constrained control of nonlinear systems

Constrained Control ofNonlinear Systems:

The Explicit Reference Governor and itsApplication to Unmanned Aerial Vehicles

Universite Libre de Bruxelles

&

Universita di Bologna

Doctoral Dissertation

Author:Marco M. Nicotra

Advisors:Prof. Emanuele Garone

Prof. Roberto Naldi

Preface

Dear reader,

This is my PhD thesis. There are many like it, but this one is mine. Its contents areoriginal and scientifically rigourous, but I’ve a taken few artistic licences1 in its exposition.Please forgive me if you are not amused by the occasional silliness, I just wanted to add apersonal touch by dropping some subtle (and many not-so-subtle) Easter eggs.

Originally, this dissertation was supposed to address the“Control of an Autonomous Tethered Aerial Vehicle for In-spection Applications with Tele-Operation Requirements”(CAT-AVIATOR). I even had a logo for it2. Unfortunately(for the logo), the thesis derailed towards the wonderful worldof nonlinear constrained control theory, and is now somethingcompletely different.

For more information about the actual subject of this thesis, there should be an overviewin the next few pages. The main message here is that I genuinely enjoyed the years I’vespent working (some may say that “working” is a strong word for it) on my PhD, and I havea very long list of people to thank for this fact.

First and foremost, my advisors: Emanuele and Roberto, who jokingly called me a slave,but actually treated me as a friend and colleague. Thank you for being always ready todiscuss ideas, explain new theory, correct my papers, offer advice for the future, or simplyto eat lots and lots of sushi together. Maybe it’s just a case of Stockholm syndrome, but I’llmiss being you PhD student.

Secondly, the departments: Thank you Michel and everybody at SAAS for being myfamily in Brussels. In particular, thank you Jingjing for putting up with my craziness in theoffice, Luis for being a trusted comrade throughout the whole journey, Tam for being theonly person to ever call me sempai, Julien for being Julien, Raffaele for bringing some moreItalian spirit to the lab, and everyone else for all the dinners, movie nights, random trips,jogging groups, GoT projections and soccer matches.Also, thank you Lorenzo and everybody at CASY for being my home away from homewhenever I was in Bologna. Between the people, the crescentine and the aperitivi, I reallywish I could have spent more time with you.

Finally, at least in terms of department-related people, I wish to thank all the studentswho have worked on various projects related to my dissertation. So, thank you Sophie,Hugo, Jerome, Tanguy, Corentin, Phuc, Laurent, Geoffrey, Kelly, and Bryan. I hope youlearned from me half as much as what I learned from you.

1In some cases, I mean it literally. In fact, I would like to thank AMU Reprints, Paws, and PhD Comicsfor graciously giving me the rights to use Calvin and Hobbes, Garfield, and PhD Comics strips in thisdissertation.

2Special thanks to Kaneda, author of Due Cuori e una Gatta, for making the logo. Sorry I didn’t use it.

I

II

Looking beyond ULB and UniBo, I’d like to thank Bruno for hosting me at CMU andwelcoming me with open arms. I had a wonderful time in Pittsburgh and really enjoyed thecampus life. Also, thank you Donatello for consistently organizing dinners at every confer-ence and bravely climbing to the top of Table Mountain along with me and Emanuele. Forwhat concerns this dissertation in particular, I wish to thank the members of my commissionPhilippe Bogaerts, Franco Blanchini, Lorenzo Marconi, Michel Kinnaert, Roberto Naldi, andEmanuele Garone for taking the time to read my PhD thesis in detail and providing manyhelpful suggestions on how to improve its contents.

As for the life outside of academia, I’d like to thank my friends in Brussels (you knowwho you are) who were always willing to provide a quick escape from the daily routine. Aparticular thanks goes to my flatmate Antonio for putting up with me and the cat for twofull years.

On a more personal note, I am grateful for my friends in Milan, who I’ve known sincehigh school and who have remained by my side despite my galavanting. Wherever I go, Iknow that you’ll always be there for me when I come back.

Last, but certainly not least, I’d like to thank my family for, well, everything. You havesupported and urged me forward throughout the years. I wouldn’t be who I am withoutyou.

Contents

Part I. The Explicit Reference Governor Framework 1

1 Introduction 3

1.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Related Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Part I Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Basic Principles 9

2.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 ERG Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Dynamic Safety Margin . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Attraction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.3 The Explicit Reference Governor . . . . . . . . . . . . . . . . . . . . . 15

2.4 Basic ERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Dynamic Safety Margin 23

3.1 Lyapunov-Based . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Optimal Threshold Values . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Off-line Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.3 Sub-Optimal Threshold Values . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Invariant Set Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.2 Returnable Set Approach . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.3 Trajectory-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.4 Feedforward Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.5 Combined Dynamic Safety Margins . . . . . . . . . . . . . . . . . . . . 40

3.3 Application to Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Lyapunov Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.2 Polyhedric Invariant set . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.3 Cylindric Invariant set . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Application to Euler-Lagrange Systems . . . . . . . . . . . . . . . . . . . . . 48

3.4.1 Optimal Threshold Values . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.2 Sub-Optimal Threshold Values . . . . . . . . . . . . . . . . . . . . . . 53

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

III

IV Contents

4 Attraction Field 594.1 Motion Planning Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Roadmap Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3 Navigation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Simply Connected Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4.1 Convex Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.2 Non-Convex Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Multiply Connected Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 714.5.1 Convex Domains with Spherical Holes . . . . . . . . . . . . . . . . . . 724.5.2 Non-Convex Domains with Non-Spherical Holes . . . . . . . . . . . . . 75

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Robustness 795.1 Robust Constrained Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 Robust ERG Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Parametric Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.1 Time-Varying Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 825.3.2 Constant Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.5 Other Dynamic Safety Margins . . . . . . . . . . . . . . . . . . . . . . . . . . 885.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Conclusions 936.1 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Future Research Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Part II. Application to UnmannedAerial Vehicles 97

7 Introduction 997.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.2 Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.1 Related Prizes and Publications . . . . . . . . . . . . . . . . . . . . . 1007.3 Part II Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8 Control Layer 1058.0.1 Quaternion Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

8.1 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.2 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.3 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078.4 Outer Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.4.1 Thrust Vectoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.5 Inner Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098.6 Interconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.6.1 Input to State Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108.6.2 State to Output Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1168.6.3 Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Contents V

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

9 Navigation Layer 1239.1 Navigation Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239.2 Maximum Attitude Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

9.2.1 Threshold Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.2.2 Attraction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1249.2.3 Vector ERG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

9.3 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.4 Maximum Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

9.4.1 Threshold Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279.4.2 Invariant Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.4.3 Attraction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289.4.4 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

9.5 Maximum Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299.5.1 Bounded Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299.5.2 Lyapunov Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309.5.3 Attraction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319.5.4 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9.6 Free-fall Avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329.7 Wall Avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9.7.1 Lyapunov Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339.7.2 Attraction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339.7.3 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9.8 Obstacle Avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.8.1 Threshold Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1349.8.2 Attraction Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.8.3 Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

10 Conclusions 13910.1 Future Research Opportunities . . . . . . . . . . . . . . . . . . . . . . . . . . 13910.2 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Part I

The Explicit Reference GovernorFramework

1

Chapter 1

Introduction

Part I of this dissertation introduces the general theory for the design and implementationof the Explicit Reference Governor, which is a novel framework for the constrained controlof nonlinear systems. Given a pre-stabilized system, the proposed scheme consists in manip-ulating the derivative of the applied reference so that the transient dynamics do not violatethe constraints as the system tends to the desired steady-state admissible equilibrium. Themain novelty of this framework is that it systematically provides a closed-form solution thatenforces both state and input constraints.

This chapter will provide a general overview of the existing literature on nonlinear con-strained control and identify the main contributions of this dissertation with respect to thestate of the art. The chapter will conclude with a brief description of the structure of PartI of this dissertation.

1.1 State of the Art

The constrained control of nonlinear systems is one of the main challenges faced by the con-trol system community. Proposed solutions can be classified based on the achieved tradeoffsbetween multiple and often contradicting requirements such as robustness, optimality, com-putational efficiency, reliability, implementation simplicity, and generality.

Currently, the most popular solution to the constrained control problem is the use ofModel Predictive Control (MPC). The common feature of MPC schemes is that the controlaction is computed by solving at each sampling time an online optimization problem on thepredicted state trajectories [1.1, 1.2, 1.3, 1.4]. Although MPC schemes typically outperformother solutions in terms of control performances, their applicability can be limited by theelevated computational cost. As a result, one of the main topics in MPC research is thedevelopment of increasingly efficient solvers for the optimization problem.For what concerns linear systems, efficient solutions and automatic code generators havebeen presented in [1.5, 1.6]. Nevertheless, the development of a fast and reliable NonlinearMPC is still an open problem [1.7, 1.8, 1.9].In the presence of weak nonlinearities, a typical approach consists in linearizing the systemdynamics and implementing a linear MPC. Stronger nonlinearities can instead be embeddedin a Linear Parameter Varying (LPV) model and addressed using a linear MPC that accountsfor time-varying parameters [1.10]. This solution simplifies the optimization problem, buthas the disadvantage of being more conservative to Nonlinear MPC.A possible way to avoid online optimization is to employ an Explicit MPC [1.11]. In thisscheme, the state space is partitioned into several regions and the optimal control actionis computed off-line in terms of a static feedback that depends on the current operating

3

4 1. Introduction

conditions. Although typically faster than standard MPC schemes, the main issue of theExplicit MPC is that the number regions of interest tends to scale unfavorably with thenumber of states. This makes consulting the lookup table increasingly impractical to thepoint where it may actually be easier to solve the optimization problem directly.

A diametrically opposite approach with respect to MPC schemes consists in using closed-form laws such as anti-windup strategies [1.12, 1.13, 1.14, 1.15]. In the presence of only inputconstraints, the basic idea of the anti-windup approach consists in saturating the input issuedby the control law and suitably modifying the internal states of the controller to avoid wind-up phenomena. Due to its simplicity, this approach has become a staple addition for PIDcontrol laws subject to actuator saturation. Nonlinear anti-windup strategies are usuallybased on feedback linearization [1.16], adaptive control [1.17], or passivity considerations[1.18].For what concerns state constraints, it has been shown that, in line of principle, the anti-windup can be used by remapping the state-space constraints into time- varying inputsaturations [1.19, 1.20]. Since the remapping operation requires a model inversion of thepredicted state trajectories, this solution is typically limited to linear systems.

A solution that is somewhat in-between MPC and anti-windup schemes is the use ofa Reference Governor (RG). The idea behind this approach is to augment a pre-stabilizedsystem with an add-on control unit that, whenever necessary, manipulates the applied ref-erence to ensure constraint satisfaction [1.21].The RG approach was initially introduced in continuous time and was somewhat similarto an anti-windup scheme [1.22]. The main difference was that the seminal RG schemeproposed in [1.22] saturated the error with respect to the reference instead of saturating thecontrol input. Since the saturation was performed prior to the computation of the controlinput, constraint satisfaction was guaranteed without having to modify the internal statesof the controller.The Reference Governor was soon reformulated in the discrete time domain in view of per-forming trajectory-based optimization. This led to the classical RG scheme [1.23, 1.24],where the reference to apply at time t is computed using a scalar interpolation betweenthe previously applied value and the desired reference. The resulting scalar optimizationproblem is formulated so that maintaining a constant reference is always a safe solution,thus ensuring that the optimization problem is always feasible.To improve performances, Command Governors formulate the optimization problem in thereference space instead of using a linear interpolation [1.25]. Unsurprisingly, this methodpresents higher computational requirements due to the increased number of optimizationvariables. Extended Command Governors provide an even further step in the tradeoff be-tween performances and computational requirements by expressing the applied reference asa constant value plus a time-vanishing term [1.26].In the presence of nonlinearities, the RG formulation is typically preferred due to its greatersimplicity. Nonlinear RG strategies ensure constraint satisfaction by either predicting thestate trajectories directly [1.27], or by using Lyapunov level-sets to avoid such computations[1.28, 1.29]. The former method provides higher output performances whereas the latterrequires less computational power. A unified framework for Reference Governors applied todiscrete-time nonlinear systems has been presented in [1.30].

Due to the different nature of these three approaches, the choice between Model Pre-dictive Control, Anti-windup and standard Reference Governors depends on the specificapplication. MPC is typically preferred in applications that prioritize high performancesand are able to sustain its elevated computational costs. The RG approach may instead bepreferable in applications where a reasonable loss of performances is acceptable in exchangeof lower computational costs. Finally, anti-windup schemes are particularly suited for ap-plications that require an intuitive and computationally inexpensive strategy for managing

1.2. Main Contributions 5

input saturations.

The objective of this dissertation is to complement existing solutions by proposing a newapproach that can enforce both state and input constraints without having to solve an onlineoptimization problem. This will hopefully be regarded as a viable control design method forapplications where the computational power is too limited (or the system dynamics are toofast) to solve a real-time optimization problem.

1.2 Main Contributions

The main contribution of this dissertation is the introduction the Explicit Reference Gover-nor (ERG), which is a simple and systematic scheme for the constrained control of nonlinearsystems. The ERG differs from classic Reference Governor schemes by being a continuous-time strategy that enforces constraint satisfaction by manipulating the derivative of theapplied reference instead of its actual value. Interestingly enough, this idea can be foundin one of the very first papers featuring the RG approach [1.31]. However, the very fewcontinuous-time RG schemes that followed since then have only considered direct manipu-lations of the applied reference [1.32, 1.33].By treating the derivative of the reference as the continuous-time equivalent of the one stepvariation, the ERG framework ensures constraint satisfaction without having to solve anonline optimization problem.

The contributions of Part I can be summarized into three main points:

1. Introduction of a general theoretical framework for the Explicit Reference Governor;

2. Development of suitable design methods for the ERG, depending on the nature of thesystem and of the constraints;

3. Extension of the ERG framework to address uncertainties and external disturbances.

The overall result is a novel framework for the constrained control of nonlinear systems thataims to be general from the theoretical viewpoint, viable from the design viewpoint, andcomplete enough to address the most common issues that arise in real-world applications.

1.2.1 Related Publications

The ERG framework has been the object of several contributions in the fields of nonlinearand constrained control theory. Preliminary versions of the ERG strategy can be found inpublications [1.34, 1.35], which introduce the general idea behind the scheme. Additionalpublications discuss the specialization of the ERG to the case of Euler-Lagrange systems[1.36], as well as its extension to cope with robustness issues [1.37]. These contributionshave been incorporated directly in the text of this dissertation.

For the sake of coherence, publications that are related to the ERG but do not featurethe ERG directly have been omitted from this dissertation. A systematic tool for buildingLyapunov and ISS-Lyapunov functions or a large and highly relevant class of second-ordersystems is presented in [1.38]. This result aids the design and analysis of the pre-stabilizingfeedback loop.An alternative algorithm for implementing the classic discrete-time scalar Reference Gover-nor is provided in [1.39] and has been used for some comparisons with the ERG.

6 1. Introduction

1.3 Part I Overview

Part I of this dissertation is structured as follows.

Chapter 2 will introduce the ERG framework. The chapter starts by defining a generalconstrained control problem and reformulating it into a reference management problem.The Explicit Reference Governor will then be introduced as the combination of two sep-arate terms: the dynamic safety margin, which limits the transient dynamics of thepre-stabilized system, and the attraction field, which generates a steady-state path ofadmissible equilibrium points.

Chapter 3 will provide several design methods for computing suitable dynamic safetymargins. The advantages and disadvantages of each approach will be illustrated with the aidof a recurring nonlinear example. The last part of the chapter will focus on the computationof dynamic safety margins for two highly relevant classes of systems: linear and Euler-Lagrange systems.

Chapter 4 will present a systematic framework for constructing suitable attractionfields. This will be done by briefly reviewing the existing literature and illustrating howexisting methods can be implemented within the ERG framework with only minor modifi-cations.

Chapter 5 will extend the ERG framework presented in the previous chapters to ensurerobustness with respect to model uncertainties and external disturbances.

Chapter 6 will conclude Part I of this dissertation by summarizing the main results,providing an experimental validation and presenting ideas for future research directions forthe ERG framework.

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[1.7] E. F. Camacho and C. Bordons, Assessment and Future Directions of Nonlinear ModelPredictive Control. Springer, 2007, ch. Nonlinear Model Predictive Control: An Intro-ductory Review, pp. 1–16.

[1.8] L. T. Biegler, Nonlinear Model Predictive Control. Springer Basel AG, 2012, ch.Efficient Solution of Dynamic Optimization and NMPC Problems, pp. 219–244.

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[1.9] G. C. Goodwin, M. E. C. Garrido, M. Seron, D. Ferris, R. Middleton, and B. Campos,“Opportunities and challenges in the application of nonlinear MPC to industrial prob-lems,” in IFAC Nonlinear Model Predictive Control Conference (IFAC Proceedings),vol. 45, no. 17, 2012, pp. 39–49.

[1.10] A. Casavola, D. Famularo, and G. Franze, “Predictive control of constrained nonlinearsystems via LPV linear embeddings,” International Journal of Robust and NonlinearControl, vol. 13, no. 3-4, pp. 281–294, 2003.

[1.11] A. Alessio and A. Bemporad, Nonlinear Model Predictive Control. Springer, 2009,ch. A Survey on Explicit Model Predictive Control, pp. 345–369.

[1.12] R. Hanus, M. Kinnaert, and J. L. Henrotte, “Conditioning technique, a general anti-windup and bumpless transfer method,” Automatica, vol. 23, no. 6, pp. 729–739, 1987.

[1.13] M. V. Kothare, P. J. Campo, M. Morari, and C. N. Nett, “A unified framework forthe study of anti-windup designs,” Automatica, vol. 30, no. 12, pp. 1869–1883, 1994.

[1.14] L. Zaccariana and A. R. Teel, “A common framework for anti-windup, bumplesstransfer and reliable designs,” Automatica, vol. 38, no. 10, pp. 1735–1744, 2002.

[1.15] L. Zaccarian and A. R. Teel, Modern Anti-windup Synthesis: Control Augmentationfor Actuator Saturation. Princeton University Press, 2011.

[1.16] G. Herrmann, P. P. Menon, M. Turner, D. G. Bates, and I. Postlethwaite, “Anti-windup synthesis for nonlinear dynamic inversion control schemes,” International Jour-nal of Robust and Nonlinear Control, vol. 20, no. 13, pp. 1465–1482, 2010.

[1.17] Q. Hu and G. P. Rangaiah, “Anti-windup schemes for uncertain nonlinear systems,”in IEE Proceedings of Control Theory Applications, 2000, pp. 321–329.

[1.18] F. Morabito, A. R. Teel, and L. Zaccarian, “Nonlinear antiwindup applied to euler-lagrange systems,” IEEE Transactions on Robotics and Automation, vol. 20, no. 3, pp.526–537, 2004.

[1.19] E. G. Gilbert and K. T. Tan, “Linear systems with state and control constraints:the theory and application of maximal output admissible sets,” IEEE Transactions onAutomatic Control, vol. 36, no. 9, pp. 1008–1020, 1991.

[1.20] O. J. Rojas and G. C. Goodwin, “A simple anti-windup strategy for state constrainedlinear control,” in IFAC World Congress (IFAC Proceeding), vol. 35, no. 1, 2002, pp.109–114.

[1.21] I. V. Kolmanovsky, E. Garone, and S. D. Cairano, “Reference and command gover-nors: A tutorial on their theory and automotive applications,” in Proceedings of theAmerican Control Conference (ACC), 2014, pp. 226–241.

[1.22] P. Kapasouris, M. Athans, and G. Stein, “Design of feedback control systems forstable plants with saturating actuators,” in Proc. of IEEE Conference on Decision andControl (CDC), 1988, pp. 469–479.

[1.23] E. G. Gilbert, I. V. Kolmanovsky, and K. T. Tan, “Discrete-time reference governorsand the nonlinear control of systems with state and control constraints,” InternationalJournal of Robust and Nonlinear Control, vol. 5, no. 5, pp. 487–504, 1995.

[1.24] E. G. Gilbert and I. V. Kolmanovsky, “Fast reference governors for systems with stateand control constraints and disturbance inputs,” International Journal of Robust andNonlinear Control, vol. 9, no. 15, pp. 1117–1141, 1999.

8 1. Introduction

[1.25] A. Bemporad, A. Casavola, and E. Mosca, “Nonlinear control of constrained linearsystems via predictive reference management,” IEEE Transactions on Automatic Con-trol, vol. 42, no. 3, pp. 340–349, 1997.

[1.26] E. G. Gilbert and C.-J. Ong, “Constrained linear systems with hard constraints anddisturbances: An extended command governor with large domain of attraction,” Auto-matica, vol. 47, no. 2, pp. 334–340, 2011.

[1.27] A. Bemporad, “Reference governor for constrained nonlinear systems,” IEEE Trans-actions on Automatic Control, vol. 43, no. 3, pp. 415–419, 1998.

[1.28] E. G. Gilbert and I. V. Kolmanosky, “Set-point control of nonlinear systems with stateand control constraints: A lyapunov-function, reference governor approach,” in Proc.of the IEEE Conference on Decision and Control (CDC), vol. 3, 1999, pp. 2507–2512.

[1.29] R. H. Miller, I. V. Kolmanovsky, E. G. Gilbert, and P. D. Washabaugh, “Control ofconstrained nonlinear systems: A case study,” IEEE Control Systems Magazine, vol. 20,no. 1, pp. 23–32, 2000.

[1.30] E. G. Gilbert and I. V. Kolmanovsky, “Nonlinear tracking control in the presence ofstate and control constraints: a generalized reference governor,” Automatica, vol. 38,no. 12, pp. 2063–2073, 2002.

[1.31] P. Kapasouris, M. Athans, and G. Stein, “Design of feedback control systems forunstable plants with saturating actuators,” in Proc. IFAC Symposium on NonlinearControl System Design, 1990, pp. 3404–3409.

[1.32] A. A. Rodriguez and Y. Wang, “Performance enhancement methods for unstablebank-to-turn (btt) missiles with saturating actuators,” International Journal of Control,vol. 63, no. 4, pp. 641–678, 1996.

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Chapter 2

Basic Principles

The objective of this chapter is to introduce a general Explicit Reference Governor (ERG)framework for the constrained control of nonlinear systems. The proposed approach is basedon the Reference Governor philosophy, meaning that constraint enforcement is achieved bypre-stabilizing the nonlinear system without taking into account the constraints and usingan add-on control unit to manipulate the applied reference in a way that ensures constraintsatisfaction.Unlike existing Reference Governor schemes, however, the ERG manipulates the derivativeof the applied reference instead of the value itself. This enables the formulation of a closed-form strategy that is effective and simple to implement.

2.1 Problem Statement

In its most general formulation, the ERG framework addresses the following control problem.

Problem 2.1. Constrained Control of a Nonlinear System: Let

x = φ(x, u), (2.1)

with x ∈ Rn and u ∈ Rm, be a nonlinear system subject to constraints

γ(x, u) ≥ 0. (2.2)

Let r ∈ Rp denote a desired reference. Determine a control law u = g(r, x) such that thefollowing conditions hold true for the largest possible set of initial conditions x(0):

1. For any piecewise-continuous signal r(t) ∈ Rp, constraints are always satisfied, i.e.γ(x(t), u(t)) ≥ 0, ∀t ≥ 0;

2. For any constant r such that γ(xr, g(r, xr)) ≥ 0, the equilibrium point1 xr ∈ Rn isAsymptotically Stable. �

Due to its generality, Problem 2.1 is quite challenging to address directly. The followingsection proposes a two-step approach aimed at simplifying the control task.

1In the literature, equilibrium points are typically denoted using the classic notation x. However, in thisdissertation it is very important to consider that the equilibrium of a closed-loop system is determined bythe reference. As such, the classic notation has been extended with a subscript that indicates the reference.

9

10 2. Basic Principles

𝑟

𝑥 System Primary

Control

ERG 𝑥 𝑢

Pre-stabilized System

𝑣

𝑥

Figure 2.1: Explicit Reference Governor and pre-stabilizing control law

2.2 Proposed Approach

In analogy to what is done in the classical Reference Governor approach, the frameworkpresented in this dissertation solves Problem 2.1 by decomposing it into two control tasksthat can be addressed separately. The first task (Problem 2.2) consists in designing aprimary control law that pre-stabilizes the system without explicitly taking into accountthe constraints. The second task (Problem 2.3) consists in designing an auxiliary controllaw that provides constraint handling capabilities to the pre-stabilized system by using theapplied reference v ∈ Rp as an input. The overall architecture of the control scheme isillustrated in Figure 2.1. The following problem states the objectives of the primary controllaw.

Problem 2.2. Pre-Stabilization: Given the nonlinear system (2.1) and an auxiliary ref-erence v ∈ Rp, find a control law

u = g(x, v), (2.3)

such that the closed loop systemx = φ(x, g(x, v)), (2.4)

is Lipschitz continuous and satisfies the following:

1. For any constant auxiliary reference v, the equilibrium point x = xv is AsymptoticallyStable, i.e. there exists an attraction basin Xv ⊆ Rn such that ∀x(0) ∈ Xv, lim

t→∞x(t) =

xv. �

Taking into account the pre-stabilizing control law (2.3), the system and the constraintsin Problem 2.1 can be mapped into

f(x, v) = φ(x, g(x, v)), c(x, v) =

[γ(x, g(x, v))γ(x, v)

],

where γ : Rn × Rp → R is an additional constraint such that γ(x, v) ≥ 0 implies x ∈ Xv.The objectives of the auxiliary control loop can then be stated as follows.

Problem 2.3. Constraint Enforcement: Given the pre-stabilized nonlinear system

x = f(x, v), (2.5)

subject to constraintsc(x, v) ≥ 0, (2.6)

and given x(0), v(0) satisfying c(x(0), v(0)) ≥ 0, find an auxiliary reference signal v(t) suchthat:

1. For any piecewise continuous signal r(t) ∈ Rp, constraints are always satisfied, i.e.c(x(t), v(t)) ≥ 0, ∀t ≥ 0;

2.3. ERG Formulation 11

2. For any constant reference r satisfying c(xr, r) ≥ 0, the applied reference v asymptot-ically tends to r. �

By combining the solutions to Problem 2.2and Problem 2.3, it can be shown that the Prob-lem 2.1 can be solved for all initial conditionssuch that c(x(0), v(0)) ≥ 0. This set of initialconditions is appropriate for applications thatdo not require aggressive maneuvers. This slightloss in generality is compensated by the fact thatsolving Problems 2.2 and 2.3 separately is mucheasier than solving Problem 2.1.

For what concerns Problem 2.2, the stabiliza-tion of unconstrained nonlinear systems is thesubject of an extensive literature (e.g. [2.1]-[2.2])and can be approached using a variety of avail-able control tools. For this reason, the designof a primary control unit that pre-stabilizes thesystem dynamics will not be addressed in thisdissertation.

The main focus of this dissertation is the in-troduction of a novel add-on control unit for

solving Problem 2.3 in closed-form. The formulation of this control unit, denoted here-after as the Explicit Reference Governor, is addressed in the following section.

Remark 2.1. Problem 2.3 assumes that r is a steady-state admissible reference. To lift thisassumption, Point 2. can be substituted with

2. For any constant reference r ∈ Rp, the applied reference v asymptotically tends to r?

satisfyingmin ‖r − r?‖s.t. c(xr? , r

?) ≥ δ, (2.7)

with δ > 0.

Following from equation (2.7), r? will be referred to as the Steady-State Admissible Pro-jection of r. Moreover, the scalar δ > 0 will be referred to as the Static Safety Marginfor reasons which will become apparent in Chapter 5. �

2.3 ERG Formulation

As stated in the previous chapter, existing Reference Governor schemes typically solve Prob-lem 2.3 by formulating an optimization problem in the discrete-time domain and assigningthe resulting value v(t). The main novelty of the Explicit Reference Governor is to show thatthe same objectives can be achieved analytically by taking advantage of certain propertiesof the continuous-time domain.

The ERG is based on two fundamental components: the dynamic safety margin andthe attraction field. Each component will be be defined rigourously in Subsections 2.3.1and 2.3.2. For the moment, the following intuitions are provided.

The dynamic safety margin ∆(x, v) represents a distance between the constraints andthe system trajectory that would emanate from the state x given a constant reference v.In other words, ∆ ≥ 0 implies that if the current reference v is maintained constant, the


𝑟

𝑥 Pre-stabilized

System

𝑣

𝑣 𝜌(𝑣, 𝑟 )

Δ(𝑥, 𝑣 )

×

Figure 2.2: ERG Scheme

system will not violate the constraints at anytime in the future. Additionally, ∆ > 0 impliesthat the current reference v can be perturbed without causing a violation of constraints.Since larger values of ∆ will admit larger perturbations of the current v, the dynamic safetymargin can be interpreted as the answer to the question “How safe is it to change the appliedreference?”

The attraction filed ρ(v, r) represents a direction along the path that leads from thecurrent reference v to the desired reference r without exiting the constraints. In other words,the attraction field can be interpreted as the answer to the question “What direction shouldthe applied reference follow?”

Based on the information provided by the dynamic safety margin and the attractionfield, the ERG framework solves Problem 2.3 by assigning

v = ∆(x, v) ρ(r, v). (2.8)

The schematic representation of the ERG is illustrated in Figure 2.2. Intuitively, equation(2.8) implies that ∆(x, v) regulates the modulus of v (i.e. “How fast the reference can move”)and ρ(v, r) determines the direction of v (i.e. “Where the reference will go”). Rigourousproof will be provided in Subsection 2.3.3.

2.3.1 Dynamic Safety Margin

This subsection provides an analytic definition of the term ∆(x, v) in equation (2.8). This isdone by identifying the sufficient conditions that ensure a correct behaviour of the ExplicitReference Governor. Suitable dynamic safety margins will be presented in Chapter 3.

Definition 2.1. Dynamic Safety Margin: Let the pre-compensated system (2.5) be sub-ject to constraints (2.6), and let x(t|x, v) denote the solution to{

˙x(t) = f(x(t), v)x(0) = x.

(2.9)

Given an applied reference v satisfying

c(xv, v) > 0, (2.10)

a continuous function ∆ : Rp × Rn → R is a “Dynamic Safety Margin” if the followingproperties hold true:

• ∆(x, v) > 0 ⇒ c(x(t|x, v), v) > 0, ∀t ≥ 0;


• ∆(x, v) ≥ 0 ⇒ c(x(t|x, v), v) ≥ 0, ∀t ≥ 0;

• ∆(x, v) = 0 ⇒ ∆(x(t|x, v), v) ≥ 0, ∀t ≥ 0;

• ∀δ > 0, ∃ε > 0 such that c(xv, v) ≥ δ ⇒ ∆(xv, v) ≥ ε. �

The first three properties of Definition 2.1 pertain to the intuition that, given a constantreference v, ∆(x, v) is a distance between the trajectory x(t) and the constraints (2.6).This is somewhat the continuous-time counterpart of the safety concept mentioned in thediscrete-time general RG framework [2.3].The final property states that if v is strictly contained in the constraints, the steady-statevalue ∆(xv, v) is strictly positive. In turn, this ensures that the system will eventually entera configuration where it is safe to change the reference. This can be linked to the strongreturnability concept mentioned in the discrete-time general RG framework [2.3].

The following Lemma illustrates how the safety requirements of the dynamic safetymargin are able to guarantee the non-violation of constraints. The interest in the strongreturnability-like requirement will be clarified in Subsection 2.3.3.

Lemma 2.1. Let ∆(x, v) be a dynamic safety margin for the pre-compensated system (2.5)subject to constraints (2.6). Given x(0), v(0) satisfying

∆(xv(0), v(0)) > 0,∆(x(0), v(0)) ≥ 0,

(2.11)

if the auxiliary reference v(t) is manipulated so that

v = ∆(x, v)ρ(v, t) (2.12)

with ‖ρ(v, t)‖ bounded ∀t ≥ 0, the trajectories v(t), x(t) will satisfy constraints (2.6) for allt ≥ 0. �

Proof. Since ‖v‖ is limited, its integral v(t) exists and is continuous [2.4]. Likewise, sincesystem (2.5) is Lipschitz, the signal x(t) is also continuous. Since the function ∆(x, v) iscontinuous and bounded by definition, the signal ∆(x(t), v(t)) is continuous in time2. Toconclude the proof, the cases ∆(t) > 0 and ∆(t) ≤ 0 will be addressed separately.

Given equation (2.9), the initial conditions of x(t|x, v) are such that

c(x(t), v(t)) = c (x(t|x(t), v(t)), v(t)) .

As a result, if follows directly from Definition 2.1 that ∆(t) > 0 implies c(x(t), v(t)) > 0.

Given ∆(0) ≥ 0, it follows from continuity that ∆(t) ≤ 0 can only be obtained for t ≥ t?,with t? satisfying ∆(t?) = 0. Following from (2.12), ∆(t?) = 0 implies v(t?) = 0. Due toequation (2.9), this ensures

c(x(t), v(t)) = c (x(t|x(t?), v(t?)), v(t?)) , ∀t ∈ [t?, t? + τ ],

where τ ≥ 0 is such that ∆(t) = 0, ∀t ∈ [t?, t? + τ ]. As a result, it follows from Definition2.1 that c(x(t), v(t)) ≥ 0 , ∀t ∈ [t?, t? + τ ]. The proof is concluded by noting that, as longas v = 0, 6 ∃τ : ∆(t? + τ) < 0. This ensures ∆(t) ≥ 0, ∀t. �

2For the sake of notational simplicity, in this proof ∆(x(t), v(t)) will be referred to as ∆(t).


2.3.2 Attraction Field

This subsection provides an analytic definition of the term ρ(v, r) in equation (2.8). This isdone by identifying the sufficient conditions that ensure a correct behaviour of the ExplicitReference Governor. Suitable attraction fields will be presented in Chapter 4.

Definition 2.2. Attraction Field: Given constraints (2.6), a piece-wise continuous func-tion ρ : Rp×Rp → Rp is an “Attraction Field” with margin δ > 0 if, for any initial conditionv(0) such that c(xv(0), v(0)) ≥ δ, the system

v = ρ(r, v) (2.13)

is such that:

• ‖ρ(v, r)‖ is bounded in any bounded domain;

• For any piecewise-continuous reference r such that c(xr, r) ≥ δ, the applied referenceis steady-state admissible, i.e. c(xv(t), v(t)) ≥ δ, ∀t ≥ 0;

• Any constant reference r such that c(xr, r) ≥ δ is Asymptotically Stable.

Moreover, an attraction field is said to be Projective if system (2.13) satisfies the furtherproperties

• For any piecewise-continuous reference r ∈ Rp, the applied reference is steady-stateadmissible, i.e. c(xv(t), v(t)) ≥ δ, ∀t ≥ 0;

• For any constant reference r ∈ Rp, its steady-state admissible projection r? satisfying(2.7) is Asymptotically Stable. �

Definition 2.2 basically states that ρ(r, v) is a vector field that generates a steady-stateadmissible trajectory that connects the currently applied reference v to the desired referencer, or its steady-state admissible projection r?.

The following Lemma states the behaviour of an attraction field when it is multiplied bya non-negative term.

