passivitymethods for the stabilization of closedsetsin ... · abstract passivity methods for the...

Passivity methods for the stabilization of closed sets in

nonlinear control systems

by

Mohamed Ibrahim El-Hawwary

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

Copyright c© 2011 by Mohamed Ibrahim El-Hawwary

Abstract

Passivity methods for the stabilization of closed sets in nonlinear control systems

Mohamed Ibrahim El-Hawwary

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2011

In this thesis we study the stabilization of closed sets for passive nonlinear control

systems, developing necessary and sufficient conditions under which a passivity-based

feedback stabilizes a given goal set. The development of this result takes us to a journey

through the so-called reduction problem: given two nested invariant sets Γ1 ⊂ Γ2, and

assuming that Γ1 enjoys certain stability properties relative to Γ2, under what conditions

does Γ1 enjoy the same stability properties with respect to the whole state space? We

develop reduction principles for stability, asymptotic stability, and attractivity which are

applicable to arbitrary closed sets. When applied to the passivity-based set stabilization

problem, the reduction theory suggests a new definition of detectability which is geo-

metrically appealing and captures precisely the property that the control system must

possess in order for the stabilization problem to be solvable.

The reduction theory and set stabilization results developed in this thesis are used to

solve a distributed coordination problem for a group of unicycles, whereby the vehicles

are required to converge to a circular formation of desired radius, with a specific ordering

and spacing on the circle.

ii

Dedication

To my parents.

iii

Acknowledgements

All gratitude is due for my supervisor Professor Manfredi Maggiore. He has been a

great mentor and a dear friend. His invaluable help and guidance served as basis for this

work and extended beyond that to have a profound effect on me, something which I will

ever be grateful for.

I want to thank Professors Bruce Francis and Mireille Broucke for their efforts as

members of my PhD committee and exam committee and for their reviews and con-

structive feedback. Also, I would like to thank members of my PhD exam committee

Professors Lacra Pavel and Raymond Kwong for their constructive remarks, and Pro-

fessor Hassam K. Khalil for serving as an external examiner for my dissertation and for

providing a valuable review.

Financial supports from the University of Toronto, Manfredi Maggiore, Government

of Ontario, Rogers Family and Ewing Rae are highly appreciated.

I want to thank all members of the Systems Control Group, at the Edward S. Rogers

Sr. Department of Electrical and Computer Engineering, who made my time in Toronto

an unforgettable experience.

Last, but not least, deepest appreciation goes to my sister, Inas, for her ultimate

support.

iv

Contents

Abstract ii

List of Figures viii

List of Notation x

1 Introduction 1

1.1 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Set stabilization applications . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Set stabilization in conventional control . . . . . . . . . . . . . . . . . . . 12

1.4 Literature on set stabilization . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Preliminaries 22

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Set stability and attractivity . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Limit sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3 Passivity-Based Set Stabilization I: Preliminaries 35

3.1 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 The set stabilization problem . . . . . . . . . . . . . . . . . . . . . . . . 39

v

3.3 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Set stabilization and the reduction problem . . . . . . . . . . . . . . . . . 42

4 Reduction Principles 46

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Seibert-Florio’s reduction theorems . . . . . . . . . . . . . . . . . . . . . 48

4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Reduction theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Proofs of reduction theorems . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.1 Proof of Theorem 4.4.1 . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.2 Proof of Theorem 4.4.6 . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5.3 Proof of Theorem 4.4.8 . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6 Cascade-connected systems . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.7 Reduction-based control design . . . . . . . . . . . . . . . . . . . . . . . 61

5 Passivity-Based Set Stabilization II: Theory 64

5.1 Γ-Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Solution of PBSSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.4 Path following for the point-mass system . . . . . . . . . . . . . . . . . . 77

6 Passivity-Based Set Stabilization III: Control Design 84

6.1 Set stabilization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Case study 1: path following for the kinematic unicycle . . . . . . . . . 89

6.3 Case study 2: stabilizing the unicycle to a circle with heading angle re-

quirement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4 Case study 3: coordination of two unicycles . . . . . . . . . . . . . . . . 100

vi

7 Circular Formation Control of Unicycles 106

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Information flow and digraphs . . . . . . . . . . . . . . . . . . . . . . . . 110

7.3 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.4 Case I: Undirected information flow graph . . . . . . . . . . . . . . . . . 117

7.4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.4.2 Global solution of CFCP . . . . . . . . . . . . . . . . . . . . . . . 128

7.5 Case II: Circulant information flow graph . . . . . . . . . . . . . . . . . . 132

7.5.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7.6 Case III: General information flow graph . . . . . . . . . . . . . . . . . . 136

7.6.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8 Conclusion 144

Appendix 147

Bibliography 150

vii

List of Figures

1.1 Circular path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Trajectory tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Simulation results comparing set stabilization and trajectory tracking for

initial condition on the desired set. . . . . . . . . . . . . . . . . . . . . . 5

1.4 Simulation results comparing set stabilization and trajectory tracking for

initial condition close to the goal set. . . . . . . . . . . . . . . . . . . . . 6

1.5 Circular Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Comparison between concepts of uniform attractivity and uniform semi-

attractivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 An illustration of the notion of local stability near Γ1 . . . . . . . . . . . 28

2.3 The equilibrium Γ1 is attractive, but not uniformly so. The circle Γ2, on

the other hand, is uniformly attractive. . . . . . . . . . . . . . . . . . . . 29

2.4 Γ1 is globally asymptotically stable relative to Γ2, and unstable relative to

Γ3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 The set Γ2 is unstable, but locally stable near Γ1. . . . . . . . . . . . . . 30

2.6 The prolongational limit set of any point on the x1 axis is the entire x2 axis. 33

4.1 Γ2 is globally asymptotically stable and Γ1 is globally exponentially stable

relative to Γ2. Yet, Γ1 is not asymptotically stable. . . . . . . . . . . . . 51

viii

4.2 Γ1 is globally attractive rel. to Γ2, Γ2 is globally asymptotically stable,

and yet Γ1 is not attractive. . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 On the left-hand side, phase portrait on O for the open-loop system (5.6).

On the right-hand side, closed-loop system (5.5) with feedback u = −y.

Note that Γ is not attractive. . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Circular path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Set stabilization for point-mass system . . . . . . . . . . . . . . . . . . . 82

6.1 Simulation results for the global path following controller in (6.6), where

C is an ellipse with major semi-axis length 2 and minor semi-axis length 1. 94

6.2 Failure of Γ-detectability in case study 2 when Γ = V −1(0). . . . . . . . . 96

6.3 Simulation results for the controller in (6.9). . . . . . . . . . . . . . . . . 99

6.4 Simulation results for the coordination controller in (6.15). . . . . . . . . 103

7.1 Digraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.2 The centre ci(xi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3 Formation on the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.4 Cyclic pursuit with uniform spacing . . . . . . . . . . . . . . . . . . . . . 116

7.5 CFCP Simulation - A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

7.6 CFCP Simulation - B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.7 CFCP Simulation - A: Case I . . . . . . . . . . . . . . . . . . . . . . . . 129

7.8 CFCP Simulation - B: Case I . . . . . . . . . . . . . . . . . . . . . . . . 130

7.9 CFCP Simulation - A: Case II . . . . . . . . . . . . . . . . . . . . . . . . 137

7.10 CFCP Simulation - B: Case II . . . . . . . . . . . . . . . . . . . . . . . . 138

7.11 CFCP Simulation - A: Case III . . . . . . . . . . . . . . . . . . . . . . . 142

7.12 CFCP Simulation - B: Case III . . . . . . . . . . . . . . . . . . . . . . . 143

ix

Notation

N Set of natural numbers

R Set of real numbers

R+ Set of nonnegative real numbers

S1 The set of real numbers modulo 2π; equivalently, the unit

circle

Rn×m Set of n×m matrices with real entries

A⊗ B Kronocker product of matrices A and B p. 22

‖x‖S Point-to-set distance of point x to set S p. 23

d(S1, S2) Maximum distance of set S1 to set S2 p. 23

cl(S) Closure of set S

N (Γ) A neighbourhood of the set Γ

Bα(x) The neighbourhood of x: y : ‖y − x‖ < α

Bα(S) The neighbourhood of S: y : ‖y‖S < α

f−1(y) The level set x : f(x) = y

f−1([a, b]) For a real-valued f , this is the set x : a ≤ f(x) ≤ b

df(x) The Jacobian matrix of f at x

LfV (x) The Lie derivative of V along f , LfV (x) = dV (x)f(x)

