4g3 − c1g3 − c2g4)z2 + (g3k3 − g4k1)z3 + ( c1g3

g4

− c1)z4

(2.2)

Some considerations are to be made on the closed loop system in 2.2. The first is that a linear controller v is

to be built for stabilizing the dynamics of z1 and z2; secondly the dynamics of z3 and z4 are to be carefully

analysed for stability because there is no direct control on them and more importantly, if not linear, they can

be subject to bifurcations.

The linear controller v is first build as an LQR controller with a feed forward action, later on a derivative action

will be used as well moving so to a PD controller. The LQR controller is build using a full information scheme,

this is reasonable even if it is not possible to measure all the state vector[z1 z2]T , indeed it is still realistic to

measure AoA (z1) and estimating the pitch angular velocity (z2)is not difficult e.g. using a Kalman filter.

2this in the aerospace industry is know as linear dynamic inversion

26

2.1. Pitch FBL control

The LQ controller is built by solving the LQ problem in the MATLAB environment, using the Bryson’s rule to

define the weight matrixes R and Q. The resulting gain values for the controller are the following:

kz1= −70.7107, kz2

= −13.8355

The controller scheme design in the Simulink/Matlab environment is shown depicted in the following image:

Figure 2.1: SS graph for AoA partial FBL with α = 0.1[rad], h = 0.01[m] and α = h = 0

In Fig.2.1 it possible to see the LQ controller3, and the subsystem consisting of the SS transformation and the

compensator for the non-linear dynamics. It should be clear that no transformation is applied to the reference

signal; since the FBL is partial and the transformation -and thus only variables z1 and z2 are of interest- consists

in no more than a simple exchange of position. The full SIMULINK Block diagram is plotted in fig.

To Workspace6

pfblalpha

To Workspace5

pfblhp

To Workspace4

pfblh

To Workspace3

t

To Workspace2

betaFBLalpha

To Workspace1

pfblalphap

To Workspace

pfblalphatrack

Signal Builder

Signal 1

Scope5

Scope4 Scope3

Scope2

Scope1

Radiansto Degrees3

R2D

Radiansto Degrees2

R2D

Radiansto Degrees1

R2D

Radiansto Degrees

R2D

Plant

beta x

Ground

Filtro riferimento

1/(s/25+1)

FBL Controller + Transformation

reference

z_1,z_2

Fsb

Beta

Clock

Figure 2.2: FBL control scheme for AoA

2.1.1 Simulations

Some simulations for the system are reported in the following figures both for α and h. The parameters used

are U = 15 ms

, a = −0.4.

3not the feedforward action which is outside of this block.

27


0 1 2 3 4 5 6 7 8 9−5

0

5

10

15

20

25

30

35

40

z1=x

2=α [deg]

z 2=x 4=

d/d

t α [d

eg/s

]

Figure 2.3: Phase Plane of the controlled AoA dynamics with the use of a LQR controller

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Time [seconds]

z 1=x 2=

α [d

eg]

Reference

α

Figure 2.4: Step response of the controlled AoA dynamics with the use of a LQR controller

28


0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

15

20

25

30

35

40

Time [seconds]

z 2=x 4=

d/dt

α [d

egre

e/s]

0 1 2 3 4 5 6 7 8 9 10−14

−12

−10

−8

−6

−4

−2

0

2

4

Time [seconds]

β [d

egre

es]

Figure 2.5: Time response and control input for α = z1 = x2 and α = z2 = x4

29


The control goal, which is to suppress LCO and possibly permit tracking of a reference signal, is achieved

on the interested variables. Before discussing the stability of the hidden dynamics it is interesting to see a

qualitative behaviour of the variables [z3 z4]. The following figure shows a phase plane of the plunge and it’s

velocity. Clearly being the relative degree r = 2 there is no direct control over it; and also having chosen the

SS transformation as shown 2.1 allows to have no control input in the hidden dynamics equations.

−0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

z3=x

1= h[m]

z 4=x 3=

d/d

t h[m

/s]

Figure 2.6: SS graph for the plunge after partial FBL on the AoA. It is clear that the variable is indirectly influenced

Another tracking examples is shown in Fig. 2.7, after appropriate filtering of the reference. In this case the

plunge phase plane also is of interest because it can be seen that not only it is qualitatively stable but it shows

a stable focus as depicted in the following figure and as will be shown further:

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

Time [seconds]

α[de

gree

]

0 1 2 3 4 5 6 7 8 9 10−40

−20

0

20

40

d/dt

α[d

egre

es/s

ec]

referenceα

Figure 2.7: Tracking graph and reference signal with null initial conditions

30


0 1 2 3 4 5 6 7 8 9 10−0.03

−0.02

−0.01

0

0.01

Time [seconds]

h[m

]

0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

Time [seconds]

d/dt

h [m

/s]

Figure 2.8: Plunge position and speed with null initial conditions when the AoA subject to reference signal as in Fig. 2.7

FBL control is highly model based and it is interesting to show a simple example of what happens when

there is uncertainty on the value of one or parameters. In the following figures a FBL law is built by considering

for the air stream velocity value at UF BL different from the real parameter U of the system. Indeed a 1g positive

acceleration is introduced a t = 5.9[s]; the following figure show the time response.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Time [seconds]

α

Reference

α

Figure 2.9: Tracking error for α give by building FBL control law on different parameters (UF BL, aF BL) from the system’s one

(U,a)

A constant tracking error and an increase of settling time with respect to Fig. 2.7 are present that the LQ

controller is not capable of rejecting. Some more information can be obtained by looking at the plot of α:

31


0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

15

20

25

30

35

40

Time [seconds]

d/dt

α

Figure 2.10: α dynamics when subject to non rejected parameter variations

Indeed when using FBL a robust linear controller is essential in order to handle unexpected parameter

variation that are not handled by the non linear dynamic compensating law. As stated earlier, for this purpose

a derivative action is introduced in the controller so to allow to reject some parameter variations. After the

introduction of a PD controller the resulting step response allows a null tracking error as can be seen in fig.İt

would have been possible also to introduce a LQI scheme or a PI linear controller, but in this occasion a

derivative action is preferred.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Time[seconds]

α [d

eg]

ReferenceFBL with uncertain parameters tracking

Figure 2.11: Null tracking error for α subject to parameter variation with a PD controller linear controller

32


2.1.2 Hidden dynamics analysis

The hidden dynamics are to be investigated; in order to accomplish the stability analysis the zero dynamics of

the system are investigated, by setting [z1 z2]T = 0 thus leading to the following system:

{

z3

z4

}

=

[

0 γ1,2

γ2,1 γ2,2

] {

z3

z4

}

(2.3)

where some constants of system 2.2 have been renamed as follows:

γ1,2 =1g4

, γ2,1 = (g3k3 − g4k1), γ2,2 =c1g3

g4− c1

2.1.2.1 Stability analysis

As qualitatively shown in Fig 2.6 there is a stable focus in [z3 z4] = (0, 0) that can be easily found solving the

system 2.3 with null solution. The eigenvalues calculated using as parameters the same as previous simulations

a = −0.4 U = 15m/s. The eigenvalues are: [−1.3122+17.1258j], [−1.3122−17.1258i] showing so a stable focus,

in accordance with what has been qualitatively shown in previous Fig.2.6. It is important to notice that the

zero dynamics system is linear, but still parameter dependant; it is necessary so to evaluate the position of the

eigenvalues for different values of a and of the velocity U as illustrated in Fig. 2.1.2.1

0

20

40

60

80

100

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−8

−6

−4

−2

0

2

4

6

8

10

12

U [ms]

a

Eigenvalues

Figure 2.12: Plot of the eigenvalues with respect to velocity U and a. Color pattern towards red indicates increasing values for the

eigenvalues

Fig.2.1.2.1 shows the how the real part of the eigenvalues is smaller than zero for all velocity values a

a < −0.55 and greater than zero for a > −0.55

Another possibility to investigate the system’s stability is to use the Lyapunov equation. This technique is not

followed since the use of the stability analysis using eigenvalues is a more synthetic way to for linear systems.