Lemma 2.2. Let ρ(r, v) be an attraction field. Given a steady-state admissible reference rand given a continuous bounded signal ∆(t) ≥ 0, ∀t, satisfying

limt→∞

∫ t

0

∆(τ)dτ =∞, (2.14)

then

v = ∆(t)ρ(r, v) (2.15)

is such that the equilibrium point v = r is asymptotically stable. �

Proof. Consider the auxiliary time scale s =∫ t

0∆(τ)dτ .

Given the composite function v(s(t)), its time derivative is v = dvds ·

dsdt . Since ds

dt = ∆(t),

it follows from equation (2.15) that dvds = ρ(r, v). As a result, it follows from Definition 2.2

that there exists a Lyapunov function W (v, r) that proves asymptotic stability of v = r inthe auxiliary time scale s.

The statement can therefore be proven by taking into account the dependency s =s(t) and showing that W (v, r) is also a Lyapunov function in the original time-scale t. Inparticular, the following must hold true:


1. dWds ≤ 0 implies3 dW

dt ≤ 0;

2. limt→∞

W (v(s(t), r) = lims→∞

W (v(s), r).

Point 1 ensures that W (v, r) is a non-increasing function in both time-scales. This conditionfollows directly from dW

dt = dWds

dsdt , since ds

dt = ∆(t) ≥ 0.Point 2 ensures that W (v, r) tends to the same asymptote in both time-scales. This conditionfollows directly from requirement (2.14) since t→∞ implies s(t)→∞. �

2.3.3 The Explicit Reference Governor

The objective of this subsection is to illustrate how the combination of the dynamic safetymargin and the attraction field simultaneously ensures constraint satisfaction and asymp-totic convergence. In particular, the following theorem combines Lemmas 2.1 and 2.2 toshow how the ERG is able to systematically solve Problem 2.3. This result is the corecontribution of the Explicit Reference Governor framework.

Theorem 2.1. Consider a pre-compensated system (2.5) subject to constraints (2.6) andlet ∆(x, v), ρ(r, v) be a dynamic safety margin and an attraction field as per Definitions 2.1and 2.2, respectively. Moreover, given an initial condition x(0), let the applied reference vat time t = 0 be such that (2.11) is satisfied. Then, the Explicit Reference Governor law(2.8), i.e.

v = ∆(x, v) ρ(r, v),

is such that:

1. for any reference signal r(t), constraints (2.6) are never violated;

2. for any constant reference r, then:

• if c(xr, r) ≥ δ, v(t) will asymptotically tend to r;

• if c(xr, r) < δ and ρ(r, v) is an projective attraction field, v(t) will asymptoticallytend to r? satisfying (2.7). �

Proof. Point 1 follows directly from Lemma 2.1 since ‖ρ(r, v)‖ is bounded.For what concerns Point 2, it follows from Lemma 2.2 that (2.8) will asymptotically convergeto the same equilibrium as (2.13) if ∆(v(t), x(t)) satisfies (2.14). To prove this condition,two cases are considered:

• if c(xr, r) ≥ δ, it follows from Definition 2.2 that v(t) is such that c(xv, v) ≥ δ.Following from Definition 2.1, the dynamic safety margin is such that ∆(t) = 0 canonly hold true for a finite time since limt→∞∆(t) ≥ ε. As a result, the time integralof ∆(t) will necessarily tend to infinity. Following from Lemma 2.2, this implies thatv(t) will asymptotically tend to r.

• if c(xr, r) < δ and ρ(r, v) is a projective attraction field, the same arguments used forthe previous point can be applied to show that v(t) will asymptotically tend to r?.

This concludes the proof. �

Remark 2.2. Please note that Theorem 2.1 assumes that, given a suitable initial statex(0), it is possible to compute an initial reference v(0) such that c(x(0), v(0)) > 0. Thisrequirement is typical in the Reference Governor literature [2.3]. A practical way to do soin many realistic scenarios is to assign v(0) so that xv(0) is sufficiently close to x(0). �

3Please note that if ρ(v, r) is a piecewise continuous function, the Lyapunov function is not smooth andits time derivative should be interpreted in the sense of Clarke’s generalized gradient. For more information,the reader is referred to [2.5].


𝑟

𝑥 System Primary

Control

ERG 𝑥 𝑢


��𝑣

𝑣

𝑥

{𝑥: 𝑉(𝑥, 𝑣) ≤ Γ(𝑣)}

{𝑥: 𝑐(𝑥, 𝑣) ≤ 0}

��𝑣

{𝑥: 𝑉(𝑥, 𝑣) ≤ Γ(𝑣)} ∪ {𝑥: 𝑐(𝑥, 𝑣) ≤ 0}

{𝑥: 𝑉(𝑥, 𝑣) = Γ(𝑣)}

𝑥𝑡

{𝑥: 𝑉(𝑥, 𝑣) = 𝑉(𝑥𝑡, 𝑣)}

Figure 2.3: Geometric interpretation of the threshold value Γ(v).

Remark 2.3. In the case where c(xr, r) < δ and ρ(r, v) is not a projective attraction field,it can be shown that v(t) will converge to the point where the trajectory generated by thevector field ρ(r, v) intersects the condition ∆(xv, v) = 0. This event is undesirable becauseit can cause the ERG to stagnate due to a possible violation of requirement (2.14). �

Theorem 2.1 represents the cornerstone of the Explicit Reference Governor approach.Indeed, given any nonlinear system subject to constraints, asymptotic convergence andconstraint satisfaction can be achieved by:

1. Pre-stabilizing the unconstrained system;

2. Determining a suitable dynamic safety margin ∆(x, v);

3. Designing a suitable attraction field ρ(r, v).

In view of making the ERG a systematic tool for the constrained control of nonlinear systems,the main challenge that remains to be addressed is how to systematically design ∆(x, v) andρ(r, v) based on the nature of the system and its constraints. This issue will be studied ingreater detail in the following chapters. To provide an intuitive insight of how the methodworks, the following section will present the most basic version of the ERG scheme.

2.4 Basic ERG

The Explicit Reference Governor was originally introduced in [2.6, 2.7] and was based onthe idea of using Lyapunov level-sets to avoid the need of explicitly predicting x(t|x, v).Indeed, given a pre-stabilized system subject to a constant reference v, any continuouslydifferentiable function that satisfies the Lyapunov asymptotic stability conditions4

V (xv, v) = 0,V (x, v) > 0, ∀x ∈ Xv,V (x, v) < 0, ∀x ∈ Xv,

defines a time-contractive set that contains x(t|x, v). As a result, given a threshold valueΓ(v) continuous in v satisfying

V (x, v) ≤ Γ(v) ⇒ c(x, v) ≥ 0, (2.16)

4Please note that this method can also be used for “weak” Lyapunov functions, meaning V (x, v) thatsatisfy V (x, v) ≤ 0 coupled with the Krasovskii-LaSalle conditions.

2.4. Basic ERG 17

it follows that Γ(v)− V (x, v) ≥ 0 implies c(x(t|x, v), v) ≥ 0, ∀t. This property is illustratedin Figure 2.3 and is also true in the case of strict inequalities, i.e. Γ(v)−V (x, v) > 0 impliesc(x(t|x, v), v) > 0, ∀t. Since Γ(v) can always be computed so that

c(xv, v) ≥ δ ⇒ Γ(v) ≥ ε, (2.17)

the original ERG was proposed using the dynamic safety margin

∆(x, v) = κ(Γ(v)− V (x, v)), (2.18)

where κ > 0 is an arbitrary scalar.

Remark 2.4. Please note that, in theory, any threshold value Γ(v) satisfying (2.16)-(2.17)is acceptable. In practice, however, it is preferable to assign Γ(v) as the largest Lyapunovlevel-set {x : V (x, v) ≤ Γ(v)} that does not violate constraints (2.6). Intuitively, this is dueto the fact that the ERG will be less conservative as the size of the set {x : V (x, v) ≤ Γ(v)}increases. �

For what regards the attraction field, the most intuitive choice is

ρ(r, v) =r − v‖r − v‖

,

which is a unitary vector that points from the auxiliary reference v to the desired referencer. Given a convex set of constraints c(xv, v) ≥ 0, this strategy is guaranteed to converge toany steady-state admissible reference r. To avoid numerical issues, it is usually preferableto introduce a smoothing radius η > 0 below which the modulus of ρ(v, r) gradually tendsto zero. This leads to the following formulation

ρ(r, v) =r − v

max{‖r − v‖ , η}, (2.19)

which clearly conserves the previous convergence properties. As a result, equation (2.19) sat-isfies all the basic requirements of Definition 2.2 for any convex set of constraints c(xv, v) ≥ 0.

As a result, the Explicit Reference Governor (2.8) can be implemented using the dynamicsafety margin (2.18) and the attraction field (2.19). The following example illustrates howthe ERG framework can be used to control a nonlinear system subject to constraints. Thisexample will be reprised and expanded in the following Chapters.

Example 2.1: Aircraft Longitudinal Dynamics

Figure 2.4: Longitudinal representation of a civil aircraft


Modeling: Consider the aircraft represented in Figure 2.4. As detailed in [2.8], itslongitudinal dynamics are

Iα = −d1L(α) cosα− µα+ d2u cosα, (2.20)

where d1, d2 are two positive scalars, µ > 0 is an estimated viscous friction coefficientand L(α) is the lift generated by the main wing. Given a certain altitude and windspeed, this force can be tabulated depending on the selected airfoil a. As illustratedin Figure 2.5, L(α) is characterized by a stall angle αS such that L(α) ≤ L(αS), ∀α.To avoid the stall of the main wing, it is necessary to impose the constraint

αS − α ≥ 0. (2.21)

0 5 10 15 200

2

4

6

8

10

12

14

16

18x 10

5

α [deg]

L [N

]

Figure 2.5: Lift generated by the main wing. The stall angle is αS = 14, 7◦.

Step 1, Pre-Stabilization: Given an auxiliary reference v, system (2.20) can bepre-stabilized using a PD control law with desired set-point compensation

u = −kP (α− v)− kDα+d1

d2L(v), (2.22)

where kP , kD > 0. The resulting closed-loop system is

Iα = −d1(L(α)− L(v)) cosα− d2kP (α− v) cosα− d2kD cosαα− µα. (2.23)

Asymptotic stability of [α, α] = [v, 0] can then be proven using the Variable GradientMethod [2.10] to construct the following Lyapunov function

V (α, α, v) =

∫ α

v

(d1(L(β)− L(v)) + d2kP (β − v)) cosβdβ +1

2Iα2. (2.24)

Indeed, (2.24) is positive-definite ∀α ≤ αS and its derivative

V (α, α, v) = −(d2kD cosα+ µ)α2

is negative semi-definite ∀α ≤ αS and satisfies the Krasovskii-LaSalle criterion.

2.4. Basic ERG 19

Step 2, Constraint Enforcement: Given the pre-stabilized system (2.23), con-straint (2.21) can be enforced by using the Explicit Reference Governor framework.Following from equations (2.8), (2.18) and (2.19), a suitable choice is

v = κ(Γ(v)− V (v, α, α))r − v

max{‖r − v‖ , η}, (2.25)

where Γ(v) is the solution to the optimization problem (2.16). Given the Lyapunovfunction (2.24) and the constraint (2.21), it will be proven in Section 3.4 that thethreshold value is

Γ(v) = V (v, αS , 0). (2.26)

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20No Reference Governor

t [s]

atta

ck a

ngle

[deg

]

rα(t)

α ≤ αS

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20Explicit Reference Governor

t [s]

atta

ck a

ngle

[deg

]

rv(t)α(t)α ≤ α

S

Figure 2.6: Step response with and without ERG

Obtained Results: The proposed scheme was validated numerically using the pa-rameters I = 4.5 106 kgm2, d1 = 4 m, d2 = 42 m, and µ = 2 106 Nms/rad. The liftfunction in Figure 2.5 was approximated by the following third order polynomial

L(α) = l0 + l1α− l3α3, (2.27)

with l0 = 2.5 105 N, l1 = 1.5 105 N/deg, l3 = 230 N/deg3. The control law (2.22) isimplemented with kP = 4.70 105 Nm/deg and kD = 1.79 105 Nms/deg. Given the


polynomial approximation (2.27) of the lift function L(α), the analyticalb expressionof the Lyapunov function (2.24) is

V (α, α, v) = 12Iα

2 + (d1l1 + d2kP )(cosα− cos v + (α− v) sinα)−d1l3

((α3−6α−v3) sinα+6v sin v+3((α2−2) cosα−(v2−2) cos v)

).

Given the initial conditions α(0) = α(0) = 0 and the desired reference r = 14 deg,Figure 2.6 compares the behaviour that is obtained using

• No RG: The reference r is applied directly to the pre-stabilized system;

• ERG: The auxiliary reference v(t) is generated using (2.25), with κ = 10−3,η = 10−2 and v(0) = 0.

In the absence of the ERG, the transient dynamics cause the main wing to stall, thusleading to an instable response. Instead, the introduction of the Explicit ReferenceGovernor successfully ensures the satisfaction of constraint (2.21) by limiting theovershoot.

aFor additional information on aerodynamics and airfoil theory, the reader is referred to [2.9].bGiven the tabulated curves of L(α), the Lyapunov function V (v, α, α) can also be computed

using numerical integration.

2.5 Summary

This chapter has presented the general Explicit Reference Governor framework for the con-strained control of nonlinear systems. The method consists in pre-stabilizing the nonlinearsystem and manipulating the derivative of the applied reference to ensure constraint en-forcement.

The ERG was decomposed in two separate terms: the dynamic safety margin and the at-traction field. For the sake of generality, each component was defined by stating the sufficientconditions to ensure constraint satisfaction and asymptotic convergence. An intuitive choicefor both terms was then provided by describing the basic ERG formulation and providingan example.

The main challenge that remains to be addressed is how to systematically determinesuitable expressions for the dynamic safety margin ∆(x, v) and the attraction field ρ(v, r).This will be the main focus of Chapter 3 and Chapter 4.

Bibliography

[2.1] A. Isidori, Nonlinear Control Systems II. Springer, 1995.

[2.2] H. K. Khalil, Nonlinear Systems. Prentice-Hall, 2002.


[2.4] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides. KluwerAcademic Publishers, 1988.

Bibliography 21

[2.5] D. Shevitz and B. Paden, “Lyapunov stability theory of nonsmooth systems,” IEEETransactions on Automatic Control, vol. 39, no. 9, pp. 1910–1914, 1994.

[2.6] M. M. Nicotra and E. Garone, “Explicit reference governor for continuous time nonlin-ear systems subject to convex constraints,” in Proc. of the American Control Conference(ACC), 2015, pp. 4561–4566.

[2.7] E. Garone and M. M. Nicotra, “Explicit reference governor for constrained nonlinearsystems,” IEEE Transactions on Automatic Control, vol. 61, no. 5, pp. 1379–1384,2016.

[2.8] M. V. Cook, Ed., Flight Dynamics Principles. Butterworth-Heinemann/Elsevier, 2007.

[2.9] I. H. Abbott and A. E. von Doenhoff, Theory of Wing Sections: Including a Summaryof Airfoil Data. Dover Publications Inc., 1959.

[2.10] D. G. Schultz and J. E. Gibson, “The variable gradient method for generating liapunovfunctions,” Transactions of the American Institute of Electrical Engineers, vol. 81, no. 4,pp. 203–210, 1962.

Chapter 3

Dynamic Safety Margin

The objective of this chapter is to provide various methods for computing the dynamic safetymargin ∆(v, x) in a systematic way. The advantages and disadvantages of each formulationwill be evaluated in terms of performance, computational requirements, and generality.

Please note that, without loss of generality, this chapter focuses on systems that aresubject to a single constraint. Indeed, the case of multiple constraints ci(x, v) can easily berecovered by assigning the overall dynamic safety margin

∆(x, v) = mini{∆i(x, v)},

where each ∆i(x, v) is the dynamic safety margin associated to a single constraint ci(x, v).By taking advantage of this superposition principle, the following sections will provide severaldifferent options for computing the dynamic safety margin associated to a single constraint.

The chapter will conclude with the study of two highly relevant classes of systems, i.e.linear and Euler-Lagrange systems, for which the nature of the trajectories is known a prioriand can be used to achieve better performances.

3.1 Lyapunov-Based

As seen in Section 2.4, the Explicit Reference Governor was originally introduced using thedynamic safety margin (2.18), i.e. ∆(x, v) = Γ(v) − V (x, v), where V (x, v) is a Lyapunovfunction and Γ(v) corresponds to the largest invariant level-set that is wholly contained inthe constraints. Although there are alternative formulations, the Lyapunov-based dynamicsafety margin constitutes a typical choice for ∆(x, v) due to its strong ties with nonlinearcontrol theory. Assuming that the Lyapunov function V (x, v) is known, this section willdiscuss several methods for computing a threshold value Γ(v).

3.1.1 Optimal Threshold Values

Although equation (2.16) defines the threshold value Γ(v) as whatever Lyapunov level-setwhich does not violate constraints, it should be noted that the choice of Γ(v) will determinethe conservativeness of the dynamic safety margin. The optimal choice for the thresholdvalue is {

Γ(v) = maxx V(x, v), s.t.V (x, v) ≤ V(x, v) ⇒ c(x, v) ≥ 0.

(3.1)

which represents the largest level-set that does not violate constraints. This interpretationwas illustrated in Figure 2.3 of the previous chapter. The main issue with the optimization

23

24 3. Dynamic Safety Margin

𝑟

𝑥 System Primary

Control

ERG 𝑥 𝑢


��𝑣

𝑣

𝑥

{𝑥: 𝑉(𝑥, 𝑣) ≤ Γ(𝑣)}

{𝑥: 𝑐(𝑥, 𝑣) ≤ 0}

��𝑣

{𝑥: 𝑉(𝑥, 𝑣) ≤ Γ(𝑣)} ∪ {𝑥: 𝑐(𝑥, 𝑣) ≤ 0}

{𝑥: 𝑉(𝑥, 𝑣) = Γ(𝑣)}

𝑥𝑡

{𝑥: 𝑉(𝑥, 𝑣) = 𝑉(𝑥𝑡, 𝑣)}

Figure 3.1: Geometric interpretation of the dual problem (3.2).

problem (3.1) is that constraints in the form A ⇒ B are typically difficult to address.Therefore, it is computationally more convenient to reformulate (3.1) using its dual form{

Γ(v) = minx V (x, v), s.t.c(x, v) ≤ 0.

(3.2)

This corresponds to finding the smallest level-set {x : V (x, v) ≤ Γ(v)} which has a non-null intersection with the set {x : c(x, v) ≤ 0}. This interpretation is illustrated in Figure3.1. The main interest in this formulation is that (3.2) is a convex optimization problem iffunctions V (x, v) and c(x, v) are convex.

The following example illustrates the implementation of the Lyapunov-based dynamicsafety margin using the above optimal threshold approach.

Example 3.1: Aircraft Control with Saturated Input

Consider the control law (2.22) presented in Example 2.1. An alternative approach toavoid instabilities is to increase the authority of the control input u by increasing thecontrol gains kp and kd. Intuitively, this would allow to recover the aircraft attitudeeven after a stall of the main wing. Although effective in theory, this solution maybe inapplicable in the presence of input saturations

0 ≤ u ≤ umax. (3.3)

Saturation Constraint: To account for the saturations, the Lyapunov-based ERG(2.25) can be extended using

Γ(v) = min(ΓS(v),Γsat1(v),Γsat2(v)), (3.4)

where ΓS is given in (2.26) and Γsat1(v), Γsat2(v) are the respective solutions to{Γ(v) = min

(α,α)V (α, α, v), s.t.

−kP (α− v)− kDα+ d1

d2L(v)− umax ≤ 0,

(3.5)

and {Γ(v) = min

(α,α)V (α, α, v), s.t.

−kP (α− v)− kDα+ d1

d2L(v) ≤ 0,

(3.6)

3.1. Lyapunov-Based 25

with V (α, α, v) given in (2.24). Since the threshold value ΓS(v) ensures the non-stalling condition α ≤ αS , (3.5)-(3.6) are two convex optimization problems that canbe solved using standard tools, e.g. Newton-Raphson.

Obtained Results: Figure 3.2 illustrates the step response and the control inputgiven kP = 4.23 106, kD = 5.39 105, umax = 4 105N , and all other parameters as inExample 2.1. The following behaviours are compared:

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20Step Response

t [s]

atta

ck a

ngle

[deg

]

0 0.5 1 1.5 2 2.5 3 3.5 4−4

−2

0

2

4

6

8

10

12x 10

5 Control Input

t [s]

elev

ator

forc

e [N

]

rα(t) No RG 1α(t) No RG 2v(t)α(t) ERGα≤α

S

u(t) No RG 1u(t) No RG 2u(t) ERG0 ≤ u ≤ u

max

Figure 3.2: Step response and control input behaviour with saturations.

• No RG 1: The control input is not subject to saturations;

• No RG 2: The control input is saturated according to (3.3);

• ERG: The auxiliary reference is generated using (2.25), (3.4).

In the absence of input saturation, the control law (2.22) successfully steers the attackangle to its desired reference r. In the presence of saturations, however, the systemresponse becomes unstable. As expected, the ERG successfully enforces the systemconstraints (2.21) and (3.3) by limiting the transient response of the pre-stabilizedsystem.

In some applications, the optimization problem (3.2) may be solved directly on-line. Al-though perfectly valid, this approach is somewhat in opposition to the overall ERG philos-ophy of being a closed-form solution. In view of formulating a computationally inexpensiveExplicit Reference Governor, the following propositions state two special cases in which theoptimization problem admits an analytical solution.

Proposition 3.1. Ellipsoidal Lyapunov and Linear Constraint: Given the ellipsoidal


Lyapunov functionV (x, v) = (x− xv)TP (x− xv), (3.7)

with P > 0, and a linear constraint

c(x, v) = aTx+ b ≥ 0, (3.8)

the solution to the optimization problem (3.2) is

Γ(v) =(aT xv + b)2

aTP−1a. (3.9)

Proof. Consider the change of coordinates x =√P (x− xv). Equations (3.7)-(3.8) can then

be rewritten asV (x, v) = xT x,

c(x, v) = aT√P−1x+ aT xv + b.

As a result, the optimization problem (3.2) becomes{Γ(v) = min xT x, s.t.

aT√P−1x+ aT xv + b ≤ 0.

(3.10)

Since (3.10) is convex, its optimum satisfies aT√P−1xOpt + aT xv + b = 0. This implies

xOpt = −√P−1a

aTP−1a(aT xv + b).

The corresponding threshold value is Γ(v) = xTOptxOpt, which is equal to (3.9). �

Proposition 3.2. Spherical Lyapunov and Distance Constraint: Given the sphericalLyapunov function

V (x, v) = α ‖x− xv‖2 , (3.11)

with α > 0, and the distance constraint

c(x, v) = ‖x− xv‖ ≤ R, (3.12)

with R > 0, the solution to the optimization problem (3.2) is

Γ(v) = αR2. (3.13)

Proof. Consider the change of coordinates x = x − xv. The optimization problem (3.2)becomes {

Γ(v) = min α xT x, s.t.R− x ≤ 0.

(3.14)

Since (3.14) is convex, its optimum must satisfy ‖xOpt‖ = R. The corresponding threshold

value Γ(v) = α ‖xOpt‖2 is given in (3.13). �

Remark 3.1. Equations (3.9) and (3.13) remain applicable also in the case of continuousreference-dependent parameters P (v), c(v), d(v), α(v), p(v), and R(v). For the sake ofnotational simplicity, this property was not specified beforehand since the proposed proofsremain unaffected. Moreover, it is worth noting that Γ(v) will be continuous if all theseparameters are continuous in v. �

The main interest in Propositions 3.1 and 3.2 is that they directly provide an analyticexpression for the optimal threshold value Γ(v). For possible cases where it is not possibleto compute the optimum value analytically, the following subsections proposes a possiblesolution for computing Γ(v) offline and storing it in manageable lookup tables.


3.1.2 Off-line Computations

The most intuitive solution for computing Γ(v) off-line is to solve (3.2) for a discrete numberof references v ∈ Rp, store the results in an p-dimensional lookup table, and perform suitableinterpolations. This solution is feasible for systems with a low-dimensional reference vector,but may become impractical as the size of v increases.

An alternative solution is to parametrise the worst-case threshold value as a function ofthe static safety margin δ > 0. This can be done by reformulating (3.2) as the followingscalar optimization problem Γ(δ) = minx,v V (x, v), s.t.

c(x, v) ≤ 0;c(xv, v) ≥ δ.

(3.15)

For a given static safety margin δ, using the worst-case threshold clearly penalizes the systemperformances. However, given multiple static safety margins δk, it is worth noting that thefollowing holds true

δk−1 ≤ c(xv, v) ≤ δk ⇒ Γ(δk−1) ≤ Γ(v) ≤ Γ(δk).

As such, a possible approach would be to solve (3.15) off-line and tabulate Γ(δ) for thechosen values of δk. Based on the current value of v, it is then possible to select the largeststatic safety margin which satisfies δk < c(xv, v) and substitute the dynamic safety margin(2.18) with

∆(x, v) = κmax{Γ(δk)− V (x, v), 0}. (3.16)

Unlike a typical lookup strategy based directly on the p-dimensional vector v, the idea ofcomputing the scalar value c(xv, v) and identifying δk makes it a one-dimensional lookupproblem. The main advantage of this strategy is that it is not subject to the curse of dimen-sionality. The main disadvantage is that it clearly provides a more conservative thresholdvalue compared to a multi-dimensional lookup table.

Remark 3.2. It is worth noting that, strictly speaking, (3.16) is not a dynamic safety mar-gin because it is discontinuous in v. Indeed, for c(xv, v) = δk, the threshold value switchesbetween Γ(δk−1) and Γ(δk). As such, Theorem 2.1 cannot be applied directly. Neverthe-less, the properties of the ERG can be recovered by studying what happens on the switchingboundaries. To do so, consider the two following cases

• v(t−) : c(xv, v) < δk ; v(t+) : c(xv, v) = δk. In this case, the threshold valueswitches from Γ(t−) = Γ(δk−1) to Γ(t+) = Γ(δk). Since Γ(δk−1) ≤ Γ(δk), the dy-namic safety margin (3.16) cannot decrease. This guarantees constraint satisfactionand asymptotic convergence as normal;

• v(t−) : c(xv, v) = δk ; v(t+) : c(xv, v) < δk. In this case, the threshold valueswitches from Γ(t−) = Γ(δk) to Γ(t+) = Γ(δk−1). Potentially, this could be problem-atic since Γ(δk−1) ≤ V (x, v) ≤ Γ(δk). However, since (3.16) implies v = 0, constraintsatisfaction is guaranteed by the fact that V (x, v) ≤ Γ(δk). Asymptotic convergence isinstead guaranteed by the fact that, given a constant reference, the time-decreasing na-ture of the Lyapunov function will eventually satisfy V (x, v) < Γ(δk−1), thus ensuring‖v(t+ τ)‖ 6= 0 after a finite time τ > 0.

This also applies to the case of a p-dimensional lookup table. �

Interestingly enough, there are a fair number of physically relevant constraints (e.g.input saturations, maximum error, maximum velocity) where the optimal threshold value


Γ(v) may be independent from the applied reference. In these cases, the threshold value Γcan be computed off-line and stored as a simple scalar.

Since the optimization problem (3.15) already introduces a degree of conservativeness,the following section provides an alternative solution to numerical methods. The proposedmethod consists in embedding the optimization problem (3.2) into a lower-approximationthat can be solved analytically. This approach will greatly extend the number of cases forwhich a threshold value Γ(v) can be computed in closed-form.

3.1.3 Sub-Optimal Threshold Values

Based on equation (2.16), the ERG framework can be implemented using any value Γ(v)such that V (x, v) ≤ Γ(v) implies c(x, v) ≥ 0. As such, the following statements extend theresults of Propositions 3.1 and 3.2 by providing sub-optimal estimates of Γ(v) for a muchlarger class of Lyapunov functions and constraints.

Proposition 3.3. (Non)Ellipsoidal Lyapunov and (Non)Linear Constraint: Giventhe Lyapunov function V (x, v) and the constraint c(x, v) ≥ 0, let there exist a continuousP (v) > 0, ∀v such that

V (x, v) ≥ (x− xv)TP (v)(x− xv), ∀v (3.17)

and continuous c(v), d(v) such that

c(v)Tx+ d(v) ≥ 0 ⇒ c(x, v) ≥ 0,

c(xv, v) ≥ δ ⇒ c(v)T xv + d(v) ≥ δ, (3.18)

with δ > 0. Then, the threshold value

Γ(v) =(c(v)T xv + d(v))2

c(v)TP (v)−1c(v)(3.19)

is such that

V (x, v) ≤ Γ(v) ⇒ c(x, v) ≥ 0,

and there exists ε > 0 such that c(xv, v) ≥ δ implies Γ(v) ≥ ε.

Proof. Due to condition (3.17), it follows that

V (x, v) ≤ Γ(v)⇒ (x− xv)TP (v)(x− xv) ≤ Γ(v).

Moreover, it follows from Proposition 3.1 that (3.19) is the largest level-set such that

(x− xv)TP (v)(x− xv) ≤ Γ(v)⇒ c(v)Tx+ d(v) ≥ 0.

As a result, it follows from (3.18) that V (x, v) ≤ Γ(v) implies c(x, v) ≥ 0, and that c(xv, v) ≥δ implies Γ(v) ≥ δ2/(c(v)TP (v)−1c(v)), which is strictly positive since δ > 0. �

Proposition 3.3 greatly extends the class of nonlinear constrained systems for which it ispossible to compute a threshold value Γ(v) analytically. This result is particularly interestingbecause it can be used for any Lyapunov function that is lower-bounded by an ellipsoidalfunction and any constraint that can be embedded in a combination of linear inequalities.The following proposition provides an even further extension by taking into account anyLyapunov function and constraint.


Proposition 3.4. Generic Lyapunov and Distance Constraint: Given the Lyapunovfunction V (x, v) and the constraint c(x, v) ≥ 0, let α(‖x− xv‖) be a class-K function thatsatisfies

α(‖x− xv‖) ≤ V (x, v), ∀x, v, (3.20)

and let R(v) be such that

‖x− xv‖ ≤ R(v) ⇒ c(x, v) ≥ 0,

c(xv, v) ≥ δ ⇒ R(v) ≥ δ, (3.21)

with δ > 0. Then, the threshold value

Γ(v) = α (R(v)) (3.22)

is such thatV (x, v) ≤ Γ(v)⇒ c(x, v) ≥ 0,

and c(xv, v) ≥ δ implies that there exists ε > 0 such that Γ(v) ≥ ε2. �

Proof. The proof is analogous to the one given in Proposition 3.3. Please note that sinceα(·) is a class-K function, α(a) ≥ α(b) implies a ≥ b for all non-negative a, b.

Following from equations (3.21)-(3.22), the scalar ε > 0 can be computed as ε = α(δ2). �

The main interest in Proposition 3.4 is that, by definition, all Lyapunov functions V (x, v)are lower bounded by a class-K function α(‖x− xv‖). Likewise, any constraint can beexpressed in terms of equation (3.21) by choosing R(v) equal to the solution of{

minx ‖x− v‖ , s.t.c(x, v) ≤ 0.

Since the analytic expressions for R(v) for the case of linear and convex quadratic constraintsis given by (3.9) and (3.13), respectively, the threshold value (3.22) represents an extremelygeneral solution to problem (2.16). Clearly, the main drawback with this solution is thatthe obtained Γ(v) is likely to be conservative.

The following example compares the behaviour of the off-line numerical method and twoanalytic threshold with respect to the optimal Γ(v) resulting from (3.2).

Example 3.2: Aircraft Control with Suboptimal Threshold Values

Consider the results presented in Example 3.1. The objective of this example is tocompare the performances of the various sub-optimal estimates proposed so far.

Lyapunov Lower-Bounds: The Lyapunov function (2.24) can be lower-boundedin several ways. The simplest solution is to note that, ∀α, v ∈ [0, αS ],∫ α

v

d1(L(β)− L(v)) cosβdβ ≥ 0,∫ α

v

d2kP (β − v) cosβdβ ≥ d2kP cosαS

∫ α

v

(β − v)dβ.

As such, a possible Lyapunov lower-bound is

V (α, α, v) ≥ 1

2

[(α− v)α

]T[d2kP cosαS 0

0 J

] [(α− v)α

]. (3.23)

Clearly, the main drawback of (3.23) is that it neglects the contribution of the liftL(α) generated by the main wing. As illustrated in Figure 3.3, it is worth noting


that

‖L(α)− L(v)‖ ≥ L(αS)− L(v)

αS − v‖α− v‖ , ∀v, α ∈ [0, αS ]. (3.24)

This implies∫ α

v

d1(L(β)− L(v)) cosβdβ ≥ d1L(αS)− L(v)

αS − vcosαS

∫ α

v

(β − v)dβ,

thus leading to another Lyapunov lower-bound

V (α, α, v) ≥ 1

2

[(α− v)α

]T[(d1L(αS)−L(v)

αS−v + d2kP

)cosαS 0

0 J

] [(α− v)α

]. (3.25)

0 5 10 15 200

2

4

6

8

10

12

14

16

18x 10

5

α [deg]

L [N

]

Figure 3.3: Linear bound of the Lift function for a given reference v = 7◦.

Since constraints (2.21) and (3.3) are linear, Proposition 3.3 can be used to computesuitable threshold values using the ellipsoidal lower bounds (3.23) and (3.25).

Switching Static Thresholds: A possible alternative to the Lyapunov lower-bound approach is to compute the threshold values associated to a sequence ofdifferent static safety margins δk. This can be done by solving (3.15), subject tothe state constraints (2.21), (3.3), and the steady-state constraint

c(xv, v) = min(αS − v ; umax − d1L(v) cos(v)) ≥ δk.

Since v ∈ [0, αS ], a possible choice for the static safety margins δk is

δk = kαSnk

k = 0, . . . , nk. (3.26)


0 1 2 3 4 5 60

2

4

6

8

10

12

14

16Auxiliary Reference

t [s]

atta

ck a

ngle

[deg

]

rv(t) Optimalv(t) Analytic 1v(t) Analytic 2v(t) Switchingv ≤ α

S

0 1 2 3 4 5 60

2

4

6

8

10

12

14

16Step Response

t [s]

atta

ck a

ngle

[deg

]

rα(t) Optimalα(t) Analytic 1α(t) Analytic 2α(t) Switchingα ≤ α

S

You Shall Not Pass!

Figure 3.4: Step Response and Applied Reference for the proposed ERG Strategies.

Obtained Results: Figure 3.4 illustrates the step response and the applied refer-ences for the system described in Example 2.1. The following behaviours are com-pared:

• Optimal Γ(v): The threshold value follows from (3.4).

• Analytic Γ(v), 1: The threshold value follows from (3.7) and (3.23);

• Analytic Γ(v), 2: The threshold value follows from (3.7) and (3.25);

• Switching Γ(v): The threshold value follows from (3.15), (3.26) with nk = 20.

As expected, all four strategies ensure constraint satisfaction. However, the responsetime differs based on the choice of Γ(v). For what regards the analytic approach,it can be noted that the performances depend on the quality of the lower bound.Indeed, the response obtained with the Lyapunov lower bound (3.25) is comparableto the one provided by the optimal threshold value. Instead, the scheme based on(3.23) is significantly slower due to the fact that it neglects L(α).For what regards the switching strategy, it is interesting to note the step behaviourof the auxiliary reference. As discussed in Remark 3.2, each plateau corresponds toa switch from δk to δk−1 which requires a finite adjustment time τ > 0. The qualityof the response can be improved by increasing nk.