LkfV (x) The k-th iterated Lie derivative of V along f p. 23

LgLfV (x) The Lie derivative d(LfV )(x)g(x)

[f, g](x) The Lie bracket of f and g, dg(x)f(x)− df(x)g(x)

x

adkfg(x) The k-th iterated Lie bracket of f and g p. 24

φu(t, x0) The solution of x = f(x) + g(x)u at time t with intial con-

dition x0, where u is either a piecewise continuous signal,

or a smooth feedback

φ(t, x0) The solution of x = f(x) at time t with initial condition x0

L+u (x0), L

−u (x0) The positive and negative limit sets of a solution φu(t, x0) p. 31

L+(x0), L−(x0) The positive and negative limit sets of a solution φ(t, x0) p. 31

J+u (x0), J

+(x0) The prolongational limit sets of φ(t, x0) and φu(t, x0) p. 31

J+u (x0, U) The prolongational limit set of φu(t, x0) relative to U p. 31

PBF Passivity-based feedback p. 39

PBSSP Passivity-based set stabilization problem p. 39

O The maximal subset of h−1(0) which is positively invariant

for the open-loop system

pp. 43, 65

xi

Chapter 1

Introduction

For many years, the problems of stabilizing equilibria and tracking reference signals have

been the main focus of control theory and practice. The control of temperature in a

building is a typical equilibrium stabilization problem, while regulating a sinusoidal volt-

age in an electric circuit is an instance of a reference tracking problem. Many advanced

engineering applications involve sophisticated control specifications that do not naturally

fit within the equilibrium stabilization or reference tracking frameworks. Controlling the

walking motion of a biped robot involves the creation and stabilization of a number

of virtual constraints embodying desired relations among the links of the robot. The

problem of making a group of robot rovers move in formation for planetary exploration

involves, among other things, stabilizing the distances between robots to desired constant

values. These specifications fit the general problem area of set stabilization, whereby one

is to design a controller stabilizing the state of a dynamic system to a goal set in the

state space. In other words, instead of stabilizing to a fixed point (equilibrium) or a

time-varying point (tracking signal), in the set stabilization problem one wishes to sta-

bilize the state to a set of points representing a desired system behaviour. Since the set

stabilization problem is a generalization of the equilibrium stabilization problem, and it

captures a large variety of advanced control specifications, one would expect that it lies

1

Chapter 1. Introduction 2

at the centre of control science and engineering. Instead, this problem has received less

attention in the control literature than reference tracking or equilibrium stabilization.

This thesis attempts to begin filling this gap by presenting a set stabilization theory for

nonlinear control systems based on the notion of passivity.

In this thesis we consider the following class of control-affine systems

x = f(x) +m∑

i=1

gi(x)ui

y = h(x)

(1.1)

where x ∈ X ⊂ Rn is the state, u = (u1, · · · , um) ∈ U = Rm is the control input and

y ∈ Y = Rm is the output. Our main objective in this thesis is to study the following

problem.

Set Stabilization Problem. Given the control system (1.1) and a closed set Γ ⊂ Rn,

find, if possible, a state feedback u(x) which renders the set Γ asymptotically stable for

the closed-loop system x = f(x) +∑m

i=1 gi(x)ui(x).

1.1 Motivating example

Consider the following simple path following problem. An unmanned aerial vehicle is

required to circle a certain region for environmental studies. The simplest model for this

system is that of a fully actuated point-mass model in R3:

Mx = f,

where M is the mass of the body, x = (x1, x2, x3) ∈ R3 is the position of the body in

an inertial coordinate frame, and f = (f1, f2, f3) ∈ R3 is the control force applied to the

body. The state space is (x, x) ∈ R3 × R3. Assume, for the sake of illustration, that

the vehicle has a GPS sensor on board, so that its inertial coordinates x are available

for feedback with perfect accuracy, and that this information is used to generate x with

perfect accuracy.


The problem we need to solve has two parts.

1. It is required to make the centre of mass move along a circular path γ of radius r,

see Figure 1.1. The plane of the path is specified by a vector, c, perpendicular to

it. Without loss of generality, we assume that the circular path is centred at the

origin.

2. On the circle γ, it is required to make the centre of mass move with a constant

forward speed, v, counter-clockwise relative to c.

ab

c

x1

x2

x3 γ

Figure 1.1: Circular path

Consider first a trajectory tracking approach to solve this problem. In this case, it is

required to obtain a temporal description of the states that can provide the behavior in

1 and 2. For instance, one such description is

xr = R col(r cos(ωt), r sin(ωt), 0)

xr = R col(−rω sin(ωt), rω cos(ωt), 0),

(1.2)

where ω = v/r, (a, b, c) is the orthonormal body coordinate system shown in Figure 1.1

and R = [a b c]. Letting e = x− xr, e = x− xr, we obtain

e =1

Mf − xr

where xr = R[−rω2 cos(ωt) − rω2 sin(ωt) 0]⊤. Choosing

f =M(xr − k1e− k2e) (1.3)


where k1, k2 > 0 are design constants, we have

e = −k1e− k2e,

and therefore the equilibrium (e, e) = (0, 0) is globally asymptotically stable. Figure 1.2

shows simulation results for the control (1.3) with the following parameters: M = r =

v = 1, and R =

1/√2 1/

√6 1/

√3

−1/√2 1/

√6 1/

√3

0 −2/√6 1/

√3

.

−2

−1

0

1

2 −1

−0.5

0

0.5

1

1.5

2−2

0

2

x1x2

x3

Figure 1.2: Trajectory tracking

Turning our attention now to a different approach, in Chapter 5 (Proposition 5.4.1)

we provide a passivity-based controller addressing this problem as one of set stabilization.

Simply put, the desired behaviour is expressed as a goal set in the state space of the system

variables, and a controller is designed to render this goal set asymptotically stable, which

roughly means that, for all initial conditions near the set, the system trajectories remain

near the set, and asymptotically approach it. We remark that the feedback presented


in Proposition 5.4.1 requires full state feedback because the desired path is expressed in

inertial coordinates. The same holds for the feedback (1.3).

The circular path in objective 1 above can be expressed as γ = x ∈ R3 : c · x =

0, ‖x‖ = r. This is a subset of the configuration space. In the state space, we wish

to stabilize the set (x, x) : c · x = 0, ‖x‖ = r, c · x = 0, x · x = 0, where the last

two identities embody the requirement that, on the circle, the velocity of the body be

tangent to the circle. With the requirement on velocity in objective 2, the goal set is

Γ =

(x, x) ∈ R6 : c · x = 0, ‖x‖ = r, x =v

rc× x

. (1.4)

The details of the set stabilizing solution are provided in Section 5.4. Here we only focus

on the qualitative differences between the reference tracking and set stabilization solu-

tions. Consider the simulation examples presented in Figures 1.3 and 1.4. In Figure 1.3

−2

0

2

−1−0.5

00.5

11.5

−1

−0.5

0

0.5

1

trackingset−stab

x1x2

x3

Figure 1.3: Simulation results comparing set stabilization and trajectory tracking for

initial condition on the desired set.

the system is initialised on the desired path with velocity v tangent to the path. The set


stabilizing controller guarantees that the solution remains on the path, while the tracking

controller makes the solution leave the path. This is due to the fact that, at time t = 0,

the state and the reference signal do not coincide.

−1

−0.5

0

0.5

1 −1−0.5

00.5

1

−1

0

1

2

trackingset−stab

x1 x2

x3

Figure 1.4: Simulation results comparing set stabilization and trajectory tracking for

initial condition close to the goal set.

In Figure 1.4 the system is initialised close to the path. The set stabilizing controller

keeps the solution close to the path while it converges to it. The tracking controller, on

the other hand, produces an overshoot. This is due to the fact that although the initial

condition is close to the path, it is not close to the reference signal.

Naturally, the simulation results presented above are dependent upon the controller

design and the choice of parameters. However, the qualitative differences we highlighted

are general consequences of these facts:

1. A tracking controller does not yield invariance of the goal set Γ, because if the

system is initialized on the circle with velocity v tangent to the path, the initial

condition does not necessarily coincide with that of the reference signal.


2. Similarly, a tracking controller does not yield stability of Γ. This is because, no

matter how close the initial condition is to Γ, the mismatch between the initial

condition and the reference signal may cause the solution to wander away from Γ

before converging to it.

3. By contrast, a set stabilizing controller yields invariance and stability of Γ. That

is to say, if the system is initialised anywhere on the circle with initial velocity v

tangent to the path, the system will remain on the circle, and it will maintain the

same speed. Moreover, initial conditions near Γ give rise to solutions that remain

near Γ while converging to it.