Secondly being the system parameter dependant the use of eigenvalues well shows how the stability varies with

respect to U and a; indeed using the Lyapunov equation would have requested to analyse a parameter varying

matrix, solution of the equation: AT P + P A < 0 with P = P T > 0.

33


2.2 Plunge FBL control

The FBL control law for plunge is built in a similar way as it was done for the AoA partial FBL control. At

first an output function is selected:

y = h(x) = x1

After this the relative degree r of the system is calculated:

Lg(h) =∂h

∂xg(x) = 0

Lf(h) =∂h

∂xf(x) = x3

Lg(Lf(h)) = g3 6= 0

The relative degree of the system is r = 2 thus allowing a partial FBL on the plunge dynamics. Since r < n

there are 2 hidden dynamics to be investigated.

Following the calculation of r, a SS transformation is introduced as following, again, the criterion shown by [5]

is used:

x 7→ z

z1 = h(x) = x1

z2 = Lf (h) = x3

z3 = x2

z4 = −g3x4 + g4x3 (2.4)

Fortunately also in this case it is possible, and easy, to find a SS transformation where Lgzi = 0, 2 ≤ i ≤ 4

making orthogonal the variables not interested by the FBL law to the control input.

The the transformation introduced in 2.4 allows a recast of the systems to the affine form of equation 2.2 where:

fµ,xα(z) =

z2

−k1z1 − (c1g4 + c2 g4

g3)z2) − (k2µ + p(z3))z3 − c2

g3z4

1g3

(g4z2 + z4)

(g4k1 − k3g3)z1 + (c1 + c2g2

4

g3− c3g3 − c4g4)z2 + (g4(k3µ + p(z3)) − g3(k4µ + q(z3)))z3 + (g4

g3c3 − c4)z4

,

g(z) =

0

g3

0

0

By selecting an appropriate control law plunge FBL can be achieved. Thus β chosen as follows:

β =+k1z1 + (c1g4 + c2 g4

g3)z2) + (k2µ + p(z3))z3 + c2

g3z4 + v

µg3

The resulting system equations after the compensator law has been selected as following:

z1

z2

z3

z4

=

z2

v1g3

(g4z2 + z4)

(g4k1 − k3g3)z1 + (c1 + c2g2

4

g3

− c3g3 − c4g4)z2 + (g4(k3µ + p(z3) − g3(k4µ + q(z3))))z3 + (g4

g3

c3 − c4)z4

(2.5)

It is now essential to build a linear controller v on an outer control loop so to stabilize the plunge dynamics.

Also in this case,a s what has happened for the IO FBL of the AoA an LQ controller is developed. By solving

the LQ problem and choosing the weight matrixes using the Bryson’s rule the following gains are obtained:

kz1= −70.7107, kz2

= −13.8355

34

2.2. Plunge FBL control

In the case of the plunge dynamics no derivative or integral action is used, but it is clear that it can easily bee

implemented.

2.2.1 Simulations

With the linear controller v deployed some simulations can be shown before analysing the zero dynamics of the

system. In the following figures some significant simulations are shown:

−0.02 0 0.02 0.04 0.06 0.08 0.1−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

h [m]

d/dt

h [m

/s]

Figure 2.13: Phase plane for a closed loop response for partial FBL on the plunge dynamics, with the following initial conditions

h = 0.1 and α = α = h = 0

0 1 2 3 4 5 6 7 8 9 10−0.02

0

0.02

0.04

0.06

0.08

0.1

Time [seconds]

h [m

]

Figure 2.14: Time response for partial FBL on the plunge dynamics, with the following initial conditions h = 0.1 and α = α = h = 0

35


0 1 2 3 4 5 6 7 8 9 10−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Time [s]

d/dt

h[m

/s]

Figure 2.15: Time response for partial FBL on the plunge dynamics, with the following initial conditions h = 0.1 and α = α = h = 0

Clearly the controller stabilises the system for the plunge dynamics for which there is no reason to have

tracking but just a regulation to zero. When dealing with partial FBL a qualitative check to the dynamics with

no control is often interesting as show in the following phase plane figure:

−5 0 5 10 15 20−300

−200

−100

0

100

200

300

α [degrees]

d/dt

α [d

egre

es/s

ec]

Figure 2.16: Phase plane for α and α for a closed loop response for partial FBL on the plunge dynamics, with the following initial

conditions h = 0.1 and α = α = h = 0

2.2.2 Hidden dynamics analysis

To analyse the system’s zero dynamics the variables z1 and z2 are set to zero, obtaining so the following

equations:{

z3

z4

}

=

{

γ1,2z4

(g4(k2µ + p(z3)) − g3(k4µ + q(z3)))z3 + γ2,2z4

}

(2.6)

36


Where γ1,2 and γ2,2 rename some constants from system 2.2 as follows:

γ1,2 =1g3

, γ2,2 =g4

g3c2 − c4

It is clear that the zero dynamics of the system are a family of non-linear equations depending on parameters

U and a.

2.2.2.1 Lyapunov stability analysis

To analyse the stability of the zero dynamics system equations 2.6 a Lyapunov theory approach is used. The

first step to simplify the process of finding a Lyapunov function is to recast the system to 2nd order ODE. This

can be achieved as follows:{

z3

z4

}

=

{

γ1,2z4

((−g4mxαb

d− g3

md

)Kα(z3) + µ(g4k2 − g3k4))z3 + γ2,2z4

}

=

{

γ1,2z4

19.94z3 − 54.57z23 + 2593.6z3

3 − 16916.5z43 + 34088.5z5

3 + γ2,2z4

}

And thus we can recast the hidden dynamics form as if it is a non linear mass-spring-damper system:

z − γ2,2z − Λ(z) = 0, Λ(z) = −4.51z + 9.86z2 − 5.87z3 + 3.82z4 − 7.71z5

We now search for a candidate Lyapunov function. A possibility is to use the energy of the system, which is

the sum of kinetic and potential energy. Thus the candidate Lyapunov function V (z) is selected as follows:

V (z) =12

z2 +∫ ζ

0

Λ(ζ) =12

z2 + 2.25z2 − 3.28z3 + 1.46z4 − 0.774z5 + 1.28z6

(a) (b)

Figure 2.17: Lyapunov function for the plunge Zero dynamics. It is clearly continuous, differentiable, null in (0, 0), positive definite

and radially unbounded

Now if we calculate ˙V (z) and use the input-output form of the zero dynamics we obtain:

z(z + Λ(z)) = γ2,2z2

Thus the system is locally asymptotically stable because γ2,2 < 0 as can bee seen from Fig.2.18.