Having proposed a variety of methods for obtaining a threshold value Γ(v) for theLyapunov-based ERG, the following section will provide alternative definitions of the dy-namic safety margin ∆(x, v) for those cases in which it is not trivial to find a Lyapunov


function or there is a need to improve performances.

3.2 Other Methods

Although ∆(x, v) = κ(Γ(v) − V (x, v)) represents a possible choice for the dynamic safetymargin, a large variety of alternative invariant sets [3.1] can be used to enforce constraints.As such, this section will briefly address possible alternatives to Lyapunov level-sets thatmay lead to improved performances or simplify the implementation of the ERG by notrequiring the construction of a Lyapunov function for the closed-loop system.

3.2.1 Invariant Set Approach

A possible extension of the Lyapunov-based approach consists in employing a referencedependent set that satisfies the constraints and is strongly invariant whenever the referenceis kept constant. The proposed set is defined as follows:

Definition 3.1. Admissible Invariant Set: Given the pre-stabilized system x = f(x, v)subject to a constant applied reference v, the set Sv is a “Admissible Invariant Set” if

x ∈ Sv ⇒ c(x, v) ≥ 0, (3.27)

and there exists τ > 0, such that ∀τ ∈ (0, τ ],

∀x ∈ ∂Sv ⇒ x+ τf(x, v) ∈ Sv \ ∂Sv, (3.28)

where ∂Sv denotes the boundary of the set Sv. �

Due to equation (3.28), if v is held constant whenever x ∈ ∂Sv, the set Sv will satisfythe strong invariance property defined in [3.2]. This implies that the state trajectories willalways remain in the interior of Sv. This is sufficient to guarantee constraint satisfactiondue to equation (3.27).Based on these intuitions, the following dynamic safety margin is proposed

∆(x, v) = dist(x, ∂Sv), (3.29)

where dist(x, ∂Sv) can be the Euclidean distance

dist(x, ∂Sv) = minz∈∂Sv

‖x− z‖ .

or any other measurement1 of the distance between the current state x and the set boundary∂Sv.

Remark 3.3. It is worth noting that the Lyapunov-based approach can be interpreted as aparticular case of the invariant set approach, with

Sv = {x : V (x, v) ≤ Γ(v)} .

As a result, the dynamic safety margin (3.29) can be seen as an extension of equation(2.18). �

1Please note that dist(a, b) does not need to satisfy all the requirements of a norm. In particular, dist(a, b)is only required to be continuous, to be zero if the Euclidean norm is 0, and to be positive if the Euclideannorm is positive.

3.2. Other Methods 33

The main interest in the invariant set approach is that Sv depends directly on the systemtrajectories and does not require the formulation of a time-decreasing scalar function. Asa result, (3.29) is typically less conservative with respect to the Lyapunov-based dynamicsafety margin.

The main drawback to this method is that the analytic construction of strongly invariantsets for nonlinear systems can be challenging. In some cases, an invariant set can be obtainedby taking advantage of known properties of the system (e.g. cone invariance for positivesystems). Alternatively, it is possible to propose a candidate set Sv satisfying (3.27) andthen verify numerically if (3.28) holds true.

3.2.2 Returnable Set Approach

A possible extension of the invariant set approach is to determine a set that ensures con-straint satisfaction without necessarily being invariant. This can be done based on thefollowing definition

Definition 3.2. Admissible Returnable Set: Given the pre-stabilized system x = f(x, v)subject to a constant applied reference v, the set Sv is an “Admissible Returnable Set” if

xv ∈ Sv \ ∂Sv, (3.30)

and there exists a set Rv ⊇ Sv such that

∀x ∈ Sv, x(t|x, v) ∈ Rv, ∀t ≥ 0, (3.31)

andx ∈ Rv ⇒ c(x, v) ≥ 0. (3.32)

Since Sv is not an invariant set, it is possible that the trajectories x(t) will exit the setboundary ∂Sv. However, it follows from equations (3.31)-(3.32) that the transient responsewill not violate the constraints if v is kept constant while x(t) /∈ Sv. Additionally, equation(3.30) ensures that the set is strongly returnable, meaning that the state trajectories willre-enter the set Sv after a finite time.Based on these considerations, the dynamic safety margin (3.29) can be modified to

∆(x, v) =

{dist(x, ∂Sv) if x ∈ Sv,0 if x /∈ Sv

(3.33)

This ensures that the ERG will wait for the state to re-enter the returnable set beforemodifying the applied reference.

Remark 3.4. It is worth noting that the invariant set in Definition 3.1 is simply a specialcase of returnable set for which Rv = Sv. �

The main advantage of the dynamic safety margin (3.33) is that Definition 3.2 is ex-tremely broad and provides a fairly flexible solution. Clearly, the main challenge resides inbeing able to identify a suitable pair of sets Sv and Rv.Interestingly enough, the returnable set approach can be used to justify the very prac-tical idea that consists in modifying the reference only when the system is “sufficientlyclose” to steady-state (i.e. only when x ∈ Sv). Indeed, if the returnable set Sv is esti-mated conservatively, the ERG becomes applicable even when the pre-stabilized systemdynamics are not known in detail. A practical approach for implementing this strat-egy consist in using estimation/learning procedures to find a function δ(ε) such that, ∀v,‖x − xx‖ ≤ ε implies ‖c(xv, v) − c(x(t|x, v), v)‖ ≤ δ(ε), ∀t. Sv can then be defined as{x : c(xv, v)− δ(‖x− xx‖) ≥ 0}.


3.2.3 Trajectory-Based Approach

Going back to the original definition of the dynamic safety margin, ∆(x, v) represents somekind of distance between the trajectory x(t|x, v) and constraint (2.6). As a result, an intuitivechoice is

∆(x, v) = minτ c(x(t|x, v), v), ∀t ≥ 0 (3.34)

where x(t|x, v) is the solution to (2.9). This satisfies Definition 2.1 since ∆(x, v) intrinsicallycontains all the information about the future distances from the constraint. Clearly, the mainlimitation with (3.34) that it is requires the state trajectory predictions along an infinitehorizon.

To make the approach implementable in practice, it is therefore necessary to formulate afinite horizon problem by determining an admissible returnable set Sv ⊂ Rn. The dynamicsafety margin (3.34) can then be reformulated as

∆(x, v) = minτ c(x(t|x, v), v), ∀t ∈ [0, T ], (3.35)

where T is a finite time instant such that x(T ) ∈ Sv.

Remark 3.5. A possible choice for the Sv is the classical Lyapunov level-set

Sv = {x : V (x, v) ≤ Γ(v)} ,

where Γ(v) is the threshold value described in Section 3.1. To avoid the computation ofa Lyapunov function for the nonlinear system, it is worth noting that x(T ) is expected tobe in close proximity to xv for sufficiently long horizons. Since the invariance conditionV (x(T ), v) ≤ Γ(v) must hold true only locally, the Lyapunov function V (x, v) can be formu-lated using the linearized system equations. �

For what regards the choice of a suitable prediction horizon T , two possible alternativesare presented.

Time-Varying Prediction Horizon: The simplest implementation approach for thetrajectory-based dynamic safety margin is to use x(t|x, v) ∈ Sv as a stopping condition forthe trajectory prediction. This has the advantage of minimizing the length of the predictionhorizon without compromising the properties of the dynamic safety margin (3.35). However,the main disadvantage is that the execution time cannot be known a-priori. This canbe problematic since if the ERG runtime may become too slow to be approximated as acontinuous-time strategy.

Hybrid Prediction/Invariance Strategy: Given a constant (possibly short) predic-tion horizon T , it becomes necessary to enforce x(T ) ∈ Sv as an additional constraint. Thiscan be done by assigning a hybrid dynamic safety margin

∆(x, v) = min(∆Traj(x, v),∆Inv(x(T ), v)) (3.36)

which combines the trajectory-based dynamic safety margin ∆Traj given in (3.35) with anyinvariance-based dynamic safety margin ∆Inv (e.g. (2.19)) computed in x(T ). This strategyhas the advantage of having a known prediction horizon, but may present lower performanceswith respect to the pure trajectory-based approach since it employs an invariance-basedconstraint at the end of the prediction horizon.


Example 3.3: Aircraft Trajectory-Based Control

Consider the control law developed in Example 3.1. The objective of this exam-ple is to implement the trajectory-based approach and to compare it with the theLyapunov-based alternative (2.18).

Trajectory Prediction: Given the current v, α and α, the predicted state trajec-tories can be computed by discretizing the dynamics of the pre-stabilized system(2.23). This can be done by choosing a sufficiently small time constant ∆T = 10−2sand using the forward Euler approximation

α1(k+1)= α1+∆T α2

α2(k+1)= α2+∆T cos α1

I(d1(L(v)−L(α1))−d2(kP (α1 − v) + kDα2)) .

As shown in Chapter 5, possible discretization errors can be accounted for by tight-ening the constraint to α ≤ αS − δ, with δ > 0. The dynamic safety can then beupdated at each time step using

∆(k + 1) = min

(∆(k),

αS − α1(k + 1)

αS,umax − u(k + 1)

umax

), (3.37)

which has the advantage of giving the same weight to the stall constraint and theinput saturation. Based on Remark 3.5, the following finite-horizon condition isconsidered

V (v, α1(K), α2(K)) ≤ Γ(v), (3.38)

where Γ(v) i the same as (3.4).

Obtained Results: Figure 3.5 illustrates the response for the system described inExample 3.1. Since the value of the trajectory-based dynamic safety margin is severalorders of magnitude smaller with respect to its invariance-based counterpart, theERG gain has been modified to κ = 103. The following behaviours are compared:

• Lyapunov ERG: The dynamic safety margin is given by (2.18), (2.26);

• Trajectory ERG: The trajectory-based approach (3.37) is implemented using(3.38) as a stopping condition;

• Hybrid ERG: The hybrid approach (3.36) is implemented with T = 0.5s.


0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12

14

16Step Response

t [s]

atta

ck a

ngle

[deg

]

rv(t) Lyapunovα(t) Lyapunovv(t) Trajectoryα(t) Trajectoryv(t) Hybridα(t) Hybridα ≤ α

S

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5x 10

5 Control Input

t [s]

elev

ator

forc

e [N

]

u(t) Lyapunovu Trajectoryu Hybridu ≤ u

max

Figure 3.5: Step Response and associated control inputs for the proposed strategies.

The simulations clearly show that the trajectory-based ERG achieves the fastestresponse by pushing the pre-stabilized system to its limits. As expected, the hybridapproach presents a response that is between the trajectory-based and the Lyapunov-based approaches.

Remark 3.6. An alternative interpretation of the hybrid dynamic safety margin (3.36) isthat it improves the performances of invariance-based strategies by introducing a finite timewindow where the system trajectories are handled directly. �

3.2.4 Feedforward Approach

In possible cases where the state vector is not available for measurements or estimation, thedynamic safety margins proposed in this chapter become inapplicable due to the fact that∆(x, v) cannot be computed without the knowledge of x. As proven in [3.3], however, mostreference governor strategies can also be implemented in a feedforward fashion by takingadvantage of the asymptotically stable nature of pre-stabilized systems.This section introduces a feedforward implementation strategy for the Lyapunov-based dy-namic safety margin. Please note that the proposed strategy can also be extended to addressother types of dynamic safety margin.

Given an unknown state vector x, a possible idea is to substitute (2.18) with

∆(v, t) = κ(Γ(v)− V (t)), (3.39)

where V (t) is a worst-case Lyapunov function such that

V (t) ≥ V (x(t), v(t)), ∀t ≥ 0. (3.40)


The construction of a suitable V (t) is addressed in the following proposition.

Proposition 3.5. Let V (x, v) be a Lyapunov function for the pre-stabilized system (2.5)subject to a constant reference v ∈ Rp. Given the system

˙V = h(V , v) + l(V , v) ‖v‖ , (3.41)

withh(V , v) ≥ max

x:V (x,v)=V∇xV (x, v)T f(x, v),

l(V , v) ≥ maxx:V (x,v)=V

‖∇vV (x, v)‖ ,

then equation (3.39) satisfies the requirements of a dynamic safety margin if h(V , v) isnegative definite and that the initial conditions satisfy V (0) ≥ V (x(0), v(0)). �

Proof. Given a time-varying reference v, the derivative of the Lyapunov function V (x, v) is

V = ∇xV (x, v)T f(x, v) +∇vV (x, v)T v.

As a result, it follows by construction that˙V (t) ≥ V (t). Due to the restrictions on the initial

conditions, equation (3.40) is satisfied at all times. Constraint enforcement is thereforeguaranteed by the fact that V (v) ≤ Γ(v) implies V (x, v) ≤ Γ(v).

Asymptotic convergence is instead guaranteed by the fact that h(V , v) is negative def-inite. Indeed, given a constant applied reference, (3.41) is strictly negative. This impliesthat (3.39) cannot remain equal to zero. �

The main advantage of (3.39) is that it ensures constraint satisfaction without requiringany knowledge of the system states. However, since it is based on the worst-case scenario,the feedforward dynamic safety margin will typically achieve reduced performances withrespect to its feedback counterpart. This is illustrated in the following example.

Example 3.4: Aircraft Feedforward Control

Consider the control law presented in Example 3.1. Since the control input (2.22) isclearly a state feedback, it would be unreasonable to assume that the state vector isunavailable for the ERG. As such, this example addresses the case where the controlinput u is a purely feedforward terma

u =d1

d2L(v). (3.42)

Strict Lyapunov Function: The main drawback with the energy-like Lyapunov

function (2.24) is that its derivative −µα2 is negative semi-definite and therefore vio-lates the requirements of Proposition 3.5. By once again using the Variable GradientMethod [3.4], the following strict Lyapunov function is proposed

V (v, α, α) =

∫ α

v

d1(L(β)− L(v)) cosβdβ +1

2

[α− vα

]T [εµ2/I εµεµ I

] [α− vα

], (3.43)

where ε ∈ (0, 1). Indeed, given a constant reference v, the resulting time derivativeis

∇xV (x, v)T f(x, v) = −εµI

(L(α)− L(v)) cosα(α− v) cosα− (1− ε)µα2,

which is negative definite ∀ε ∈ (0, 1).


Worst-Case Lyapunov Dynamics: The first step for finding h(V , v) and g(V , v)is to provide a linear upper bound between the state vector and the two componentsof V . Following from (3.24), the effect of a time-varying state can be bounded using

∇xV (x, v)T f(x, v) ≤ −[α− vα

]T[d1

εµI

L(αS)−L(v)αS−v cos(αS) 0

0 (1− ε)µ

][α− vα

],

(3.44)Likewise, the effect of a time-varying reference is

∇vV (x, v)T v = −∫ α

v

d1L′(v) cosβdβ −

[α− vα

]T[εµ2/Iεµ

]v.

Since sin(α) − sin(v) ≤ cos(v)(α − v),∀α, v ∈ (0, π/2), the following upper-boundholds true

∥∥∇vV (x, v)T v∥∥ =

∣∣∣∣∣[α− vα

]T[d1L

′(v)cos(v) + εµ2/Iεµ

]∣∣∣∣∣ |v| . (3.45)

The second step consists in providing a linear upper bound between the state vectorand the Lyapunov function value. As illustrated in Figure 3.6, it is worth notingthatb

‖L(α)− L(v)‖ ≤ L′(0) ‖α− v‖ , ∀v, α ∈ (0, αS). (3.46)

As a result, the Lyapunov function (3.43) is upper bounded by

V (α, α, v) ≤ 1

2

[(α− v)α

]T[d1L

′(0) + εµ2/I εµεµ I

] [(α− v)α

]. (3.47)

By combining (3.44), (3.47), it can be shown that

∇xV (x, v)T f(x, v) ≤ −λmin

(√P (v)

−TQ(v)

√P (v)

−1)V ,

∀x : V (x, v) = V , where

P (v) =1

2

[d1L

′(0) cos(v) + εµ2/I εµεµ I

],

and

Q(v) =

[d1

εµI

L(αS)−L(v)αS−v cos(αS) 0

0 (1− ε)µ

].

Likewise, by combining (3.45), (3.25), it can be shown that

∇vV (x, v)T v ≤∣∣∣∣P (v)−T

[d1L


]∣∣∣∣√V ‖v‖ ,∀x : V (x, v) = V , where

P (v) =1

2

[d1

L(αS)−L(v)αS−v cos(αS) + εµ2/I εµ

εµ I

].


0 5 10 15 200

2

4

6

8

10

12

14

16

18x 10

5

α [deg]

L [N

]

Figure 3.6: Linear bound of the Lift function for a given reference v = 7◦.

As a result, the feedforward dynamic safety margin in equation (3.39) can be imple-mented using the worst-case Lyapunov dynamics

˙V = −h(v)V + l(v)

√V ‖v‖ . (3.48)

where

h(v) = λmin

(√P (v)

−TQ(v)

√P (v)

−1)and

l(v) =

∣∣∣∣P (v)−T[d1L


]∣∣∣∣ .


0 10 20 30 40 50 60 70 800

5

10

15

20Output Response

t [s]

atta

ck a

ngle

[deg

]

rv(t) Feedbackα(t) Feedbackv(t) Feedforwardα(t) Feedforwardα ≤ α

S

0 10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5x 10

5 Lyapunov Behaviour

t [s]

Lyap

unov

Fun

ctio

n V

alue

Measured ValueUpper Approximation

Figure 3.7: Step Response and Lyapunov values for the proposed strategies.

Obtained Results: Figure 3.7 illustrates the step response and the Lyapunov func-tion dynamics for the system described in Example 2.1 with kP = kD = 0. Thefollowing behaviours are compared:

• Feedback ERG: Constraints enforcement is ensured using the Lyapunov feed-back (2.18);

• Feedforward ERG: Constraints enforcement is ensured using the Lyapunovfeedforward (3.39).

As expected, the feedforward approach successfully ensures constraint satisfactionbut presents significantly lower performances with respect to the feedback imple-mentation strategy. This is due to the conservative nature of the worst-case approx-imation V which tends to be much larger than the actual value of the Lyapunovfunction. Nevertheless, the motivation behind the feedforward approach is that itprovides a feasible solution without any knowledge of the state vector.

aThis control law is representative for aircrafts which are not fly-by-wire.bGiven the polynomial approximation (2.27), the slope of the lift function is L′(v) = l1 − 3l3v2,

which is strictly positive for v ∈ (0, αS).

3.2.5 Combined Dynamic Safety Margins

This chapter has presented several alternatives for computing a dynamic safety margin∆(x, v) for a given constraint. In the presence of multiple dynamic safety margins for the

3.3. Application to Linear Systems 41

same constraint, it is worth noting that it is always possible to choose the least restrictiveone by assigning

∆(x, v) = maxj{∆j(x, v)}.

This concludes the general overview on how to design suitable dynamic safety margins for anunspecified nonlinear system. The following sections will instead provide design strategiesfor more specific, but still highly relevant, classes of systems.

3.3 Application to Linear Systems

Although the ERG framework has been proposed and developed as a tool for nonlinearsystems, it is worth noting that many control applications prefer linear systems theory dueto its greater simplicity. As a result, the objective of this section is to specialize the ERGframework to the case of linear systems subject to linear constraints.

To this end, consider the state-space model of a pre-stabilized2 system

x = Ax+Bv, (3.49)

where A ∈ Rn×n is a Hurwitz matrix and B ∈ Rn×p. Given a constant reference v, thesystem will asymptotically tend to the equilibrium point xv = −A−1Bv. By applyingbasic linear systems theory (see e.g. [3.5]), the following subsections will illustrate how theperformances of the ERG framework can be improved by selecting suitable invariant sets.

3.3.1 Lyapunov Selection

As seen in Section 3.1, the ERG is generally implemented using the the Lyapunov-baseddynamic safety margin ∆(x, v) = κ(Γ(v) − V (x, v)). Interestingly enough, linear systemsare characterized by an infinite choice of quadratic Lyapunov functions

V (x, v) = (x+A−1Bv)T P, (x+A−1Bv), (3.50)

where P > 0 must satisfy the Lyapunov equation3

ATP + PA ≤ 0. (3.51)

The selection of P can therefore be seen as an opportunity to improve the ERG performances.Given the linear constraints (3.8), i.e. c(x, v) = aTx+ b ≥ 0, a possible approach consists inchoosing P so as to maximise the volume of the level-set

Sv = {x : V (x, v) ≤ Γ(v)} . (3.52)

This is done in the following proposition.

Proposition 3.6. Given the linear system (3.49) subject to the linear constraint (3.8), theLyapunov function (3.50) that maximises the volume of (3.52) can be obtained by solving

min log det(P ), s.t.ATP + PA ≤ 0P ≥ aaTP > 0

(3.53)

Moreover, the optimal solution is independent on v. �

2Note that equation (3.49) represents the dynamics of a pre-stabilized system. As such, the theory isalso valid for nonlinear that have been pre-stabilized using feedback linearization.

3Since A is Hurwitz, the LaSalle criterion can be applied to show that (3.50) is a Lyapunov function evenin the case ATP + PA = 0.


Proof. For a fixed v, consider the error coordinates x = x+A−1Bv and consider the ellipsoidE =

{x : xTPx ≤ g

}, where g > 0 can be assigned arbitrarily. This ellipsoid is symmetric

with respect to the origin. Constraint (3.8) can then be re-written in error coordinates

aT x− aTA−1Bv + b ≥ 0,

and can be made symmetric with respect to the origin by enforcing

xTaaT x ≤ h(v)2,

where h(v) = b − aTA−1Bv. Given the singular ellipsoid C ={x : xTaaT x ≤ h(v)2

}, the

inclusion E ⊆ C is satisfied if

P ≥ g

h(v)2aaT . (3.54)

Since h(v)2 is a positive scalar, it is possible to assign g = h(v)2 without any loss of generality.The remainder of the proof follows directly from [3.6, p. 414]. �

Remark 3.7. It is worth noting that Proposition 3.6 remains applicable even in the case ofreference-dependent constraints in the form

aTx+ b(v) ≥ 0.

Indeed, modifying the scalar b(v) is equivalent to introducing a rigid translation of the coor-dinate system and not modify the solution of (3.53).The case of a reference-dependent vector a(v) is instead more difficult to address because thesolution of (3.53) would be a function of the applied reference v. �

By taking advantage of Proposition 3.6, the Lyapunov-based ERG for linear systemssubject to linear constraints can be designed by solving the linear matrix inequality (3.53)to obtain V (x, v) and applying Proposition 3.1 to compute the threshold value Γ(v). Theadvantages of using this approach rather than selecting an arbitrary Lyapunov function willbe illustrated in Examples 3.5 and 3.6.

3.3.2 Polyhedric Invariant set

As discussed in Subsection 3.2.1, the dynamic safety margin can be implemented using anykind of invariant set. In the particular case of linear systems, a possible alternative to theLyapunov level-sets is to study the eigenvalue decomposition of system (3.49).

Since the system response strongly depends on the nature of the eigenvalues, this sub-section will focus on the case of real eigenvalues. The extension to systems with complexconjugate poles will be presented in Subsection 3.3.3.

Lemma 3.1. Let system (3.49) be characterized by a Hurwitz state-matrix with real eigen-values, and let T be an invertible matrix such that

T−1AT = diag{λ1, . . . , λn}. (3.55)

Given a constant reference v and given the change of coordinates z = T−1(x − xv), thefollowing hold true

1. sgn(zi(t)) = sgn(zi(0)), ∀t ≥ 0, ∀i ∈ {1, . . . , n};

2. |zi(t)| ≤ |zi(0)|, ∀t ≥ 0, ∀i ∈ {1, . . . , n}. �


Proof. System (3.49) can be re-written as T z = ATz. Following from equation (3.55), thisimplies zi = λizi, ∀i ∈ {1, . . . , n}. Each component of vector z is therefore characterized bythe exponential behavior zi(t) = zi(0)eλit, which is constant in sign and time-decreasing inmodulus. �

Lemma 3.1 basically states that, given a constant reference, the trajectories of the systemare always contained in a polyhedric set aligned with the eigenvectors. By taking advantageof this invariance property, the following proposition provides a suitable dynamic safetymargin.

Proposition 3.7. Let system (3.49) be characterized by a Hurwitz state-matrix with realeigenvalues. Given the linear constraints

a(v)Tx+ b(v) ≥ 0,

the function4

∆(z, v) = β(v) +∑i∈I

αi(v)zi, (3.56)

with z = T−1(x+A−1Bv), α = TTa(v), β = b− aTA−1Bv, and I= {i ∈ {1, . . . , n} : βizi <0}, is a dynamic safety margin. �

Proof. Referring to equation (3.56), it is straightforward to see that ∆(x, v) is continuous ifa(v) and b(v) are continuous. Moreover, since x = xv implies z = 0, ∆(z, v) will be strictlypositive whenever b − aTA−1Bv > 0. Referring to Definition 2.1, the following propertiesare sufficient to prove the statement

1. for any constant v, ∆(z, v) = 0 implies a(v)Tx+ b(v) ≥ 0;

2. for any constant v, ∆(z, v) > 0 implies a(v)Tx+ b(v) > 0;

3. for any constant v, ∆(z(t), v) ≥ ∆(z(0), v).

Points 1) and 2) follow directly from the definition of z, α, β and I since

a(v)Tx+ b(v) = β(v) + α(v)T z ≥ β(v) +∑i∈I

αi(v)zi.

Point 3) follows from Lemma 3.1 which ensures that the sign of αizi(t) is constant and themodulus of αizi(t) is a time-decreasing function. This concludes the proof. �

The following example illustrates the advantages and possible limitations of the poly-hedric approach when compared to the Lyapunov-based approach.

Example 3.5: Second Order System with Real Eigenvalues

Modeling: Consider the linear system x = u subject to the control input u =

−ω2(x − v) − 2ζωx, where ω > 0 is the angular frequency and ζ > 1 is a dampingratio that ensures real eigenvalues. The system is subject to the input saturationconstraint ‖u‖ ≤ umax.

4For the sake of notational simplicity, the dependency on v will be omitted since the dynamic safetymargin is studied under the assumption that v is constant.


0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

45

t

v(t)

, x 1

(t)

Case ζ = 2

0 1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

45

t

v(t)

, x 1

(t)

Case ζ = 1.1

ERG−TrajERG−Lyap1ERG−Lyap2ERG−Poly

ERG−TrajERG−Lyap1ERG−Lyap2ERG−Poly

Figure 3.8: Applied reference (dashed) and output response (solid) for two differentvalues of the damping ratio ζ

Obtained Results: Figure 3.8 illustrates the step response for r = 42, ω = 10 s−1,umax = 100, and ζ = 2 (top) ζ = 1.1 (bottom). The following behaviours arecompared:

• Trajectory ERG: Constraints enforcement is ensured using the trajectory-based dynamic safety margin (3.35) with T = 8. This represents the optimalERG response;

• Lyapunov ERG-1: Constraints enforcement is ensured using the Lyapunov-based dynamic safety margin (2.18). The Lyapunov function was arbitrarilychosen as V = 1

2kp(x− v)2 + 12 x

2;

• Lyapunov ERG-2: Constraints enforcement is ensured using the Lyapunov-based dynamic safety margin (2.18). The Lyapunov function was assigned bysolving the optimization problem (3.53);

• Polyhedric ERG: Constraints enforcement is ensured using the dynamicsafety margin (3.56).

As expected, the Lyapunov-based dynamic safety margin provides better perfor-mances when the Lyapunov function is aligned with the constraints. As for the poly-hedric approach, it is interesting to note that the behaviour depend on the dampingratio. Indeed, the polyhedric ERG outperforms the Lyapunov ERG for ζ = 2 and isoutperformed for ζ = 1.1.


ζ 10 5 3.5 2 1.5 1.2 1.1 1.05 1.01Poly 0.1% 0.6% 2.8% 20% 45% 108% 185% 297% 770%Lyap 1.8% 14% 34% 88% 102% 102% 99% 97% 95%

Table 3.1: Settling time comparisons for the Lyapunov and polyhedric approaches

Table 3.1 compares the performances of the Lyapunov and polyhedric ERG withrespect to the optimal solution. This is done by computing the performances index

ϕ =Ti − TOptTOpt

,

where TOpt is the settling time of the trajectory-based ERG.

Interestingly enough, the polyhedric ERG performances are almost optimal for largedamping values, i.e. ζ ≥ 3.5, but rapidly deteriorate as the damping coefficientapproaches the critical value ζ = 1.As for the Lyapunov-based strategy, its performances tends to be inferior to thepolyhedric approach for large damping values. As ζ approaches the critical dampingvalue, however, the Lyapunov-based approach remains fairly consistent in its delay.

As detailed in Example 3.5, the polyhedric approach may or may not be better than theLyapunov-based approach depending on the system at hand. This is likely due to the factthat the transformation matrix T may become ill-conditioned5 when there are two similareigenvalues.

The following subsection completes the results of Proposition 3.7 by taking into accountthe presence of complex conjugate eigenvalues.

3.3.3 Cylindric Invariant set

The following Lemma addresses the properties of the eigenvalue decomposition in the caseof a state matrix A with both real and complex conjugate eigenvalues.

Lemma 3.2. Let system (3.49) be characterized by a Hurwitz state-matrix with l real eigen-values λi and m complex-conjugate pairs of eigenvalues σi ± jωi. Given the correspondingeigenvectors vi, ∀i ∈ {1, . . . , l} and wi,1, wi,2, ∀i ∈ {1, . . . ,m}, let

T =

[v1, . . . , vl,

w1,1 + w1,2

2,w1,1 − w1,2

2, . . . ,

wm,1 + wm,22

,wm,1 − wm,2

2

](3.57)

be an invertible matrix. Given given the change of coordinates z = T−1(x − xv) and thevectors

ξi =

[zl+2i−1

zl+2i

], (3.58)

the following hold true for any constant reference v:

1. sgn(zi(t)) = sgn(zi(0)), ∀t ≥ 0, ∀i ∈ {1, . . . , l};

2. |zi(t)| ≤ |zi(0)|, ∀t ≥ 0, ∀i ∈ {1, . . . , l};

3. ‖ξi(t)‖ ≤ ‖ξi(0)‖, ∀t ≥ 0, ∀i ∈ {1, . . . ,m}. �

5Please note that this is not necessarily the case. Indeed, the state matrix A = −I has two eigenvaluesin λ = −1 and yet the transformation matrix T = I is clearly invertible.


Proof. System (3.49) can be re-written as T z = ATz. Due to equation (3.57), the followingJordan decomposition applies

z = diag

{λ1, . . . , λl,

[σ1 ω1

−ω1 σ1

], . . . ,

[σm ωm−ωm σm

]}.

Since the fist l eigenvalues are real, points 1. and 2. follow directly from Lemma 3.1. As forthe remaining modes, the following behaviour applies

ξi(t) =

[cos(ωit) sin(ωit)− sin(ωit) cos(ωit)

]ξi(0)eσit, ∀i ∈ {1, ...,m}.

This is sufficient to prove point 3. since σi < 0, ∀i ∈ {1, ...,m} �

Lemma 3.2 basically defines a n-dimensional cylinder obtained by combining the poly-hedric set defined by the real eigenvalues and the ellpsoidal sets generated by the complexconjugate eigenvalues. By taking advantage of this invariance property, Proposition 3.7 canbe extended as follows.

Proposition 3.8. Let system (3.49) be characterized by a Hurwitz state-matrix and let(3.57) be an invertible transformation matrix. Given the linear constraints

a(v)Tx+ b(v) ≥ 0,

and given z, α, β, I as in (3.56), ξi equal to (3.58), and

γi

[βl+2i−1

βl+2i

],

then, the function

∆(z, v) = β(v) +∑i∈I

αi(v)zi −m∑i=1

‖γi‖ ‖ξi‖ , (3.59)

is a dynamic safety margin. �

Proof. The proof is analogous to the one of Proposition 3.7. The only thing worth notingis that Lemma 3.2 ensures ‖γi‖ ‖ξi(t)‖ ≤ ‖γi‖ ‖ξi(0)‖. �

Example 3.6: Second Order System with Complex Conjugate Eigenvalues

Modeling: Consider the second order system described in Example 3.5 subject tothe constraint x ≤ xmax and characterized by a damping coefficient ζ ∈ (0, 1).

Obtained Results: Figure 3.9 illustrates the step response for r = 42, ω = 10 s−1,xmax = 42, and ζ = 0.1 (top) ζ = 0.9 (bottom). The following behaviours arecompared:

• Trajectory ERG: Constraints enforcement is ensured using the trajectory-based dynamic safety margin (3.35) with T = 5. This represents the optimalERG response;

• Lyapunov ERG-1: Constraints enforcement is ensured using the Lyapunov-based dynamic safety margin (2.18). The Lyapunov function was arbitrarilychosen as V = 1

2kp(x− v)2 + 12 x

2;

• Lyapunov ERG-2: Constraints enforcement is ensured using the Lyapunov-based dynamic safety margin (2.18). The Lyapunov function was assigned by


solving the optimization problem (3.53);

• Cylindrical ERG: Constraints enforcement is ensured using the dynamicsafety margin (3.59).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

45

t

v(t)

, x 1

(t)

Case ζ = 0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

35

40

45

t

v(t)

, x 1

(t)

Case ζ = 0.9

ERG−TrajERG−Lyap1ERG−Lyap1ERG−Cylndr

ERG−TrajERG−Lyap1ERG−Lyap1ERG−Cylndr

Figure 3.9: Applied reference (dashed) and output response (solid) for two differentvalues of the damping ratio ζ

As expected, the Lyapunov-based dynamic safety margin provides better perfor-mances when the Lyapunov function is aligned with the constraints. As for thecylindrical approach, the behaviour is once again dependent on the damping ratio.In fact, the cylindrical ERG outperforms the Lyapunov ERG for ζ = 0.1 (pleasenote that, although the rise time is similar, the cylindrical ERG leads to very littleresidual oscillations) and is outperformed for ζ = 0.9.

ζ 0.05 0.1 0.25 0.5 0.7 0.9 0.95 0.99Cylndr 6.5% 14% 39% 87% 119% 167% 212% 445%Lyap 411% 144% 139% 84% 138% 155% 149% 144%

Table 3.2: Settling time comparisons for the Lyapunov and cylindrical approaches

Table 3.1 compares the performances of the Lyapunov and cylindrical ERG withrespect to the optimal solution. The most interesting thing to note is that thecylindrical ERG seems to perform very well for small damping values, i.e. ζ ≤ 0.1.For damping values close to ζ = 0.7, the cylindrical ERG and the Lyapunov ERGprovide similar responses. As ζ tends to the critical damping value ζ = 1, theperformances of the cylindrical approach degenerate in a similar way to Example3.5.


It is worth noting that for higher order systems subject to multiple constraints, it maybe challenging to predict which approach will perform better between the Lyapunov and theCylindrical ERG. As a result, it is typically convenient to combine the two approaches bydefining the dynamic safety margin

∆(x, v) = max(∆L(x, v),∆C(x, v)), (3.60)

where ∆L(x, v) is the Lyapunov-based dynamic safety margin (2.18) with the optimal Lya-punov matric (3.53), and ∆P (x, v) is the cylindrical dynamic safety margin (3.59). Byselecting the less restrictive dynamic safety margin, equation (3.60) will automatically selectthe best approach.