The qualitative differences above have important practical implications. The lack of

stability of Γ when using a reference tracking controller makes the autopilot unsafe, as

the aircraft may hit the ground or other obstacles during transient. Further, the lack of

invariance of Γ makes the aircraft display an unnatural sensitivity to disturbances, as any

disturbance that slows down the aircraft along the path may cause it to leave the path.

Various path following techniques in the literature (e.g., [41, 87, 25]) modify the tracking

controller (1.3) by reparametrizing the reference signal xr(t) as xr(θ(t)) and controlling

θ(t) in such a way that the distance between x(t) and xr(θ(t)) is minimized. While these

approaches mitigate the problems listed above, they do not eliminate them because they

do not yield invariance of Γ.

The feedback (1.3) solving the tracking problem can also be derived by posing the

path following problem as one of output regulation [20, 34, 33], whereby an exosystem

w = Sw is defined that generates the sinusoidal reference signals in (1.2). In this context,

the controller in (1.3) is a full-information output regulator (i.e., a regulator using x and w

as feedback variables) which stabilizes a controlled invariant subspace (x, w) : x = Pw

in the augmented state space of the plant and exosystem, where the matrix P is found

by solving the regulator equations. The output regulation problem for the point-mass


system is intrinsically linear, which explains why the feedback (1.3) is linear. On the

other hand, the set Γ in (1.4) does not have have the structure of a vector space1, and

hence the stabilization of Γ is an intrinsically nonlinear problem. It is therefore not

surprising that the set stabilizing controller presented in Chapter 5 is nonlinear.

In classical regulator theory, the controller is required to enforce the internal stability

property: when the reference signal is identically zero, the closed-loop system is asymp-

totically stable. If in place of the full-information output regulator (1.3) one designs an

error feedback output regulator (i.e., a dynamic controller using the tracking error e as

feedback variable), the internal stability property induces structural stability. In this

context, structural stability means that the regulation property e(t) → 0 is preserved

under small variations of the plant parameters. The implication of structural stability on

the path following problem is that the point-mass system controlled by an error feedback

output regulator will asymptotically approach the circle even when the plant parameter

M is perturbed. On the other hand, a state feedback controller which asymptotically

stabilizes the set Γ in (1.4) yields a different kind of structural stability. Namely, if the

point-mass parameters are perturbed, the controller will no longer stabilize Γ, but it will

stabilize a set Γ “close to” Γ. Moreover, if the perturbation preserves the invariance of Γ,

then the asymptotic stability of Γ is preserved. The implication is that the point-mass

system controlled by a set stabilizer may2 converge to a new path γ close to the circle γ

when the mass M is perturbed.

1.2 Set stabilization applications

As mentioned earlier, many modern engineering applications have control goals which do

not fit within the equilibrium stabilization or reference tracking frameworks, but instead

fit the set stabilization setting. In this section we review some of the applications of the

1Indeed, Γ is diffeomorphic to S1 × S1 × R4, where S1 denotes the unit circle.2It so happens that the controller presented in Chapter 5 is robust against small variations of M .


set stabilization problem.

Path following

In the example of Section 1.1, a fully actuated vehicle was required to follow a circular

path in the configuration space, expressed as γ = x ∈ R3 : c · x = 0, ‖x‖ = r, with the

additional requirement that the speed on the path be a desired constant. This control

specification is an instance of the path following problem, which involves making the

output of a system approach and follow a path γ in the output space with additional

specifications for the motion on the path, such as speed regulation or stability. Referring

to system (1.1), if the path is expressed in implicit form as γ = y ∈ Y : ϕ(y) = 0, then

making the output of the system approach γ corresponds to making the state trajectory

approach the set Γ = x ∈ X : ϕ(h(x)) = 0.

Generally, the set Γ may not be controlled invariant, i.e., there may be initial condi-

tions on Γ such that for any control signal the corresponding solution x(t) leaves Γ, and

therefore the output signal y(t) leaves the curve γ. For instance, if the vehicle in the

example of Section 1.1 is initialised on the circle but its initial velocity is not tangent

to it, then no matter what external force one applies to it, the vehicle will leave the

circle. In light of the above, the path following problem entails the stabilization of the

largest subset Γ of Γ with the property of being controlled invariant. In the example of

Section 1.1, this is the set (x, x) : c ·x = 0, ‖x‖ = r, c · x = 0, x · x = 0. Requirements

on the motion on the path translate into additional constraints in the definition of Γ.

Robotics is the typical application domain of path following, with such examples as the

design of autopilots for autonomous vehicles (terrestrial, aerial, or marine) [59, 97, 92],

rocket launch control and orbital control of satellites [101], and motion control [36].

Control tasks for such systems are varied. In addition to path following, it is often

required to address complex tasks such as manipulator force control, and obstacle or

collision avoidance.


Coordination, Consensus, and Synchronization

Consider a collection of control systems which may or may not be coupled to each other,

xi = f i(x) + gi(x)ui, i = 1, . . . , m

yi = hi(x).

(1.5)

The collective state is x = col(x1, . . . , xm). The coordination problem entails designing

feedbacks ui(x), i = 1, . . . , m, such that certain relations are asymptotically satisfied for

the outputs yi,

ϕj(y1, y2, · · · , ym) = 0, j = 1, . . . , p.

Similarly to path following, the coordination problem can be formulated as the stabi-

lization of the largest controlled invariant subset Γ of x : ϕj(h1(x), h2(x), · · · , hm(x)) =

0, j = 1, . . . , p.

Often associated with system (1.5), there is an information flow structure modelled

by a graph which represents, for each system in the collection, which other systems pass

information to it. In this case, if the feedbacks solving the coordination problems are to

respect the information flow structure, one speaks of distributed coordination.

The coordination problem is rather general and has several important special cases.

When the goal relations between the outputs are y1 = · · · = ym, i.e., when it is desired

that all outputs converge to each other, then one speaks of consensus, agreement,

or output synchronization. Such problems arise, for instance, in the area of multi-

agent systems [27, 74], in which case each system in (1.5) represents a vehicle, and the

output yi represents the physical location of the vehicle on the plane or in Euclidean

three-space. Another special type of distributed coordination problem in the area of

multi-agent systems is the formation stabilization problem, whereby one wants the

vehicles to converge to and maintain a formation. In this case, the asymptotic relation

between outputs might be expressed as a requirement on distances, e.g., ‖yi − yj‖ = d.

In addition to achieving the coordination requirement, additional control objectives are


typically imposed. For instance, in addition to maintaining a formation, it is typically

desired that the vehicle follow a path, in which case a path following requirement is

imposed on top of the coordination problem.

As an application of the theory developed in this thesis, in Chapter 7 we address a

circular formation control problem for a network of kinematic unicycles: A group

of n kinematic unicycles with a certain information flow structure is required to follow

a circle, with specified radius, and also to acquire a certain desired formation on the

circle, as illustrated in Figure 1.5. In Chapter 7 we provide a precise formulation of this

Figure 1.5: Circular Formation

problem and show its equivalence to the stabilization of an unbounded goal set. Then,

using the theory developed in this thesis, we solve the problem in complete generality.

The fact that the goal set in the circular formation control problem is unbounded

should come as no surprise, as even in the simplest coordination problem, that of state

agreement, the goal set Γ = x : x1 = · · · = xm is unbounded. The unboundedness

of the goal set is one of the major theoretical challenges of coordination, one that is

addressed by the theory developed in this thesis.


Control of oscillations and biped locomotion

The control of oscillations arises in those engineering applications where one wishes to

induce in the system an oscillatory behavior which represents the periodic repetition of a

task. This occurs, for instance, in the biped locomotion problem, whereby the repetitive

task corresponds to walking. From the control theoretic point of view, such control of

oscillations entails stabilizing a set in the state space which is either a closed curve or a

surface representing virtual constraints which induce an oscillatory motion. Such virtual

constraint perspective figures prominently in Jessy Grizzle’s work on biped locomotion,

e.g. [70, 100]. The reader is referred to the book by Fradkov [31] for more details on this

subject.

1.3 Set stabilization in conventional control

Besides being the natural setting to formulate complex control goals, set stabilization

also arises in conventional control problems, even though the solution to such problems

has not traditionally been approached from the set stabilization perspective.

In the state estimation problem one is to design a dynamical system (an observer)

with input (u, y) and state x with the property that the set

Γ = (x, x) : x = x,

is asymptotically stable. One can think of an observer as a dynamic feedback stabilizing

Γ.