37


Figure 2.18: Derivative of the Lyapunov function of Fig.2.18.

A possibility to show that the system is globally stable, is to use the La Salle invariant set theorem and so

use the so called sector conditions as shown in [1]. The sector conditions, given a 2nd order ODE in the form

of y + d(y) + ky = 0, are the following:

• d(y)y > 0, x > 0

• k(y)y > 0, x > 0

• d(0) = k(0) = 0

if the conditions are full filled then the system is globally stable because being the biggest invariant set for which

V = 0 the equilibrium point in (0, 0). In this case as can be seen in the following figures the condition are full

filled thus the zero dynamics are globally stable.

A second possibility to show that the system is globally stable is to use the Babashin − Krasovskii theo-

rem;indeed from what can be seen in earlier figures, the condition on radial unboundedness is respects, and

having ˙V (z) = γ2,2z2 < 0 we can conclude that the system is globally stable.

2.2.2.2 Bifurcation analysis

For an accurate stability evaluation, since the system in 2.6 is parametric, it is important to conduct a bifurcation

analysis. The first step is to calculate the equilibrium point of the system. By posing [z3 z4] = (0, 0) it is

trivial to obtain [z3 z4] = z = (0, 0). The equilibrium point can move from being stable to un stable depending

on the values of parameters U and a.

Following [1], the first step to evaluate the possibility of bifurcation for the consider system is by using the

bifurcation existence conditions as follows:

f(z, ζ) = (0, 0) (2.7)∂f

∂x(x, ζ) = (0, 0)

Where f is the vectorial function describing the system in 2.6 and z, ζ are respectively the equilibrium point

considered and the corresponding value of the parameter - or parameter vector - for which the bifurcation

occurs. Bifurcations can so occur for this system as shown by the previous calculations. Indeed the following a

numerical evaluation is shown first with respect to parameter µ = U2 and then with respect to a

38


0 100 200 300 400 500 600 700 800

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

µ

z 3

Figure 2.19: Bifurcation diagram with respect to µ and with a = −0.4

0 100 200 300 400 500 600 700 800−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

µ

z 3


39


0 100 200 300 400 500 600 700 800−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

µ

z 3


0 100 200 300 400 500 600 700 800−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

µ

z 3


It is important to notice that Fig.2.19,2.20,2.21,2.22 show a pitchfork bifurcation but it is not a symmetric

bifurcation. Indeed this sort of bifurcation is also know as an imperfect bifurcation or perturbated bifurcation as

showed in [13] and [4]. In These kind of bifurcation the normal form of the bifurcation, that in the case of a

standard pitchfork is x = µx − x3, becomes x = µx + βx2 − x3, therefor adding a quadratic term. Although

these bifurcations are perturbated it is still possible to show a critical velocity µc after which the uncontrolled

plunge dynamics equilibrium is unstable.

Bifurcation diagrams are show with respect to a, varying µ as shown in the Fig.2.23 and 2.24 :

40


−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

a

z 3

Figure 2.23: Bifurcation diagram with respect to a and with µ = 115

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0

−0.1

−0.05

0

0.05

0.1

0.15

a

z 3

Figure 2.24: Bifurcation diagram with respect to a and with µ = 115

41

Chapter 3

MRAC Minimum Controller Synthesis

control for flutter

In this chapter an MRAC adaptive controller will be applied to the plant defined in chapter 1 using the Minimum

Controller Synthesis algorithm (MCS)1. We will show briefly how the MCS algorithm works before showing the

results, and the extensions of the MCS and simulations.

3.1 MCS algorithm

The MCS strategy is based upon an extension of Landau’s MRAC approach [6] as shown in [11]. The MCS

control strategy aims to track asymptotically a reference model as defined in 3.3 trough a gain adaptation

law. One of the most important characteristics of the control strategy is that there is no assumption on the

knowledge of the plant parameters. The only assumption that is held is that the plant structure is known and

fully controllable in a canonical form like follows:

x = Ax(t) + Bu(t) (3.1)

where x(t) ∈ Rn,A(t) ∈ ℜn×n,u(t) ∈ ℜ and vector B ∈ ℜn×1 are:

A =

0 1 0 · · · 0

0 0 1 · · · 0...

......

. . ....

−a1 −a2 −a3 · · · −an

B =

0

0...

b

(3.2)

Given the plant as in 3.1, it is necessary to define a reference model as follows2:

xm = Amxm(t) + Bmr(t) (3.3)

where

Am =

0 1 0 · · · 0

0 0 1 · · · 0...

......

. . ....

−am1 −am2 −am3 · · · −amn

Bm =

0

0...

1

(3.4)

The control law used in the MCS strategy is the following:

uMCS(t) = K(t)x(t) + KR(t)r(t) (3.5)

1from now on the MCS algorithm will be referred to as if it a control strategy with a slight name abuse2note that the reference model is supposed to be stable, as so, the matrix Am is a Hurwitz matrix

43

MRAC Minimum Controller Synthesis control for flutter

where:

K(t) = α

∫ t

0

ye(τ)xT (τ)dτ + βye(t)xT (t) KR(t) = α

∫ t

0

ye(τ)r(τ)dτ + βye(t)r(t) (3.6)

K(0) = 0, KR(0) = 0, xe(t) = xm(t) − x(t)

In the equation 3.6 the terms ye and is defined as:

ye(t) = Cexe(t), Ce =[

0 0 0 . . . 1]

P

and P is the solution of the Lyapunov equation

P Am + ATmP = −Q P > 0, P = P T

With the control law defined in 3.5 the error system shown in figure 3.1, is asymptotically stable, as proven

in [1]

Figure 3.1: MCS scheme

The MCS controller can be applied to non linear plants, as it has been shown in [12] that MCS control strategy

rejects non linear disturbances3 and vanishing disturbances towards the input direction.

3.1.1 MCS extensions

Some extension of the MCS control strategy have been proposed in literature. Part of the extensions focus on

conjugating optimal control with the MCS control strategy, while others aim to modify the control law so obtain

specific proprieties of the controlled system.

3.1.1.1 MCS-LQ

In this extension of the MCS strategy the reference model is an optimally controlled linear model. The reference

model, that in applications is typically derived from the non linear model of the plant, is controlled with an

optimal LQ controller so to reach desired proprieties of optimality. This scheme, show in figure 3.2 allows to

increase the LQ robustness because the MCS strategy compensates the mismatch between plant and the LQ

optimal trajectory.

3note that in this propriety well fits with the use of the MCS strategy that is used in this report. Indeed in aeronautics often

not only parameters are unknown but non linear effects cannot be correctly modeled due to order reduction and other factors

44

3.1. MCS algorithm

Figure 3.2: LQMCS scheme

3.1.1.2 MCSI

This extension of the MCS strategy modifies the control law by inserting an integral action (MCSI). This

extension in necessary in those cases in which it is mandatory to have a null tracking error. The MCS control

law is modified as in equation 3.7:

uMCSI(t) = K(t)x(t) + KR(t)r(t) + KI(t)xI(t) (3.7)

In equation 3.7 the terms KI(t) and xI(t) are defined:

xI(t) =∫

(r − y) KI(t) = α

∫ t

0

ye(τ)xI (τ)dτ + βye(t)xI(t), KI(0) = 0 (3.8)

The stability of the MCSI strategy has been proven in [1].