3.4 Application to Euler-Lagrange Systems

Euler-Lagrange systems are a class of nonlinear systems that appear in a large number ofmeaningful engineering problems. Due to their intrinsic properties, these systems are oftenregulated using passivity-based arguments where the idea is to design the control input sothat the energy of the closed-loop system is a time-decreasing function [3.7]. This approachhas the great advantage of exploiting the intrinsic physical properties of the system and istherefore very robust and fairly simple to implement.

The objective of this section is to show how the ERG framework can be used to augmenta passivity-based control law with a passivity-based constraint enforcement strategy. To doso, the following theorem summarizes the fundamental properties of a pre-stabilized Euler-Lagrange system.

Theorem 3.1. Consider the following pre-stabilized Euler-Lagrange system

M(q)q + C(q, q)q = −G(v, q)−D(q, q, v), (3.61)

where matrices M(q) and C(q, q) satisfy

M(q) > 0 ∀q (3.62)

M(q, q)− 2C(q, q) = Σ(q) ∀q, q, (3.63)

where Σ(q) is a skew-symmetric matrix, vector G(v, q) satisfies

G(v, v) = 0 ∀v (3.64)

G(v, q)T (q − v) > 0 ∀v, ∀q 6= v (3.65)

∂Gi∂qj

=∂Gj∂qi

∀v, q, (3.66)

and vector D(q, q, v) satisfies

D(q, 0, v) = 0 ∀v, q (3.67)

D(q, q, v)T q > 0 ∀v, q, ∀q 6= 0. (3.68)

Then, given a constant reference v, it follows that [q, q] = [v, 0] is an asymptotically stableequilibrium point and that

V (q, q, v) =

∫ q

v

G(v, ξ)Tdξ +1

2qTM(q)q (3.69)

is a Lyapunov function. �

3.4. Application to Euler-Lagrange Systems 49

Proof. Following from (3.66), G(v, q) is a conservative vector field which admits a scalarpotential field

P(v, q) =

∫ q

v

G(v, ξ)Tdξ. (3.70)

Moreover, due to conditions (3.64)-(3.65), P(v, q) is positive-definite. Likewise, it followsfrom (3.62) that

K(q, q) =1

2qTM(q)q (3.71)

is positive-definite. As a result, (3.69) is a Lyapunov candidate function. Given a constantreference v and taking into account (3.63), the time derivative of V (q, q, v) is

V (q, q, v) = −D(q, q, v)Tq, (3.72)

which is negative semi-definite due to conditions (3.67)-(3.68). Asymptotic stability followsfrom the Krasovskii-LaSalle principle since q = 0 leads to the accelerationM(q)q = −G(v, q),which is non-zero for all q 6= v. �

Remark 3.8. It is worth noting that all the requirements of system (3.61) have a veryclear physical interpretation. Indeed, it follows from equations (3.62)-(3.63) that M(q) andC(q, q) are the classical mass and Coriolis matrices. As such, K(q, q, v) in (3.71) is thekinetic energy stored by the system.Following from (3.64)-(3.66), the term G(v, q) can be interpreted as a “virtual spring” thatconnects the current position q to the reference v. As such, P(v, q) in equation (3.70) is thepotential energy stored by the control action.Finally, it follows from (3.67)-(3.68) that D(q, q, v) can be interpreted as a “virtual damper”which always opposes movement. As such, this term is responsible for dissipating the totalenergy (3.69) of closed-loop system. �

Theorem 3.1 provides a Lyapunov function that is suitable for any system in the form(3.61). As discussed in Remark 3.8, this Lyapunov function has the great advantage of havinga very clear physical interpretation. To maintain this property, it is therefore advantageousto implement the Lyapunov-based Explicit Reference Governor explained in Section 3.1.Indeed, given the dynamic safety margin ∆(q, q, v) = Γ(v)−V (q, q, v), the resulting ERG isguaranteed to enforce

V (v(t), q(t), q(t)) ≤ Γ(v(t)), ∀t, (3.73)

where Γ(v) is the solution to the optimization problem{Γ(v) = minq,q V (q, q, v), s.t.c(q, q, v) ≤ 0.

(3.74)

In other words, the ERG will manipulate the auxiliary reference so that the pre-stabilizedsystem will never have enough energy to violate the constraint c(q, q, v) ≥ 0.

In addition to the physical interpretation, the main interest in focusing on Euler-Lagrangesystems is that the Lyapunov function (3.69) has a very specific structure that can beexploited to compute the threshold value Γ(v). The following subsections will provide furtherdetails on how to solve the optimization problem (3.74) in the presence of a pre-stabilizedEuler-Lagrange system.

3.4.1 Optimal Threshold Values

In analogy with Subsection 3.1.1, this subsection characterizes under what conditions theoptimization problem (3.74) can be solved efficiently. The following proposition details aspecific case in which the optimal solution can be obtained analytically.


Proposition 3.9. Let system (3.61) satisfying (3.62)-(3.68) be subject to the constraint

qi ≤ d(v) or qi ≥ d(v). (3.75)

Then, the solution to the optimization problem (3.74) is

Γ(v) =

∫ d(v)

vi

G(v, vi|ξ)dξ, (3.76)

where

vi|ξ = [v1, . . . , vi−1, ξ, vi+1, . . . vn]T.

Proof. Equation (3.76) is equal to the Lyapunov function (3.69) evaluated in

qi = d(v)qj = vj ∀j 6= iqj = 0 ∀j.

Due to conditions (3.64)-(3.65), it follows that

P(v, q) ≥∫ d(v)

vi

G(v, vi|ξ)dξ, ∀v, ∀q : qi = d(v).

Likewise, it follows from (3.62) that K(q, q) ≥ 0, ∀q, q. As a result,

V (q, q, v) ≥ Γ(v), ∀v, q, ∀q : qi = d(v).

This concludes the proof. �

Proposition 3.9 basically states that any constraint on a single degree of freedom can beenforced by ensuring that the total energy of the system is inferior to the potential energy ofthe constraint itself. Although this result seems relatively simple, the proposition is highlyrelevant because it is fairly common to find constraints in the form (3.75). Indeed, theoptimal threshold value presented in Example 2.1 was obtained using equation (3.76). Thefollowing example provides another application of the results given in Proposition 3.9.

Example 3.7: Overhead Crane

Consider the overhead crane illustrated in Figure 3.10. Given a desired referencer, design a control law such that the horizontal coordinate x asymptotically tendsto r while ensuring that the sway angle θ never violates the security requirement|θ| ≤ θM .


Figure 3.10: Side view of an overhead crane.

Modeling: The state-space model can be obtained using the Euler-Lagrange ap-proach. To do so, define the potential energy

EP = mgL cos θ (3.77)

and the kinetic energy

EK =1

2mx2 +

1

2mL2θ2. (3.78)

Given the Lagrangian L = EK−EP , the dynamic model can be obtained by computing

d

dt

∂L

∂qi=∂L

∂qi+ Fi,

where Fi is the external force acting on the i-th degree of freedom. Given Fx = uand Fθ = 0, the following model is obtained:[

M+m −mL cos θ−ml cos θ mL2

][x

θ

]+

[0 −mL sin θθ0 0

][x

θ

]=

[0

−mgL sin θ

]+

[u0

]Step 1, Pre-Stabilization: The system can be pre-stabilized using a standard PDcontrol law on the chariot position, i.e.

u = −kP (x− v)− kDx.

The resulting closed-loop system is in the form (3.61) with

M(q) =

[M +m −mL cos θ−ml cos θ mL2

]G(v, q) =

[kp(x− v)mgL sin θ

]C(q, q) =

[0 −mL sin θθ0 0

]D(q, q, v) =

[kdx0

].

Given the equilibrium point (x, θ, x, θ) = (v, 0, 0, 0), asymptotic stability can provenusing the Lyapunov function (3.69), which becomes

V (q, q, v) =1

2kP (x− v)2 +mgL(1− cos θ) +

1

2qTM(q)q. (3.79)

Please note that x = 0 implies D(q, q, v) = 0 regardless of θ. Although condition(3.68) is not verified in this particular case, it has been proven [3.8] that the Lyapunovfunction (3.79) remains applicable.


Step 2, Constraint Enforcement: Given the pre-stabilized system, it follows fromProposition 3.9 that the constraint |θ| ≤ θM can be enforced by computing thethreshold vale

Γ(v) = mgL(1− cos θM)

and applying the basic ERG scheme v = κ(Γ(v)− V (q, q, v)) r−vmax(‖r−v‖,η) .

0 5 10 15 20 25 30−2

0

2

4

6

8

10

12

t [s]

Pos

ition

[m]

Cart Dynamics

0 5 10 15 20 25 30−150

−100

−50

0

50

100

t [s]

Load

Sw

ay [d

eg]

Load Dynamics

v(t) No RGx(t) No RGv(t) Passive ERGx(t) Passive ERG

α(t) No RGα(t) Passive ERG|α(t)|≤α

max

Figure 3.11: Behaviour of the cart and the load of the controlled overhead crane.

Obtained Results: The proposed scheme was validated numerically using the pa-rameters M = 10 kg, m = 3 kg, L = 2 m, kP = 42.12 N/m, and kD = 46.8 Ns/m. Thesystem initialized in the origin and is provided with the desired reference r = 10 m.The maximum admissible sway is θM = 45 deg. Figure 3.11 compare the behaviourthat is obtained using


• Passive ERG: The auxiliary reference v(t) is generated using the ExplicitReference Governor proposed in this section, with κ = 102, η = 1, and v(0) =x(0).

As expected, the passive ERG successfully guarantees passive constraint enforcementby limiting the total energy of the closed loop system.

To extend the range of problems for which it is numerically feasible to find the exactoptimum, the following proposition states under what conditions the optimization problem(3.74) is convex.


Proposition 3.10. Let the Lyapunov function (3.69) satisfy

M(q) = M > 0, (3.80)

JG(v, q) > 0, ∀v, q (3.81)

where JG(v, q) is the Jacobian of G(v, q) with respect to6 q. Then, if c(q, q, v) is a convexfunction, the optimization problem (3.74) is convex. �

Proof. Since the function c(q, q, v) is convex, the statement can be proven by showing thatV (q, q, v) is a convex function for any fixed v. This can be done by computing the Hessianmatrix with respect to q, q, i.e.

HV (q, q, v) =

[JG(v, q) 0

0 M

].

Following from conditions (3.80)-(3.81), HV (q, q, v) > 0, ∀v, q. Therefore, the Lyapunovfunction (3.69) is convex. �

Proposition 3.10 provides sufficient conditions for checking if the optimization problem(3.74) in numerically tractable. However, the main limitation for applying this result is thatit requires a constant mass matrix. The following subsection overcomes this limitation byproviding suboptimal estimates of the threshold value Γ(v) for any mass matrix.

3.4.2 Sub-Optimal Threshold Values

In analogy to Subsection 3.1.3, this subsection provides a few strategies for estimating asub-optimal Lyapunov threshold Γ(v). In particular, the following proposition provides asub-optimal formulation of the problem (3.74) that is convex for any mass matrix M(q).

Proposition 3.11. Let the Lyapunov function (3.69) satisfy (3.81) and

M(q) ≥M, ∀q, (3.82)

with M > 0. Then, if c(q, q, v) is a convex function, the optimization problem{Γ(v) = minq,q

∫ qvG(v, ξ)Tdξ + 1

2 qTMq, s.t.

c(q, q, v) ≤ 0(3.83)

is convex and its solution is such that

V (q, q, v) ≤ Γ(v)⇒ c(q, q, v) ≥ 0.

Proof. Following from Proposition 3.9, (3.83) is a convex optimization problem. Moreover,the optimization problem is such that∫ q

v

G(v, ξ)Tdξ +1

2qTMq ≤ Γ(v) ⇒ c(q, q, v) ≥ 0.

To conclude the proof, it is sufficient to note that∫ q

v

G(v, ξ)Tdξ +1

2qTMq ≤ Γ(v) ≤ V (q, q, v)

due to equations (3.69) and (3.82). �

6Please note that the optimization problem is defined for any fixed value of v.


Proposition 3.11 has the great advantage of being applicable to the majority of pre-stabilized Euler-Lagrange systems. Indeed, the control actionG(v, q) can usually be designedto satisfy the requirement (3.81). Moreover, condition (3.82) can always be satisfied bychoosing

M = minq{λ1{M(q)}} In, (3.84)

where λ1{A} is the smallest eigenvalue of a given matrix A and In is an identity matrix ofsize n. The following proposition provides a closed-form estimate of Γ(v) for those cases inwhich it is preferable to avoid solving an on-line optimization problem.

Proposition 3.12. Let the Lyapunov function (3.69) satisfy (3.82) and

JG(v, q) ≥ Φ(v), ∀v, q, (3.85)

with Φ(v) > 0, ∀v. Then, given c1(v), c2(v) and d(v) such that

c1(v)T q + c2(v)T q + d(v) ≥ 0 ⇒ c(q, q, v) ≥ 0,

c(v, v, 0) ≥ δ ⇒ c(q, q, v) ≥ δ, (3.86)

the threshold value

Γ(v) =1

2

(c1(v)T v + d(v))2

c1(v)TΦ(v)−1c1(v) + c2(v)TM−1c2(v), (3.87)

is such thatV (q, q, v) ≤ Γ(v)⇒ c(q, q, v) ≥ 0,

and ∃ε : c(v, v, 0) ≥ δ ⇒ Γ(v) ≥ ε.

Proof. Following from conditions (3.82), (3.85), the Lyapunov function (3.69) is such that

V (q, q, v) ≥ 1

2(q − v)TΦ(v)(q − v) +

1

2qTMq.

The rest of the proof follows directly from Proposition 3.3. �

The following example illustrates the use of Proposition 3.12 for performing constrainedcontrol of robotic manipulators. The example also addresses how to deal with multipleconstraints.

Example 3.8: Robotic Manipulator

Given a generic robotic manipulator

M(q)q + C(q, q)q = −W (q) + τ,

subject to the input constraints|τi| ≤ τi,

design a control law that reaches the desired reference r without violating the con-straints.

Step 1, Pre-Stabilization: The robotic manipulator can be pre-stabilized using aPD with desired gravity compensation [3.9]

τ = W (v)− (KW +KP )(q − v)−KD q,

with KP ,KD > 0 andKW ≥ JW (q). (3.88)


Following from Theorem 3.1, asymptotic stability of the point (q, q) = (v, 0) can beproven using the Lyapunov function

V (q, q, v) =

∫ q

v

(W (ξ)−W (v))Tdξ +1

2(q−v)T(KW +KP )(q−v) +

1

2qTM(q)q, (3.89)

where

P(v, q) =

∫ q

v

(W (ξ)−W (v))Tdξ +1

2(q−v)T(KW +KP )(q−v).

Step 2, Constraint Enforcement: Given a generic mass matrix M(q), finding theoptimal threshold value for each constraint may require the solution of a non-convexoptimization problem. However, by taking advantage of the mass matrix lower-bound M satisfying (3.82), it is always possible to compute analytic estimates of eachindividual threshold Γi(v). Following from Proposition 3.12 , a possible approach isto take advantage of the control requirement (3.88) which implies

J (W (q)−W (v) + (KW +KP )(q − v)) ≥ KP .

Since the Lyapunov function (3.89) can be lower bounded by the quadratic form

V (q, q, v) ≥ 1

2(q − v)TKP (q − v) +

1

2qTMq.

This leads to the following threshold value

Γi(v) =1

2

(τi − |Wi(v)|)2

cT1 K−1P c1 + cT2 M

−1c2, (3.90)

where cT1 = [KW + KP ]i and cT2 = [KD]i. Please note that if KP and KD arediagonal matrices (which is a fairly typical design choice), equation (3.90) can befurther simplified into

Γi(v) =1

2

(τi − |Wi(v)|)2

[KW +KP ]2ii / [KP ]ii + [KD]2ii / [M ]ii.

Figure 3.12: Representation of a two-link planar robot.

Modeling and Control: To perform the simulations, consider the planar two-linkrobot illustrated in Figure 3.12. As shown in [3.10], the dynamic model of a two-link


planar robot is described by the mass and Coriolis matrices

M(q) =

[µ1 + µ2 + 2ν cos q2 µ2 + ν cos q2

µ2 + ν cos q2 µ2

]C(q, q) =

[−ν sin q2q2 −ν sin q2(q1 + q2)ν sin q2q1 0

],

whereµ1 = J1 +m1r

21 +m2l

21 µ2 = J2 +m2r

22 ν = m2l1r2,

and the weight vector

W (q) =

[η1 cos q1 + η2 cos(q1 + q2)

eta2 cos(q1 + q2)

],

withη1 = m1gr1 +m2gl1 η2 = m2gl2.

The proposed gain matrices for the PD with desired gravity compensation are

KW =

[η1 + η2 η2

η2 η2

]KP =

[kp1 00 kp2

]KD =

[kd1 00 kd2

].

By solving the integral in equation (3.89), the resulting Lyapunov function for theclosed-loop system is

V (q, q, v) =1

2(q − v)TKP (q − v) + qTM(q)q + η1S(v1, q1) + η2S(v1+v2 , q1+q2),

where S(a, b) = sin b − sin a − cos a(b − a) + 12 (b − a)2 is a positive semi-definite

function.

0 1 2 3 4 5 620

30

40

50

60

70

80

90

t [s]

q 1 [deg

]

0 1 2 3 4 5 6−140

−120

−100

−80

−60

−40

−20

t [s]

q 2 [deg

]

No RGERGq

1 ≥ 40

No RGERGq

2 ≤ −45

Figure 3.13: Applied reference (dashed) and output response (solid) of the two links.

3.5. Summary 57

Obtained Results: The parameters used for the simulation are m1 = 4, m2 = 3,l1 = 0.4, l2 = 0.3, kp1 = 65, kp2 = 45, kd1 = 3.22, kd2 = 1.34, κ = 103, η = 10−3.The system is subject to the constraints q1 ≥ 45, q2 ≤ 35, ‖u1‖ ≤ 35, ‖u2‖ ≤ 25.Figures 3.13 and 3.14 compare the behaviour that is obtained using


• ERG: The Lyapunov-based ERG is implemented using (3.90).

As shown by the simulations, the ERG strategy successfully enforces the systemconstraints.

0 1 2 3 4 5 6−50

0

50

t [s]

u 1 [Nm

]

Input Dynamics, Link 1

0 1 2 3 4 5 6−30

−20

−10

0

10

20

30

40

t [s]

u 2 [Nm

]

Input Dynamics, Link 2

No RGERG||u

1|| ≤ 35

No RGERG||u

2|| ≤ 25

Figure 3.14: Torques applied to the two robot joints.

3.5 Summary

This chapter has provided several alternatives for systematically constructing a dynamicsafety margin. The advantages and disadvantages of each approach are as follows:

• Trajectory-Based approaches are able to achieve optimal performances and are rel-atively simple to implement. However, these schemes have moderately high computa-tional requirements and strongly depend on the accuracy of the system model;

• Lyapunov-Based approaches are highly systematic, present limited computationalcosts and are inherently robust due to their conservative nature. However, the maindisadvantage is that their performances strongly depend on the quality of the Lyapunovfunction, which may be difficult to find in some cases;


• Invariance-Based approaches are extremely flexible due to their general formulation.As a result, they can potentially achieve high performances with relatively limitedcomputational expense. The main challenge consists in being able to identify a suitableinvariant set;

• Returnability-Based approaches rely on the most basic definition of stability. Thesemethods have the advantage of being relatively simple to implement in practice, buthave the disadvantage of being very conservative;

• Feedforward-Based approaches have the advantage of not needing any measurementof the system states. Since they rely on the worst-case scenario, these methods willachieve fairly limited performances.

Specific dynamic safety margins for two highly relevant classes of systems, i.e. linear andEuler-Lagrange, have also been proposed. The effectiveness of the proposed methods areillustrated with the aid of several numerical simulations.

Bibliography

[3.1] F. Blanchini, “Set invariance in control,” Automatica, vol. 35, no. 11, pp. 1747–1767,1999.

[3.2] T. C. Gard and V. Lakshmikantham, “Strongly flow-invariant sets,” Applicable Anal-ysis: An International Journal, vol. 10, no. 4, pp. 285–293, 1980.

[3.3] E. Garone, F. Tedesco, and A. Casavola, “Sensorless supervision of linear dynamicalsystems: The feed-forward command governor approach,” Automatica, vol. 47, no. 7,pp. 1294–1303, 2011.

[3.4] D. G. Schultz and J. E. Gibson, “The variable gradient method for generating liapunovfunctions,” Transactions of the American Institute of Electrical Engineers, vol. 81, no. 4,pp. 203–210, 1962.

[3.5] P. J. Antsaklis and A. N. Michel, A Linear Systems Primer. Birkhauser, 2007.

[3.6] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press,2004.

[3.7] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez, Passivity-based Controlof Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications.Springer, 1998.

[3.8] Y. Fang, E. Zergeroglu, W. E. Dixon, and D. M. Dawson, “Nonlinear coupling controllaws for an overhead crane system,” in Proc. of the IEEE International Conference onControl Applications, 2001, pp. 639–644.

[3.9] R. Kelly, “Pd control with desired gravity compensation of robotic manipulators: areview,” International Journal of Robotics Research, vol. 16, no. 5, pp. 660–672, 1997.

[3.10] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo, Robotics: Modelling, Planningand Control. Springer, 2009.

Chapter 4

Attraction Field

The objective of this chapter is to provide a systematic design framework for the attractionfield ρ(v, r). Unlike the problem addressed in the previous chapter, this problem is relativelywell-known and is the object of an extensive literature.

Indeed, by defining the domain of admissible steady-state equilibria

D = {v : c(xv, v) ≥ δ} , (4.1)

the construction of a suitable attraction field is equivalent to the generation of a continuouspath that connects any point v ∈ D to any desired reference r ∈ D without exiting D. Thisis known as the motion planning problem.The following section will provide a brief overview of the existing results and identify ap-propriate solutions for generating ρ(v, r).

4.1 Motion Planning Overview

Motion planning algorithms have the general objective of generating a path that connectsa point v to a desired reference r without exiting the constraints. Existing method canroughly be divided into two main categories: roadmap methods and potential field methods[4.1, 4.2].

Roadmap methods consist in building a network of waypoints connected by collision freepaths. This reduces the path-planning problem to the selection of a suitable sequence ofintermediate waypoints. This selection is typically done using graph theory or similar toolsand often assumes that the structure of the environment is not known a priori [4.3, 4.4, 4.5].Section 4.2 illustrates how to construct an attraction field starting from an existing roadmap.For the reader’s convenience, an intuitive (and non comprehensive) roadmap generationmethod is briefly summarized.

Potential field methods consist in generating a combination of attraction and repulsionforces to guide a robot to a target without colliding into obstacles.The Artificial Potential Field (APF) method was originally introduced in [4.6] and only relieson local knowledge. Although quite popular in the field of mobile robotics, a well-knownlimitation of the APF approach is the existence of undesired local minima which may causethe system to stagnate [4.7]. Possible solutions to this problem consist in complementingthe algorithm with suitable escaping mechanisms [4.8].An alternative solution consist in designing the potential field so that the stagnation pointsare all locally unstable. This is known as the Navigation Function (NF) method which wasoriginally introduced in [4.9]. Although the method typically assumes global knowledge of

59

60 4. Attraction Field

the constraints, several solutions for navigating in unknown environments have also beenproposed [4.10, 4.11].

Due to its elegance and simplicity, the NF philosophy was selected as the preferred solu-tion for generating the attraction field ρ(v, r). Therefore, a brief summary of the method willbe provided in Section 4.3. Sections 4.4 and 4.5 will then propose some minor modificationswhich are required to incorporate the existing NF approach into the ERG framework.

The possibility of using hybrid roadmap/potential field solutions will also be discussed.

4.2 Roadmap Methods

Given a connected domain D, roadmap methods provide a succession of waypoints gk suchthat the straight line connecting two consecutive waypoints gk to gk+1 is strictly containedin D. The current position v and the desired objective r are respectively defined as theinitial and final waypoints of the succession.

A possible method for generating the roadmap consists in performing the following steps:

1. Divide the domain D into multiple overlapping convex domains Di;

2. Build a graph where each node represents a domain Di and two nodes i− j are linkedif and only if Di ∩ Dj 6= ∅;

2

1

3

4

11

5

7

6

12

8

10

0

9

1

2

3 4

5

6

7

8

9

10

11 12

𝒗 𝒈𝟏

𝒈𝟐 𝒈𝟑 𝒈𝟒

𝒓

𝒓

𝒗

𝒈𝟏

𝒈𝟐 𝒈𝟑 𝒈𝟒

Figure 4.1: Illustrative example a roadmap generation algorithm.Upper Left: Original non-convex domain; Lower Left: Partition into overlapping convexsubdomains; Right: Creation of a connectivity graph and waypoint selection.

4.3. Navigation Functions 61

3. Given v ∈ Dv and r ∈ Dr, determine a sequence (possibly the shortest) of domains onthe graph that connects Dv and Dr;

4. Given the succession of domains i− j that lead from Dv to Dr, assign each waypointby choosing gk ∈

∫{Di ∩ Dj}.

An illustration of this method is provided in Figure 4.1. Additional information and alter-native methods can be found in [4.1, 4.2]. Given a succession of waypoints, regardless ofhow they are obtained, Definition 2.2 can be satisfied by choosing the attraction field

ρ(v, gk) =gk − v

max{‖gk − v‖ , η}, (4.2)

which ensures that v will enter an arbitrarily small neighborhood of gk in finite time. Sincethe segment gk, gk+1 is strictly contained in D, there exists a finite time in which the segmentv, gk+1 will also strictly belong to the domain D. As soon as this happens, equation (4.2)can switch to the next waypoint gk+1. Asymptotic convergence is then proven recursively.

Although the is no particular challenge in generating an attraction field starting from anexisting roadmap, it is worth noting that the generation of the roadmap itself is typicallya computationally intensive task. Due to the analytical philosophy of the ERG, furtherinvestigations on the subject will be omitted in favor of the NF approach.

4.3 Navigation Functions

The Navigation Function is a motion planning tool which was originally presented in [4.9].The objective of this section is to provide a brief summary of the method and identifypotential obstacles to its implementation within the ERG framework. The following theoremis the general basis for the classical NF approach.

Theorem 4.1. Given a domain D which must not be violated, let the Navigation FunctionN : D → R be

1. Smooth, i.e. N(v) is a class C2 function;

2. Uniformly bounded, i.e. ∀v ∈ ∂D, N(v) = Nmax;

3. Polar in r ∈ D \ ∂D, i.e. N(v) has a unique minimum in v = r;

4. Morse, i.e. ∀v : ∇N(v) = 0, the Hessian ∇2N(v) is nonsingular.

Then, for almost any initial condition v(0) ∈ D, the system v = −∇N(v), will asymptoticallyconverge to r without exiting D. �

Proof. An intuitive sketch of the proof reported in [4.9] can be summarized as follows:

Condition 2 guarantees constraint satisfaction. Indeed, since N(v) is maximum on theconstraint boundary, following its anti-gradient will ensure v(t) ∈ D, ∀t.

In the absence of saddle points1, condition 3 guarantees asymptotic stability. Indeed, sinceN(v) does not have any local minima besides r ∈ D, following its anti-gradient ensureslimt→∞ v(t) = r.

In the presence of saddle points, condition 4 ensures that their basin of attraction is a set ofmeasure zero. This implies limt→∞ v(t) = r for “almost any” initial condition v(0) ∈ D. �

1A saddle point s is a point where the gradient is zero and the Hessian matrix has at least one positiveand one negative eigenvalue.


Due to its simplicity and closed-form nature, the Navigation Function is a solid startingpoint for constructing an attraction field ρ(v, r). Nevertheless, before the method can beimplemented in the ERG framework, there are a few limitations which should be addressed:

• Condition 1 can be relaxed to a class C1 functions if∇2N(v) exists wherever∇N(v) = 0[4.12].

• Condition 2 introduces a dependency between all the points belonging to the bound-ary ∂D. Since this requirement complicates the construction of N(v), it would bepreferable to find a weaker formulation;

• Condition 3 does not take into account the case r /∈ D. As such, the theory should beextended so that, ∀r ∈ Rp, v(t) asymptotically tends to r? defined in (2.7);

• Condition 4 can be relaxed. A weaker formulation is that all the points satisfying∇N(v) = 0 must be isolated [4.13];

• Condition 4 implies the existence of a set of measure zero that converges to the sad-dle points instead of the global minimum. In practical applications, this is typicallydeemed irrelevant since the undesired trajectories will be destabilized by the systemnoise. However, since the ERG implements v = ρ(v, r) numerically, the possibility ofgetting stuck in a saddle point becomes much more problematic.

To overcome these limitations, the following definition provides a generalization of theclassical Navigation Function considered in Theorem 4.1.

Definition 4.1. Navigation Function: Given the domain (4.1), a class C1 functionN : D × Rp → R is a “Navigation Function” if it is

• Invariant, i.e.∇vN(v, r)T ∇c(xv, v) ≤ 0, v ∈ ∂D; (4.3)

• Polar in r? satisfying2 (2.7), i.e. ∀v ∈ D : ∇vN(v, r) = 0,

∇2vN(v, r) > 0 ⇔ v = r?. (4.4)

• Practically Morse, i.e. ∀v ∈ D\{r?} : ∇vN(v, r) = 0, there exists a sufficiently smallε > 0 such that ∇vN(v + εu, r) 6= 0,∀u ∈ Rp : ‖u‖ = 1. �

The following sections of this chapter will illustrate how to adapt the results of Theorem4.1 to the navigation function proposed in Definition 4.1. Since the behaviour of the Navi-gation Function varies depending on the topology of the domain D, the following definitionsare recalled:

Definition 4.2. Simply Connected Domain: A domain D is “Simply Connected” if:

• any two points in the set D can be connected by a continuous path;

• given any closed curve3 belonging to the set D, its interior also belongs to D. �

Definition 4.3. Multiply Connected Domain: A domain D is “n-Multiply Connected”if:

• any two points in the set D can be connected by a continuous path;

2For the sake of convenience, it is recalled that equation (2.7) requires r? to be the solution of min ‖r − r?‖subject to c(xr? , r

?) ≥ δ.3Given a domain of dimensions p, a “curve” is to be interpreted as a surface of dimensions p− 1.

4.4. Simply Connected Constraints 63

Figure 4.2: Simply Connected domain (left), and a 2-Multiply Connected domain (right).

• it can be described the difference between a simply connected set S0 and n distinct4

simply connected sets Si ⊂ S0. �

Roughly speaking, a domain is simply connected if it does not contain holes. Instead, adomain with n holes is n-multiply connected. Given these definitions, the following resultis reported:

Theorem 4.2. Given a simply connected domain D, there exists a navigation functionN(v) such that ∇N(v) = 0 if and only if v = r?. Given a n-multiply connected domainD, any navigation function N(v) must contain at least n saddle points vSi 6= r? such that∇N(vSi) = 0 for i = 1, . . . , n. �

Proof. See [4.9, Corollary 2.3]. �

By taking advantage of this result, Section 4.4 will address the properties of the proposedNavigation Function in the absence of saddle points. Section 4.5 will then address thepresence of saddle points. In addition to purely theoretical results, both sections will providesuitable attraction fields for some relevant classes of domains D.

4.4 Simply Connected Constraints

The objective of this section is to adapt the classical NF approach summarized in Section4.3 so that it can be implemented within the ERG framework. Given the Definition 4.1,Theorem 4.1 can be modified as follows:

Theorem 4.3. Given a simply connected domain D, let N : D × Rp → R be a navigationfunction such that ∇vN(v, r) = 0⇔ v = r?. Then, ∀v(0) ∈ D and ∀r ∈ Rp, the system

v = −∇vN(v, r), (4.5)

will asymptotically converge to r? without exiting D. �

4Si ∩ Sk = ∅, ∀i = 1, . . . , n, ∀k = 1, . . . , n, k 6= i.


Proof. Following from equations (4.3) and (4.5), v(t) cannot exit the domain D.To prove asymptotic stability, consider the navigation function N(v, r). Following from(4.5), its derivative

N(v, r) = −∇vN(v, r)T∇vN(v, r).

is negative definite ∀v ∈ D; v 6= r?. As a result, v(t) will asymptotically tend to r? for allinitial conditions v(0) ∈ D. �

Remark 4.1. Please note that, due to the absence of saddle points, the practically Morserequirement in Definition 4.1 is not necessary for the proof of Theorem 4.3. This requirementwill be motivated in Section 4.5. �

As a direct result of Theorem 4.3, an attraction field ρ(v, r) can be computed by buildinga suitable NF and then choosing

ρ(v, r) = −∇vN(v, r). (4.6)

Theorem 4.3 is typically used by proposing a Navigation Function first and then computingits derivative [4.9]. However, it is interesting to note that it can also be used to validatean existing attraction field by showing that it is the gradient of a navigation function. Bytaking advantage of this intuition, the following subsections will design suitable attractionfields for certain classes of simply connected domains.

4.4.1 Convex Domains

Convex domains constitute a highly relevant class of simply connected domains. Theirdefinition is as follows:

Definition 4.4. Convex Domain: A domain D is “Convex” if, for any two points in D,the straight line between them is wholly contained in D. �

The main advantage of convex domains is that, any convex function defined in a convexdomain contains only one stationary point, which is also the minimum. This propertyensures that, if ρ(v, r) is the gradient of a convex function and ρ(r?, r) = 0, r? will be theonly local minimum in the domain D.

Since the sum of convex functions is also a convex function, the problem can be ap-proached by decoupling attraction field as

ρ(v, r) = ρr(v, r) + ρ0(v), (4.7)

where ρr(r, v) is a vector field which points towards the reference, and ρ0(v) is a vector fieldwhich points away from the boundary ∂D. The two terms can then be designed separately.

For what regards the attraction term ρr(r, v), it is convenient to choose a smooth functionwith a norm that is lower or equal to one. Both requirements are satisfied by the basicattraction field introduced in Chapter 2, i. e.

ρr(r, v) =r − v

max{‖r − v‖ , η}. (4.8)

Given a smoothing margin η > 0, the vector field (4.8) has a unitary norm for ‖v − r‖ ≥ η,and continuously tends to zero for v → r.

For what regards the repulsion term ρ0(v), its direction should be parallel to ∇vc(xv, v).To dominate the attraction term (4.8) in proximity of the constraints, its modulus should


be equal to one whenever c(xv, v) = δ.Assuming that the constraint effects should be local rather than global, it is reasonable todefine an influence margin ζ > δ outside of which the constraint repulsion is zero. Assumingthat there exists a scalar ∇cδ > 0 such that, ∀v : c(xv, v) ∈ [0, ζ],

(ζ − c(xv, v)) ‖∇vc(xv, v)‖ = (ζ − δ)∇cδ ⇔ c(xv, v) = δ, (4.9)

the following repulsion term is proposed:

ρ0(v) = max

{ζ − c(xv, v)

ζ − δ, 0

}∇vc(xv, v)

∇cδ. (4.10)

Remark 4.2. It is worth noting that the classical NF approach assumes c(xv, v) to be theEuclidean distance between v and the constraint boundary. Since no other types of constraintfunctions are not even considered, the requirement (4.9) is more general than the existingtheory.If c(xv, v) does not satisfy (4.9), the proposed approach can be recovered by using the Eu-clidean distance between v and the level-set {v : c(xv, v) = 0} as an alternative constraintfunction c(v). Since the domain is convex, c(v) is fairly simple to compute. �

Having designed its two components separately, the following proposition states that(4.7) is an attraction field for any set of convex constraints.