In adaptive control [35, 5], one considers a plant with a vector of unknown param-

eters θ and designs a dynamic feedback with state θ that makes the set

Γ = (x, θ) : x = 0

either attractive or asymptotically stable.


When the plant is a time-varying system, the problem of stabilizing the origin is

sometimes (see, e.g., [94]) approached by considering the time t as an extra state with

dynamics t = 1, in which case the control objective becomes the stabilization of the set

Γ = (x, t) : x = 0.

The output regulation problem [20, 34, 33], can be formulated as that of design-

ing a dynamic output feedback making the output of an augmented plant (original plant

plus exosystem) converge to zero while guaranteeing boundedness of the state trajecto-

ries. This is an instance of the more general output stabilization problem, whereby

a feedback is sought making the output of the plant converge to zero (see, e.g., [56]) and

guaranteeing boundedness of the state trajectories. Solving this problem involves making

the largest controlled invariant subset of the zero level set of the output an attractive set

(see, e.g., [56]). To guarantee robustness against noise and uncertainties, an additional

requirement3 of stability is sought for said controlled invariant set, turning output sta-

bilization (and hence output regulation) into a set stabilization problem (with the extra

requirement of boundedness of state trajectories). A typical approach for solving the

output stabilization problem is input-output feedback linearization. If successful, this

technique yields stabilization of the largest controlled-invariant subset of the zero level

set of the output function. This technique, however, has limitations because it requires

the satisfaction of certain relative degree conditions, see Remark 5.4.2. In fact, this

technique would fail in addressing any of the set stabilization examples presented in this

thesis.

The design technique of sliding mode control [96, 88] is mainly used to fulfill

objectives of equilibrium stabilization and tracking. However, the main step in designing

a switching controller involves stabilizing a sliding surface of codimension one on which

the system exhibits a desirable behaviour (stability).

3In the linear time invariant setting, the attractivity of the largest controlled invariant subspacecontained in the zero level set of the output is equivalent to its asymptotic stability.


1.4 Literature on set stabilization

In this section we review the literature on set stabilization. The main approaches are

based on control Lyapunov functions, passivity, and geometric control.

Passivity

A passive system is one that stores energy, may have dissipative components, and ex-

changes its energy with the outside world by transferring power through input-output

ports. The origin of this notion can be traced back to the second half of the 1950’s,

with work on linear passive network theory [105]. Throughout the 1950’s and 1960’s,

research on passivity focused on input-output operator descriptions of LTI systems (see,

e.g., [3, 23]). In his seminal 1972 work [103, 104], Jan Willems revolutionised the field

by developing a theory of dissipative systems based on the state space perspective. The

notion of dissipativity invented by Willems generalizes that of passivity. In [37] and [38],

Hill and Moylan used Willems’ approach to develop equilibrium stabilization results for

nonlinear control systems. They also provided results for stability of feedback inter-

connections of these systems. In their milestone 1991 paper [14], Byrnes, Isidori and

Willems generalized the equilibrium stabilization results of Hill-Moylan identifying zero-

state detectability (reviewed in Chapter 3) as being the key property for passivity-based

stabilization. They also answered the fundamental question of when can a nonlinear sys-

tems be rendered passive by state feedback, with necessary and sufficient geometric and

dynamical conditions. Recently, passivity has been used for set stabilization of nonlinear

systems. In a number of papers [82, 81, 83, 84, 85], Shiriaev and coworkers applied the

results of stabilization of passive systems by Byrnes et al. to stabilize compact invari-

ant sets of passive systems, with application to the control of oscillations of mechanical

systems. The theory of Shiriaev and co-workers is a straightforward adaptation of the

results of Byrnes-Isidori-Willems and does not introduce any novel conceptual insight.

The control law used by Shiriaev for stabilization of sets, and by Byrnes et al. for


stabilization of equilibria of nonlinear passive systems is closely related to a control

algorithm developed in the 1970’s known as speed-gradient method [29]. It is also related

to the so-called Jurdjevic-Quinn control [44]. In [32], Fradkov and co-authors studied the

speed-gradient control method in the context of set stabilization. In this technique,

the control objective is described by a certain, possibly time-dependent, function of the

system variables Q(x, t), where it is required to insure, using controls, that this function

approaches zero. The essence of the algorithm is to change the control u in the direction

of decrease of Q(x, t). Since Q does not explicitly depend on u, u is chosen in the direction

of decrease of Q, hence the name speed-gradient.

Passivity is appealing for system analysis and control design, for it gives an insight of

the energy of the system and its interactions. This aids in understanding the dynamics

of the system and addressing the nonlinearities. It also provides a valuable tool for

control design if one can manipulate this energy through control inputs. This point of

view has been particularly successful in the research on stabilization of Euler-Lagrange

control systems and, more generally, port-Hamiltonian systems, [62, 98, 64], and is a very

promising area of research.

In Chapter 3 we present a more detailed account on the literature on passivity-based

set stabilization. Our main results in Chapter 5 extend the results mentioned above in

two directions. First, in our setting the goal set Γ is only required to be closed, and not

necessarily bounded. Second, rather than requiring Γ to coincide with the zero level set

of the storage function, we merely require Γ to be a subset of it. This leads, in particular,

to a generalization of the equilibrium theory of Byrnes-Isidori-Willems in that we do not

require the storage function to be positive definite. Moreover, this feature of our theory

gives greater flexibility in control design, as we demonstrate in Chapters 6 and 7.


Control Lyapunov functions

The control Lyapunov function (CLF) technique is a classical equilibrium stabilization

tool in which the objective is to look for Lyapunov-like functions which can be made

to decrease with appropriate choice of controls. Artstein [4] showed that the existence

of a smooth CLF implies smooth stabilizability. In [90], Sontag improved this result by

providing a universal formula for the stabilizing controller.

Control Lyapunov functions have been used to answer a crucial question in nonlinear

control theory: What is the relationship between asymptotic controllability and feedback

stabilization? In [89], Sontag showed that asymptotic controllability is equivalent to

the existence of a continuous CLF. Using this result, Clarke et al. in [17] showed the

equivalence between asymptotic controllability and the existence of a sample and hold

feedback stabilizing an arbitrarily small neighborhood of the origin. These ideas have

been extended in [2] by Albertini and Sontag to show that, for a time varying control

system, uniform global asymptotic controllability to a closed (not necessarily compact)

subset of the state space is equivalent to the existence of a continuous CLF with respect

to the set. In another work [47], Kellet and Teel showed that uniform global asymptotic

controllability to a closed (not necessarily compact) set implies the existence of a locally

Lipschitz CLF (see also related work by Rifford [72, 73]). Using this CLF, they were

able to construct a stabilizing feedback based on a sample and hold scheme, similarly

to the work in [17] for the equilibrium case. In conclusion, their work established the

equivalence between asymptotic controllability and feedback stabilization with respect

to closed sets, analogously to the equilibrium case.

It is worth noting that the results mentioned above address the case when the system is

uniformly globally asymptotically controllable, and stabilization is addressed in a uniform

sense. These papers do not address the case of (non-uniform) asymptotic stability and

stabilization, which is our main focus in this thesis.

Techniques based on control Lyapunov functions are of theoretical interest in that, as


mentioned earlier, they are used to investigate the relationship between asymptotic con-

trollability and feedback stabilization, but they have severe practical limitations. Find-

ing a control Lyapunov function is generally a difficult problem, and feedbacks based on

control Lyapunov functions are usually unnecessarily complicated. In contrast, storage

functions in passivity-based stabilization often arise from the physics of the problem, as in

the port-Hamiltonian framework, or are naturally deduced from the control specification,

as in all applications presented in this thesis. The practical advantage of passivity-based

stabilization over the control-Lyapunov function approach is reflected in the wealth of

applications of passivity-based control.

Geometric control

An important approach for solving the problem of set stabilization is to view it from a

geometric perspective. The goal set is typically assumed to have the geometric structure

of an embedded submanifold, and coordinate transformations are sought to gain insight

on the nature of the set stabilization problem.

These tools were used by Banaszuk and Hauser in [8] to investigate the path following

problem. The objective is to drive the system trajectories to approach and traverse a

specified path. Their main approach to the solution of the problem is the stabilization

of the dynamics transverse to the path using feedback linearization of these dynamics,

an approach that has since become known as transverse feedback linearization (TFL). In

their work, and for a class of single input nonlinear systems, they considered the special

case when the path is a simple closed curve.