3.1.1.3 EMCS

EMCS stands for Extended MCS and it is a modified control of the standard MCS law so to reject generic

disturbance.

The MCS law presented in 3.5 is modified by adding a commuting control action:

uEMCS(t) = K(t)x(t) + KR(t)r(t) + Nsgn(ye) (3.9)

where the value of N must respect the following propriety as shown in [1]:

N >1b

max{|d|} (3.10)

It is important to observe that in control law 3.9 it is necessary to have a measurement, or at least an estimation,

of the disturbance d. The main advantage of using the control law in 3.9 is so have a more robust controller.

This can be simply understood by recalling the main results of commuting controllers such a Sliding controllers.

3.1.1.4 NEMCS

Based upon 3.9, and based upon the fact the value N is set by a measurement of the disturbance amplitude,

the NEMCS introduces a gain varying law for the commuting action.

uNEMCS(t) = K(t)x(t) + KR(t)r(t) + Kn(t)sgn(ye) (3.11)

In equation 3.11 Kn(t) is defined as follows:

Kn(t) = γ

∫ t

0

ye(t) (3.12)

the proof of stability for this controller, with some interesting experimental results, is given in [7]

45


3.1.1.5 LQ-NEMCSI

The LQ-NEMCSI an extension of the LQ-MCS and NEMCS control strategy where the control laws are modified

so to have integral action and good rejection capabilities following so an optimal trajectory from the linear model.

The control law is defined as follows:

uNEMCSI(t) = K(t)x(t) + KR(t)r(t) + Kn(t)sgn(ye) + KI(t)xI(t) (3.13)

It has been show in [7] to be asymptotically stable.

3.2 MCS control synthesis and simulations

In this section MCS and some extensions of MCS control strategy are deployed to control the plant defined in

1.7. The main goal is the suppression of LCO and tracking of a given reference signal.

The system defined in 1.12 is defined with x ∈ ℜ4 and with β ∈ ℜ and does not result in the affine canonical

form. This poses a first problem in applying the MCS strategy. Indeed to control such a system a possible

solution is to add a second control surface and defining two MCS controllers each one working on a separate

affine system and rejecting the effects of the remaining part of the system. Another possible solution, marly

qualitative, is to use only on control input for allow tracking on AoA and show that the plunge dynamics is

stable. In this report two different controllers are deployed one for AoA and one for plunge dynamics.

As a first step we can rewrite the system defined in 1.12 f and g functions in so to separate clearly what is seen

as a disturbance from the controller:

fµ,xα(x) =

x3

x4

−k1x1 − (k2µ − 2.82mxαbd

)x2 − c1x3 − c2x4 + d3(x, t)

−k3x1 − (k4µ + 2.82md

)x2 − c3x3 − c4x4 + d4(x, t)

, g(x) =

0

0

g3

g4

(3.14)

where d3(x, t) and d4(x, t) as the following:

d3(x, t) = p(x2) −2.82mxαb

d, d4(x, t) =

2.82m

d+ q(x2)

The following step is to define a reference model for both the plunge and AoA dynamics. This is accomplished

by selecting Am and Bm as the following matrixes:

Am =

[

0 1

1000 −75

]

, Bm =

[

0

1

]

(3.15)

The step response of this model is shown in Fig.3.3 and has no overshooting and a settling time of approximately

0.3[seconds].

46

3.2. MCS control synthesis and simulations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.2

0.4

0.6

0.8

1

Time [s]

Am

plitu

de [d

eg] −

[m]

ReferencePosition

Figure 3.3: Reference model step response

Under the assumption of a single control surface and thus controlling only the AoA dynamics4 the closed

loop equations can be given for a MCS controller as in 3.16:

x1

x2

x3

x4

=

x3

x4

−k1x1 − (k2µ − 2.82mxαbd

)x2 − c1x3 − c2x4 + d3(x, t) + g3µβMCS

−k3x1 − (k4µ + 2.82md

)x2 − c3x3 − c4x4 + d4(x, t) + g4µβMCS

(3.16)

Similarly by changing the control law for β it is possible to write the closed loop equations for the various MCS

extensions.

If using two control surfaces by recalling system defined in 1.10 it is possible, with some manipulation, to redefine

in a SS representation the closed loop equations as in equation 3.17:

x1

x2

x3

x4

=

x3

x4

−k1x1 − (k2µ − 2.82mxαbd

)x2 − c1x3 − c2x4 + d3(x, t) + g13µβ1MCS+ g23µβ2MCS

−k3x1 − (k4µ + 2.82md

)x2 − c3x3 − c4x4 + d4(x, t) + g14µβ1MCS+ g24µβ2MCS

(3.17)

As stated previously since the system in not in canonical form two different controllers are deployed one for

controlling the plunge dynamics and one for controlling the AoA as can be seen in Fig.?? thus the two control

surface model is used. Clearly the is no necessity for the plunge dynamics to have tacking capabilities and the

plunge controller has no feed forward, and the control surface used is smaller than the one used for the AoA

dynamics thus it is just a stabilising controller. Also, in a realistic application there is no real need to have

plunge control that introduces a cost growth, due to the introduction of a seconds control surface; indeed in

the aerospace industry the plunge dynamics typically is not controlled but some action are taken so to reduce

or cancel undesired behaviours. From now on only simulations for the AoA are shown, but the reader should

keep in mind that all the simulation have on the plunge dynamical a gently tuned stabilising controller. Only

one simulation is displayed in fig.3.4a with the relative feedback gains n fig.3.4b.

4it is no difficult to imagine that this is the main goal for control, plunge can be controlled with structural stiffness or wight

distribution

47


Riferimento filtrato

Filtered_rif

Radiansto Degrees3

R2D

Radiansto Degrees2

R2D

Radiansto Degrees1

R2D

Radiansto Degrees

R2D

Plant

beta1

beta2

xMCS h

rif

x

beta1

MCS alpha

rif

x

Beta1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time [seconds]

h [m

]

(a) Controlled plunge dynamics with MCS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−30

−20

−10

0

10

20

30

Time [seconds]

Fee

dbac

k ga

ins

(b) Feedback gains for controlled plunge dynamics

3.2.0.6 Simulations

Some simulations are shown using the equations defined in the previous sections. At the beginning of this

chapter it is stated that no assumption is made on the numerical values of the plant parameters; this on one

side means that the MCS controller works as an identifier other than as a controller - as can be seen in [1], and

on the other side means that in order to allow a correct identification of the plant parameters a transient is

necessary so to allow the gains to adapt. In this report this phase will be shown by using long simulations but,

as stated in [7], a more industrial approach would be to use as parameters from linear models of the system as

a rough first estimation. This, that might seems a minor aspect for the use of the MCS control, turns out to

be a limitation. Indeed, as will be shown further, if no knowledge of the plant parameters is assumed the gain

adaptation law will start with a peak that will be reflected on the control input. This can result in unexpected

behaviours on the real plant that can lead to instability, LCO, or chaos because of un modelled dynamics. Due

to these reasons the reference signal for the AoA dynamics is periodic square wave at a frequency of 0.2[Hz].