Proposition 4.1. Let constraints c(xv, v) be a class C1 function such that the domain (4.1)is convex and (4.9) holds true. Then, given a sufficiently small smoothing margin η > 0,the vector field (4.7)-(4.8), and (4.10) satisfies Definition 2.2, ∀r : c(xr, r) /∈ (ζ, δ]. �

Proof. The vector field (4.8) is the anti-gradient of the potential function

Pr(v, r) =

‖v − r‖ , if ‖v − r‖ ≥ η;

1

2

‖v − r‖2 + η2

η, if ‖v − r‖ < η.

(4.11)

Likewise, the vector field (4.10) is the anti-gradient of the potential function

Pc(v) =

0, if c(xv, v) ≥ ζ;1

2

(c(xv, v)− ζ)2

ζ − δ, if c(xv, v) ∈ [0, ζ).

(4.12)

Given N(v, r) = Pr(v, r) + P0(v), the statement can be proven by showing that N(v, r)satisfies all the requirements of Definition 4.1.

For what regards the border invariant property, consider

∇vN(v, r)T ∇c(xv, v) = − (ρr(v, r) + ρ0(v))T ∇c(xv, v). (4.13)

Following from (4.8), ‖ρr(v, r)‖ ≤ 1. Moreover, it follows from (4.9)-(4.10) that ‖ρ0(v)‖ = 1,∀v ∈ ∂D. Since ρ0(v) is always parallel to ∇c(xv, v), (4.13) satisfies condition (4.3), thusproving that N(v, r) is border invariant.

To prove that N(v, r) is polar in r?, it is sufficient to prove ∇vN(r?, r) = 0. Indeed,since D is convex and (4.11)-(4.12) are also convex, N(v, r) admits only one local minimumin the domain D. Consider the following two cases:

r : c(xr, r) ≥ ζ. In this case, it follows from (4.8) and (4.10) that ρr(r, r) = 0 andρ0(r) = 0. As a result, ∇vN(r, r) = 0. Since the desired reference r is contained within thesafety margin δ, r? = r is the solution to (2.7).


r : c(xr, r) < δ. In this case, r /∈ D is not an admissible solution to (2.7). To ensure∇vN(r?, r) = 0, r? ∈ D must be such that ρr(r

?, r) = −ρ0(r?). Given a sufficiently smallsmoothing margin η > 0, it follows from (4.8) that ‖ρr(v, r)‖ = 1, ∀v ∈ D. Following from(4.9)-(4.10), ‖ρ0(v)‖ = 1 can only be satisfied for v ∈ ∂D. Moreover, since ρ0(v) is alwaysdirected along the constraint gradient, ρr(r

?, r) = −ρ0(r?) can only be achieved for

r? ∈ ∂D :r − r?

‖r − r?‖= − ∇vc(xv, v)

‖∇vc(xv, v)‖.

Since ∂D is convex, r? is the solution to equation (2.7).

Since ∇vN(v, r) = 0 ⇔ v = r?, no additional effort is needed to prove that N(v, r)is practically Morse. Since N(v, r) is a Navigation Function and D is a simply connecteddomain, the rest of the proof follows from Theorem 4.3. �

Remark 4.3. Proposition 4.1 does not address the case r : c(xr, r) ∈ [δ, ζ). Indeed, eventhough r ∈ D, the proposed law does not ensure ∇vN(r, r) = 0 because ρr(r, r) = 0 whereasρ0(r) 6= 0. In other words, if r belongs to the region contained between the safety margin δand the influence margin ζ, v(t) will asymptotically tend to r? 6= r.Nevertheless, two things are worth mentioning. First of all, the region can be made arbitrarilysmall by choosing a sufficiently small margin ζ − δ > 0. More importantly, it can be shownthat if

‖ρ0(v)‖ < 1, ∀v : c(xv, v) ∈ [δ, ζ),

the error between the desired r and the obtained r? will satisfy ‖r − r?‖ ≤ η. Since η > 0can be chosen arbitrarily, Definition 4.1 is “practically” satisfied ∀r ∈ Rp.Please note that (4.3) is satisfied ∀r ∈ Rp, therefore constraint satisfaction is guaranteedregardless of the parameters η and ζ. �

The following example illustrates the behaviour of the proposed attraction field for thecase where D is a sphere in Rn. Although seemingly trivial, this case is highly relevantbecause any simply connected domain D is topologically equivalent to an Rn sphere. Thiswill be discussed in Subsection 4.4.2.

Example 4.1: Circular Domain

Consider the constraint c(xv, v) = R − ‖v‖ ≥ 0 with R > 0. Given a safety marginδ ∈ (0, R), the domain D is a circle of radius R − δ and is therefore convex. Givenan influence margin ζ ∈ (δ,R), equation (4.9) is satisfied for ∆cδ = 1.


−40 −30 −20 −10 0 10 20 30 40

−30

−20

−10

0

10

20

30

40

|| v || = R|| v || = R − δ|| v || = R − ζ

Figure 4.3: Steady-state trajectories given a circular constraint.

Given the parameters R = 30, δ = 5, ζ = 10, η = 3, the Figure (4.3) showsthe behaviour of v(t) initialized in v(0) = [−10 ; 5] and subject to three differentreferences: r1 = [4 ; 8], r2 = [15 ; 16], and r3 = [23 ; 42].

As illustrated in Figure 4.3, in the three simulations v(t) asymptotically tend tor?1 = [4 ; 8], r?2 = [14.5 ; 15.5] and r?3 = [12 ; 21.9]. As expected, r?1 and r?3 are thesolutions to the optimization problem (2.7). As for r?2 , the correct solution wouldhave been r?2 = r2. As discussed in Remark 4.3, the error ‖r?2 − r2‖ = 0.71 is inferiorto the smoothing margin η.

Remark 4.4. The proposed theory implicitly assumes that c(xv, v) is a smooth function,otherwise its gradient is not defined everywhere in D. However, the proposed theory can beextended to the case where c(xv, v) is a non-smooth function given by the combination ofmultiple smooth constraints cj(xv, v).

In this case, given J independent safety margins δj > 0 and influence margins ζj > δj,∀j = 1, . . . , J , the vector field ρ0(v) in equation (4.10) can be modified into

ρ0(v) =

J∑j=1

max

{ζj − (cTj v + dj)

ζj − δj, 0

}cj .

This preserves the properties of Proposition 4.1 for most r? ∈ D. The only exception iswhen r? belongs to the region where two or more influence margins overlap. Indeed, sincethe superposition of independent repulsion fields will have a smoothing effect on ∂D, theproposed solution will inevitably cause distortions in proximity of the edges. However, thiseffect can be made arbitrarily small by reducing the margin ζj − δj > 0. �

The following example addresses the behavior of the proposed attraction field in the casewhere D is given by the combination of multiple planar surfaces.


Example 4.2: Polyhedral Domain

Consider the case where c(xv, v) is given by the following combination of linear con-straints cTj v + dj ≥ 0, where

cT1 = [−1 0 ] d1 = 20cT2 = [ 1 0 ] d2 = 0cT3 = [ 0 1 ] d3 = 0cT4 = [ 1 − 1 ] d4 = 12cT5 = [−1 − 1 ] d5 = 32.

Given the safety margins δ = 0.5 and the influence margins ζ = 1.5, the Figure 4.4shows the behaviour of v(t) initialized in v(0) = [5 ; 10] and subject to three differentreferences: r1 = [4 ; 8], r2 = [15 ; 16], and r3 = [23 ; 42].

As illustrated in Figure 4.4, in the three simulations the applied reference v(t) asymp-totically tends to r?1 = [4; 8], r?2 = [14.97; 15.97] and r?3 = [10.3; 21.11]. As expected,r?1 is the correct solution to the optimization problem (2.7). In analogy to Example4.1 the error ‖r?2 − r2‖ = 0.04 is inferior to the margin η. As for r?3 , the correctsolution would have been r?3 = [10 ; 21.3] which belongs to the influence region ofconstraints 4 and 5. As discussed in Remark 4.4, the error ‖r?3 − r?3‖ = 0.35 is inferiorto ζ − δ = 1.

−15 −10 −5 0 5 10 15 20 25 30 35−5

0

5

10

15

20

25

30

35

40

45

|| v || = Rv(0)r1

r2

r3

Figure 4.4: Steady-state trajectories given a polyhedric constraint.

4.4.2 Non-Convex Domains

Given a non-convex domain D, the vector field (4.7) may generate multiple local minima.As such, the associated potential would not be a navigation function, due to the violationof requirement (4.4). As detailed in [4.14], this issue can be overcome by finding a suitablehomeomorphism Φ : D → D such that D is a convex domain. Indeed, given the imagesv = Φ(v) and r = Φ(r), equation (4.7) will provide an attraction field ρ(v, r) with onlyone local minimum. The obtained results can then be translated back to the domain D by


choosing

ρ(v, r) = Φ−1(ρ(Φ(v),Φ(r))). (4.14)

Remark 4.5. The main drawback of equation (4.14) is that the Φ−1 : D → D may notconserve optimality if the desired reference is not steady-state admissible. In other terms,given r? equal to the solution of (2.7) in the domain D, v(t) will converge to Φ−1(r?) whichis, in general, different from the desired r?. A possible solution to this problem is to computer? directly in the domain D and then substitute equation (4.14) with

ρ(v, r) = Φ−1(ρ(Φ(v),Φ(r?))).

However, this approach requires solving problem (2.7) online. �

The construction of a suitable homeomorphism Φ for a generic D is a topological problemwhich goes far beyond the scope of this dissertation. Further details on the subject can befound in [4.14]. For the sake of completeness, it is worth mentioning that the following classof non-convex sets admits a fairly simple solution.

Definition 4.5. Starred Domain: A domain D is “Starred” if there exists a focus pointo ∈ D \ ∂D such that, ∀v ∈ D, the straight line between o and v is wholly contained in D.�

By taking advantage of the fact that the focus point o views the rest of the domain as ifit were convex, D can be distorted into a unitary sphere in Rn using the homeomorphism

Φ(v) =‖v − o‖‖v∂ − o‖

, (4.15)

where v∂ is the intersection between the boundary ∂D and the line directed from o to v.

Example 4.3: Star Domain

Let constraint c(xv, v) define an equilateral seven-pointed star with an outer radiusrout = 3 and an inner radius rin = 1. Given the parameters R = 30, δ = 5, ζ = 10,η = 3, the following figures compare the results achieved using:

• Convex ρ(v, r) : The attraction field is generated using (4.7) directly;

• Homeomorphic ρ(v, r) : The attraction field is generated using (4.14)-(4.15);

• Roadmap ρ(v, r) : The attraction field is generated using (4.2) with r = 0.


−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

ConvexDiffeomorphicRoadmap

Figure 4.5: Steady-state trajectories for v(0) and r belonging to a convex subdomain.

As illustrated in Figure 4.5, if v(0) and r belong to a convex subdomain, the directimplementation of (4.7) generates a suitable steady-state path without any need ofintermediate waypoints. For what regards the homeomorphic approach, it is inter-esting to note that the resulting trajectory is not a straight line due to the spacialdeformation introduced by (4.14).

As illustrated in Figure 4.6, if v(0) and r do not belong to a convex subdomain, thedirect implementation of (4.7) converges to a local minimum which is not necessarilyr. Instead, the other two approaches successfully converge to the desired reference.

Finally, Figure 4.7 illustrates what happens if the reference is outside the admissibleregion. In this case, the convex strategy converges to a local minimum differentfrom r?. The attraction field based on the homeomorphism (4.15) converges to thesolution of (2.7) in the distorted domain D. This can be seen by the fact that theangular error between r and the achieved r? is equal to zero. Instead, the attractionfield based on the roadmap method successfully converges to the solution of (2.7).

4.5. Multiply Connected Constraints 71

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

ConvexDiffeomorphicR

Figure 4.6: Steady-state trajectories for v(0) and r not in a convex subdomain.

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

ConvexDiffeomorphicRoadmap

Figure 4.7: Steady-state trajectories for a non-admissible r.

4.5 Multiply Connected Constraints

As stated in Section 4.3, the main issue with the classical NF approach applied to a multiplyconnected domain is that there will always exist a set of initial conditions which converge tothe saddle points instead of r?. In robotic applications, this problem is usually disregardedbecause the system noise is deemed sufficient to destabilize the saddle points. However,since the ERG implements the dynamics of v(t) internally, the objective of this section is toillustrate how the saddle points can be destabilized in the absence of external noises.

Clearly, the simplest solution would be to introduce an artificial noise in the attractionfield. Although effective, this is likely to degrade the control performances. As such, a morestructured approach is deemed preferable.


Following from Theorem 4.2, it is impossible to construct a navigation function suchthat v = −∇vN(v, r) asymptotically converges to r? for any initial condition. To overcomethis issue, the following theorem proposes a novel solution in which the anti-gradient of thenavigation function is complemented with a non conservative vector field that destabilizesthe saddle points without compromising constraint satisfaction and asymptotic convergence.

Theorem 4.4. Let c(xv, v)δ define a multiply connected domain D that can be described bythe difference between a simply connected domain D0 and nh simply connected domains Di,with Di ⊂ D0 for i = 1, . . . , nh. Given the navigation function N : D×Rp → R let there existonly nh saddle points. Moreover, let each saddle point be is strictly contained in a simplyconnected domain Bi satisfying Bi ⊂ D0, Bi ⊇ Di, for i = 1, . . . , nh, and Bi ∩ Bk = ∅, fork 6= i. Then, given the vector fields ρi(v) satisfying

ρi(v)T∇vc(xv, v) = 0, ∀v ∈ ∂D, (4.16)

ρi(v)T∇vN(v, r) ≤ 0, ∀v ∈ D,∀r ∈ Rp, (4.17)

and‖ρi(v)‖ 6= 0, ∀v ∈ Bi \ ∂Bi‖ρi(v)‖ = 0, ∀v /∈ Bi \ ∂Bi,

(4.18)

the system

v = −∇vN(v, r) +

nh∑i=1

ρi(v), (4.19)

will asymptotically converge to r? without exiting D, for all initial conditions v(0) ∈ D andfor all r /∈

⋃nhi=1 Bi. �

Proof. Following from equations (4.3), (4.16), and (4.19), v(t) cannot exit the domain D.To prove asymptotic stability, consider the navigation function N(v, r). Following from(4.17), (4.19), its derivative

N(v, r) = −∇vN(v, r)T∇vN(v, r) +∇vN(v, r)T ρi(v)

is negative semi-definite. For all points v satisfying∇vN(v, r) = 0, it follows from (4.19) thatv = ρ(v). Due to equation (4.18), the only point that satisfies ∇vN(v, r) = 0 and ρ(v) 6= 0simultaneously is v = r?. Therefore, it follows from the Krasovskii-LaSalle principle thatv(t) will asymptotically tend to r? for all initial conditions v(0) ∈ D. �

Remark 4.6. It is worth noting that Theorem 4.4 does not address the case r? ∈ Bi. Indeed,due to equation (4.18), the point r? ∈ Bi has the unusual property of being globally attractive,i.e. limt→∞ v(t) = r?, ∀v(0) ∈ D, without being an equilibrium, i.e. v(r?, r) 6= 0.If necessary, this issue can be overcome by imposing ρ(v) = 0 in a neighborhood of r?. �

The following subsections will illustrate how the results of Theorem 4.4 can be appliedto the construction of a suitable attraction field ρ(v, r) for a relevant class of admissibledomains.

4.5.1 Convex Domains with Spherical Holes

In analogy with Subsection 4.4.1, this subsection will address the construction of the at-traction field ρ(v, r) for a specific class of multiply connected constraints. The extension ofthese results will then addressed in the following subsection.

The basic idea behind the proposed solution is to define the attraction field ρ(v, r) asthe sum of a conservative vector field ρC(v, r) obtained using a Navigation Function, and a


non-conservative vector field ρ(v) that destabilizes the saddle points. The conservative termρC(v, r) will be the same as (4.7), meaning

ρC(v, r) = ρr(v, r) +

nh∑i=0

ρi(v), (4.20)

where ρr(v, r) is given in (4.8) and ρi(v) are separate attraction fields generated by eachboundary of the domain D. In particular, the outer boundary ∂D0 is equal to the set{v : c0(xv, v) = δ0}. The nh holes ∂Di are instead defined by the set {v : ci(xv, v) = δi}. Foreach constraint, the repulsion field (4.10) is defined separately using independent influencemargins ζi > δi. This implies

ρi(v) = max

{ζi − c(xv, v)

ζi − δi, 0

}∇vc(xv, v)

∇cδi. (4.21)

The main focus of this section is to build a suitable non-conservative term ρ(v) satisfying(4.16)-(4.18). To do so, the first step is to determine the influence region Bi. Since each ofthe repulsion terms (4.21) is zero for ci(xv, v) > ζi, it follows from (4.20) that the saddlepoints must belong to the sets

Bi = {v : ci(xv, v) ≤ ζi}. (4.22)

Assuming that Bi ∩ Bk = ∅, ∀i = 0, . . . , nh, ∀k 6= i, requirement (4.16) can be satisfied bychoosing ρ(v) perpendicular to ∇vc(xv, v). Requirement (4.17) can instead be satisfied bychoosing the direction of ρ(v) that does not oppose ρC(v, r). Finally, requirement (4.18) canbe satisfied by choosing the magnitude of ρi(v) in the same way as (4.21). Based on theseconsiderations, the following is proposed

ρi(v) = max

{ζi − ci(xv, v)

ζi − δi, 0

}ˆsgn(ρr(v, r)

T∇c⊥i (v)))∇c⊥i (v), (4.23)

where

ˆsgn (a)) =

{1 if a ≥ 0−1 if a < 0,

(4.24)

and ∇c⊥i (v) is an arbitrarily chosen unitary vector perpendicular to ∇vci(xv, v).

Proposition 4.2. Given the constraints ci(xv, v) ≥ 0, let the safety margins δi > 0 be suchthat the set D0 = {v : c0(xv, v) ≥ δ0} is convex and the sets

Di = {v : ci(xv, v) ≤ δi} i = 1, . . . , nh

are spherical and satisfy Di ⊂ D0 for i = 1, . . . , nh. Moreover, let the influence marginsζi > δi, i = 0, . . . , nh be such that the sets (4.22) satisfy Bi∩Bk = ∅, for k 6= i. Then, givena sufficiently small smoothing margin η > 0, the vector field

ρ(v, r) = ρC(v, r) +

nh∑i=1

ρi(v) (4.25)

with ρC(v, r), ρ(v) as in (4.20), (4.23) satisfies Definition 2.2, ∀r such that c0(xr, r) /∈(ζ0, δ0], and ci(xr, r) ≥ ζi with i = 1, . . . , nh. �

Proof. The first part of the proof consists in demonstrating that the conservative vector field(4.20) is the anti-gradient of a navigation function satisfying Definition (4.1). To this end,consider the potential function

N(v, r) = Pr(v, r) +

nh∑i=0

Pi(v) (4.26)


with Pr(v, r) as in (4.11) and Pi(v) analogous to (4.12) for i = 0, . . . , nh. As proven inProposition 4.1, the function Pr(v, r) + P0(v) presents only one local minimum r? ∈ D0

satisfying (2.7).

Consider what happens with the addition of subsequent potential functions Pi(v). SincePi(v) = 0 for v /∈ Bi, the local properties outside the influence set Bi remain unaltered.Within the influence set, the gradient of (4.26) is

∇vN(v, r) =r − v

max{‖r − v‖ , η}+∇vci(xv, v)

‖∇vci(xv, v)‖.

Given a sufficiently small η > 0, it follows from (4.9) that ∇vN(v, r) = 0 if and only ifv ∈ ∂Di. Since Di is convex and r /∈ Bi, there exists only one point si ∈ Bi satisfying∇vN(si, r) = 0. Moreover, since Di is a sphere, si is a saddle point.As a result, N(v, r) is characterized by nh saddle points si and one local minimum r?, whichsatisfies (2.7). Since the border invariance property (4.3) can be proven using the sameconsiderations as Proposition 4.1, N(v, r) satisfies all the requirements of Definition (4.1).

The second part of the proof consists in showing that the non-conservative vector field∑nhi=1 ρi(v) satisfies all the requirements of Theorem 4.4. This follows by construction since

∇c⊥i ensures requirement (4.16), ˆsgn(ρC(v, r)T∇c⊥i (v))

)implies (4.17) and

max

{ζi − ci(xv, v)

ζi − δi, 0

}satisfies (4.18). As such, the statement follows directly from Theorem 4.4. �

Remark 4.7. It is worth noting that Proposition 4.2 does not address r : ci(xv, v) < δi fori = 1, . . . , nh and r : ci(xv, v) ∈ [δi, ζi) for 0 = 1, . . . , nh.The first case, i.e. r ∈ Di, implies r? ∈ ∂Di. This leads to the same condition discussedRemark 4.6 where r? is globally attractive but is not an equilibrium.The second case, i.e. r ∈ Bi \ Di, entails r? 6= r, even though r is the solution of (2.7).As discussed in Remark 4.3, this error can be made arbitrarily small due to the fact that‖r − r?‖ < η. �

Example 4.4: Sphere world Domain

Consider the constraints ‖v‖ ≤ R0, ‖v−p1‖ ≥ R1, and ‖v−p2‖ ≥ R2, with R0 = 25,p1 = [0 − 8]T , p2 = [6.5 8]T , R1 = R2 = 5. Given the safety margin δ = 2, theset D in equation (4.1) is described by an external circular boundary containing twodistinct circular holes. Figure 4.8 depicts the results obtained using the followingstrategies:

• Conservative ρ(v, r): The attraction field is generated using equation (4.7);

• Undirected ρ(v, r): The attraction field is generated using equation (4.25) and

the nonconservative fields ρi(v) = ‖ρi(v)‖∇c⊥i ;

• Directed ρ(v, r): The attraction field is generated using equation (4.25) andthe nonconservative fields given in (4.23).


−30 −20 −10 0 10 20 30−30

−20

−10

0

10

20

30

rDirectedUndirectedConservativec(v)=0

Figure 4.8: Steady-state trajectories in a sphere-world domain.

To illustrate the issues that may arise using purely conservative vector fields, v(0) =[0 −20]T was purposely chosen within the set of initial conditions that asymptoticallytend to a saddle point. As expected, the conservative vector field (4.7) is unable toreach r?. Instead, the two non-conservative strategies successfully destabilize thestagnation points.The main difference between the undirected and the directed ρ(v, r) is that ˆsgn(a) = 1may lead to N(v, r) > 0 whereas ˆsgn(a) as in (4.24) guarantees N(v, r) < 0. Referringto Figure 4.8, this is represented by the fact that the directed approach will choosethe shortest path to circumvent each obstacle.

4.5.2 Non-Convex Domains with Non-Spherical Holes

Given a non-convex outer domain D0 with non-spherical holes Di, the conservative vectorfield (4.20) may not be the anti-gradient of a navigation function satisfying Definition 4.1.In line of principle, this problem can be addressed in a similar way to Subsection 4.4.2.

Indeed, for any multiply connected domain D, it has been proven in [4.9] that therealways exists a homeomorphism Φ which distorts the original domain D into a suitabledomain D. The desired attraction field can be reconstructed using (4.14). Examples on howto construct a suitable homeomorphism for can be found in [4.14].

A possible alternative consists in inscribing the holes Di inside spherical domains ˆDk.This fairly simple solution allows to recover all the properties given in Subsection (4.5.1) as

long as r /∈ ˆDk.For what concerns a non-convex outer domain D0, a possible approach is to combine theroadmap method with the NF approach [4.10]. The idea of this approach consists in applyingthe roadmap method to the external domain D0, without accounting for the holes.Then, given a partition into locally convex windows, the presence of holes Di within eachwindow is addressed using a local Navigation Function. In the ERG framework, this isequivalent to substituting the roadmap attraction field (4.2) with (4.25).The advantage of this approach is that roadmap algorithms become more computationally


efficient in the absence of holes and that Navigation Functions are easier to implement inlocally convex regions.

4.6 Summary

This chapter has addressed the systematic design of the attraction field term in the ERGframework. This was done by linking it to the well-known motion planning problem andtaking advantage of the existing literature.After a brief discussion on roadmap methods and how they can be integrated in the ERGframework, the chapter mostly focused on the Navigation Function approach due to itsanalytic nature. To allow for its implementation in the ERG framework, some modificationsto the classical NF method have been presented.

This chapter, combined with Chapter 2 and Chapter 3, concludes the systematic char-acterization of the Explicit Reference Governor framework.

Bibliography

[4.1] J. C. Latombe, Robot Motion Planning. Springer Science & Business Media, 2012.

[4.2] P. Raja and S. Pugazhenthi, “Optimal path planning of mobile robots: A review,”International Journal of Physical Sciences, vol. 7, no. 9, pp. 1314–1320, 2012.

[4.3] A. Stentz, “Optimal and efficient path planning for partially-known environments,”in Proc. of the IEEE International Conference on Robotics and Automation (ICRA),vol. 4, 1994, pp. 3310–3317.

[4.4] L. E. Kavraki, P. Svestka, J. C. Latombe, and M. H. Overmars, “Probabilisticroadmaps for path planning in high-dimensional configuration spaces,” IEEE Trans-actions on Robotics and Automation, vol. 12, no. 4, pp. 566–580, 1996.

[4.5] J. J. Kuffner and S. M. LaValle, “RRT-connect: An efficient approach to single-querypath planning,” in Proc. of the IEEE International Conference on Robotics and Au-tomation (ICRA), vol. 2, 2000, pp. 995–1001.

[4.6] O. Khatib, “Real-time obstacle avoidance for manipulators and mobile robots,” inProc. of International Conference on Robotics and Automation (ICRA), vol. 2, 1985,pp. 500–505.

[4.7] Y. Koren and J. T. Borenstein, “Potential field methods and their inherent limitationsfor mobile robot navigation,” in Proc. of the IEEE International Conference on Roboticsand Automation (ICRA), vol. 2, 1991, pp. 1398–1404.

[4.8] P. Vadakkepat, K. C. Tan, and W. Ming-Liang, “Evolutionary artificial potential fieldsand their application in real time robot path planning,” in Proc. of the Congress ofEvolutionary Computation, 2000, pp. 256–263.

[4.9] D. E. Koditschek and E. Rimon, “Robot navigation functions on manifolds with bound-ary,” Advances in Applied Mathematics, vol. 11, no. 4, pp. 412–442, 1990.

[4.10] O. Brock and O. Khatib, “High-speed navigation using the global dynamic windowapproach,” in Proc. of the IEEE International Conference on Robotics and Automation,vol. 1, 1999, pp. 341–346.

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[4.11] G. Oriolo, G. Ulivi, and M. Vendittelli, “Real-time map building and navigation forautonomous robots in unknown environments,” IEEE Transactions on Systems, Man,and Cybernetics, vol. 28, no. 3, pp. 316–333, 1998.

[4.12] H. G. Tanner and A. Boddu, “Multiagent navigation functions revisited,” IEEETransactions on Robotics, vol. 28, no. 6, pp. 1346–1359, 2012.

[4.13] C. I. Connolly, J. B. Bruns, and R. Weiss, “Path planning using laplace’s equation,” inProceeding of the IEEE International Conference on Robotics and Automation (ICRA),1990.

[4.14] E. Rimon and D. E. Koditschek, “The construction of analytic diffeomorphisms forexact robot navigation on star worlds,” Transactions of the American MathematicalSociety, vol. 327, no. 1, pp. 71–116, 1991.

Chapter 5

Robustness

The objective of this chapter is to extend the ERG framework by addressing its robustnessin the presence of parametric uncertainties and external disturbances. Interestingly enough,it will be shown that the framework itself does not require any major alteration to berobust. To provide a better context, the following section will illustrate how other nonlinearconstrained control laws typically address the same issue.

5.1 Robust Constrained Control

The constrained control of nonlinear systems is a notoriously challenging task. In the pres-ence of uncertainties and external disturbances, the introduction of robustness requirementswill usually penalize the other performance parameters.

For what concerns Model Predictive Control, a considerable amount of research is beingbeen devoted to robust MPC theory. Existing results mostly focus on linear systems whichalready provide a fair amount of challenges [5.1]. Results on robust nonlinear MPC can beroughly divided in three separate approaches.The first approach consists in analysing the robustness of the already existing nonlinearMPC schemes. This enables the choice of suitable terminal costs and terminal regions toensure robust constraint satisfaction [5.2, 5.3]. The main advantage is that the structureof the MPC does not change, meaning that the computational burden is the same. Thedisadvantage is that the chosen cost will likely penalize the performances.The second approach consists in designing an MPC that solves the optimization problemunder the worst-case scenario. This approach, known as the min-max MPC [5.4, 5.5, 5.6],ensures high performances but also leads to a significant increase in the computational re-quirements since the determination of the worst-case scenario can be time consuming.Finally, the third approach consists in pre-stabilizing the nonlinear system using a primaryfeedback loop and then applying an MPC to the applied reference. This enables the con-struction of a positively invariant uncertainty tube centered on the trajectory of the nominalsystem. The nominal trajectory is then restricted so that the tube is entirely within theconstraints [5.7]. Although appealing for linear systems [5.8], the main issue with nonlinearsystems is that the computation of uncertainty tubes is a fairly challenging task that mayprove too conservative or too computationally intensive.In conclusion, the systematic generation of a Robust Nonlinear MPC for the general case islikely to remain a very active field of research for the foreseeable future.

As for anti-windup strategies, most robustness results are centered on linear systems[5.9, 5.10, 5.11, 5.12]. In some nonlinear applications, robust anti-windup schemes havebeen proposed using adaptive linear control [5.13]. To the author’s best knowledge, the

79

80 5. Robustness

systematic design of a robust nonlinear anti-windup scheme for state and input constraintsis still an open problem.

Robust Reference Governor schemes ensure constraint satisfaction by suitably restrictingthe set of admissible references [5.14, 5.15, 5.16]. This is done by taking advantage of thedisturbance-invariant set introduced in [5.17], which is also the basis for the tube-MCP [5.18].As per the non-robust case, Robust RG schemes tend to be more computationally efficientthan tube-MPC due to the fact that they solve a scalar optimization problem instead ofa multi-variable optimization problem. Clearly, this is compensated by the fact that tube-MPC typically achieves better performances.For what concerns nonlinear systems, [5.19] suggests that a similar approach can be usedby constructing a suitable disturbance-invariant set. To the author’s best knowledge, thisline of research was not investigated further.

5.2 Robust ERG Formulation

The following section briefly reiterates the steps of Chapter 2 to account for possible non-idealities. Indeed, since Theorem 2.1 is applicable as long as ∆(x, v) and ρ(v, r) satisfyDefinitions 2.1 and 2.2, the robustness issue can be reformulated in terms of finding a suit-able Dynamic Safety Margin and Attraction Field.

Since the ERG is an add-on control unit, it is necessary to characterize the robustnessof the pre-stabilized system before addressing the robustness of the ERG itself.

Definition 5.1. Robust Pre-stabilized System: Consider a pre-stabilized system

x = f(x, v, µ, d), (5.1)

where x(t) ∈ Rn is the state vector, v(t) ∈ R is the vector of applied references, µ ∈M is aconstant vector of uncertain parameters, and d(t) ∈ Rq is a vector of external disturbances.Moreover, let xv be the nominal steady state associated to a nominal parameter µ ∈M anda constant reference v. The pre-stabilized system (5.1) is said to be “robust” if

• Given a constant reference v, the trajectories are ultimately bounded ∀µ ∈ M and∀d(t) : ‖d‖∞ ≤ dmax, i.e. lim

t→∞‖x(t)− xv‖ ≤ R, where R ≥ 0 is bounded and

depends on the set M and the dmax. �

Definition 5.1 assumes that the primary control loop solves Problem 2.2 for µ = µ and‖d‖∞ = 0 and is robust in the presence of parametric uncertainties and external distur-bances. In analogy to Chapter 2, this dissertation will not address the design of the primarycontrol law. For what concerns the ERG, the following problem statement extends Problem2.3 in the presence of uncertain parameters and bounded external disturbances.

Problem 5.1. Robust Constraint Enforcement: Given the robust pre-stabilized system(5.1) subject to constraints

c(x, v, µ) ≥ 0, (5.2)

find an auxiliary reference signal v(t) such that, ∀µ ∈M:

1. Constraints are always satisfied, i.e. c(x(t), v(t), µ) ≥ 0, ∀t ≥ 0;

2. There exists a Static Safety Margin δ > 0 such that, for any constant desiredreference r satisfying c(xr, r, µ) ≥ δ, the equilibrium point v = r is AsymptoticallyStable, i.e. lim

t→∞v(t) = r.

5.2. Robust ERG Formulation 81

Moreover, a robust constraint enforcement strategy is Projective if it satisfies the furtherproperties ∀µ ∈M:

1. Constraints are always satisfied, i.e. c(x(t), v(t), µ) ≥ 0, ∀t ≥ 0;

2. For any constant reference r, limt→∞

v(t) = r?, where r? is the solution to{min(r − r?)T (r − r?)s.t. c(xr? , r

?, µ) ≥ δ, (5.3)

and δ > 0 is a suitably chosen Static Safety Margin. �

The main intuition for solving Problem 5.1 is that the ERG remains valid as long as∆(x, v) and ρ(v, r) satisfy Definition 2.1 and Definition 2.2 with δ > δmin. Since ρ(v, r)does not depend on the state of the system, the attraction field is unaffected by parametricuncertainties and external disturbances. As for the dynamic safety margin ∆(x, v), it isnecessary to extend Definition 2.1 to account for robustness.

Definition 5.2. Robust Dynamic Safety Margin: Let (5.1) be a robust pre-stabilizedsystem subject to constraints (2.6), and let x(µ, d, t|x, v) denote the solution to{

˙x(t) = f(x(t), v, µ(t), d(t))x(0) = x.

(5.4)

Given an applied reference v satisfying (2.10), a continuous function ∆ : Rp ×Rn → R is a“Robust Dynamic Safety Margin” if there exists a δmin ≥ 0 such that:

• ∆(x, v) > 0⇒ c(x(µ, d, t|x, v), v, µ) > 0, ∀t≥0, ∀µ(t)∈M, ∀d(t) : ‖d‖∞≤dmax;

• ∆(x, v) = 0⇒ c(x(µ, d, t|x, v), v, µ) ≥ 0, ∀t≥0, ∀µ(t)∈M, ∀d(t) : ‖d‖∞≤dmax;

• ∆(x, v) = 0⇒ ∆(x(µ, d, t|x, v), v) ≥ 0, ∀t≥0, ∀µ(t)∈M, ∀d(t) : ‖d‖∞≤dmax;

• ∀δ > δmin, ∃ε > 0 such that c(xv, v, µ) ≥ δ ⇒ ∆(xv, v) ≥ ε. �

Although subtle, the main difference between Definition 2.1 and Definition 5.2 is thefact that the static safety margin δ cannot be chosen arbitrarily. Indeed, the lower boundδmin must ensure the Dynamic Safety Margin properties even in the presence of parametricuncertainties and external disturbances. The following statement extends the results ofTheorem 2.1 to account for robustness.