In [60], Nielsen and Maggiore extended Hauser’s work to the stabilization of arbitrary

embedded submanifolds of the state space for single-input systems, and in [61] they fur-

ther extended the results to the case of multi-input systems. In [36], transverse feedback

linearization was used to solve a path following problem for an experimental magneti-

cally levitated positioning system, and in [18] to present a solution to the path following


problem for the planar/vertical take-off and landing (PVTOL) aircraft model.

The application of geometric control in set stabilization was also investigated by Frad-

kov and co-authors in [32]. Their approach to the problem involves using transformations

to describe the system in coordinates on and off the set, i.e., to decompose the system

dynamics into transversal and tangential ones. Under certain conditions, stabilizing the

transversal dynamics is equivalent to stabilizing the goal set. The stabilizing controls

in [32] are designed mainly based on the linearized models of transversal dynamics.

The previous geometric schemes are based on finding coordinate and feedback trans-

formations allowing one to decompose the system dynamics in components tangential and

transversal to the set. The theory only guarantees that, under suitable conditions, the

transformations in question exist locally around a point, and not necessarily in a neigh-

borhood of the goal set. Also, in order to find the required coordinate transformation

one may have to solve a set of partial differential equations.

1.5 Thesis outline

The body of this thesis consists of six chapters.

• Chapter 2: Preliminaries

We provide essential preliminaries needed in the thesis, including the notation, the

definitions of stability, notions of limit sets and prolongational limit sets, and their

implications.

• Chapter 3: Passivity-Based Stabilization I: Preliminaries

We start our investigation of the set stabilization problem for passive systems. After

reviewing the main definitions and tools of passivity, we formalise the passivity-

based set stabilization problem (PBSSP), and review the state of the art. Finally,

we establish the connection between PBSSP and the following reduction problem.


Given two nested invariant sets Γ1 ⊂ Γ2, and assuming that Γ1 enjoys certain

stability properties relative to Γ2, under what conditions does Γ1 enjoy the same

stability properties with respect to the whole state space?

• Chapter 4: Reduction Principles

After reviewing the literature on the reduction problem, and showing its relevance

to other areas of nonlinear control theory, we present three novel reduction theorems

for stability, attractivity and asymptotic stability of closed sets which generalize all

available results. We then introduce a general perspective for control design based

on reduction theory which is used later, in Chapter 6, to provide a passivity-based

stabilizing procedure.

• Chapter 5: Passivity-Based Stabilization II: Theory

The reduction principles of Chapter 4 inspire a new notion of detectability, Γ-

detectability, which generalizes existing detectability notions. After providing suf-

ficient geometric conditions for Γ-detectability, we leverage the reduction theorems

of Chapter 4 to solve the passivity-based set stabilization problem. Even in the

case when the goal set is an equilibrium, our results generalize the state of the art

of passivity-based stabilization theory.

• Chapter 6: Passivity-Based Stabilization III: Control Design

Having solved PBSSP in Chapter 5, in this chapter we present a synthesis proce-

dure for passivity-based stabilization of closed sets. The main idea is to use part

of the control freedom to enforce detectability, while the remaining part is used

for passivity-based stabilization. To illustrate the procedure, we present three case

studies concerning the path following problem for one kinematic unicycle, maneu-

vering of one unicycle, and coordination for two unicycles.

• Chapter 7: Circular Formation Control of Unicycles


In this chapter we use the theory developed in this thesis to solve a challenging

distributed coordination problem: make a group of unicycles converge to a circular for-

mation of desired radius, with specified spacing and ordering on the circle. We show that

this problem has an intrinsic reduction aspect in that it can be broken down into two

tasks: circular path following and formation stabilization. Using this insight, we leverage

our reduction and set stabilization theory, and apply the passivity-based control design

approach of Chapter 6.

1.6 Thesis contributions

The main contributions of this thesis can be summed up as follows.

1. Chapter 4 provides solution to the reduction problem for closed and unbounded

sets.

• Reduction principle for stability, Theorem 4.4.1.

• Reduction principle for attractivity, Theorem 4.4.6.

• Reduction principle for asymptotic stability, Theorem 4.4.8 .

• Application of reduction principles for cascade connected systems, Corol-

lary 4.6.1.

2. Chapter 5 provides a solution to the passivity-based set stabilization problem.

• New notion of Γ-detectability, Section 5.1.

• Sufficient conditions for Γ-detectability, Proposition 5.1.6.

• Necessary and sufficient conditions for passivity-based set stabilization, The-

orem 5.2.2.

3. Chapter 6 provides a novel passivity-based control design procedure for set stabi-

lization.


• Set stabilization procedure, Section 6.1.

• Case study 1: path following for the kinematic unicycle, Proposition 6.2.5.

• Case study 2: stabilizing the unicycle to a circle with heading angle require-

ment, Proposition 6.3.2.

• Case study 3: coordination of two unicycles, Proposition 6.4.2.

4. Chapter 7 solves the circular formation control problem for n-unicycles.

• Case I: undirected flow graph, Proposition 7.4.3.

• Global solution of the circular formation control problem, Proposition 7.4.5.

• Case II: circulant information flow graph, Proposition 7.5.3.

• Case III: general information flow graph, Proposition 7.6.1.

Chapter 2

Preliminaries

This chapter presents the preliminary notions and definitions used in this thesis. In

Section 2.1 we introduce some notation, which is summarised in the table on page x.

In Section 2.2 we present various set stability definitions, mainly taken from [10], and

introduce a new notion of uniform boundedness. Finally, in Section 2.3 we review notions

of limit sets due to Birkhoff [13] and prolongational limit sets due to Ura [95], and we

present some of their implications.

2.1 Notation

In the sequel, N denotes the set of natural numbers, R denotes the set of real numbers,

and R+ denotes the positive real line [0,+∞). If k ∈ N, then Rk denotes the Cartesian

product R × · · · × R, k times. Similarly, if n, k ∈ N and S ⊂ Rn, Sk denotes the k-fold

Cartesian product S × · · · × S. If n,m ∈ N then Rn×m denotes the set of real-valued

n×m matrices. If A ∈ Rn×m is a matrix with elements aij and B ∈ Rp×q, A⊗B denotes

the Kronecker product of A and B which is the np×mq matrix

A⊗ B =

a11B · · · a1mB

.... . .

...

an1B · · · anmB

.

22

Chapter 2. Preliminaries 23

We denote by col(x1, · · · , xk) the column vector [x1 · · · xk]⊤ where ⊤ denotes transpose.

If x and y are two column vectors then col(x, y) := [x⊤ y⊤]⊤.

Given a nonempty set S ⊂ Rn, a point x ∈ Rn, and a vector norm ‖ · ‖ : Rn → R,

the point-to-set distance ‖x‖S is defined as ‖x‖S := inf‖x− y‖ : y ∈ S. Given two

subsets S1 and S2 of Rn, the maximum distance of S1 to S2, d(S1, S2), is defined as

d(S1, S2) := sup‖x‖S2: x ∈ S1. We denote by cl(S) the closure of the set S, and by

N (S) a generic open neighbourhood of S, that is, an open subset of X containing S.

Throughout this thesis we will let X ⊂ Rn be either an open subset or a smooth

submanifold of Rn. Being a subset of Rn, X inherits a norm from Rn, which we will

denote ‖ · ‖ : X → R+. For a constant α > 0, a point x ∈ X , and a set S ⊂ X , define

the open sets Bα(x) = y ∈ X : ‖y − x‖ < α and Bα(S) = y ∈ X : ‖y‖S < α.

A function f : Rn → Rm is said to be of class Ck, or a Ck function, if all the partial

derivatives ∂kf/∂xi1∂xi2 · · ·∂xik exist and are continuous, where each i1, i2, · · · , ik is an

integer between 1 and n. If y is a point in the image of f , we denote f−1(y) = x : f(x) =

y. Moreover, if f is real-valued, we denote by f−1([a, b]) the set x : a ≤ f(x) ≤ b.

Given a function f : Rn → Rm, df : Rn → Rm×n denotes the Jacobian of f . If f

is a vector field defined on X and V : Rn → Rm is a C1 function, we use the following

standard notation for the Lie derivative of V along f at x ∈ X :

LfV (x) = dV (x)f(x).

The k-th iterated Lie derivative of V along f , k ≥ 1, is defined as

LkfV (x) = LfL

k−1f V (x) = d(Lk−1

f V )(x)f(x),

L0fV (x) = V (x).

Given a second vector field g defined on X , the Lie derivative of V with respect to f ,

and then with respect to g is given by

LgLfV (x) = d(LfV )(x)g(x).