A single period of the reference signal is shown in Fig3.4. If necessary a first order filter is used so to slow

down the reference signal with a bandwidth of approximately 10 rads

. If other reference signals are used it will

be clearly stated.

48


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Time [s]

Am

plitu

de [d

eg] −

[m]

Reference

Figure 3.4: Reference signal period

Simulations are shown on wide time range so to allow the gains to settle and there are performed on the

basic MCS, some extension, and some possible advanced use cases and problems are also addressed.

The first simulations are conducted under the assumption of a two control surfaces, only the AoA dynamics,

since a simply stabilising controller is on the plunge dynamics, as stated previously.

3.2.0.7 MCS controller

An MCS controller, as show in Fig. ?? , is so deployed using the following parameters α = 15000, β = α10 and

the results are show in the following figures:

Ce

Beta1

1

State−Space

x’ = Ax+Bu y = Cx+Du

Radiansto DegreesR2D

MatrixMultiply

MatrixMultiply

ye KN(t)*sgn(ye)

r

ye

Ki*int

r

ye

Kr

ye

x

K0

K*u

x2

rif

1

49


0 50 100 150−2

0

2

4

6

8

10

12

Time [s]

Ang

le o

f atta

ck α

[deg

]

ReferenceAngle of attack α

Figure 3.5: Reference signal and output for a MCS controller on the AoA dynamics

0 50 100 150−100

−80

−60

−40

−20

0

20

Time [s]

β [d

eg]

Figure 3.6: Control signal for a MCS controller on the AoA dynamics

In Fig.3.5 it can be seen how the controller adapts the gains after a transient. It is also important to notice

that there happens to be no null tracking error because of the lack of an integral action

50


0 50 100 150−8

−6

−4

−2

0

2

4

6

8

Time [s]

Err

or [d

eg]

Error

Figure 3.7: Tracking error for a MCS controller on the AoA dynamics

In Fig.3.7 the tracking error is shown; during the transient in which the gains are adapting the error is

consistent but gradually decreases as the controller adapts the gain. The tracking error does not go to zero

specially in the rise phase of the signal, this problem will be solved, at least in stationary conditions by the use

of a MCSI controller.

When using an adaptive control scheme, like the MCS, one of the most important things to check is the values

of the gains. This is crucial in the use of a MCS control scheme since, as for instance in the case of saturations

o persistent disturbances, the gain can assume very high values leading so the system to instability. In this case

so the feed forward and feedback gains are show in Fig.3.8 and Fig3.9.

0 50 100 150−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Time [s]

KR

Figure 3.8: Feed Forward gains for the single MCS controller on the AoA dynamics

51


0 50 100 150−5

−4

−3

−2

−1

0

1

2

Time [s]

K

Figure 3.9: Feed back gains for the single MCS controller on the AoA dynamics

In Fig.3.9 it is important to notice two details: the first regards the initial peak of the gains this often in

practical applications has to be saturated since can cause big initial control actions, the second aspect regards

the gains that do not settle at a fixed value. What really happens is that the gains do settle at a fixed value but

over a much bigger period of time as illustrated in Fig.3.10. The reason for this behaviour is to be researched int

the persistent disturbances the controller is faced to contrast due not only to the uncontrolled plunge dynamics

but also to the non linearities. As will be shown later other control actions will allow to reduce the settling time

for the gains, and their values.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000−10

−5

0

5

10

15

Time [s]

K

Figure 3.10: Feed back gains for the single MCS controller on the AoA dynamics

Before showing the results obtained with a MCSI controller a closer look to the control signal and relative

feedback gain is of interest so to show what happens on a short period of time. Indeed the gain grows briefly

for the single maneuver but the reassest on a lower value. The reason for such a behaviour is because the

MCS control action does not only have a integral law for the gain adaptation but also a faster proportional law

52


weighed by the parameter β .

77 78 79 80 81 82 83 84

−35

−30

−25

−20

−15

−10

−5

0

5

10

Time [s]

β [d

eg]

76 77 78 79 80 81 82 83 84 85−2

−1.5

−1

−0.5

0

0.5

1

Time [s]

K

Figure 3.11: Control signal and relative Feedback gain for a single maneuver

3.2.0.8 MCSI controller

An implementation of an MCSI increases the performance of the closed loop system as show in the following

figures.

0 10 20 30 40 50 60 70 80 90 100−1

0

1

2

3

4

5

6

7

8

9

Time [s]

α (x

2) [d

eg]

Figure 3.12: Reference signal and output for a single MCSI controller on the AoA dynamics

53


0 10 20 30 40 50 60 70 80 90 100−8

−6

−4

−2

0

2

4

6

8

Time [s]

Tra

ckin

g er

ror

[deg

]

Figure 3.13: Tracking error for a single MCSI controller on the AoA dynamics

Differently from what happens in Fig.3.13 the tracking error is null specially after the transients for the rise

time of the reference signal.

0 10 20 30 40 50 60 70 80 90 100−120

−100

−80

−60

−40

−20

0

20

40

Time [s]

β [d

eg]

Figure 3.14: Control signal for a single MCSI controller on the AoA dynamics

In Fig. 3.14 the control signal is shown; the major difference with the Fig.3.6 results in smaller peak values

of the control signal but a closer look shows in Fig.3.18a, also a smoother control action. The feedback and feed

forward gains for the MCSI control settle in a much shorter time with respect to what happens in the MCS

controller (a good assessment can be seen from Fig. 3.10 after more or less 3000[s] ). This is due to the integral

action show in Fig.3.17 that helps a faster stabilisation. The gains for feed forward, feedback and integral action

can be seen respectively in Fig.3.16, 3.15 and Fig.3.17.

54


5 10 15 20 25 30 35 40 45 50 55

−5

−4

−3

−2

−1

0

Time [s]

K

Figure 3.15: Feed back gain for the single MCSI controller on the AoA dynamics

0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

Time [s]

KR

Figure 3.16: Feed Forward gains for the single MCSI controller on the AoA dynamics

55


10 20 30 40 50 60 70 80 90

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time [s]

KI

Figure 3.17: Integral gain KI for the single MCSI controller on the AoA dynamics

Before showing other controller implementations a comparison is helpful to show the increase of performance

between the MCS ad MCSI control strategy as can be seen in the following figures:

56


68.5 69 69.5 70 70.5 71 71.5 72

0

1

2

3

4

5

6

7

8

Time [s]

α (x

2) [d

eg]

MCSIMCSReference

71 71.2 71.4 71.6 71.8 72 72.2 72.4 72.6

7

7.2

7.4

7.6

7.8

8

8.2

8.4

8.6

8.8

9

Time [s]

α (x

2) [d

eg]

MCSIMCSReference

69 70 71 72 73 74 75

−6

−4

−2

0

2

4

6

Time [s]

Tra

king

err

ors

MC

S V

S M

CS

I [de

g]

MCSMCSI

73.5 74 74.5 75 75.5 76 76.5 77 77.5 78 78.5

−25

−20

−15

−10

−5

0

5

10

15

20

Time [s]

Con

trol

sig

nals

[deg

]

MCSIMCS

Figure 3.18: MCS vs MCSI controller on AoA dynamics

3.2.0.9 EMCSI controller

An EMCSI controller is implemented in this section and a first comparison result is shown with respect to the

MCSI controller in the first seconds of simulation.