Theorem 5.1. Consider a robust pre-compensated system (5.1) subject to constraints (2.6)and let ρ(r, v) and ∆(x, v), be an attraction field and a robust dynamic safety margin as perDefinitions 5.2, 2.2. Then, given any initial conditions x(0), v(0) satisfying

∆(x(0), v(0)) ≥ 0c(xv(0), v(0), µ) ≥ δ,

with δ > δmin, ERG law (2.8), i.e. v = ∆(x, v) ρ(r, v), ensures that:

1. for any reference signal r(t), constraints (2.6) are never violated ∀µ ∈ M and ∀d(t) :‖d‖∞ ≤ dmax;

2. for any constant reference r, then:

• if c(xr, r, µ) ≥ δ, v(t) will asymptotically tend to r;

• if c(xr, r, µ) < δ and ρ(r, v) is a projective attraction field, v(t) will asymptoti-cally tend to r? satisfying (2.7). �

82 5. Robustness

Proof. The proof is analogous to the one given in Theorem 2.1. �

Having extended the ERG framework to account for robustness, the question that re-mains is how the dynamic safety margins presented in Chapter 3 are affected by the changefrom Definition 2.2 to Definition 5.2. This chapter will mostly focus the Lyapunov-basedfeedback approach. Other dynamic safety margins will be briefly discussed in Section 5.5.

The following section will show how the Lyapunov-based dynamic safety margin can bemodified to account for parametric uncertainties µ ∈ M. Section 5.4 will then illustratehow the lower-bound δmin ≥ 0 can account for bounded external disturbances.

5.3 Parametric Uncertainties

Let the pre-stabilized system (5.1) be characterized by parametric uncertainties µ ∈M andno external disturbance. The robustness of the ERG can be characterized depending onwether the parameter µ remains constant or changes in time.

5.3.1 Time-Varying Parameters

In the best-case scenario, the Lyapunov-based dynamic safety margin (2.18) can be usedwithout any modifications if V (x, v) is a common Lyapunov function such that

∇xV (x, v)T f(x, v, µ) < 0, ∀x ∈ X (v), ∀v ∈ D, ∀µ ∈M. (5.5)

The main advantage of providing a common Lyapunov function is that the resulting ERG willbe robust with respect to a time-varying signal µ(t) ∈ M. Systematic tools for generatinga common Lyapunov functions have been proposed in the literature for switching linearsystems [5.20, 5.21] and linear systems with time-varying uncertainties [5.22, 5.23].

Example 5.1: Linear System with Time-Varying Parametric Uncertainty

Consider the second order system x = u cos(a) where u is a control input and a isa time-varying unknown parameter satisfying |a| ≤ amax. Given a reference v, thecontrol input is assigned using a standard PD approach u = −kp(x−v)−kdx, leadingto the closed-loop system x = A(a)x+B(a)v, where

A(a) =

[0 1

−kp cos(a) −kd cos(a)

]B(a) =

[0

kp cos(a)

].

Given kp = 100, kd = 8, and amax = π/6, Figure 5.1 illustrates the behaviour of thefollowing ERG strategies:

• Basic ERG: The ERG is implemented using the quadratic Lyapunov function(3.50). The Lyapunov matrix P was obtained by solving the optimizationproblem (3.53) subject to the constraint A(0)TP + PA(0) ≤ 0;

• Robust ERG: The ERG is implemented using the quadratic Lyapunov func-tion (3.50). The Lyapunov matrix P was obtained by solving the optimiza-tion problem (3.53) subject to the constraints A(0)TP + PA(0) ≤ 0 andA(amax)TP + PA(amax) ≤ 0.

5.3. Parametric Uncertainties 83

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

45

t

v(t)

, x 1

(t)

x1 ≤ 42

Basic ERGRobust ERG

Figure 5.1: Second order system with time-varying parametric uncertainties.

Since the basic ERG was designed without taking into account the parametric uncer-tainties, the proposed V (x, v) is not a Lyapunov function in the presence of paramet-ric uncertainties. As illustrated in Figure 5.1, this implies that constraint satisfactionis not guaranteed.Instead, the robust ERG successfully enforces constraint satisfaction by simply as-signing the Lyapunov matrix P so that A(a)TP + PA(a) ≤ 0, ∀a : |a| ≤ amax. Thisensures that (3.50) is a common Lyapunov function that accounts for time-varyingparametric uncertainties.

Unfortunately, finding a common Lyapunov function in the general case of nonlinearsystems can be a challenging task. To address other cases where the parameter µ is time-varying, it is typically necessary to add the assumption that µ(t) can be measured and hasa maximum rate of change µmax.In this case, the ERG can be implemented without modification if it is possible to find aparameter-dependent Lyapunov function V (x, v, µ) such that

∇xV (x, v, µ)T f(x, v, µ) < −∣∣∇µV (x, v, µ)T µ

∣∣ (5.6)

∀x ∈ X (v), ∀v ∈ D, ∀µ ∈M, ∀ |µ| ≤ µmax.

The cases of unknown time varying parameters, or known parameters with an excessiverate of change, will not be addressed in this dissertation. Instead, the following subsectionwill focus on the case where µ is constant, but its value is uncertain.

5.3.2 Constant Parameters

In many scenarios, it is reasonable to assume that µ is a physical constant and its valueis known within an uncertainty bound. In this case, the problem of having a parameter-dependent Lyapunov function is that the Lyapunov-based dynamic safety margin (2.18)cannot be computed with certainty. To overcome this issue, consider the lower and anupper bounds for the family of all possible V (x, v, µ), meaning

VL(x, v) ≤ V (x, v, µ) ≤ VU (x, v), ∀x ∈ X (v), ∀v ∈ D, ∀µ ∈M. (5.7)

84 5. Robustness

Although VL(x, v) and V (x, v) are not necessarily time-decreasing functions, the fact thatthey bound a time-decreasing function can be used to construct a suitable dynamic safetymargin. This is shown in the following proposition.

Proposition 5.1. Let V (x, v, µ) be a Lyapunov function for any constant v ∈ Vδ, µ ∈ M.Moreover, let there exist two class-K functions VL(x, v) and VU (x, v) satisfying (5.7). Then,given

Γ(v) =

{minx VL(x, v)s.t. c(x, v) ≤ 0,

(5.8)

the function∆(x, v) = κmax(Γ(v)− VU (x, v), 0), (5.9)

with κ > 0, is a robust dynamic safety margin. �

Proof. Equation1 (5.7), combined with the fact that V (x(t), v, µ) is a Lyapunov function,implies the following

VL(t) ≤ V (t) ≤ V (0) ≤ VU (0), ∀t ≥ 0.

Due to equation (5.8), VL(x(t), v) ≤ Γ(v) ensures c(x(t), v) ≥ 0, ∀t ≥ 0. Therefore,

VU (x(0), v) ≤ Γ(v) ⇒ c(x(t), v) ≥ 0, ∀t ≥ 0.

Since the same can be proven using strict inequalities, properties 2 and 3 of Definition 5.2are satisfied.Properties 1, 4 and 5 follow directly from the fact that VL(x, v) and VU (x, v) are two class-Kfunctions and that x(t) asymptotically tends to xv. Indeed, it follows from (5.8) that

∀δ > 0, ∃ε > 0 : c(xv, v) ≥ δ ⇒ Γ(v) ≥ ε.

Since VU (x(t), v) asymptotically tends to VU (xv, v) = 0, (5.9) will asymptotically tend tothe finite value ε > 0. As a result, the minimum safety margin is δmin = 0. This concludesthe proof. �

Remark 5.1. It is worth noting that the threshold margin Γ(v) proposed in (5.8) can becomputed and approximated using all the methods presented in Section 3.1. This is valid aslong as Γ(v) is obtained using the Lyapunov lower bound VL(x, v). �

Remark 5.2. Looking at equations (5.7)-(5.9), it can be seen that if the set M tends to asingle value, the robust dynamic safety margin (5.9) will tend to the non-robust formulation(2.18). As a result, if the parametric uncertainties tend to zero, the performances of therobust ERG will be the same as the classic ERG. �

Example 5.2: Aircraft Control with Parametric Uncertainties

Modeling and Control: Consider the aircraft presented in Example 2.1, given thelift function L(α), let the parameters of the polynomial approximation (2.27) beuncertain.In particular, the following uncertainty bounds are considered

l0 ≤ l0 ≤ l0l1 ≤ l1 ≤ l1l1 ≤ l1 ≤ l3.

(5.10)

1For the sake of notational simplicity, functions such as V (x(t), v, µ) will be noted as V (t).

5.3. Parametric Uncertainties 85

Given the nominal parameters l0, l1, and l3, the control input (2.22) is substitutedwith

u = −kP (α− v)− kDα+d1

d2(l0 + l1v − l3v3).

This leads to the closed-loop system

Iα = −d1(L(α)−L(v)) cosα−d2kP (α−v) cosα−d2kD cosαα−µα+∆L(v), (5.11)

where ∆L(v) = l0 + l1v − l3v3 − L(v) represents the steady-state error of the liftcompensation. In this example, it is assumed that ∆L(v) = 0. This is equivalentto stating the aircraft behaviour is well-known at steady-state but its dynamics areuncertain.Lyapunov Bounds: As shown in Example 2.1, asymptotic stability of the referencev can be proven using the energy-like Lyapunov function (2.24). Following fromequation (3.25), and taking into account

L(αS)− L(v) = l1(αS − v)− l3(α3S − v3) ≥ l1(αS − v)− l3(α3

S − v3),

the Lyapunov function can be lower-bounded by

1

2

(d1

(l1 − l3

α3S − v3

αS − v

)+ d2kP

)cosαS(α− v)2 +

1

2Iα2 ≤ V (α, α, v). (5.12)

In analogy to equation (3.47), and taking into account L′(0) = l1 ≤ l1, the followingLyapunov upper-bound appliesa

V (α, α, v) ≤ 1

2

(d1l1 + d2kP

)cos(v)(α− v)2 +

1

2Iα2. (5.13)

Both Lyapunov bounds (5.12)-(5.13) apply for any parameter belonging to the un-certainty region (5.10). As a result, the robust ERG can be implemented by usingthe Lyapunov lower bound (5.12) to compute the threshold values Γ(v), and theLyapunov upper bound to compute the robust dynamic safety margin (5.9).

Obtained Results: The system is simulated using the parameters l1 = 7.3 ·106Nm/rad and l3 = 5 · 107Nm/rad3. Figure 5.2 compares the following behaviours:

• Ideal ERG: The ERG is implemented using the correct parameters;

• Nominal ERG: The ERG is implemented using the nominal parameters l1 =8.6 ·106Nm/rad and l3 = 4.3 ·107Nm/rad3, without accounting for uncertainty;

• Robust ERG: The ERG is implemented using the same nominal parametersand taking into account the uncertainty bounds l1 = 6.9 · 106Nm/rad, l1 =9.5 · 106Nm/rad, l3 = 4 · 107Nm/rad3, and l3 = 5.2 · 107Nm/rad3.

As previously shown, ERG successfully enforces system constraints in the ideal case.In the presence of parametric uncertainties, however, the stall constraint is violatedin the time interval t ∈ [1.7, 2.2] s.Interestingly enough, it can be seen that the nominal ERG issued a constant ref-erence during the time interval t ∈ [1.4, 2.3] s. This implies that it had detected arisk and was maintaining a “safe” configuration. Due to the conservative nature ofLyapunov functions, this may prevent constraint violation for a small mismatch inthe parameters. As illustrated in this example, however, there are no guarantees.

86 5. Robustness

Instead, the robust ERG successfully ensures constraint satisfaction if the system pa-rameters belong to the estimated range. Notably, its behaviour is quite similar to thenominal ERG. The only exception is that the safe configuration v = 0 is guaranteedto ensure constraint satisfaction.

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16Explicit Reference Governor

t [s]

atta

ck a

ngle

[deg

]

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16Robust Explicit Reference Governor

t [s]

atta

ck a

ngle

[deg

]

rv(t) Robustα(t) Robustα ≤ α

S

rv(t) Idealα(t) Idealv(t) Nominalα(t) Nominalα ≤ α

S

Figure 5.2: System Response in the presence of parametric uncertainties.

As could be expected, comparisons with the ideal ERG show that the robust ERGtakes more time to converge (t = 6 s versus t = 2.6 s). This loss of performance isfairly acceptable considering that the size of the uncertainty bounds, i.e. li − li, is

30% of the nominal values li.

aEquation (3.47) refers to the strict Lyapunov function (3.43), a few trivial modifications aretherefore necessary to account for the energy-like Lyapunov function (2.24).

The following section will address the robustness of the ERG with respect to externaldisturbances. For the sake of simplicity, this will be done without taking into accountparametric uncertainties.

5.4 External Disturbances

Let the pre-stabilized system (5.1) be characterized by a bounded external disturbanced ∈ Rq and no parametric uncertainties. In this scenario, the main issue with the Lyapunov-based dynamic safety margin (2.18) is that V (x, v) will no longer asymptotically tend tozero. However, the following proposition shows that standard Input-to-State Stability (ISS)arguments [5.24] can be used to ensure robustness without any modifications to equation(2.18). Intuitively, this is done by restricting the set of acceptable steady-states so that theISS invariant level-set centered in xv is always strictly contained in the constraints.

5.4. External Disturbances 87

Proposition 5.2. Let the robust pre-stabilized system (5.1) be subject to a bounded externaldisturbance d ∈ Rq, and let V (x, v) be an ISS-Lyapunov function satisfying

V (x, v) ≥ χ(‖d‖∞) ⇒ V (x, v, d) ≤ 0, (5.14)

where χ : R→ R is a class-K function. Then, given the minimum static safety margin δmin

satisfying

c(xv, v) ≥ δmin ⇒ Γ(v) ≥ χ(‖d‖∞), (5.15)

equation (2.18) is a robust dynamic safety margin. �

Proof. Given a safety margin δ > δmin, it follows from (5.14) that, for any constant v ∈D, the ISS-Lyapunov function V (x, v) is strictly time-decreasing, even in the presence ofdisturbances. Moreover, ∀v ∈ D, equation (5.15), guarantees that the invariant level-set{x : V (x, v) ≤ χ(‖d‖∞)} is strictly contained in the threshold set {x : V (x, v) ≤ Γ(v)}. Asa result, (2.18) satisfies all the requirements of Definition 5.2. �

Interestingly enough, given a pre-stabilized system subject to external disturbances, theLyapunov-based ERG is able to ensure constraint satisfaction by simply choosing a suitablestatic safety margin δ.

The following example illustrates the behaviour of the Lyapunov-based ERG in thepresence of external disturbances.

Example 5.3: Aircraft Control with External Disturbances

Input-to-State Stability: Let the aircraft presented in Example 2.1 be subject toa bounded wind disturbance ‖fw‖∞ ≤ fmax

Iα = −d1L(α) cosα− µα+ d2u cosα+ fw.

Given the same control input as before (2.22), it follows from [5.25] that Input-to-State Stability can be proven by extending the Lyapunov function (2.24) with

V (α, α, v) = U1(α, v) +1

2xTv

[ε(µ+ kD cosαS)2/I ε(µ+ kD cosαS)ε(µ+ kD cosαS) I

]xv,

with xv = [α−v α]T

, ε ∈ (0, 1), and

U1(α, v) =

∫ α

v

(d1(L(β)− L(v)) + d2kP (β − v)) cosβdβ+

+εµ+ kD cosαS

I

∫ α

v

kDβ(cosβ − cosαS)dβ.

88 5. Robustness

0 1 2 3 4 5 6 7 80

2

4

6

8

10

12

14

16Output Response

t [s]

atta

ck a

ngle

[deg

]

0 1 2 3 4 5 6 7 8−2

0

2

4

6

8

10x 10

4 External Disturbance

t [s]

win

d ef

fect

[Nm

]

rNo WindWith Windα ≤ α

S

Figure 5.3: System Response in the presence of an external disturbance.

Given the polynomial approximation of the lift function (2.27), the function U1(α)can be expressed analytically. For what concerns the computation of the thresholdvalues, it is worth noting that U1(α) can be lower-bounded using the same quadraticfunctions as Example 3.2, meaning

U1(α, v) ≥ cosαS2

(d1

L(αS)−L(v)αS−v + d2kP

)(α− v)2.

As a result, Proposition 3.3 can be applied as before.

Obtained Results: Figure 5.3 compares the following behaviors:

• No Wind: The ERG is implemented using the ISS-Lyapunov function, in theabsence of external disturbances;

• With Wind: The same ERG is implemented in the presence of disturbances.

As shown in the simulations, the ERG successfully ensures constraint satisfactioneven in the presence of an external disturbance. This is achieved without introducingany modifications to the ERG scheme.

5.5 Other Dynamic Safety Margins

Having addressed the robustness of the Lyapunov-based dynamic safety margin, the objec-tive of this section is to very briefly address the robustness of the dynamic safety marginspresented in Section 3.2.

5.6. Summary 89

Invariant Set Approach: To implement the invariant set approach in a robust frame-work, Definition 3.1 should be extended by substituting equation (3.28) with

∀x ∈ ∂Sv, x+ τf(x, v, µ, d) ∈ Sv \ ∂Sv, ∀µ ∈M, ∀d : ‖d‖ = dmax.

Although this condition is more challenging to verify with respect to (3.28), it is worthnoting that a robust invariant set is still entirely determined by the behavior at the border∂Sv. This implies that the invariant set approach remains relatively feasible to address.

Returnable Set Approach: To implement the returnable set approach in a robustframework, Definition 3.2 should be extended by substituting equation (3.31) with

∀x ∈ Sv, x(µ, d, t|x, v) ∈ Rv, ∀t ≥ 0, ∀µ ∈M, ∀d : ‖d‖ ≤= dmax.

Since this condition requires the computation of all the possible trajectories that exit andre-enter Sv, the theoretical computation of a robust returnable set is much more challengingwith respect to the non-robust case.Interestingly enough, it is worth noting that if the the sets Sv and Rv are determinedexperimentally, the resulting strategy will be inherently robust.

Trajectory-Based Approach: Clearly, the trajectory-based dynamic safety margin isthe strategy that is most susceptible to robustness issues. In the presence of parametricuncertainties and external disturbances, a robust trajectory-based scheme would require thecomputation of all possible trajectories x(µ, d, t|x, v), ∀µ ∈ M and ∀d : ‖d‖∞ ≤ dmax. Inmost cases, the resulting computational complexity is deemed hard to justify within theERG framework.

Feedforward Approach: The feedforward approach is basically a Lyapunov-baseddynamic safety margin in which the Lyapunov function is not measured and is insteadinferred using a worst-case model. As such, the feedforward approach can be applied in arobust framework by modifying equation 3.41 so that the worst-case V also accounts forµ ∈M and ‖d‖∞.

5.6 Summary

This chapter has extended the ERG framework by taking into account the possible presenceof parametric uncertainties and external disturbances. In particular, it has been shownthat constraint satisfaction and asymptotic convergence can be ensured with little to novariations to the Lyapunov-based dynamic safety margin ∆(x, v) combined with a suitablechoice of the static safety margin δ > 0.The simplicity of this extension possibly represents one of the main strengths of the ExplicitReference Governor when addressing the constrained control of nonlinear systems.

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[5.25] M. M. Nicotra, R. Naldi, and E. Garone, “Sufficient conditions for the stability of aclass of second order systems,” Systems and Control Letters, vol. 84, pp. 1–6, 2015.

Chapter 6

Conclusions

Part I of this dissertation has presented the Explicit Reference Governor: a new and sys-tematic framework for the constrained control of nonlinear systems. The ERG formulationhas been introduced in general terms by focusing on the conditions that are sufficient toensure constraint satisfaction and asymptotic convergence.The two main components of the ERG scheme were investigated in dedicated chapters. Sev-eral alternatives for the systematic construction of each term were presented and classifiedbased on the properties of the system and of the constraints. Examples throughout thedissertation provided an intuitive interpretation of the proposed theoretical framework.The ERG scheme was then extended to ensure its robustness with respect to uncertainparameters and external disturbances.

6.1 Experimental Validation

Although this dissertation mostly presents theoretical results, it is worth noting that thesimple and robust nature of the ERG framework should ensure a relatively painless transitionto practical scenarios.

To confirm this intuition, the Explicit Reference Governor was applied to the invertedpendulum on cart of the Service d’Automatique et d’Analyse des Systemes (SAAS) of theUniversite Libre de Bruxelles (ULB). The experimental setup is illustrated in Figure 6.1.

Figure 6.1: Experimental set-up of the inverted pendulum on cart

93

94 6. Conclusions

0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

t [s]ca

rt p

ositi

on [m

]

0 5 10 15 20 25−100

−50

0

50

100

150

200

t [s]

pend

ulum

ang

le [d

eg]

r(t)p(t)

α(t)

||π/2−α|| ≤ ∆αmax

Figure 6.2: Response of the closed-loop system to a step variation of the reference

Modeling and Control: Under the reasonable assumption that the cart dynamics arenot influenced by the pendulum oscillations, the inverted pendulum on cart is described bythe following nonlinear model{

(M +m) p+ cpp = F23 lθ − p cos θ − ag sin θ + 2cθ

ml θ = 0

where p is the cart position, θ is the pendulum angle, F is the actuator force, M, m arethe masses of the cart and of the pendulum, l is the length of the pendulum, g is theacceleration of gravity cp, cθ are the friction coefficients of the cart rail and of the pendulum

hinge. Defining x =[p p θ − π θ

]T, the linearized dynamic model was identified as

˙x =

0 1 0 00 −1.92 0 00 0 0 10 −0.96 38.9 −1.21

x+

0

22.080

11.04

F.The control input F was then designed using a standard Linear Quadratic Regulator. Thisled to the linear state feedback F = [−1.27 −0.93 24.88 3.50](x−xv) where xv = [1 0 0 0]v,where v is the reference applied to the cart position. Local Asymptotic Stability of theequilibrium poi xv was proven using the Lyapunov function

V = xT

6.08 3.30 −66.1 −6.653.30 2.52 −53.3 −5.44−66.1 −53.3 1369 124−6.65 −5.44 124 14.9

x. (6.1)

However, as illustrated in Figure 6.1, the nonlinear closed-loop system could be destabilizedby providing a sufficiently large step variation of the cart reference v.

Constraint Enforcement: The instability of the controlled system is clearly caused bythe nonlinearities introduced by an excessive deviation of the inverted pendulum from the

6.2. Future Research Opportunities 95

0 5 10 15 20 250.2

0.3

0.4

0.5

0.6

0.7

0.8

t [s]ca

rt p

ositi

on [m

]

0 5 10 15 20 25160

165

170

175

180

185

190

195

200

t [s]

pend

ulum

ang

le [d

eg]

r(t)v(t)p(t)

α(t)

||π/2−α|| ≤ ∆αmax

Figure 6.3: Response of the closed-loop system integrated with the Explicit Reference Gov-ernor

upright position. As such, an ERG was tasked with maintaining the state of the closed-loopsystem within its basin of attraction.The maximum angular error α≤15◦ was assigned taking into account parametric uncertain-ties (e.g. dynamic friction) and unmodeled dynamics (e.g. static friction, actuator inertia).The Lyapunov-based dynamic safety margin was then computed using (2.18) and (3.9). Thisled to Γi (xv) = 210, i = 1, 2.

Obtained Results: The combined ERG and LQR were implemented in real-time usinga desktop pc running a MATLAB script with a sampling rate of 1kHz. The mean executiontime of the ERG routine was 0.72ms. As illustrated in Figure 6.1, the ERG was ableto handle successfully the abrupt change in the desired cart objective. Even though theexperimental setup is far from ideal, no additional adjustments were required to account forrobustness.

In the author’s opinion, this experimental validation demonstrated the great simplicity1

of the Explicit Reference Governor. The proposed framework is therefore believed to be ofgreat interest for many applications that require a simple, systematic, and computationallyinexpensive add-on tool for nonlinear constraint enforcement.

6.2 Future Research Opportunities

This dissertation has introduced the ERG framework and has strived to provide a com-plete and systematic discussion. Nevertheless, there are still many opportunities for furtherresearch.

Due to the generality of Definitions 2.1 and 2.2, the dynamic safety margins and attrac-tion fields presented in this dissertation are not an exhaustive list of all the possible choices.

1The experiment took a total of two hours from start (i.e. recovering an existing report that detailed thelinearized model and the LQR gain) to finish (i.e. the acquisition of both curves). This included writing theERG code from scratch on the laboratory computer.

96 6. Conclusions

Indeed, the development of new dynamic safety margins (or alternative threshold values) ispossibly one of the main research directions for the ERG since it may lead to better tradeoffsbetween performances, computational effort, generality, and robustness.As for the attraction field, there is still room for investigating the relationship with themotion planning literature. In a sense, given any motion planning algorithm that providesa steady-state admissible path from v to r, the ERG can be interpreted as a tool that lim-its the reference speed to ensure that the transient dynamics cannot cause a violation ofconstraints.

In addition to improving the existing theory, there are many future lines of enquiry forthe ERG framework. Possible extensions (and ideas on how to perform them) include

• Explicit Command Governor: in analogy to the Command Governor extensionsfor classical Reference Governors, the ERG could be extended to an Explicit CommandGovernor by computing each component of the vector v using different gains basedon the constraints. Preliminary results based on this intuition will be discussed inSubsection 9.2.3.

• Multi-Agent Systems: due to its simplicity, the ERG should be well-suited forextension to a multi-agent scenario where each agent has to coordinate with the otheragents while satisfying its own constraints.

• Output Regulation: the ERG framework currently assumes full knowledge of thestate vector. In the presence of a state observer, the ERG must be modified to takeinto account the combined system and observer dynamics. Please note that the thefeedforward approach detailed in Subsection 3.2.4 bypasses this issue by avoiding theneed of a known state vector.

• Passive Systems: a first step in specialising the ERG framework to passive systemshas already been taken by studying its application to Euler-Lagrange systems. Dueto its energy-based interpretation, it is worth noting that the ERG is a good candi-date for augmenting Port-Hamiltonian systems with a systematic solution for handlingconstraints.

• Time-Varying Constraints: the ERG framework has been introduced under theassumption of constant constraints. However, it is reasonable to assume that theERG framework will remain applicable as long as the dynamics of the pre-stabilizedsystem are faster than the rate of change of the constraints. This intuition shouldbe formalized by quantifying the rate at which the dynamic safety margin ∆(x, v)increases due to the pre-stabilizing feedback and the rate at which it decreases due tothe moving constraints. If these two contributions lead to a non-decreasing ∆(x, v), itcan be proven that the ERG is able to ensure constraint satisfaction.

Part II

Application to UnmannedAerial Vehicles

97

Chapter 7

Introduction

Part II of this dissertation illustrates how the ERG can be used to perform the constrainedcontrol of Unmanned Aerial Vehicles (UAVs). This application was the driving force thateventually led to the development of the Explicit Reference Governor. Indeed, the highlynonlinear nature and the relatively limited computational capabilities of UAVs emphasisedthe interest in a simple and systematic framework for handling constraints.

7.1 State of the Art

Recent advancements in the field of Unmanned Aerial Vehicles1 have lead to the availabilityof inexpensive aerial robots with a growing range of applications. As a result, a wide varietyof control laws for UAVs have been proposed in the literature [7.1].

Perhaps unsurprisingly, linear control schemes are still quite current in the domain dueto their robustness and implementation simplicity. Recent examples include PID schemes[7.2, 7.3, 7.4], linear quadratic regulators [7.5], and H∞ controllers [7.6, 7.7].Nevertheless, an increasing number of nonlinear control laws have also been proposed toextend the stability domain. Systematic methods such as dynamic inversion [7.8, 7.9] andbackstepping [7.10] have the advantage of providing a Lyapunov function for the closed-loop system. However, it is worth noting that their model-based nature makes them fairlysusceptible to robustness issues.An alternative approach consist in using a cascade control strategy that controls the UAVattitude and position using two distinct control loops and taking advantage of a timescaleseparation principle to stabilize the UAV [7.11, 7.12, 7.13]. Since the position dynamics canbe treated as a linear system and the attitude dynamics can be addressed using a variety ofdedicated control laws [7.14], this type of approach has the advantage of being robust andsimple to implement.

Model Predictive Control schemes have also been proposed for Unmanned Aerial Vehicles.Due to the limited onboard computational power, most of the proposed schemes are limitedto solving an unconstrained optimization problem [7.15, 7.16, 7.17, 7.18]. These control lawstypically achieve better performances with respect to closed-form solutions, and are typicallyable to avoid input saturations given a suitable choice of the cost function. Linear MPCschemes that enforce hard constraints on the control inputs have recently been proposed[7.19, 7.20].For what concerns other constraints, e.g. obstacle avoidance, the current trend in MPC

1Please note that this dissertation focuses on underactuated Vertical Take-Off and Landing (VTOL) typeUAVs such as quadrotors, helicopters, planar multirotors, ducted fans.

99

100 7. Introduction

schemes is to use a ground station to generate a suitable trajectory and communicate it tothe UAV [7.21, 7.22, 7.23].

Reference Governor schemes for Unmanned Aerial Vehicles have recently been investi-gated [7.24, 7.25]. Due to the reduced computational complexity, these methods show greatpromise for onboard implementation. Nevertheless, it is worth noting that solving the op-timization problem on standard UAV hardware is a challenging task that typically leaveslittle room for other computations.

By applying the Explicit Reference Governor to a pre-stabilized Unmanned Aerial Ve-hicle, the objective of this dissertation is to provide a constraint handling strategy with alimited computational footprint.

7.2 Main Contributions

Part II of this dissertation illustrates how to implement the ERG framework to perform theconstrained control of an Unmanned Aerial Vehicle subject to a wide variety of physicallyrelevant constraints.The UAV is pre-stabilized using the cascade control structure presented in [7.11]. Thiscontrol structure has been selected because its simplicity and robustness makes it one ofthe most popular approaches to control UAVs. To the author’s best knowledge, existingproofs of stability do not provide a Lyapunov function for the pre-stabilized system. In viewof implementing the ERG framework, an alternative proof of stability is proposed whichexplicitly constructs a Lyapunov function.Suitable dynamic safety margins and attraction fields will be designed separately for eachconstraint. This will be done by taking advantage of several results presented in Part I ofthis dissertation.

The contributions of Part II can be summarized in two main categories

1. Formulation of an analytic Lyapunov function for UAVs subject to the cascade controllaw [7.11];

2. Systematic design of a suitable ERG strategy that enforces a wide variety of typicalUAV constraints.

The overall result is a constrained control strategy that can be used as a simple add-on scheme for providing constraint handling capabilities to any UAV that has been pre-stabilized using a cascade control law.

7.2.1 Related Prizes and Publications

Most of the author’s earlier publications focused on applications where an UAV had to in-teract with the environment and was therefore subject to constraints. Initial solutions reliedon designing suitable input saturation strategies to control an UAV carrying a suspendedload [7.26], or an actuated tethered airfoil [7.27].

The use of a scalar Reference Governor to perform the constrained control of a tetheredUAV was investigated in [7.28]. This publication was awarded at the 19th IFAC WorldCongress with an Honorable Mention for the Young Author Prize.Although not detailed in this dissertation, publication [7.28] (later extended in [7.29]) wasfundamental to the conception of the Explicit Reference Governor because it emphasisedthe effectiveness of the Reference Governor approach in the context of nonlinear systems.

7.3. Part II Overview 101

Later contributions in the field of Unmanned Aerial Vehicles take advantage of the ERGframework. Preliminary results implementing the ERG in a multi-agent scenario are re-ported in [7.30]. This paper assumes that the attitude dynamics are ideal. A seminalversion of the constrained control scheme presented in this dissertation can be found inpublication [7.31], which develops a robust ERG able to deal with a bounded attitude erroras well as wind disturbances.

7.3 Part II Overview

Part II of this dissertation is structured following a multi-layer approach known as theGuidance, Navigation and Control (GNC) architecture. This scheme is typical of aerospacesystems and is remarkably close to the ERG philosophy.

Chapter 8 will address the problem of pre-stabilizing the UAV dynamics by means ofa primary control loop. This will be done by implementing a standard cascade approach.In view of implementing the ERG, the main focus of this chapter will be the constructionof a suitable Lyapunov function for the obtained system.

Chapter 9 will illustrate how to design an Explicit Reference Governor for the pre-stabilized UAV. This will be done by formulating a series of physically relevant constraintsand proposing a suitable dynamic safety margin and attraction field for each.

Chapter 10 will conclude Part II of this dissertation by summarizing the main resultsand providing future research directions for Unmanned Aerial Vehicles that have been aug-mented with an ERG.

Bibliography

[7.1] M.-D. Hua, T. Hamel, P. Morin, and C. Samson, “Introduction to feedback control ofunderactuated VTOL vehicles: A review of basic control design ideas and principles,”IEEE Control Systems, vol. 33, no. 1, pp. 61–75, 2013.

[7.2] J. M. Pflimlin, P. Binetti, P. Soueres, T. Hamel, and D. Trouchet, “Modeling andattitude control analysis of a ducted-fan micro aerial vehicle,” Control EngineeringPractice, vol. 18, no. 3, pp. 209–218, 2010.

[7.3] P. Pounds, R. Mahoney, and P. Corke, “Modelling and control of a large quadrotorrobot,” Control Engineering Practice, vol. 18, no. 7, pp. 691–699, 2010.

[7.4] K. Alexis, G. Nikolakopoulos, and A. Tzes, “Autonomous quadrotor position and at-titude PID/PIDD control in GPS-denied environments,” International Review of Au-tomatic Control, vol. 4, no. 3, pp. 421–430, 2011.

[7.5] S. Bouabdallah, A. Noth, and R. Siegwart, “PID vs LQ control techniques applied toan indoor micro quadrotor,” in Proc. of the IEEE/RSJ International Conference onIntelligent Robots and Systems, vol. 3, 2004, pp. 2451–2456.

[7.6] S. Mammar and G. Duc, “Loop shaping h∞ design: Application to the robust sta-bilization of a helicopter,” Control Engineering Practice, vol. 1, no. 2, pp. 349–356,1993.

[7.7] M. La Civita, G. Papageorgiou, W. Messner, and T. Kanade, “Design and flight testingof a gain-scheduled h-infinity loop shaping controller for wide-envelope flight of a robotichelicopter,” in Proc. of the American Control Conference (ACC), 2003, pp. 4195–4200.

102 7. Introduction

[7.8] T. J. Koo and S. Sastry, “Output tracking control design of a helicopter model based onapproximate linearization,” in Proc. of the IEEE Conference on Decision and Control(CDC), vol. 4, 1998, pp. 3635–3640.

[7.9] D. Das, K. Subbarao, and F. Lewis, “Dynamic inversion with zero-dynamics stabili-sation for quadrotor control,” IET Control Theory and Applications, vol. 3, no. 3, pp.303–314, 2008.

[7.10] E. Frazzoli, M. A. Daleh, and E. Feron, “Trajectory tracking control design for au-tonomous helicopters using a backstepping algorithm,” in Proc. of the American ControlConference (ACC), 2000, pp. 4102–4107.