If f and g are two vector fields defined on X , [f, g](x) denotes their Lie bracket, which

is the vector field on X defined as

[f, g](x) = dg(x)f(x)− df(x)g(x).

By adkfg(x) we denote the k-th iterated Lie bracket of the vector fields f and g, where

ad0fg(x) = g(x),

adfg(x) = [f, g](x),

adkfg(x) = [f, adk−1

f g](x).

Consider the control-affine system

x = f(x) +m∑

i=1

gi(x)ui

y = h(x)

(2.1)

with state space X ⊂ Rn, set of input values U = Rm and set of output values Y = Rm.

We assume that f and gi, i = 1, . . .m, are smooth vector fields on X , and that h : X → Y

is a smooth mapping. Given either a smooth feedback u(x) or a piecewise-continuous

open-loop control u(t) : R+ → U , we denote by φu(t, x0) the unique solution of (2.1) with

initial condition x0. By φ(t, x0) we denote the solution of the open-loop system x = f(x)

with initial condition x0. Given an interval I of the real line and a set S ∈ X , we denote

by φu(I, S) the set φu(I, S) := φu(t, x0) : t ∈ I, x0 ∈ S. The set φ(I, S) is defined

analogously.

2.2 Set stability and attractivity

We introduce here the basic notions of set stability and attractivity used in this thesis.

Consider the dynamical system

Σ : x = f(x), x ∈ X . (2.2)


Let Γ ⊂ X be a closed set which is positively invariant for Σ, that is, such that all

solutions of Σ originating in Γ remain in Γ for all positive time in their maximal interval

of existence.

Definition 2.2.1 (Set stability and attractivity).

(i) Γ is stable for Σ if for every ε > 0 there exists a neighbourhood N (Γ) such that

φ(R+,N (Γ)) ⊂ Bε(Γ).

(ii) Γ is an attractor1 for Σ if there exists a neighbourhood N (Γ) such that, for all

x0 ∈ N (Γ), limt→∞ ‖φ(t, x0)‖Γ = 0.

(iii) Γ is a global attractor for Σ if it is an attractor with N (Γ) = X .

(iv) Γ is a uniform semi-attractor for Σ if for all x ∈ Γ, there exists λ > 0 such that,

for all ε > 0, there exists T > 0 yielding φ([T,+∞), Bλ(x)) ⊂ Bε(Γ).

(v) Γ is a [globally] asymptotically stable for Σ if it is stable and attractive [globally

attractive] for Σ.

(vi) Γ is almost globally asymptotically stable if it is asymptotically stable and its

region of attraction equals X minus a set of zero measure.

When Γ is a compact positively invariant set, the concepts of stability, attractivity,

and asymptotic stability defined above can be equivalently restated using familiar ε-δ

definitions below.

Definition 2.2.2 (Compact set stability and attractivity). If Γ is compact then,

(i) Γ is stable for Σ if for every ε > 0 there exists δ > 0 such that φ(R+, Bδ(Γ)) ⊂

Bε(Γ).

1In [11], what we call an attractor is referred to as semi-attractor.


(ii) Γ is an attractor for Σ if there exists δ > 0 such that, for all x0 ∈ Bδ(Γ),

limt→∞ ‖φ(t, x0)‖Γ = 0.

(iii) Γ is a global attractor for Σ if it is an attractor with arbitrary δ.

(iv) Γ is [globally] asymptotically stable for Σ if it is stable and attractive [globally

attractive] for Σ.

When the set Γ is unbounded, the notions in Definitions 2.2.1 and 2.2.2 are no longer

equivalent. For instance, the ε-δ notion of attractivity requires that the domain of attrac-

tion of Γ contains a tube of radius δ, whereas the notion of attractivity in Definition 2.2.1

does not, and in fact if Γ is unbounded the width of its domain of attraction may shrink

to zero at infinity.

In the literature on Lyapunov-based set stabilization and stability, various researchers

(e.g., [48], [2], [53]) have used a global version of the following notion of uniform asymp-

totic stability.

Definition 2.2.3 (Uniform set stability and attractivity).

(i) Γ is uniformly stable for Σ if for every ε > 0 there exists δ > 0 such that

φ(R+, Bδ(Γ)) ⊂ Bε(Γ).

(ii) Γ is a uniform attractor for Σ if there exists λ > 0 such that, for all ε > 0, there

exists T > 0 yielding φ([T,+∞), Bλ(Γ)) ⊂ Bε(Γ).

(iii) Γ is uniformly asymptotically stable for Σ if it is uniformly stable and uniformly

attractive.

The notion of uniform asymptotic stability in Definition 2.2.3 is stronger than that

of asymptotic stability in Definition 2.2.1. In particular, Definition 2.2.1 requires the ex-

istence of a neighbourhood N (Γ) in the notions of stability and attractivity. This neigh-

bourhood does not necessarily contain a neighbourhood Bδ(Γ), as in Definition 2.2.3, if


Γ is unbounded. Even in the case when Γ is compact, where N (Γ) contains a neighbour-

hood Bδ(Γ), Definition 2.2.3 is stronger than Definition 2.2.1 because uniform attractivity

is stronger than attractivity.

When Γ is compact, the notions of uniform semi-attractivity in Definition 2.2.1 and

uniform attractivity in Definition 2.2.3 are equivalent. Figure 2.1 illustrates the difference

between these two notions.

ΓΓ

x0x0

x

φ(T, x0)φ(T, x0)ε

ε λλ

uniform attractor uniform semi-attractor

Figure 2.1: Comparison between concepts of uniform attractivity and uniform semi-

attractivity

Definition 2.2.4 (Relative set stability and attractivity). Let Γ1 ⊂ X be positively

invariant and Γ2 ⊂ X be such that Γ1∩Γ2 6= ∅. We say that Γ1 is stable relative to Γ2

for Σ if, for any ε > 0, there exists a neighbourhood N (Γ1) such that φ(R+,N (Γ1)∩Γ2) ⊂

Bε(Γ1). Similarly, one modifies all other notions in Definition 2.2.1 by restricting initial

conditions to lie in Γ2.

Definition 2.2.5 (Local stability and attractivity near a set). Let Γ1 and Γ2,

Γ1 ⊂ Γ2 ⊂ X , be closed positively invariant sets. The set Γ2 is locally stable near Γ1 if

for all x ∈ Γ1, for all c > 0, and all ε > 0, there exists δ > 0 such that for all x0 ∈ Bδ(Γ1)

and all t > 0, whenever φ([0, t], x0) ⊂ Bc(x) one has that φ([0, t], x0) ⊂ Bε(Γ2). The set

Γ2 is locally attractive near Γ1 if there exists a neighbourhood N (Γ1) such that, for

all x0 ∈ N (Γ1), φ(t, x0) → Γ2 at t→ +∞.

The definition of local stability can be rephrased as follows. Given an arbitrary ball

Bc(x) centred at a point x in Γ1, trajectories originating in Bc(x) sufficiently close to Γ1


Bδ(Γ1)Bε(Γ2)

x ∈ Γ1Γ2

Bc(x)

Figure 2.2: An illustration of the notion of local stability near Γ1

cannot travel far away from Γ2 before first exiting Bc(x); see Figure 2.2. It is immediate

to see that if Γ1 is stable, then Γ2 is locally stable near Γ1, and therefore local stability

of Γ2 near Γ1 is a necessary condition for the stability of Γ1.

The example below illustrates some of the stability notions above and their rela-

tionships. The reduction principles studied in Chapter 4 will explore more relationships

between the concepts of stability and attractivity of a set, local stability and attractivity

near a set, and relative stability and attractivity.

Example 2.2.6. Let us now illustrate some of the stability notions introduced so far.

Consider the system (taken from [11]) on R2\0 expressed in polar coordinates (r, θ) as

r = r(1− r)

θ = sin2(θ/2).

The point Γ1 = (1, 0) is an equilibrium, and the set Γ2 = (x1, x2) : x21 + x22 = 1

is invariant. The phase portrait of the system, shown in Figure 2.3, illustrates that Γ1

is globally attractive. However, Γ1 is not uniformly attractive. To see why this is the

case, fix ε > 0 and take a sequence of initial conditions on the unit circle with angles

θi > 0, θi → 0. Let xi(t) be the corresponding sequence of solutions, and let Ti > 0

be the smallest time such that xi([Ti,+∞)) ⊂ Bε(Γ1). Since the sequence of initial

conditions approaches the equilibrium, it follows by continuity of solutions with respect

to initial conditions that Ti → ∞, proving that Γ1 is not uniformly attractive. The same


reasoning allows us to conclude that Γ1 is unstable. On the other hand, the unit circle Γ2

is globally asymptotically stable and uniformly attractive, because in polar coordinates

the subsystem with state r is decoupled from the subsystem with state θ, and it has an

asymptotically stable equilibrium at r = 1.