The dimensioning of the parameter N has been carried out with in a qualitative manner as follows. In equation

3.14, all values of variables x1 and x3 have been substituted with what is a reasonable physical value for them.

The function d4 has been evaluated a 1.5[rad] - which is a high value for the angle of attack). Then by following

the rules showed in 3.10 N results approximately 15000. Thus by leaving the parameters α and β the same

as what has been done with the MCS and MCSI simulations from Fig.3.19, it is possible to see that time to

settle good performance is much lower with respect to the MCSI. The reason for such behaviour is to seek in

the commuting action that introduced by the EMCSI controller, that as stated before make the system more

robust.

57


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

1

2

3

4

5

6

7

8

Time [s]

α (x

2) [d

eg]

MCSIReferenceEMCSI

Figure 3.19: Reference signal and output for a single EMCSI controller on the AoA dynamics

Similarly for what has been done for the MCS and MCSI controller the gains are plotted so to show that

the not go grow indefinitely.

0 50 100 150−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [s]

K

Figure 3.20: EMCSI Feedback gains for AoA dynamics with one control surface

58


0 50 100 150−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

Time [s]

KR

0 50 100 150−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Time [s]

Kn s

gn(y

e)

Figure 3.21: EMCSI gains for AoA dynamics with one control surface

The down side of this control scheme is to be searched in the commuting action introduced on the actuator

and show in Fig.3.22.

32 34 36 38 40 42 44

−30

−20

−10

0

10

20

30

Time [s]

β [d

eg]

Figure 3.22: Control signal for a single EMCSI controller on the AoA dynamics

A final simulation with respect to the MCSI controller is shown in Fig.3.23

59


78.9 79 79.1 79.2 79.3 79.4 79.5 79.6 79.7−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Time [s]

α (x

2) [d

eg]

ReferenceEMCSIMCSI

Figure 3.23: Controller tracking comparison between MCSI and EMCSI controller on the AoA dynamics after a longer simulation

3.2.0.10 NEMCSI controller

For this controller only major results are shown as a comparison with the EMCS and the values of the gains.

In Fig.3.24 it is possible to see after the gain have reached a stable value a performance increase.

101.5 102 102.5 103 103.5 104 104.5

0

1

2

3

4

5

6

7

8

Time [s]

α (x

2) [d

eg]

NEMCSIReferenceEMCSI

101.2 101.3 101.4 101.5 101.6 101.7 101.8 101.9 1027.5

8

8.5

9

Time [s]

α (x

2) [d

eg]

NEMCSIReferenceEMCSI

Figure 3.24: Output performance comparison between NEMCSI and EMCSI

The gains are reported, with exception for KN reported in Fig.3.26, are reported in Fig. 3.25.

60


10 20 30 40 50 60 70 80

−1.5

−1

−0.5

0

0.5

Feedback Gain KFeedback Gain KFeedforward gains K

R

Figure 3.25: NEMCSI controller gains on the AoA dynamics after a gain assessment, with exception of KN reported in Fig.3.26

10 20 30 40 50 60 70

−6

−4

−2

0

2

4

6

x 10−3

Time [s]

KN

Figure 3.26: NEMCSI controller KN gain on the AoA dynamics after a gain assessment

3.2.0.11 LQ-NEMCSI controller

This extension of the MCS strategy uses an LQ controlled linear model as a reference model for the plant

and using the control laws seen for he NEMCI. An LQ model is o synthesised by solving the LQ problem on

the reference model as in equation 3.3; the selection of the weight matrixes is done using the Bryson rule.

The simulation results obtained, and showed in comparison to the other MCS extensions, are show on in the

following figures.

61


100 100.5 101 101.5 102 102.5 103 103.5 104 104.5 105−1

0

1

2

3

4

5

6

7

8

9

Time [s]

α (x

2) [d

eg]

ReferenceLQNEMCSINEMCSIEMCSI

Figure 3.27: LQ-NEMCSI controller performance

10 20 30 40 50 60 70 80 90

−4

−3

−2

−1

0

1

2

3

Time [s]

Gai

ns

Feedforward gainFeedback gainFeedback gain

10 20 30 40 50 60 70 80

−10

−8

−6

−4

−2

0

2

4

6

8

10

Integral gain K

I(t)

Figure 3.28: Gain plot for LQ-NEMCSI

3.2.0.12 Gain-locking

It is important to show that the MCS control strategy is not free of problems. One of the major ones is to be

searched in the infinite value that the controller gains can reach. A very simple complication of the system allows

to show how this can severely affect the performance of the controller. A saturation can be introduced to model

the actuator limitation for high values of the angles, the behaviour is shown in the following figure. Saturation

on the control input β is introduced and as can be seen in Fig.3.29a the values of the gains grows indefinitely.

This happens because the controller reads a null tracking error that cannot compensate fully because of the

saturation o the actuators; the result of this a fast growth of the gains. To solve this issue, that is also known

as gain drifting, a gain locking criterion has to be introduced. As for shown example a possible solution is to

lock the gains after that the tracking error is below 1% and 1[s] has passed after the assigned settling time.

62


340 345 350 355 360 365 370 375−20

−15

−10

−5

0

5

10

15

20

Time [s]

β [d

eg]

340 345 350 355 360 365 370 3750

1

2

3

4

5

6

7

8

Time [s]

α (x

2) [d

eg]

Referenceα

0 50 100 150 200 250 300 350 400 450−120

−100

−80

−60

−40

−20

0

20

40

Time [s]

Gai

ns

Feedback GainFeedback GainFeedforward gain

340 345 350 355 360 365 370 375 380

−100

−50

0

50

Time [s]

Gai

ns

Feed back gainFeed back gainFeed forward gain

3.2.0.13 Velocity variation rejection and comparison

One of the main characteristics of the MCS control scheme is it’s capacity to face parameter variations if the

variation of the parameter is slow than the gain adaptation law. Two interesting case are here shown, the first

shows a 2g deceleration of the air stream and sudden re-acceleration in a steady state, the second case will show

only a 2g deceleration during a transient of a manoeuver.

The first velocity profile that is used is the shown in fig.3.29.

63


120 122 124 126 128 130 132 134 136 138 140

5

10

15

20

25

Time [seconds]

U [m

/s]

Aistream Velocity

Figure 3.29: Airstream velocity variation

The results of the simulation carried on all the controllers are shown in the following figure

128.5 129 129.5 130 130.5 131

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

ReferenceMCSINEMCSIMCSLQNEMCSI

Figure 3.30: Output for the various controllers under Fig.3.29 airstream parameter variation

The feedback gains are displayed as well for the MCS controller and the LQ-NEMCSI.