[7.11] A. Isidori, L. Marconi, and A. Serrani, “Robust nonlinear motion control of a heli-copter,” IEEE Transactions on Automatic Control, vol. 48, no. 3, pp. 413–426, 2003.

[7.12] J. M. Pflimlin, T. Hamel, P. Soueres, and R. Mahoney, “A hierarchical control strategyfor the autonomous navigation of a ducted fan flying robot,” in Proc. of the IEEEInternational Conference on Robotics and Automation (ICRA), 2006, pp. 2491–2496.

[7.13] L. Marconi and R. Naldi, “Robust full degree-of-freedom tracking control of a heli-copter,” Automatica, vol. 43, no. 11, pp. 19 009–1920, 2007.

[7.14] N. A. Chaturvedi, A. K. Sanyal, and N. H. McClamroch, “Rigid-body attitude con-trol,” IEEE Control Systems, vol. 31, no. 3, pp. 30–51, 2011.

[7.15] R. Franz, M. Milam, and J. Hauser, “Applied receding horizon control of the caltechducted fan,” in Proc. of the American Control Conference (ACC), vol. 5, 2002, pp.3735–3740.

[7.16] H. J. Kim, D. H. Shim, and S. Sastry, “Nonlinear model predictive tracking con-trol for rotorcraft-based unmanned aerial vehicles,” in Proc. of the American ControlConference (ACC), vol. 5, 2002, pp. 3576–3581.

[7.17] S. Bertrand, T. Hamel, and H. Piet-Lahanier, “Performance improvement of an adap-tive controller using model predictive control: Application to an uav model,” in Proc.of the IFAC Symposium on Mechatronic Systems, 2006, pp. 770–775.

[7.18] P. Bouffard, A. Aswani, and C. Tomlin, “Learning-based model predictive controlon a quadrotor: Onboard implementation and experimental results,” in Proc. of IEEEInternational Conference on Robotics and Automation, 2012, pp. 279–284.

[7.19] K. Alexis, G. Nikolakopoulos, and A. Tzes, “Model predictive quadrotor control:attitude, altitude and position experimental studies,” IET Control Theory and Appli-cations, vol. 6, no. 12, pp. 1812–1827, 2012.

[7.20] M. Hofer, M. Muehlebach, and R. D’Andrea, “Application of an approximate modelpredictive control scheme on an unmanned aerial vehicle,” in Proc. of the IEEE Inter-national Conference on Robotics and Automation (ICRA), 2016.

[7.21] A. Richards and J. P. How, “Robust variable horizon model predictive control forvehicle maneuvering,” International Journal of Robust and Nonlinear Control, vol. 16,no. 7, pp. 333–351, 2006.

[7.22] D. Mellinger and V. Kumar, “Minimum snap trajectory generation and control forquadrotors,” in IEEE International Conference on Robotics and Automation (ICRA),2011, pp. 2520–2525.

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[7.23] M. Hehn and R. D’Andrea, “Real-time trajectory generation for interception maneu-vers with quadrocopters,” in IEEE/RSJ International Conference on Intelligent Robotsand Systems, 2012, pp. 4979–4984.

[7.24] N. E. Kahveci and I. V. Kolmanovsky, “Constrained control of UAVs using adaptiveanti-windup compensation and reference governors,” SAE Technical Paper, 2009.

[7.25] W. Lucia, M. Sznaier, and G. Franze, “An obstacle avoidance and motion planningcommand governor based scheme: The Qball-X4 quadrotor case of study,” in IEEEConference on Decision and Control (CDC), 2014, pp. 6135–6140.

[7.26] M. M. Nicotra, E. Garone, and R. N. and, “Nested saturation control of an uavcarrying a suspended load,” in Proc. of the American Control Conference (ACC), 2014,pp. 3585–3590.

[7.27] S. Eeckhout, M. M. Nicotra, R. Naldi, and E. Garone, “Nonlinear control of anactuated tethered airfoil,” in Proc. of the Mediterranean Conference of Control andAutomation (MED), 2014, pp. 1412–1417.

[7.28] M. M. Nicotra, R. Naldi, and E. Garone, “Taut cable control of a tethered UAV,” inProc. of the IFAC World Congress, 2014, pp. 3190–3195.

[7.29] M. M. Nicotra, R. Naldi, and E. Garone, “Nonlinear control of atethered UAV: thetaut cable case,” Automatica, to appear.

[7.30] M. M. Nicotra, M. Bartulovic, E. Garone, and B. Sinopoli, “A distributed explicitreference governor for constrained control of multiple uavs,” in IFAC Workshop onDistributed Estimation and Control in Networked Systems (IFAC Proceedings), vol. 48,no. 22, 2015, pp. 156–161.

[7.31] M. M. Nicotra, R. Naldi, and E. Garone, “A robust explicit reference governor forconstrained control of unmanned aerial vehicles,” in Proc. of the American ControlConference (ACC), 2016, pp. 6284–6289.

Chapter 8

Control Layer

The objective of this chapter is to present a primary control law for a Vertical Take-Off andLanding (VTOL) type UAV and provide a Lyapunov function for the pre-stabilized system.These results will be used as a starting point for the next chapter which addresses the designof the ERG.

For the reader’s convenience, the following subsection provides a brief summary of quater-nion algebra. Additional information about quaternions can be found in [8.1].

8.0.1 Quaternion Algebra

In analogy to how complex numbers C can be used to perform operations in R2, operationsin R3 can be performed using hypercomplex numbers H. A hypercomplex number h ∈ H isa number in the form h = a+ ib+ jc+ kd, where a, b, c, d are real numbers whereas i, j andk are three distinct complex numbers subject to the following multiplication rules

ii = −1 jj = −1 kk = −1ij = −k jk = −i ki = −jik = −j ji = −k kj = −i.

(8.1)

Any R3 vector w = [w1 w2 w3]T can be represented using the equivalent H expressionw = iw1 + jw2 + kw3. A rotation of θ degrees around a unitary vector u = [u1 u2 u3] canthen be performed by defining the unitary quaternion

q = cosθ

2+ (iu1 + ju2 + ku3) sin

θ

2

and computing w′ = q w q∗, where q∗ is the complex conjugate of q. By taking into accountthe quaternion rules (8.1), the product w′ = q w q∗ in H can be reformulated in R3 usingthe matrix product w′ = R(q)w, with

R(q) = I3 + 2q∧I q∧I + 2qRq

∧I , (8.2)

where qR = cos θ2 is the real component of q, qI = [u1 u2 u3]T sin θ2 is the vector of its

imaginary components, and the operator ∧ : R3 → R3×3 denotes the cross-product matrix

p∧ =

−0 −p3 −p2

−p3 −0 −p1

−p2 −p1 −0

.105

106 8. Control Layer

8.1 Dynamic Model

The objective of this section is to describe the general state-space dynamic model that iscommon to most1 VOTLs. To this end, let p ∈ R3 denote the position of the UAV in theglobal reference frame, let ω ∈ R3 denote its angular velocity in the body reference frame,and let the unitary quaternion2 q ∈ H; qq∗ = 1 describe the rigid rotation from the bodyreference frame to the global reference frame. The quaternion q can be decomposed into areal component qR ∈ R and a vector of imaginary components qI ∈ R3.

As detailed in [8.2], the control inputs acting on a VTOL can usually be expressed interms of a resulting trust force T ∈ R and torque vector τ ∈ R3. These control inputs arealways defined in the body reference frame. Given an UAV of mass m ∈ R; m > 0 andinertia matrix J ∈ R3×3; J = JT ; J > 0, the Newton-Euler equations of motion lead to thefollowing dynamic model:

mp = mg ·e3 − T ·R(q)e3[qRqI

]=

1

2E(q)ω

Jω = −ω∧Jω + τ,

(8.3)

where g ≈ 9.81m/s2 is the gravitational acceleration, e3 = [0 0 1]T , R(q) is the rotationmatrix in equation (8.2) and E(q) is the quaternion differential kinematics matrix

E(q) =

[−qTI

qRI3 + q∧I

]. (8.4)

System (8.3) is entirely described by the state vector x =[pT pT qR q

TI ω

T]T ∈ R13 and

the control input vector u =[T τT

]T ∈ R4. The following section characterizes the set ofconfigurations which can be maintained at steady-state and formulates the objective of thischapter.

8.2 Control Objectives

The objective of the control layer is to stabilize the UAV in a given equilibrium point xv. Tocharacterize the set of all achievable points of equilibrium, it is necessary to identify underwhich conditions system (8.3) remains stationary. This leads to the following steady-staterequirements

p = 0; ω = 0; R(q)e3 = e3.

Since a vector is rotationally invariant only with respect to itself, the only steady-stateadmissible attitude q is given by an arbitrary rotation around the axis e3. As for the positionp, the UAV model does not introduce any restraints on the equilibrium coordinates. As aresult, the UAV is able to maintain the equilibrium position p = v and the equilibriumattitude qR = cos(ψ/2), qI = sin(ψ/2)e3 for all v ∈ R3 and ψ ∈ [−π, π). The controlobjectives of the pre-stabilizing control law can therefore be states as follows.

Problem 8.1. Given a constant reference position v ∈ R3 and yaw angle ψ ∈ [−π, π), theobjective of the control layer is to pre-stabilize the UAV dynamics (8.3) so that

xv =[vT 0T cosψ2 sinψ2 e

T3 0T

]T(8.5)

1This thesis focuses on the more “typical” VOTL configurations (e.g. quadrotors, planar multirotors,ducted fans) where the thrust vector is always opposite to the body axis e3. Although fully actuatedconfigurations (e.g. non-planar multirotors, tilt-rotors) are not addressed in this thesis, it is worth notingthat their Lyapunov functions are generally easier to determine with respect to the underactuated case.

2q∗ denotes the complex-conjugate of q.

8.3. Control Strategy 107

𝑥, �� 𝑞, 𝜔

𝜏

𝑣, 𝜓 Position

Dynamics Attitude

Dynamic

Inner

Loop

Outer

Loop 𝑞𝐶

𝑇

𝑞 𝑥

Control Law UAV

𝜒𝑂𝑢𝑡

𝜑𝑂𝑢𝑡 𝜑𝐼𝑛

𝜒𝐼𝑛

𝑉𝑂𝑢𝑡

𝑉𝐼𝑛

��

𝜔𝐶

Figure 8.1: Representation of the inner/outer loop control scheme.

is an Asymptotically Stable equilibrium point. �

In view of using a Lyapunov-based ERG for ensuring constraint enforcement, the follow-ing objective must also be addressed.

Problem 8.2. Determine a Lyapunov function for the pre-stabilized system. �

8.3 Control Strategy

As stated in the introduction, the stabilization of an UAV is a well-known problem that hasbeen widely addressed in the control literature. As such, there are many available solutionsto Problem 8.1. One of the most popular control strategy consists in introducing the virtualcontrol input qC ∈ H; qCq

∗C = 1 and stabilizing the UAV position and attitude with two

separate control loops as illustrated in Figure 8.1. As detailed in [8.3], the design principlefor this control scheme is based on the following steps:

1. Design an outer loop to control the UAV position using qC as an input. This is doneunder the assumption that q = qC ;

2. Design an inner loop to control the UAV attitude using qC as a reference. This isdone under the assumption that qC = 0;

3. After dropping both assumptions, determine under what conditions the interconnec-tion of the two control loops is Asymptotically Stable.

It is worth noting that [8.3] and related articles have proven asymptotic stability of thisscheme using the Small Gain Theorem [8.4]. Although this implies that a Lyapunov functionexists, to the best of the author’s knowledge its actual formulation has never been computed.To solve Problem 8.2, it is therefore necessary to perform a more in-depth stability analysiswith respect to the existing literature.

8.4 Outer Loop Control

The objective of the outer loop is to control the position of the UAV so that it asymptoticallytends to a constant v ∈ R3 under the assumption that the attitude can be used as a controlinput. Given q = qC , the position dynamics of the state-space model (8.3) become

mp = mg ·e3 − T ·R(qC)e3, (8.6)


where −T·R(qC)e3 is the thrust vector expressed in the global reference frame. Since both Tand qC are control inputs, the thrust vector can be assigned as a standard PD with gravitycompensation,

T ·R(qC)e3 = m (kp (p− v) + kdp+ g ·e3) , (8.7)

where kp, kd are positive scalars. The following proposition provides a strict Lyapunovfunction for the resulting closed-loop system.

Proposition 8.1. Let system (8.6) be subject to the control law (8.7) with kp, kd > 0.

Given a constant reference v, then[pT pT

]T=[vT 0T

]Tis a Globally Asymptotically Stable

equilibrium point and

VOut(p, p, v) =1

2

[p− vp

]T[(kp+εk2

d)I3 εkdI3εkdI3 I3

] [p− vp

], (8.8)

with ε ∈ (0, 1) is a strict Lyapunov function. �

Proof. Given ε < 1, equation (8.8) is a Lyapunov candidate function for the equilibriumpoint p = v and p = 0. By taking its time derivative and substituting (8.6)-(8.7), it followsthat

VOut(p, p, v)=−[p− vp

]T[εkpkdI3 0

0 (1− ε)kdI3

][p− vp

],

which is negative definite ∀ε ∈ (0, 1). �

The Lyapunov function (8.8) will be used in Section 8.6 to characterize the ISS propertiesof the outer loop when the assumption q = qC is dropped. Before addressing the inner loopcontrol, the following subsection illustrates how to compute T and qC to satisfy equation(8.7), while simultaneously imposing the desired yaw ψ ∈ [−π, π).

8.4.1 Thrust Vectoring

Given the control law (8.7), it is necessary to determine a suitable the thrust T and controlattitude qC , which provide the thrust vector. To this end, consider the components of thedesired thrust vectorFxFy

Fz

= −mkp

px−vxpy−vypz−vz

−mkd pxpypz

−mg 0

01

. (8.9)

Since ‖R(qC)e3‖ = 1, ∀qC ∈ H, the required thrust can be obtained directly from themodulus

T =√F 2x + F 2

y + F 2z . (8.10)

As for qC , it is useful to define α ∈ [−π, π] as the angle between the axis e3 and the vectorT ·R(qC)e3, i.e.

α = arctan

√F 2x + F 2

y

Fz. (8.11)

The minimum rotation between e3 and R(qC)e3 is therefore equal to qα ∈ H, qαq∗α = 1,with3

qα,R = cosα

2, qα,I =

sin α2√

F 2x + F 2

y

Fy−Fx

0

.3Please note that the solution is well-posed even if F 2

x + F 2y = 0. Indeed, in this case it follows that

α = 0, which implies qα,R = 1 and qα,I = 0.

8.5. Inner Loop Control 109

Although qα is such that R(qC)e3 = R(qα)e3, it is worth noting that this does not implyqC = qα. Indeed, given any qψ ∈ H, qψq∗ψ = 1 satisfying

qψ,R = cosψ

2, qψ,I = sin

ψ

2·e3,

it follows that e3 = R(qψ)e3. The control quaternion qC can therefore be expressed as thecombination of an arbitrary rotation ψ around the yaw axis e3 and the minimum rotationα that aligns e3 with the desired thrust vector. This is given by the following product in H

qC = qαqψ,

which can be computed in matrix form by taking advantage of the multiplication rules (8.1).This leads to [

qC,RqC,I

]=

[qα,R −qTα,Iqα,I qα,RI3 + q∧α,I

] [qψ,Rqψ,I

].

8.5 Inner Loop Control

The objective of the outer loop is to control the attitude of the UAV under the assumptionthat the desired control attitude qC remains constant. Given the attitude error quaternion

q = qq∗C , (8.12)

the attitude dynamics in (8.3) can be rewritten in error coordinates[

˙qR˙qI

]=

1

2E(q)ω


(8.13)

and stabilized using a PD control law in the quaternion space

τ = −hpqI − hdω, (8.14)

where hp, hd are positive scalars. The following proposition provides a strict Lyapunovfunction for the resulting closed-loop system.

Proposition 8.2. Let system (8.13) be subject to the control law (8.14) with hp, hd > 0.

Then,[qR; qTI ; ωT

]T=[1; 0T ; 0T

]Tis an Asymptotically Stable equilibrium point and

VIn(qR, qI , ω) = 2hp(1− qR) +1

2

[qIω

]T[4ηhdI3 2ηJ

2ηJ J

] [qIω

], (8.15)

is a strict Lyapunov function, ∀η ∈ (0,min{hd/(µ(J) + hd/2hp), hd/λ3{J}}), where

µ(J) = λ3{J} cos arctan∆λ{J}λ3{J}

+ ∆λ{J} sin arctan∆λ{J}λ3{J}

, (8.16)

∆λ{J} = λ3{J} − λ1{J}, and λ1{J}, λ3{J} are, respectively, the smallest and largesteigenvalue of matrix J . �

Proof. Given η < hd/λ3{J}, equation (8.15) is a Lyapunov candidate function for the equi-librium point qR = 1, qI = 0, and ω = 0. By taking its time derivative and substituting4

(8.13)-(8.14), it follows that

VIn(qR, qI , ω)=−[qIω

]T[2ηhpI3 ηhd(qR − 1)I3

ηhd(qR − 1)I3 hdI3 − ηJ(qRI3 − q∧I )

][qIω

]. (8.17)

4Performing this step requires the use of the following properties: a× (a ·b) = 0 and a · (b×c) = c · (a×b),∀a, b, c ∈ R3, where × and · are the vector and scalar products, respectively.


Using the modulus of qI and ω, equation (8.17) can be upper-bounded by

VIn ≤ −[

sin |α|2‖ω‖

]TQIn(α)

[sin |α|2‖ω‖

],

where

QIn(α) =

2ηhp ηhd

(cos |α|2 − 1

)ηhd

(cos |α|2 − 1

)hd − η

(λ3{J} cos |α|2 + ∆λ{J} sin|α|

2

). (8.18)

Since cos |α|2 ≥ 0 and λ3{J} cos |α|2 + ∆λ{J} sin |α|2 ≤ µ(J), ∀α ∈ [−π, π], it follows from(8.18) that QIn(α) > QIn, where

QIn =

[2ηhp ηhdηhd hd − ηµ(J)

]. (8.19)

As a result, (8.17) is negative definite ∀η ∈ (0, hd/(µ(J) + hd/2hp)]. �

The Lyapunov function (8.15) will be used in the following section to characterize theISS properties of the inner loop when the assumption qC = 0 is dropped.

8.6 Interconnection

Having designed the inner and outer loops separately, the final step consists in determiningunder what conditions the interconnection is asymptotically stable. This will be done usingan approach similar to [8.3] which uses the Small Gain Theorem to prove that the inter-connection is asymptotically stable if the inner loop is sufficiently fast. Instead of using theclassical formulation [8.4] of the Small Gain Theorem, however, asymptotic stability will beproven using the Lyapunov formulation developed in [8.5]. This will enable the solution ofProblem 8.2.

Due to its complexity, the problem will be addressed in four separate lemmas which willthen be combined in one final statement. Referring to Figure 8.2, the following outline isproposed:

• Subsection 8.6.1: Lemmas 8.1 and 8.2 will address the ISS properties of each controlloop and provide the asymptotic gains χ between the disturbances and the Lyapunovfunctions. In this context, q will act as a disturbance for the outer loop, whereas ωCwill act as a disturbance for the inner loop.

• Subsection 8.6.2: Lemmas 8.3 and 8.4 will determine the maximum gains ϕ betweenthe Lyapunov functions and the output signals. In this context, qC will be treated asan output of the outer loop, whereas q will be treated as an output of the inner loop.

• Subsection 8.6.3: Proposition 8.3 will apply the Small Gain Theorem to identify thestability conditions and to construct a Lyapunov function for the overall system.

8.6.1 Input to State Gains

For what regards the outer loop behaviour in the presence of a non-zero attitude errorq = qq∗C , consider the UAV model (8.3). Given q = qqC , the position dynamics can berewritten as

mp = mg ·e3 − T ·R(q)R(qC)e3.

8.6. Interconnection 111

𝑥, �� 𝑞, 𝜔

𝜏

𝑣, 𝜓 Position

Dynamics Attitude

Dynamic

Inner

Loop

Outer

Loop 𝑞𝐶

𝑇

𝑞 𝑥

Control Law UAV

𝜒𝑂𝑢𝑡

𝜑𝑂𝑢𝑡 𝜑𝐼𝑛

𝜒𝐼𝑛

𝑉𝑂𝑢𝑡

𝑉𝐼𝑛

��

𝜔𝐶

Inner

Loop

Outer

Loop

Figure 8.2: Schematic representation of the gains addressed in this section.

By substituting the control law (8.7), the closed-loop system becomes

p = −kpR(q)(p− v)− kdR(q)p+ g(I3 −R(q))e3. (8.20)

The following lemma extends the results of Proposition 8.1 by characterizing the ISS prop-erties of the outer loop in the presence of an attitude error.

Lemma 8.1. Let system (8.20), with kp, kd > 0, be subject to a constant applied referencev and a bounded attitude error ‖α‖∞ ≤ ∆α satisfying

0 ≤ ∆α < arccos

(b−√b2 − aca

), (8.21)

with

a = (kp − εk2d)2; b = k2

p+2εkpk2d(1−ε)+ε2k4

d; c = (kp + εk2d)2.

Then, the Lyapunov function in equation (8.8) satisfies the asymptotic gain VOut ≤ χOut(‖α‖∞),where5

χOut(‖α‖∞) = maxσ∈[ 0 , d ]

{[σ

µ(σ)

]T[WTPW WTPMMTPW MTPM

][σ

µ(σ)

]}, (8.22)

5For the sake of notational simplicity, the dependency on ‖α‖∞ has been omitted for all the right termsof equations (8.22)-(8.28).


with

d(‖α‖∞) = (1− cos ‖α‖∞) g (8.23)

µ(σ, ‖α‖∞) = sgn(WT PM

)√σ(d− σ)

WTW

MTM(8.24)

P (‖α‖∞) =1

2

√QOut

−T[

(kp+εk2d) εkd

εkd 1

]√QOut

−1, (8.25)

W (‖α‖∞)=√QOut

−T[εkd1

], (8.26)

M(‖α‖∞)= Ker(WT ), (8.27)

and

QOut(‖α‖∞) =1

2

[2εkpkd cos ‖α‖∞ (kp+εk2

d)(cos ‖α‖∞−1)(kp+εk2

d)(cos ‖α‖∞−1) 2kd(cos ‖α‖∞−ε)

]. (8.28)

Proof. Given ε < 1, (8.8) is an ISS-Lyapunov candidate function. By taking its time deriva-tive and substituting (8.20), it follows that

VOut=−1

2

[p− vp

]TQ(q)

[p− vp

]+

[p− vp

]T[εkdI3I3

]g(I3−R(q))e3, (8.29)

where

Q(q) =

[εkpkd(R(q)T +R(q)) kp(R(q)T−I3)+εk2

d(R(q)−I3)kp(R(q)−I3) + εk2

d(R(q)T−I3) kd(R(q)T +R(q)−2εI3)

].

Using the modulus of p− v and p, equation (8.29) can be upper bounded by

VOut≤−[‖p− v‖‖p‖

]TQOut

[‖p− v‖‖p‖

]+

[‖p− v‖‖p‖

]T[εkd1

]d, (8.30)

with QOut = QOut(‖α‖∞) and d = d(‖α‖∞) given in equations (8.28) and (8.23), respec-tively. To prove ISS, it is sufficient to note that restriction (8.21) is such that QOut(‖α‖∞) >0, ∀ ‖α‖∞ ≤ ∆α.

The asymptotic gain can now be computed by finding χOut(‖α‖∞) such that VOut ≥χOut(‖α‖∞) implies VOut ≤ 0. To do so, consider the change of coordinates ζ =

√QOut [ ‖p− v‖ ‖p‖ ]

T.

Using (8.26), equation (8.30) can be rewritten as

VOut ≤ −ζT ζ + ζTWd. (8.31)

Since W and M in equations (8.26)-(8.27) describe an orthonormal basis, the vector ζ canbe decomposed into

ζ = σW + µM. (8.32)

As a result, (8.31) becomes

V ≤ −µ2MTM − σ(σ − d)WTW, (8.33)

which is non-negative for σ ∈ [ 0, d ] and

µ ∈

[√σ(d− σ)

WTW

MTM,

√σ(d− σ)

WTW

MTM

].


0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

ε

∆αm

ax(ε

) [d

eg]

Optimization Problem

∆α

max(ε)

max(∆αmax

)

Figure 8.3: Typical behaviour of the maximum attack angle as a function of ε for ζ = 0.7.

At this point, consider the Lyapunov function (8.8). By substituting ζ as in (8.32), it followsthat

VOut =

[σµ

]T[WTPW WTPMMTPW MTPM

][σµ

].

To conclude the proof, it is sufficient to note that (8.22) is the largest value of VOut suchthat VOut is non-negative. �

Remark 8.1. It is worth noting that the scalar parameter ε ∈ (0, 1) in the Lyapunov function(8.8) can be chosen freely and can be used to maximise the upper bound on ∆α. Givenkd = 2ζ

√kp, where ζ > 0 is the damping ratio of the outer loop, the coefficients of inequality

(8.21) become

a = (1− 4ζ2ε)2 b = 1 + 8ζ2ε(1− ε) + 16ζ4ε2 c = (1 + 4ζ2ε)2.

This implies that the bound on ∆α is independent from kp. Given a fixed value of thedamping ratio6, it is possible to maximise the restriction (8.21) by solving

maxε∈(0,1)

arccos

(b+√b2 + ac

a

).

As illustrated in Figure 8.3, this is a convex scalar optimization problem which can be solvedoff-line. �

Remark 8.2. Given suitable parameters kp, kd, ε, the optimization problem (8.22) is themaximization of a smooth scalar function VOut(σ) in a bounded interval σ ∈ [ 0 , d(‖α‖∞) ].As shown in Figures 8.4 and 8.5, the function χOut(‖α‖∞) can be easily be computed off-lineand tabulated in a one-dimensional lookup function. �

6The damping ratio ζ is usually chosen during the control design phase based on other considerations.


0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

14

16

σ

χ Out

(σ ,

∆α)

Scalar Optimization Problem

V

Out(σ,∆α)

χOut

(∆α)

Figure 8.4: Typical behaviour of VOut(σ, ‖α‖∞) for a given ‖α‖∞.

0 10 20 30 40 500

1

2

3

4

5

6

7

8x 10

4

∆α

χ Out

(∆α)

InterpolationTo infinity,and beyond!

Figure 8.5: Typical behaviour of χOut(‖α‖∞).


For what regards the Input to State Stability of the inner loop, the presence of a time-varying control attitude qC modifies the attitude error dynamics (8.13) into

[˙qR˙qI

]=

1

2E(q) (ω − ωC)


(8.34)

where ωC is the angular velocity of the control attitude. The following lemma extendsthe results of Proposition 8.2 by characterizing the ISS properties of the inner loop for anon-constant qC .

Lemma 8.2. Let system (8.34) be subject to the control law (8.14) with hp, hd > 0. Then,

there exists an asymptotic gain χIn such that VIn ≥ χIn(‖ωC‖∞) implies VIn ≤ 0. Moreover,given η ∝ hd and hd ∝

√hp, the asymptotic gain χIn remains bounded for arbitrarily large

hp. �

Proof. Given η < hd/λ3{J}, (8.15) is an ISS-Lyapunov candidate function. By taking itstime derivative and substituting (8.34) and (8.34), it follows that

VIn = −[qIω

]T[2ηhpI3 ηhd(qR − 1)I3

ηhd(qR − 1)I3 hdI3 − ηJ(qRI3 − q∧I )

][qIω

]+ . . .

VIn =−[qIω

]T[(hp + 2ηqRhd) I3ηJ (qRI3 + q∧I )

]ωC .

Using the modulus of qI and ω, the following upper-bounded is obtained

VIn ≤ −[

sin |α|2‖ω‖

]TQIn(α)

[sin |α|2‖ω‖

]+

[sin |α|2‖ω‖

]TDIn(α) ‖ωC‖∞

with QIn(α) as in (8.18) and

DIn(α) =

[hp + 2ηhd cos |α|2

η(λ3{J} cos |α|2 + ∆λ(J) sin |α|2

)].To further simplify the study, the following bounds are applied: QIn(α) ≥ QIn, ∀α ∈ [−π, π],where QIn is given in (8.19) and is positive-definite ∀η ∈ (0, hd/(µ(J) + hd/2hp)), and‖DIn(α)‖ ≤

∥∥DIn

∥∥ , ∀α ∈ [−π, π], with

DIn =

[hp + 2ηhdηµ(J)

]. (8.35)

As a result, it follows that

VIn ≤ −[

sin |α|2‖ω‖

]TQIn

[sin |α|2‖ω‖

]+

[sin |α|2‖ω‖

]TDIn ‖ωC‖∞, (8.36)

which is sufficient to prove ISS. To characterize the asymptotic gain, consider the change of

coordinates ζ =√QIn

[sin |α|2 ‖ω‖

]T. Equation (8.36) can then be rewritten as

VIn ≤ −ζT ζ + ζT√QIn

−TDIn ‖ωC‖∞ ,

which implies

‖ζ‖ ≥∥∥∥∥√QIn−T DIn

∥∥∥∥ ‖ωC‖∞ ⇒ VIn ≤ 0. (8.37)


To compute the invariant level-set, consider the following upper bound on the Lyapunovfunction (8.15)

VIn ≤ 2hp sin2 |α|2

+1

2

[sin |α|2‖ω‖

]T[4ηhd 2ηλ3{J}

2ηλ3{J} λ3{J}

][sin |α|2‖ω‖

],

which is valid ∀α ∈ [−π, π]. By substituting ζ, (8.37) implies VIn ≤ λ3{PIn} ‖ζ‖2, where

PIn =1

2

√QIn

−T [4(hp + ηhd) 2ηλ3{J}2ηλ3{J} λ3{J}

]√QIn

−1

. (8.38)

As a result, the asymptotic gain is

χIn(‖ωC‖∞) = λ3{PIn}∥∥∥∥√QIn−T DIn

∥∥∥∥2

‖ωC‖2∞ . (8.39)

To study the behaviour of χIn(‖ωC‖∞) for increasing hp, consider η ∝ hd and hd ∝√hp.

Given (8.19), (8.35), the following proportionalities hold true

QIn ∝[√

hphp hphp

√hp

]DIn ∝

[hp√hp

]. (8.40)

By applying the following properties of 2× 2 matrices

√A = 1√

Tr(A)+2√

detA

[a11 +

√detA a12

a21 a22 +√

detA

]A−1 = 1

detA

[a22 −a21

a12 a11

],

it can be shown that √QIn

−1

∝ 4√hp

[1hp

1hp

1hp

1√hp

]. (8.41)

Therefore, given equations (8.38) and (8.40), the following holds true√QIn

−TDIn ∝ 4

√hp

[11

]PIn ∝ 1√

hp

[1 11 1

]. (8.42)

Since

χIn ∝ λ3{PIn}∥∥∥∥√QIn−T DIn

∥∥∥∥2

=

√hp√hp

= 1,

the asymptotic gain χIn(‖ωC‖∞) remains bounded for arbitrarily large hp. �

Remark 8.3. It is worth noting that the assumption hd ∝√hp presented in Lemma 8.2 is

a fairly reasonable design choice. Indeed, for linear systems in the form x = −hpx − hdx,it is customary to assign hd = 2ζ

√hp where ζ is the damping ratio and

√hp is the natural

frequency. Although nonlinear systems behave differently, this assumption is generally a goodrule of thumb. �

Having determined the asymptotic gains between the disturbances and the value of theLyapunov function, the following subsection will characterize the maximum gain betweenthe value of the Lyapunov function and the output of each loop.

8.6.2 State to Output Gains


As shown in Subsection 8.4.1, the control atti-tude qC can be computed directly from the de-sired outer loop control law (8.7). Since the in-ner loop considers the angular velocity ωC asa disturbance, it is necessary to characterize themaximum gain between the outer loop Lyapunovfunction VOut and the resulting derivative of thecontrol attitude. This is done in the followinglemma.

Lemma 8.3. Given the outer loop control law(8.7), subject to a constant reference position vand yaw angle ψ, there exists a maximum gainϕOut between the Lyapunov function (8.8) andthe output ωC . Moreover, ‖ωC‖ is bounded forarbitrarily large values of VOut. �

Proof. Given a constant yaw angle ψ, the angu-lar velocity ωC is equal to the derivative of the minimal angle α between the axis e3 and thevector T ·R(qC)e3. Given equation (8.11), it follows that

ωC =FxyFz − FzFxyF 2xy + F 2

z

, (8.43)

with Fxy=√F 2x+F 2

y , and Fx, Fy, Fz given in (8.9). The derivative of Fz is

Fz = mkpkd(pz − vz) +m(k2d − kp)pz,

which is proportional to |pz − vz| and |pz|. Likewise, it can be shown that Fxy ∝ |px − vx|+|py − vy|+ |px|+ |py|. Since (8.8), (8.43) are continuous functions of p−v and v, there existsa maximum gain ϕOut such that

‖ωC‖ < ϕOut(VOut).

To prove that ‖ωC‖ is bounded, consider what happens for arbitrarily large VOut and,therefore, for arbitrarily large ‖p− v‖ and ‖v‖. Given (8.43), it follows that

ωC ∝(|px − vx|+ |py − vy|+ |px|+ |py|) · (|pz − vz|+ |pz|)|px − vx|2 + |py − vy|2 + |pz − vz|2 + |px|2 + |py|2 + |pz|2

.

As a result, ‖ωC‖ is bounded for any combination of state variables going to infinity. Sinceit is also continuous, ϕOut(VOut) admits a global maximum. �

Although Lemma 8.3 does not explicitly state the gain ϕOut(VOut), its existence andglobal boundedness will be sufficient for the remainder of this thesis. The following lemmawill instead provide a more detailed characterization of the maximum gain between the innerloop Lyapunov function and the output α.

Lemma 8.4. Given the Lyapunov function (8.15), the output α satisfies the maximum gain|α| ≤ ϕIn(VIn), where

ϕIn(VIn) = 2 arccos

(1− VIn

2(hp + η(hd − ηλ3{J}))

). (8.44)


Proof. The Lyapunov function (8.15) is minimum with respect to the vector ω for ω = −2ηqI .By substituting this value in (8.15), it follows that

VIn ≥ 2hp(1− qR) + 2η qTI (hdI3 − ηJ) qI .

By taking into account qTI JqTI ≤ λ3{J} ‖qI‖2 and sin2(x) ≥ (1− cos(x)), ∀x ∈ [−π/2, π/2],

the following lower bound holds true

VIn ≥ 2(hp + η (hdI3 − ηλ3{J}))(

1− cosα

2

), ∀α ∈ [−π, π].

The statement is then proven by inverting this inequality to obtain (8.44). �

Having determined the input to state and the state to output gains of each loop sepa-rately, the final step remaining is to study the behaviour of the interconnection between thetwo loops.

8.6.3 Small Gain Theorem

The following statement combines all the previous lemmas to construct a Lyapunov functionfor an UAV controlled with a cascade approach. This is the main result of the chapter.