−1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

x1

x2

Γ1

Γ2

Figure 2.3: The equilibrium Γ1 is attractive, but not uniformly so. The circle Γ2, on the

other hand, is uniformly attractive.

Next, consider the linear system

x1 = −x1

x2 = x2,

which has a saddle point at the origin. The phase portrait is displayed in Figure 2.4.

The set Γ2 = (x1, x2) : x2 = 0 is clearly unstable, but Γ1 = (0, 0) is globally

asymptotically stable relative to Γ2. Vice versa, the set Γ3 = (x1, x2) : x1 = 0 is

globally asymptotically stable and Γ1 is unstable relative to Γ3.

Finally, consider the system

x1 = −x1(1− x22)

x2 = x2,

and let Γ1 = (0, 0) and Γ2 = (x1, x2) : x1 = 0. The phase portrait of the system

in Figure 2.5 illustrates that Γ2 is unstable. At the same time, Γ2 is locally stable near


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

Γ1

Γ2

Γ3

Figure 2.4: Γ1 is globally asymptotically stable relative to Γ2, and unstable relative to

Γ3.

Γ1. The figure illustrates how, given any ε > 0 and c > 0, there exists δ > 0 such

that solutions originating in Bδ(Γ1) cannot exit the set Bε(Γ2) as long as they remain in

Bc((0, 0)).

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

x1

x2 Γ1

Γ2

Bǫ(Γ2)

Bc((0, 0))

Bδ(Γ1)

Figure 2.5: The set Γ2 is unstable, but locally stable near Γ1.

Definition 2.2.7 (Local uniform boundedness (LUB)). The system Σ is locally

uniformly bounded near Γ if for each x ∈ Γ there exist positive scalars λ and m such

that φ(R+, Bλ(x)) ⊂ Bm(x).


Remark 2.2.8. If Γ is a stable compact set, then Σ is locally uniformly bounded near

Γ. By setting λ = δ in Definition 2.2.2-(i), we get φ(R+, Bλ(Γ)) ⊂ Bε(Γ), which implies

φ(R+, Bλ(x)) ⊂ Bε(Γ) for any x ∈ Γ. Since Γ is compact, one can find m > 0 such that

φ(R+, Bλ(x)) ⊂ Bε(Γ) ⊂ Bm(x) for any x ∈ Γ.

The next lemma, proved in the Appendix, clarifies the relationship between uniform

semi-attractivity and asymptotic stability.

Lemma 2.2.9. Let Γ be a closed set which is positively invariant for Σ in (2.2), and

let U ⊃ Γ be a closed set. If Γ is a uniform semi-attractor [relative to U ], then it is

asymptotically stable [relative to U ]. Furthermore, if Σ is locally uniformly bounded

near Γ, then Γ is asymptotically stable [relative to U ] if, and only if, it is a uniform

semi-attractor [relative to U ].

2.3 Limit sets

In order to characterize the asymptotic properties of bounded solutions of a dynamical

system, we will use the well-known notion of limit set, due to G. D. Birkhoff (see [13]),

and that of prolongational limit set, due to T. Ura (see [95]).

Consider the control-affine system (2.1). Given a smooth feedback u(x) and a point

x0 ∈ X , the positive limit set (or ω-limit set) of the closed-loop solution φu(t, x0) is

defined as

L+u (x0) := p ∈ X : (∃tn ⊂ R+) tn → +∞, φu(tn, x0) → p.

The positive limit set of the open-loop solution φ(t, x0), defined in an analogous way,

is denoted L+(x0). The negative limit sets (or α-limit sets) L−u (x0) and L−(x0) of

φu(t, x0) and φ(t, x0), respectively, are defined using time sequences diverging to −∞.

We let L+u (S) :=

⋃

x0∈SL+u (x0) and L

+(S) :=⋃

x0∈SL+(x0).

The significance of limit sets as pertains to the asymptotic behavior of solutions lies

in the next result, due to Birkhoff.


Theorem 2.3.1 (Birkhoff [13]). Consider the dynamical system Σ in (2.2). For any

x0 ∈ X , the limit sets L+(x0), L−(x0) are closed and invariant. Moreover, if φ(R+, x0)

[φ(R−, x0)] is a bounded set, then L+(x0) [L−(x0)] is nonempty, compact, connected,

invariant, and ‖φ(t, x0)‖L+(x0) → 0 as t→ +∞ [‖φ(t, x0)‖L−(x0) → 0 as t→ −∞].

The result above has the following immediate corollary, which highlights the usefulness

of limits sets in assessing the property of set attractivity.

Corollary 2.3.2. Let Γ ⊂ X be a closed and positively invariant set for Σ in (2.2), and

suppose that Σ has the property that there exists a neighbourhood N1(Γ) such that all

solutions originating in N1(Γ) are bounded. Then, Γ is an attractor if and only if there

exists a neighbourhood N2(Γ) ⊂ N1(Γ) such that L+(N2(Γ)) ⊂ Γ.

Ura’s notion of prolongational limit set, introduced below, deals with uniform conver-

gence of solutions and allows one to characterize uniform semi-attractivity of sets. The

prolongational limit set J+u (x0) of a closed-loop solution φu(t, x0) is defined as

J+u (x0) := p ∈ X : (∃(xn, tn) ⊂ X × R+), xn → x0, tn → +∞, φu(tn, xn) → p.

If U ⊂ X , the prolongational limit set of φu(t, x0) relative to U is defined as

J+u (x0, U) := p ∈ X : (∃(xn, tn) ⊂ U × R+), xn → x0, tn → +∞, φu(tn, xn) → p.

The corresponding prolongational limit sets of an open-loop solution φ(t, x0) are denoted

by J+(x0) and J+(x0, U). We let

J+u (S) :=

⋃

x0∈S

J+u (x0), J

+u (S, U) :=

⋃

x0∈S

J+u (x0, U)

J+(S) :=⋃

x0∈S

J+(x0), J+(S, U) :=

⋃

x0∈S

J+(x0, U).

Obviously, L+u (x0) ⊂ J+

u (x0) and L+(x0) ⊂ J+(x0). Moreover, if x0 ∈ U , then

L+u (x0) ⊂ J+

u (x0, U) ⊂ J+u (x0), L+(x0) ⊂ J+(x0, U) ⊂ J+(x0).


Example 2.3.3. To illustrate the difference between limit sets and prolongational limit

sets, consider again the linear system with a saddle point at the origin

x1 = −x1

x2 = x2.

Let x0 = (x1, 0) be an arbitrary initial condition on the x1 axis. Since limt→∞ φ(t, x0) =

(0, 0), we have that L+(x0) = (0, 0). On the other hand, the prolongational limit set

includes limits of sequences φ(tn, xn), where the sequence xn converges to x0 but does

not necessarily lie in the x1 axis. Since φ2(t, x0) → 0 (φ2(t, x0) is the second component

of φ(t, x0)) for any initial condition x0, such limits will lie on the x2 axis. In fact, as

illustrated in Figure 2.6 the prolongational limit set J+(x0) is the entire x2 axis. This

example illustrates the fact that J+(x0) contains asymptotic information about not just

the solution φ(t, x0), but also about the flow in a neighbourhood of x0.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

xn

φ(tn, xn)

x0

Figure 2.6: The prolongational limit set of any point on the x1 axis is the entire x2 axis.

The following result will be useful in the sequel.

Proposition 2.3.4 (Theorem II.4.3 and Lemma V.1.10 in [11]). Consider the dynamical

system Σ in (2.2). For any x ∈ X , J+(x) is closed and invariant. Moreover, for any

ω ∈ L+(x), J+(x) ⊂ J+(ω).


The results in Proposition 2.3.4 still hold if one replaces J+(x) by J+(x, U), with U ⊂

X . The next result establishes the earlier mentioned relationship between prolongational

limit sets and uniform semi-attractivity.

Proposition 2.3.5. Suppose that Σ in (2.2) is locally uniformly bounded near a closed

and positively invariant set Γ. Let U ⊂ X be a closed set, Γ ⊂ U . Then, for each x

in some neighbourhood of Γ [and x ∈ U ], J+(x) 6= ∅ [J+(x, U) 6= ∅]. Moreover, Γ is a

uniform semi-attractor [relative to U ] for Σ if there exists a neighbourhood N (Γ) such

that J+(N (Γ)) ⊂ Γ [J+(N (Γ), U) ⊂ Γ].