64


126 128 130 132 134 136 138 140−5

−4

−3

−2

−1

0

1

2

3

4

5

Time [seconds]

Fee

dbak

gai

ns M

CS

Figure 3.31: Feedback gain for the MCS controller under Fig.3.29 airstream parameter variation

128.5 129 129.5 130 130.5 131 131.5 132

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time [seconds]

Fee

bbac

k G

ains

LQ

−N

EM

CS

I

Figure 3.32: Feedback gain for the LQ-NEMCSI controller under Fig.3.29 airstream parameter variation

65


127 128 129 130 131 132 133 134

−4

−3

−2

−1

0

1

2

3

x 10−4

Time [seconds]

NE

MC

SI K

N(t

)

Figure 3.33: Kn(t) gain for the LQ-NEMCSI controller under Fig.3.29 airstream parameter variation

As can be seen from Fig.3.31,3.32 and 3.33, while the MCS controller can only compensate the variation

of airstream speed with a Feedback action in the LQ-NEMCSI the variation is compensated main by the

discontinuous action KN(t)sgn(ye) and the integral action (here not shown).

The seconds parameter variation wave form used is displayed in fig.3.34.

0 50 100 150

6

8

10

12

14

16

18

20

22

24

26

Time [seconds]

Air

stre

am v

eloc

ity [m

/s]

Aistream Velocity U

Figure 3.34: Airstream velocity variation

The output for the MCS and LQ-NEMCSI controller is shown in Fig.3.35

66


126 127 128 129 130 131 132 133 134

−2

−1

0

1

2

3

4

5

6

7

8

Time [seconds]

α

ReferenceMCSLQ−NEMCSI

Figure 3.35: Output for the MCS and LQ-NEMCSI controllers under Fig.3.34 airstream parameter variation

3.2.0.14 Hybrid parameter variation

A simulation that is worth to be displayed is one with a hybrid system. Although this of no physical interest

in this type of system (U or a vary with continuity), it interesting to show how the controller reacts to such

variation. To achieve a hybrid system a sudden variation (a 20 ms

velocity decrease) of U in a null time is

introduced, at t = 126.5[s] as shown in fig.3.36

0 20 40 60 80 100 120 140 160 180 200

5

10

15

20

25

Time [s]

U a

irstr

eam

vel

ocity

Figure 3.36: Airstream velocity "hybrid" variation

The output of the system is shown in 3.37. The controller does take some time to handle the variation but

the over all behaviour is good, especially considering the tracking error is below 1[degree] approximately. The

gains are also plotted for such variation. A fine tuning allows better performances.

67


126 128 130 132 134 136 138 140 142 144

0

1

2

3

4

5

6

7

8

Time [seconds]

α [d

egre

es]

Reference

α

Figure 3.37: Feedback gains for the chaos recovery simulation

120 125 130 135 140

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time [s]

KN

(t)

(a) KN (t) gain variation for the closed loop system under

going parameter variation in fig.3.36

120 125 130 135 140 145 150

−1.5

−1

−0.5

0

0.5

Time [seconds]

Fee

dbac

k ga

ins

K

(b) Feedback K gain variation for the closed loop system

under going parameter variation in fig.3.36

120 125 130 135 140 145

−60

−40

−20

0

20

40

60

Time [seconds]

Inte

gral

gai

n K

I

(c) Integral gain variation for the closed loop system un-

der going parameter variation in fig.3.36

68


3.2.0.15 Chaos recovery

As mentioned in chapter 1 we proposed to show how the MCS strategy - or an extension - can control and

recover from chaos. In Fig. 3.38, the system is left to it’s chaotic behaviour for 50 seconds an then the MCS

controller is switched on. The system is simulated with no saturations.

−4 −2 0 2 4 6 8 10−80

−60

−40

−20

0

20

40

60

80

α [deg]

d/dt

α [d

egre

es/s

ec]

Figure 3.38: Chaos recovery

The system after the chaotic evolution is to follow the reference in Fig.3.4. After a slow transient, the chaotic

vector field is removed an tracking is well achieved. A major aspect to consider in the simulation it very high

value for control signal as shown in Fig.3.39 which has to be saturated.

60 80 100 120 140 160

−200

0

200

400

600

800

1000

1200

1400

Time [seconds]

β [d

eg]

Figure 3.39: Control signal for the simulation on chaos recovery

A saturation is thus introduced and the control signal is displayed in Fig. 3.40 as well with the phase plane

in Fig.3.41.

69


0 20 40 60 80 100 120 140 160 180−30

−20

−10

0

10

20

30

Time [seconds]

β [d

egre

es]

Figure 3.40: Control signal with saturation for the simulation on chaos recovery

−4 −2 0 2 4 6 8 10 12−80

−60

−40

−20

0

20

40

60

80

α

d/dt

α

Figure 3.41: Chaos recovery

Finally the gains for the simulation with the saturation are also shown in the following figures:

70


0 20 40 60 80 100 120 140 160 180−5

0

5

10

15

20

Time [seconds]

Fee

dbac

k G

ains


0 20 40 60 80 100 120 140 160 180−250

−200

−150

−100

−50

0

Time [seconds]

Fee

dfor

war

d G

ain


71


72

References

[1] Mario Di Bernardo. Non linear dynamics and control lectures. University Federico II, Naples Italy, 2012.

[2] Bigoni D. e Noselli G. Experimental evidence of flutter and divergence instabilities induced by dry friction.

Journal of mechanics and physics of solids, 2011.

[3] Y. C. Fung. An Introduction to the Theory of Aeroelasticity. John Wiley and Sons New York, 1955.

[4] M. Golubitsky. Bifurcation theory. Ohio State University, 2011.

[5] Alberto Isisdori. Nonlinear Control systems. Springer, 1995. from page 137 to 144.

[6] I. D. Landau. Adaptive Control: The Model Reference Approach. Marcel-Dekker, 1979.

[7] Umberto Montanaro Mario di Bernardo, Alessandro di Gaeta and Stefania Santini. Synthesis and experi-

mental validation of the novel lq-nemcsi adaptive strategy on an electronic throttle valve. IEEE TRANS-

ACTIONS ON CONTROL SYSTEMS TECHNOLOGY, 2010.

[8] O’Neil and W. Strganac. An experimental investigation of nonlinear aeroelastic respons. AIAA Journal of

Aircraft, 1995.

[9] C. Gilliatt O’Neil and T. W. Strganac. Investigations of aeroelastic response for a system with continuous

structural nonlinearities. In 37th AIAA Structures, Structural Dynamics and Materials Conference, Salt

Lake City, Utah, 1996.

[10] Nicola Schiavoni Paolo Bolzern, Riccardo Scattolini. Fondamenti di controlli automatici. Mc Graw-Hill,

3rd edition, 2008.

[11] D. P. Stoten and H. Benchoubane. Empirical studies of a mrac algorithm with minimal controller synthesis.

International Journal of Control, 1990.

[12] D. P. Stoten and H. Benchoubane. Robustness of a minimal controller synthesis algorithm. International

Journal of Control, 1990.

[13] S. Strogaz. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering.

Perseus Books, 1994.