Proposition 8.3. Consider system (8.3) subject to the control laws (8.7), with kp > 0, kd >0, and (8.14), with hp > 0 and hd ∝

√hp. Then, given a constant reference position v ∈ R3

and yaw angle ψ ∈ [−π, π], the equilibrium point (8.5) is semi-Globally Asymptotically Stablefor sufficiently large hp. Moreover,

V (p, p, qR, qI , ω, v, ψ) = max (VOut(p− v, p), χOut(ϕIn(VIn(q, ω)))) (8.45)

with VOut, VIn, χOut, and ϕIn given in (8.8), (8.15), (8.22), and (8.44) is a Lyapunovfunction for all initial conditions such that V (0) ≤ χOut(∆α), with ∆α satisfying (8.21). �

Proof. Consider the interconnection illustrated in Figure 8.2. Following from Lemma 8.3,‖ωC‖∞ < ϕOut(VOut) is bounded regardless of VOut. Additionally, it follows from Lemmas8.2 and 8.4 that the attitude error α will asymptotically satisfy ‖α‖ ≤ ϕIn(χIn(‖ωC‖∞)),where χIn remains bounded whereas ϕIn becomes arbitrarily small for arbitrarily large hp.As a result, there exists a sufficiently large hp such that

‖α(t)‖ ≤ ∆α, ∀t ≥ τ,

where τ ≥ 0 is a finite time instant. Without loss of generality7, consider the case τ = 0.Since the requirements of Lemma 8.1 hold true, the outer loop will satisfy the asymptoticgain VOut ≤ χOut(‖α‖∞). The stability of the interconnected loops is therefore proven usingthe Small Gain Theorem since

χOut(ϕIn(χIn(ϕOut(VOut)))) < VOut

holds true for sufficiently large hp. As a result, it follows from [8.5] that (8.45) is a Lyapunovfunction. To conclude the proof, it is sufficient to note that V (0) ≤ χOut(∆α) impliesϕIn(VIn(t)) ≤ ∆α, ∀t ≥ 0. �

The main contribution of Proposition 8.3 is that it provides an overall Lyapunov functionfor the interconnected closed-loop system. To the author’s best knowledge, this result isnot found in the literature since its construction is not only difficult, but also unnecessaryfor what regards the proof of stability. Although marginally important for what regardsthe control layer (which simply requires its existence), the Lyapunov function (8.45) isa fundamental starting point for the Navigation layer which will enforce constraints byaugmenting the pre-stabilized UAV with an Explicit Reference Governor.

7If τ >0, it is sufficient to define a new timescale t= t−τ and study the system for t≥0.

8.7. Simulations 119

0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

15

20

25

30

35

40

45

t [s]

Pos

ition

[m]

Step Response

vx

vy

vz

xyz

0 1 2 3 4 5 6 7 8 9 100

50

100

150

200

t [s]

Atti

tude

ang

le [d

eg]

ψ2 acos(q)

R

Figure 8.6: Step Response of the pre-stabilized system.

8.7 Simulations

The objective of this section is to illustrate the behaviour of the control layer by showingthat

• The control law (8.7), (8.14) is successful at pre-stabilizing the system for kp, kd > 0,hd ∝

√hp, and sufficiently large hp > 0;

• The Lyapunov function (8.45) is, indeed, a time-decreasing function.

Numerical simulations were performed for an UAV with mass m = 2 kg and moment ofinertia J = diag([0.0082 0.0082 0.164])kgm2. The system is initialized in p(0) = [0 0 0]T ,p(0) = [0 0 0]T , q(0) = [1 0 0 0]T , and ω(0) = [0 0 0]T .

Figure 8.6 illustrates the behaviour of the pre-stabilized system given the applied refer-ence position v = [42 0 0] and yaw angle ψ = π. The control gains are kP = 1.10 N/mkg,kD = 1.47 Ns/mkg, hP = 89.92Nm/rad, and hD = 7.55Nms/rad. The proposed control lawsuccessfully meets the desired objectives.

Figures 8.7 and 8.8 illustrate the behaviour of the outer loop Lyapunov function VOut(t)and of the rescaled inner loop Lyapunov function χOut(φIn(VIn(t))).


0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

4

5

6

7x 10

6

t [s]

Lyap

unov

val

ue

Initial Behaviour

χ

Out(φ

In(V

In))

VOut

Figure 8.7: Detail of the initial response of the Lyapunov function.

0 2 4 6 8 100

500

1000

1500

t [s]

Ove

rall

Beh

avou

r

Overall Behaviour

χ

Out(φ

In(V

In))

VOut

Figure 8.8: Behaviour of the two components of the Lyapunov function.

8.8. Summary 121

As illustrated in Figure 8.7, χOut(φIn(VIn(t))) is initially several orders of magnitudehigher than VOut and is strictly time-decreasing as long as the condition χOut(φIn(VIn(t))) ≥VOut(t) is satisfied. This condition is met at time t ≈ 0.27 s, after which χOut(φIn(VIn(t)))becomes several orders of magnitude smaller than VOut(t).As illustrated in Figure 8.8, VOut(t) becomes strictly time-decreasing as soon as VOut(t) ≥χOut(φIn(VIn(t))) is verified.

This supports the statement of Proposition 8.3 that max{VOut, χOut(VIn)} is a Lyapunovfunction. Interestingly enough, the two functions are not always time-decreasing when takensingularly. Indeed, VOut(t) shows a very slight increase for t < 0.27 s and χOut(φIn(VIn(t)))presents a local peak in the time interval t ∈ (1, 1.5) s. This justifies the need for using acombination of the two ISS Lyapunov functions.

8.8 Summary

This chapter has described the stabilization of an Unmanned Aerial Vehicle by means of arelatively standard cascade control scheme. The stability of the closed-loop system has beenstudied in great detail and a strict Lyapunov function has been proposed for the overallsystem.

The proposed Lyapunov function will be used as a starting point to design an ExplicitReference Governor for the closed-loop system.

Bibliography

[8.1] M. D. Shuster, “A survey of attitude representation,” The Journal of the AstronauticalSciences, vol. 41, no. 4, pp. 439–517, 1993.

[8.2] M.-D. Hua, T. Hamel, P. Morin, and C. Samson, “Introduction to feedback control ofunderactuated VTOL vehicles: A review of basic control design ideas and principles,”IEEE Control Systems, vol. 33, no. 1, pp. 61–75, 2013.

[8.3] A. Isidori, L. Marconi, and A. Serrani, “Robust nonlinear motion control of a heli-copter,” IEEE Transactions on Automatic Control, vol. 48, no. 3, pp. 413–426, 2003.

[8.4] Z.-P. Jiang, A. R. Teel, and L. Praly, “Small-gain theorem for iss systems and appli-cations,” Mathematics of Control, Signals and Systems, vol. 7, no. 2, pp. 95–120, 1994.

[8.5] Z.-P. Jiang, I. M. Y. Mareels, and Y. Wang, “A lyapunov formulation of the nonlinearsmall-gain theorem for interconnected iss systems,” Automatica, vol. 32, no. 8, pp. 1211–1215, 1996.

Chapter 9

Navigation Layer

The objective of this chapter is to design an Explicit Reference Governor for an UAV thathas been pre-stabilized using a cascade control approach. This will be done by determiningsuitable dynamic safety margins and attraction fields for a variety of physically relevantconstraints.

9.1 Navigation Objectives

Given a pre-stabilized UAV, the objective of the navigation layer is to enforce a set of stateand input constraints by suitably manipulating the auxiliary reference of the closed-loopsystem. As such, the problem statement addressed in this chapter is the same as Problem2.3, where the generic system x = f(x, v) is replaced with equations (8.13), (8.14), and(8.20). As for c(x, v), the following state and input constraints are considered:

• Maximum Attitude Error: |α| ≤∆α, with ∆α satisfying (8.21). This constraintensures that (8.45) satisfies the requirements of Proposition 8.3 at all times;

• Maximum Torque: τ ≤ τmax, with τ given by (8.14). This constraint can be usedto avoid actuator saturations;

• Maximum Thrust: T ≤ Tmax, with T as in (8.9)-(8.10) and Tmax > mg. Thisconstraint can be used to avoid actuator saturations;

• Free-fall Avoidance: Fz ≥mg−Tmin, with Tmin ∈ (0,mg). This constraint ensuresthat the UAV propellers are not subject to stall phenomena due to the updraft of air;

• Wall Avoidance: cTp+ d ≥ 0. This constraint models a planar surface that theUAV must not trespass;

• Obstacle Avoidance: ‖p− p0‖ − R ≥, with R > 0. This constraint models aspherical surface that the UAV must circumvent.

The following sections will address each constraint separately by constructing suitable dy-namic safety margins and attraction fields. Numerical examples will demonstrate the validityof the ERG that progressively takes into account all of the constraints.

9.2 Maximum Attitude Error

The limit on the attitude error |α| ≤ ∆α ensures that (8.45) is a strict Lyapunov functionand is fundamental for ensuring asymptotic stability and enforcing all the other constraints.

123

124 9. Navigation Layer

9.2.1 Threshold Value

To implement the Lyapunov-based dynamic safety margin, it is necessary to identify athreshold value associated to the constraint |α| ≤ ∆α. By taking advantage of the resultsin the previous chapter, the following is proposed:

Proposition 9.1. Given the Lyapunov function (8.45) and given ∆α satisfying (8.21), thethreshold value

Γ∆α = χOut(∆α) (9.1)

is such that V (p, p, qR, qI , ω, v, ψ) ≤ Γ∆α ensures |α| ≤ ∆α. �

Proof. The statement follows directly from the proof of Proposition 8.3 since V (τ) ≤χOut(∆α) implies |α| ≤ ϕIn(VIn(t)) ≤ ∆α, ∀t ≥ τ . �

Since the threshold value Γ∆α does not depend of the current reference, it can be com-puted off-line and stored in memory.


To compute the attraction field, it is necessary to express the constraint |α| ≤ ∆α at steady-state. Since the steady-state attitude error is |α| = 0, the system will always satisfy thebound ∆α > 0 at steady-state. As a result, the standard attraction field (2.19), i.e.

ρ(r, ψr, v, ψ) =

[r − vψr − ψ

]/max

(∥∥∥∥[ r − vψr − ψ

]∥∥∥∥ , η) , (9.2)

is a suitable solution.

9.2.3 Vector ERG

It is worth noting that the basic ERG[v

ψ

]= κ (Γ∆α − V (p, p, qR, qI , ω, v, ψ)) ρ(r, ψr, v, ψ) (9.3)

presented in Part I of this dissertation does not distinguish between v and ψ, meaning thatthe behaviour of the two references is coupled. However, since r ∈ R3 and ψr ∈ [−π, π)represent a position and an angle, respectively, it is reasonable to assume that v and ψ willhave a significantly different effect on the system dynamics. As a result, it may be interestingto decouple the two references based on their effect on the Lyapunov function.

Since v has a direct influence on VOut(p, p, v), whereas ψ has a direct influence onVIn(qR, qI , ω), a tempting choice would be

v = κv (Γ∆α − VOut(p, p, v)) ρv(r, v)

ψ = κψ (∆α− φIn(VIn(qR, qI , ω))) ρψ(ψr, ψ),(9.4)

with

ρv(r, v) =r − v

max (‖r − v‖ , ηv)(9.5)

and

ρψ(ψr, ψ) =ψr − ψ

max (|ψr − ψ| , ηψ). (9.6)

9.3. Numerical Validation 125

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

t [s]

y [m

]

Position

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

t [s]

ψ [d

eg]

Attitude

r(t)Scalar v(t)Scalar y(t)Vector1 v(t)Vector1 y(t)Vector2 v(t)Vector2 y(t)

ψr(t)

Scalar ψ(t)Scalar α(t)Vector1 ψ(t)Vector1 α(t)Vector2 ψ(t)Vector2 α(t)

Figure 9.1: Output response of the scalar and vector ERG.

Indeed, v ensures Γ∆α ≥ VOut(p, p, v) whenever VOut ≥ χOut(φIn(VIn)), whereas ψ ensures1

Γ∆α ≥ χOut(φIn(VIn)) whenever χOut(φIn(VIn)) ≥ VOut.Due to the interactions between VOut and VIn, however, v and ψ will also present an

indirect influence on VIn and VOut, respectively. As a result, (9.4) may lead to constraintviolations since v 6= 0 when χOut(φIn(VIn)) = Γ∆α and ψ 6= 0 when VOut = Γ∆α. Toaccount for this problem, a possible solution is to substitute (9.4) with

v = κv (Γ∆α − VOut(p, p, v)) min(

∆α−φIn(VIn(qR,qI ,ω))∆ΓIn

, 1)ρv(r, v)

ψ = κψ (∆α− φIn(VIn(qR, qI , ω))) min(

Γ∆α−VOut(p,p,v)∆ΓOut

, 1)ρψ(ψr, ψ),

(9.7)

where ∆ΓIn > 0 and ∆ΓOut > 0 are positive scalars. This ensures that v and ψ will behaveindependently as long as ∆α−φIn(VIn(qR, qI , ω)) > ∆ΓIn and Γ∆α−VOut(p, p, v) > ∆ΓOut.However, the two references will once again be coupled as soon as one of the two loops is inproximity of a constraint violation.

Remark 9.1. Equation (9.7) is a promising first step in the development of a generaltheory for the vector ERG. Indeed, the idea of addressing each component of the referencein a separate manner is analogous to the vector Reference Governor approach in [9.1]. �

9.3 Numerical Validation

Given the pre-stabilized UAV presented in Section 8.7 subject to a step response from theorigin to r = [0 80 0] and ψr = π, the behaviour of the following three strategies arecompared:

1The function χOut was simplified on both sides of the equation by taking into account equation (9.1).


0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

t [s]

α err [d

eg]

Attitude Error

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

t [s]

α err [d

eg]

Close−up

αerr

≤ ∆α

ScalarVector1Vector2

Figure 9.2: Attitude errors achieved by the scalar and vector ERG.

• Scalar ERG: The ERG is implemented using equation (9.3) with κ = 103, η = 0.1;

• Vector ERG 1: The ERG is implemented using equation (9.4) with κv = 103,κψ = 102, ηv = 0.1, and ηψ = π/20;

• Vector ERG 2: The ERG is implemented using equation (9.7) with κv = 103,κψ = 102, ηv = 0.1, ηψ = π/20, ∆ΓIn = 2·10−2, and ∆ΓOut = 10−4.

As illustrated in Figure 9.1, the response of the position dynamics is virtually indistin-guishable among all three strategies. However, the attitude response clearly shows that thescalar ERG is substantially slower at imposing the yaw angle.This loss of performance is due to the formulation of the scalar ERG which causes all thecomponents of the applied reference to converge simultaneously. Since the vector ERG doesnot impose this behaviour, the position reference v is no longer a bottleneck for the yawreference ψ.

For what concerns the two vector ERG strategies, it is interesting to note that the twooutput responses are basically the same in terms of position and attitude. However, as shownin Figure 9.2, the two strategies are substantially different since (9.4) causes a violation ofthe maximum attitude error constraint for t ∈ [0, 0.2). Instead, the vector ERG formulation(9.7) successfully enforces the constraint at all times.

Based on these simulations, the vector ERG formulation (9.7) has been selected as themost suitable strategy to enforce the maximum attitude error constraint.

9.4. Maximum Torque 127

ω [rad/s]

α err [r

ad]

Torque Saturation Threshold

−0.05 0 0.05

−4

−3

−2

−1

0

1

2

3

4

x 10−3

VIn

Γτ|τ| ≤ τ

max

Figure 9.3: Lyapunov level-sets in the plane α− ω and threshold value estimation.

9.4 Maximum Torque

Although the torque saturation constraint ‖τ‖ ≤ τmax can be enforced using a Lyapunov-based ERG, this section shows that the required threshold is too conservative for practicalimplementation. As a result, an alternative solution based on the invariant set approachwill be presented.


Since the maximum torque constraint depends only on the inner loop variables, the corre-sponding threshold value can be obtained by studying the Lyapunov lower-bound

V (p, p, qR, qI , ω, v, ψ) ≥ χOut(ϕIn(VIn(q, ω))),

where VIn(q, ω) ≥ V In(α, ‖ω‖), with

V In = 2hp

(1− cos

|α|2

)+

1

2

[sin |α|2‖ω‖

]T[4ηhd −2ηλ3{J}

−2ηλ3{J} λ1J

][sin |α|2‖ω‖

].

The threshold value can then be found by solving the optimization problem (3.2) in therange |α| ∈ [0, 2 arcsin(τmax/hp)], ‖ω‖ ∈ [0, τmax/hd) and subject to constraint

hp sin|α|2

+ hd ‖ω‖ ≤ τmax.

As illustrated in Figure 9.3, finding the threshold value Γτ is relatively simple. Unfortunately,however, Figure 9.3 also illustrates the limit of using the Lyapunov-based dynamic safetymargin. Indeed, given τmax = 0.2 [Nm], the largest attitude error satisfying VIn ≤ Γτ is α =1.9 10−3[rad]. Clearly, this value is very restrictive and likely to penalize the performances.


9.4.2 Invariant Approach

Since the Lyapunov-based dynamic safety margin is too conservative, an alternative solutionconsists in using the invariant-based approach proposed in Subsection 3.2.1. Due to thecomplexity of the system, however, the analytic formulation of an invariant set Sv is a fairlychallenging task.

This subsection investigates a more practical solution that infers a invariant set Sv basedon the following hypothesis.

Hypothesis 9.1. Let the closed-loop system (8.3), with (8.14) and (8.7), be subject toconstant references v, ψ. Then, for any point satisfying

V (p, p, qR, qI , ω, v, ψ) ≤ Γ∆α

‖τ‖ = τmax,(9.8)

the state trajectories are such that τT τ ≤ 0. �

Although reasonable for typical system and control parameters, this hypothesis must bevalidated numerically on a case-by-case scenario. Given a specific pre-stabilized UAV, thevalidation process requires the computation of the flow lines along the surface satisfying(9.8). This (computationally intensive) step can be performed as part of the design phaseof the ERG.

If Hypothesis 9.1 is verified, it follows that the closed surface

∂Sv =

{x :

V (x, v) ≤ Γ∆α

‖τ‖ = τmax

}⋃{x :

V (x, v) = Γ∆α

‖τ‖ ≤ τmax

}is the boundary of an admissible invariant set Sv. By taking into account the fact thatV ≤ Γ∆α is ensured by the previous section, the invariant-based ERG can therefore beimplemented using τmax − ‖τ‖ as a distance between the current state and the boundary∂Sv. This leads to the following dynamic safety margin

∆τ (p, p, qR, qI , ω, v, ψ) = κτ (τmax − ‖τ‖), (9.9)

with κτ > 0.

The main advantage of the invariant-based approach (9.9) is that it ensures constraintsatisfaction by simply checking the current value of the torque and comparing it to thesaturation limit. In addition to being computationally inexpensive, this solution provideshigh performances since the set Sv is well-aligned with the constraints.


In analogy to Section 9.2, the steady-state torque ‖τ‖ = 0 will always be inferior to τmax > 0.As a result, the attraction fields (9.5)-(9.6) do not require any modifications.

9.4.4 Numerical Validation

The following simulation compares the behaviours obtained using the following strategies:

• Previous ERG: The Vector ERG2 of the previous section is implemented withoutmodifications;

• Lyapunov ERG: The ERG is implemented accounting for the torque constraint usingthe Lyapunov-based approach;

9.5. Maximum Thrust 129

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

t [s]

p [m

]

Position

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

t [s]

ψ [d

eg]

Attitude

ry(t)

Previous vy(t)

Previous y(t)Lyapunov v

y(t)

Lyapunov y(t)Invariant v

y(t)

Invariant y(t)

ψr(t)

Previous ψ(t)Previous α(t)Lyapunov ψ(t)Lyapunov α(t)Invariant ψ(t)Invariant α(t)

Figure 9.4: Output response of the Lyapunov-based and invariant-based ERG.

• Invariant ERG: The ERG is implemented accounting for the torque constraint usingthe invariant-based dynamic safety margin in equation (9.9).

As illustrated in Figure 9.4, the Lyapunov-based approach is so conservative that v ≈ 0.Instead, the output response of the invariant-based approach is very similar to the strategythat does not account for the torque constraint.The control input ‖τ‖ is reported in Figure 9.5. As expected, the invariant-based ERGsuccessfully enforces the torque constraint at all times.

9.5 Maximum Thrust

This section addresses the input saturation constraint T ≤ Tmax, which is not linear in thestate variables since T = m ‖kp(p− v) + kdp+ ge3‖. To apply the Lyapunov-based ERG, itwill therefore be necessary to find a linear constraint that implies T ≤ Tmax.

9.5.1 Bounded Accelerations

A possible approach to provide a linear constraint is to make a distinction between thesteady-state thrust mge3 and the dynamic feedback mkp(p− v) +mkdp). This can be doneby taking advantage of the triangular inequality

T ≤ ‖mkp(p− v) +mkdp‖+mg.

To enforce the maximum thrust constraint, it is therefore sufficient to ensure

[kpeT kde

T ]

[p− vp

]≤ Tmax −mg

m, ∀e ∈ R : ‖e‖ = 1. (9.10)


0 5 10 15 200

0.2

0.4

0.6

0.8

1

t [s]

τ [N

m]

Torque

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

t [s]

τ [N

m]

Close−up

||τ|| ≤ τmax

Previous ||τ(t)||Lyapunov ||τ(t)||Invariant ||τ(t)||

Figure 9.5: Input torque required by the Lyapunov-based and invariant-based ERG.

This is equivalent to limiting the maximum acceleration of the UAV in any direction. Themain interest with equation (9.10) is that it defines a rotationally invariant constraint thatis linear for any given unitary vector e.

The next step to implement a Lyapunov-based ERG is to select a suitable ellipsoidalLyapunov function and compute the threshold value using the analytic estimate (3.19).


As discussed in Section 8.4, the outer loop can be modeled as a linear system subject toa bounded2 time-varying uncertainty |α| ≤ ∆α. As a result, the Lyapunov-based dynamicsafety margin can be applied by finding a common Lyapunov function for perturbed outerloop dynamics. A possible option is to simply use the outer loop Lyapunov function (8.8).Since the system is linear, however, it may be advantageous to combine the results in Sections3.3 and 5.3 by seeking a common Lyapunov function in the quadratic form

VT =

[p− vp

]TPT

[p− vp

], (9.11)

with PT > 0. Following from equation (8.20), the common Lyapunov function must satisfythe Lyapunov equation[

0 I3kpR(q) kdR(q)

]TPT + PT

[0 I3

kpR(q) kdR(q)

]≤ 0,

for all q ∈ H such that 2 arccos(qR) ≤ ∆α. By taking advantage of the rotational symmetryof the system and defining

PT =

[PT,11I3 PT,12I3PT,21I3 PT,22I3

],

2The bound on α is ensured by the presence of the ERG.

9.5. Maximum Thrust 131

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

t [s]

T [N

]

Thrust

T ≤ Tmax

Previous T(t)Lyapunov1 T(t)Lyapunov2 T(t)

Figure 9.6: Required Thrust.

it follows from Proposition (3.53) that the optimal Lyapunov function can be obtained bysolving

min log det(PT ), s.t.

A(0)T PT + PTA(0) ≤ 0;

A(∆α)T PT + PTA(∆α) ≤ 0;

PT ≥ cT cTT ,

(9.12)

where

A(α) =

[0 1

kp cos(α) kd cos(α)

], (9.13)

and

cT = [kp kd]T .

Given the ellipsoidal Lyapunov function (9.11), the threshold value

ΓT =(Tmax −mg)2

m2cTT PT cT

follows directly from Proposition 3.3.


It is worth noting that, at steady-state, the UAV must be able to generate the thrustT = mge3. As a result, the input saturation constraint must necessarily satisfy Tmax > mg,meaning that the actuators must at least be able to sustain the weight of the UAV. Assumingthat the UAV was designed accordingly, the system will always satisfy the input constraintat steady-state. As a result, the attraction fields (9.5)-(9.6) do not require any modifications.


The following simulation compares the behaviours obtained using the following strategies:


0 5 10 15 20 25 30 35 400

20

40

60

80

100

t [s]

p [m

]

Position

0 5 10 15 20 25 30 35 400

50

100

150

200

t [s]

ψ [d

eg]

Attitude

ψr(t)

Previous ψ(t)Previous α(t)Lyapunov1 ψ(t)Lyapunov1 α(t)Lyapunov2 ψ(t)Lyapunov2 α(t)

ry(t)

Previous vy(t)

Previous y(t)Lyapunov1 v

y(t)

Lyapunov1 y(t)Lyapunov2 v

y(t)

Lyapunov2 y(t)

Figure 9.7: Output response of the two Lyapunov-based strategies.

• Previous ERG: The Invariant ERG of the previous section is implemented withoutmodifications;

• Lyapunov ERG 1: The ERG is implemented accounting for the maximum thrustconstraint using the Lyapunov function (8.8);

• Invariant ERG 2: The ERG is implemented accounting for the maximum thrustconstraint using the Lyapunov function (9.11).

As illustrated in Figure 9.6, the response of the system subject to the previously designedERG does not violate the maximum thrust constraint. This is likely due to the fact that theother constraints (in particular |α| ≤ ∆α) are more restrictive. Ideally, the ERG strategiesthat enforce the maximum thrust constraint should therefore be able to achieve the samebehaviour.

As shown in Figure 9.7, however, the ERG that employs the Lyapunov function (8.8)incurs in a significant loss of performances. Instead, the ERG which employs the opti-mal Lyapunov function (9.11) is able to guarantee constraint satisfaction while ensuring aresponse that is comparable to the previous behaviour.

9.6 Free-fall Avoidance

The free-fall avoidance constraint Fz ≥ 0 can be addressed in the same way as the maximumthrust constraint by ensuring ‖mkp(p− v) +mkdp‖ ≤ mg. Since the maximum thrustconstraint is typically Tmax ≈ 2mg, the free-fall avoidance constraint can be treated asredundant.

9.7. Wall Avoidance 133

9.7 Wall Avoidance

The wall avoidance constraint cT p+d ≥ 0 is the first constraint in this chapter that takes intoaccount the surroundings of the UAV. Since the position p is one of the outer loop states, thecomputation of the dynamic safety margin will be similar to the maximum thrust constraint.


In analogy to Section (9.5), the wall constraint cT p + d ≥ 0 only takes into account outerloop variables. As a result, the Lyapunov-based dynamic safety margin can be computed bytreating the outer loop as a linear system with a bounded time-varying uncertainty |α| ≤ ∆α.As a result, the following Lyapunov function is proposed

VW =

[p− vp

]TPW

[p− vp

], (9.14)

where

PW =

[PW,11I3 PW,12I3PW,21I3 PW,22I3

],

and PW > 0 is the solution tomin log det(PW ), s.t.

A(0)T PW + PWA(0) ≤ 0;

A(∆α)T PW + PWA(∆α) ≤ 0;

PW ≥ cW cTW ,

(9.15)

with A(α) given in (9.13) and

cW = [1 0]T .

This provides a Lyapunov function which is specifically designed for limiting the maximumposition error ‖p− v‖. Following from Proposition 3.3, the threshold value associated to awall constraint cT p+ d ≥ 0 is

ΓW (v) =(cT v + d)2

cTW P−1W cW

. (9.16)


Unlike the previous constraints addressed in this chapter, the wall constraint cT p + d ≥ 0can potentially be violated at steady-state p = v. As a result, the attraction field (9.5) isno longer sufficient to ensure the correct behaviour of the ERG.

Since the domain D = {v : cT v + d ≥ 0} is convex, it follows from Proposition 4.1 thata suitable choice3 for the attraction field is to extend the attraction field (9.5) with

ρv,W (v, r) = ρv(v, r) + max

(ζ − (cT v + d)

ζ − δ, 0

)c, (9.17)

where ζ > δ is the influence region of the wall constraint and δ > 0 is the static safetymargin.

3Equation (9.17) assumes that ‖c‖ = 1. Otherwise, the wall constraint should be rescaled as c = c/ ‖c‖and d = d/ ‖c‖.


−50 0 500

10

20

30

40

50

60

70

80

90

100

x [m]

y [m

]

rx,y

(t)

vx,y

(tf)

px,y

(t)

cTp+d ≤ 0

Figure 9.8: Output response in the presence of a wall constraint.


Figure 9.8 illustrates the behaviour of the controlled system in the x−y plane given the wallconstraint cT = [1 1 0]/

√2 and d = 60/

√2. The ERG was implemented using the influence

margin ζ = 4m and the static safety margin δ = 2m. As expected, the UAV navigatestowards the best steady-state admissible projection of the desired reference.

As illustrated in Figure 9.9, the simulation also accounts for a wind perturbation that isapproximately 10% of the UAV weight. As discussed in Chapter 5, the choice of a sufficientlylarge static safety margin δ is sufficient to ensure the robustness of the ERG even in thepresence of external disturbances. This can be done by taking advantage of the fact that(9.14) is an ISS-Lyapunov function for the outer loop dynamics.

9.8 Obstacle Avoidance

The obstacle avoidance constraint ‖p− p0‖−R ≥ 0 defines a non-convex admissible region.Therefore, it is typically regarded as a relatively challenging constraint. Interestingly enough,this section will show that the ERG framework has no particular difficulties in handling thisconstraint.


Given a fixed reference v, it can be shown using triangular inequalities that

‖p− p0‖ ≥ ‖p0 − v‖ − ‖p− v‖ .

As a result, ‖p− p0‖ −R ≥ 0 can be enforced by simply ensuring

(p0 − v)T

‖p0 − v‖(p− v) ≤ R− ‖p0 − v‖ . (9.18)

9.8. Obstacle Avoidance 135

0 5 10 15 20 25 30 35 40−3

−2

−1

0

1

2

3

t [s]

Fw

[N]

F

x

Fy

Fz

Figure 9.9: Wind disturbances acting on the UAV.

The main interest in equation (9.18) is that it defines a reference-dependent virtual wallc(v)T p + d(v) ≤ 0 that guarantees the non-violation of the obstacle. The dynamic safetymargin of the previous subsection can therefore be used by choosing

c(v) =(p0 − v)T

‖p0 − v‖

and

d(v) = ‖p0 − v‖ −(p0 − v)T

‖p0 − v‖v −R.


For what concerns the attraction field, it is worth noting that the domain D = v : cT + d ≥ δ∪ v : ‖v − p0‖ −R ≥ δ contains a spherical hole. Following from Proposition 4.2, the attrac-tion field (9.17) should be extended as follows

ρv,O(v, r) = ρv,W (v, r) + max

(ζ − (‖p0 − v‖ −R)

ζ − δ, 0

)((p0 − v)T

‖p0 − v‖+ ρO(v)

), (9.19)

withρO(v) = ˆsgn

(ρTv,W (v, r)(p0 − v)⊥)

)(p0 − v)⊥. (9.20)

Remark 9.2. Since (p0−v) ∈ R3, it is worth noting that the perpendicular vector (p0−v)⊥

is not uniquely defined. A possible strategy for choosing (p0 − v)⊥ is to select an arbitraryvector (e.g. the altitude axis e3) and assign

(p0 − v)⊥ = Ker([(p0 − v) e3]T ). (9.21)

Although reasonable in many scenarios, this strategy has the disadvantage of being singularif (p0 − v) is parallel to e3.


−50 0 500

10

20

30

40

50

60

70

80

90

100

x [m]

y [m

]

rx,y

(t)

vx,y

(tf)

px,y

(t)

c(xv,v) ≥ 0

Figure 9.10: Output response in the presence of a wall and obstacle constraint.

A second strategy consists in defining

(p0 − v)⊥ = aKer1(p0 − v)T +√

1− a2 Ker2(p0 − v)T , (9.22)

where Keri represents the i-th column vector of the kernel and a ∈ [0, 1] can be selectedrandomly4. �


Figure 9.10 illustrates the behaviour of the controlled system in the x − y plane given anobstacle of radius R = 8m centered in p0 = [0 23 0]T . For ease of representation, (p − v)⊥

was assigned using (9.21). As expected, the UAV successfully circumvents the obstacle andnavigates towards the best steady-state admissible projection of the desired reference.

9.9 Summary

This chapter has described the step-by-step development of an Explicit Reference Governorfor Unmanned Aerial Vehicles. This has been done by addressing several different constraintsand proposing suitable dynamic safety margins and attraction fields for each constraint.Numerical validations of the performance and robustness of the scheme have been provided.

As illustrated in this chapter, the ERG successfully provides constraint-handling capa-bilities to the standard cascade control scheme. This is achieved without any need of onlineoptimization. The proposed scheme has the advantage of being computationally efficientand simple to implement. For what concerns the performances, it has been shown that anappropriate choice of the ERG components provides a reasonable output response. As such,this chapter illustrates the potential of the ERG framework as a systematic approach forensuring constraint satisfaction.

4Please note that equation (9.20) is designed to guarantee convergence for whatever choice of (p0 − v)⊥

Bibliography 137

Bibliography

[9.1] E. G. Gilbert, I. V. Kolmanovsky, and K. T. Tan, “Discrete-time reference governorsand the nonlinear control of systems with state and control constraints,” InternationalJournal of Robust and Nonlinear Control, vol. 5, no. 5, pp. 487–504, 1995.

Chapter 10

Conclusions

Part II of this dissertation has shown how the Explicit Reference Governor can be used asan add-on tool to provide constraint handling capabilities to an Unmanned Aerial Vehicle.The UAV was stabilized using an established control strategy and the Lyapunov functionfor the controlled system has been proposed. The ERG was then designed and validated fora variety of physically relevant constraints.

10.1 Future Research Opportunities

This dissertation has applied the ERG to a simple free-flying UAV. Due to the great interestin these systems, there are still many opportunities for further research. Possible improve-ments include finding a less conservative Lyapunov function, addressing new constraints,obtaining better thresholds or applying different dynamic safety margins. Future lines ofenquiry could also focus on using the ERG to perform application-specific tasks such as

• Physical Interactions: the ERG could be used to manage contact forces that arisewhen the UAV docks, operates and undocks with a rigid surface. This is a major issuethat arises in aerial service robotics.

• Tethered Flight: in some cases, it may be interesting to employ an UAV that isphysically connected to a ground station by means of a cable. The cable can beused to exchange energy and/or data to extend the flight-time and computationalcapabilities of the UAV. The ERG could then be employed to limit the tensile forceacting on the cable and avoid entanglements.

• Teleoperation: a possible way to improve human-UAV interactions is to provide thepilot with a tactile feedback of the constraints. This could be done using an ERG andgenerating a force that is proportional to the difference between v and r.

• Swarming: due to its simplicity, the ERG should be well-suited for extension toscenarios with multiple UAVs that must collaborate between themselves while takinginto account possible constraints.

10.2 Closing Remarks

Part I of this dissertation has described the Explicit Reference Governor framework for therobust constrained control of nonlinear systems. The flexibility of the proposed frameworkwas then demonstrated in Part II by applying the ERG to an Unmanned Aerial Vehicle.

139

140 10. Conclusions

These systems represent an excellent example of the target application of the proposedframework due to their relatively strong nonlinearities and limited computational power.

Based on the obtained results, the author hopes that the Explicit Reference Governormay find its place among existing constrained control strategies as a robust closed-formalternative that does not strive for optimal performances but provides a simple, reliable andsystematic solution for enforcing constraint satisfaction.

constrained control of nonlinear systems

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