73

REFERENCES

74

List of Figures

1.1 Lift (L), resistance (R) and pitch moment (P) of a foil in a steady airstream . . . . . . . . . . . . 10

1.2 Typical Cl,Cm and Cr behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Foil subject to elastic recall K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Foil subject to elastic recall K in opposite direction after overshoot . . . . . . . . . . . . . . . . . 11

1.5 The tail of the Piper during an LCO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 The Ziegler Column 2 DoF arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Schematic representation of the 2 DoF mechanical arm. vp is the plate velocity . . . . . . . . . . 13

1.8 Experimental data and model prediction for the Ziegler Column . . . . . . . . . . . . . . . . . . 13

1.9 Configuration of the nonlinear 2-D prototypical aeroelastic wing . . . . . . . . . . . . . . . . . . 14

1.10 Wing cross-section representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.11 Wing cross-section schematic representation showing a and mid-chord b . . . . . . . . . . . . . . 15

1.12 Two trailing edge control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.13 Simulink block diagram for the system’s open loop dynamics . . . . . . . . . . . . . . . . . . . . 18

1.14 Open loop responses on AoA for the system with α = 0.1[rad], h = 0.1[m] and α = h = 0 and

U = 15, 17, 20, 19, 40, a = −0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.15 Open loop responses on plunge for the system with α = 0.1[rad], h = 0.1[m] and α = h = 0 and

U = 15, 17, 20, 19, 40, a = −0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20


a = −0.1, −0.3, −0.4, −0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


a = −0.1, −0.3, −0.4, −0.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1 SS graph for AoA partial FBL with α = 0.1[rad], h = 0.01[m] and α = h = 0 . . . . . . . . . . . 27

2.2 FBL control scheme for AoA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Phase Plane of the controlled AoA dynamics with the use of a LQR controller . . . . . . . . . . . 28

2.4 Step response of the controlled AoA dynamics with the use of a LQR controller . . . . . . . . . . 28

2.5 Time response and control input for α = z1 = x2 and α = z2 = x4 . . . . . . . . . . . . . . . . . 29

2.6 SS graph for the plunge after partial FBL on the AoA. It is clear that the variable is indirectly

influenced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Tracking graph and reference signal with null initial conditions . . . . . . . . . . . . . . . . . . . 30

2.8 Plunge position and speed with null initial conditions when the AoA subject to reference signal

as in Fig. 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.9 Tracking error for α give by building FBL control law on different parameters (UF BL, aF BL) from

the system’s one (U,a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.10 α dynamics when subject to non rejected parameter variations . . . . . . . . . . . . . . . . . . . 32

2.11 Null tracking error for α subject to parameter variation with a PD controller linear controller . . 32

2.12 Plot of the eigenvalues with respect to velocity U and a. Color pattern towards red indicates

increasing values for the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

75

LIST OF FIGURES

2.13 Phase plane for a closed loop response for partial FBL on the plunge dynamics, with the following

initial conditions h = 0.1 and α = α = h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.14 Time response for partial FBL on the plunge dynamics, with the following initial conditions

h = 0.1 and α = α = h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.15 Time response for partial FBL on the plunge dynamics, with the following initial conditions

h = 0.1 and α = α = h = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.16 Phase plane for α and α for a closed loop response for partial FBL on the plunge dynamics, with

the following initial conditions h = 0.1 and α = α = h = 0 . . . . . . . . . . . . . . . . . . . . . . 36

2.17 Lyapunov function for the plunge Zero dynamics. It is clearly continuous, differentiable, null in

(0, 0), positive definite and radially unbounded . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.18 Derivative of the Lyapunov function of Fig.2.18. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.19 Bifurcation diagram with respect to µ and with a = −0.4 . . . . . . . . . . . . . . . . . . . . . . 39




2.23 Bifurcation diagram with respect to a and with µ = 115 . . . . . . . . . . . . . . . . . . . . . . . 41

2.24 Bifurcation diagram with respect to a and with µ = 115 . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 MCS scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 LQMCS scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Reference model step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 Reference signal period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5 Reference signal and output for a MCS controller on the AoA dynamics . . . . . . . . . . . . . . 50

3.6 Control signal for a MCS controller on the AoA dynamics . . . . . . . . . . . . . . . . . . . . . . 50

3.7 Tracking error for a MCS controller on the AoA dynamics . . . . . . . . . . . . . . . . . . . . . 51

3.8 Feed Forward gains for the single MCS controller on the AoA dynamics . . . . . . . . . . . . . . 51

3.9 Feed back gains for the single MCS controller on the AoA dynamics . . . . . . . . . . . . . . . . 52

3.10 Feed back gains for the single MCS controller on the AoA dynamics . . . . . . . . . . . . . . . . 52

3.11 Control signal and relative Feedback gain for a single maneuver . . . . . . . . . . . . . . . . . . . 53

3.12 Reference signal and output for a single MCSI controller on the AoA dynamics . . . . . . . . . . 53

3.13 Tracking error for a single MCSI controller on the AoA dynamics . . . . . . . . . . . . . . . . . . 54

3.14 Control signal for a single MCSI controller on the AoA dynamics . . . . . . . . . . . . . . . . . . 54

3.15 Feed back gain for the single MCSI controller on the AoA dynamics . . . . . . . . . . . . . . . . 55

3.16 Feed Forward gains for the single MCSI controller on the AoA dynamics . . . . . . . . . . . . . . 55

3.17 Integral gain KI for the single MCSI controller on the AoA dynamics . . . . . . . . . . . . . . . 56

3.18 MCS vs MCSI controller on AoA dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.19 Reference signal and output for a single EMCSI controller on the AoA dynamics . . . . . . . . . 58

3.20 EMCSI Feedback gains for AoA dynamics with one control surface . . . . . . . . . . . . . . . . . 58

3.21 EMCSI gains for AoA dynamics with one control surface . . . . . . . . . . . . . . . . . . . . . . . 59

3.22 Control signal for a single EMCSI controller on the AoA dynamics . . . . . . . . . . . . . . . . . 59

3.23 Controller tracking comparison between MCSI and EMCSI controller on the AoA dynamics after

a longer simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.24 Output performance comparison between NEMCSI and EMCSI . . . . . . . . . . . . . . . . . . . 60

3.25 NEMCSI controller gains on the AoA dynamics after a gain assessment, with exception of KN

reported in Fig.3.26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.26 NEMCSI controller KN gain on the AoA dynamics after a gain assessment . . . . . . . . . . . . 61

3.27 LQ-NEMCSI controller performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.28 Gain plot for LQ-NEMCSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.29 Airstream velocity variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

76

LIST OF FIGURES

3.30 Output for the various controllers under Fig.3.29 airstream parameter variation . . . . . . . . . . 64

3.31 Feedback gain for the MCS controller under Fig.3.29 airstream parameter variation . . . . . . . . 65

3.32 Feedback gain for the LQ-NEMCSI controller under Fig.3.29 airstream parameter variation . . . 65

3.33 Kn(t) gain for the LQ-NEMCSI controller under Fig.3.29 airstream parameter variation . . . . . 66

3.34 Airstream velocity variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.35 Output for the MCS and LQ-NEMCSI controllers under Fig.3.34 airstream parameter variation . 67

3.36 Airstream velocity "hybrid" variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.37 Feedback gains for the chaos recovery simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.38 Chaos recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.39 Control signal for the simulation on chaos recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.40 Control signal with saturation for the simulation on chaos recovery . . . . . . . . . . . . . . . . . 70

3.41 Chaos recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70



77

List of Tables

1.1 System SS variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 System SS parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

79

professor mario di bernardo faculty of engineering, naples

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