This proposition is proved in the Appendix. It can be shown that the condition

J+(N (Γ)) ⊂ Γ [J+(N (Γ), U) ⊂ Γ] is also necessary for uniform semi-attractivity. An

analogous result holds for compact sets without the local uniform boundedness assump-

tion, see Proposition V.1.2 in [11].

Chapter 3

Passivity-Based Set Stabilization I:

Preliminaries

The notion of passivity for state space representations of nonlinear systems, pioneered by

Willems in the early 1970’s, [103, 104], was instrumental for much research on nonlinear

equilibrium stabilization. Key contributions in this area were made in the early 1980’s

by Hill and Moylan in [37, 38, 39, 40], and later by Byrnes, Isidori, and Willems, in

their landmark paper [14]. More recently, in a number of papers [83, 82, 81], Shiriaev

and Fradkov addressed the problem of stabilizing compact invariant sets for passive

nonlinear systems. Their work is a direct extension of the equilibrium stabilization results

by Byrnes, Isidori, and Willems in [14].

The passivity paradigm is particularly successful for stabilization because it provides

a useful interpretation of the control design process in terms of energy exchange, a view

which makes the control design more intuitive, and allows one to naturally handle in-

terconnections of dynamical systems. This view is at the centre of much research on

stabilization of Euler-Lagrange control systems and, more generally, port-Hamiltonian

systems; we refer the reader to the books by Ortega et al. [62], A. J. van der Schaft [98],

and the paper [64].

35

Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 36

In this chapter we begin an investigation of the set stabilization problem for passive

systems which will lead us to the development, in Chapter 5, of results generalizing the

equilibrium theory of [14], as well as the theory in [83, 82, 81]. The enabling insight

in our development is the realisation that at the heart of the stabilization problem by

passivity-based feedback there lies a so-called reduction problem for a dynamical system.

This chapter is organised as follows. In Section 3.1 we present the basic concepts of

dissipativity and passivity from [103, 104, 37]. Section 3.2 presents the passivity-based

set stabilization problem, and Section 3.3 reviews the state of the art on passivity-based

stabilization. Finally, Section 3.4 presents the link between the set stabilization and the

reduction problems.

3.1 Passivity

The notion of passivity was pioneered by Willems, in the early 1970’s, in [103] and

[104]. In [103] Willems introduced the general theory of dissipative dynamical systems.

Dissipativity, a more general concept which encompasses passivity, was defined in terms

of an inequality involving the storage function and the supply rate. The theory was

introduced for dynamical systems with state space models.


x = f(x) +

m∑

i=1

gi(x)ui

y = h(x)

(3.1)

with state space X ⊂ Rn, set of input values U = Rm, and set of output values Y = Rm.

The set U of admissible inputs consists of all U-valued piecewise continuous functions

defined on R.

Definition 3.1.1 (Controlled Invariant Set). A set O ⊂ X is said to be controlled

invariant for (3.1) if there exists a smooth feedback u(x) which makes it invariant for

the closed-loop system x = f(x) +∑m

i=1 gi(x)ui(x).


For system (3.1), a supply rate is a function w : U ×Y → R such that for any u ∈ U

and for any x0 ∈ X , the output y(t) = h(φu(t, x0)) is such that w satisfies∫ t

0

|w(u(τ), y(τ))|dτ <∞,

for all t in the maximal interval of existence of φu(·, x0). Using the supply rate we have

the following definition of dissipativity.

Definition 3.1.2 (Dissipative System, [103]). System (3.1) is said to be dissipative

with supply rate w if there exists a nonnegative function V : X → R+, called a storage

function, such that for all u ∈ U and all x0 ∈ X , V satisfies

V (φu(t, x0))− V (x0) ≤∫ t

0

w(u(τ), y(τ))dτ

for all t in the maximal interval of existence of φu(t, x0), where y(t) = h(φu(t, x0)). This

inequality is called the dissipation inequality.

By choosing a certain form for the supply rate w, the definition of dissipativity is

specialised to that of passivity.

Definition 3.1.3 (Passive System, [37]). A system of the form (3.1) is said to be

passive if it is dissipative with supply rate w(u, y) = y⊤u. In other words, system (3.1)

is passive if there exists a nonnegative storage function V : X → R+ such that for all

u ∈ U and all x0 ∈ X , V satisfies

V (φu(t, x0))− V (x0) ≤∫ t

0

y(τ)⊤u(τ)dτ.

for all t in the maximal interval of existence of φu(t, x0).

It is possible to give a differential characterization of the dissipation inequality as

follows.

Proposition 3.1.4 ([37]). Let V : X → R+ be a C1 function. Then, system (3.1) is

passive with storage function V if and only if

LfV (x) ≤ 0

LgV (x) = h(x)⊤(3.2)


for every x ∈ X , where LgV denotes the row vector [Lg1V · · · LgmV ].

This proposition generalizes the classical Kalman-Yakubovich-Popov (KYP) lemma

for linear time invariant passive systems, and for this reason system (3.1) with a candidate

storage function V is said to possess the KYP property if (3.2) holds.

Example 3.1.5. The proposition above makes it easy to check whether, given a candi-

date storage function, a system is passive. To illustrate, consider a one degree-of-freedom

controlled mechanical system with configuration variable q, massM(q), and potential en-

ergy P (q) ≥ 0. The energy function is H(q, q) = 12M(q)q2+P (q), and the dynamics read

as

M(q)q +1

2M ′(q)q2 +

dP

dq= u,

where u is the control input. Letting p be the momentum, p =M(q)q, the energy in (q, p)

coordinates becomes H(q, p) = 12[M(q)]−1p2 +P (q) and the dynamics take the canonical

Hamiltonian form

q =∂H

∂p

p = −∂H∂q

+ u.

Consider the output function y = q = p/M(q). The resulting system is passive with

storage H(q, p). To see that, let f = col(∂H/∂p,−∂H/∂q) and g = col(0, 1). It is readily

seen that LfH = 0 and LgH = ∂H/∂p = q. Therefore the KYP property holds and the

system is passive.

Now suppose that the system is affected by dissipation, so that the model becomes

M(q)q +1

2M ′(q)q2 +R(q)q +

dP

dq= u,

where R(q) ≥ 0. In (q, p) coordinates, we have

q =∂H

∂p

p = −∂H∂q

− R(q)∂H

∂p+ u.


The system with input u and output y = q is still passive with storage function H , since

LfH = −R(

∂H

∂p

)2

≤ 0, LgH = ∂H/∂p = q.

More generally still, consider the class of port-Hamiltonian systems (see, e.g., [63])

x = [J(x)−R(x)]dH(x)⊤ + g(x)u

y = dH(x)g(x),

where x ∈ Rn, H : Rn → R+ is a C1 function, J(x) = −J(x)⊤ : Rn → Rn×n, and

R(x) : Rn → Rn×n is a positive semidefinite matrix-valued function. It is readily verified

that such a system is passive with storage function H . We have thus established that

the class of port-Hamiltonian systems is included in that of passive systems. A rich class

of electromechanical systems, including robots, electric motors, and nonlinear circuit

networks, falls within the port-Hamiltonian framework, and is therefore amenable to

passivity-based control design.

3.2 The set stabilization problem

We assume throughout the rest of this chapter that (3.1) is passive with a C1 nonnegative

storage function V : X → R+.

Definition 3.2.1. A function u = −ϕ(x), ϕ : X → U , is said to be a passivity-based

feedback (PBF) with respect to the output h(x) if it enjoys the two properties

(∀x ∈ h−1(0)) ϕ(x) = 0,

(∀x ∈ X ) (h(x) 6= 0 =⇒ h(x)⊤ϕ(x) > 0).

(3.3)

The simplest example of PBF is the negative output feedback u = −Kh(x), K > 0,

commonly used in the literature on passivity-based control. Now the main theoretical

problem of this thesis.

Passivity-Based Set Stabilization Problem (PBSSP). Consider the passive sys-

tem (3.1) with storage function V , and let the goal set Γ ⊂ V −1(0) = x ∈ X : V (x) =


0 be closed and positively invariant for the open-loop system x = f(x). Given a PBF

u = −ϕ(x), find conditions guaranteeing that Γ is [globally] asymptotically stable for the

closed-loop system.

Remark 3.2.2. The objective in PBSSP is not the synthesis of a stabilizing feedback,

but rather the derivation of conditions under which a given passivity-based feedback

stabilizes Γ. Thus, PBSSP is a problem of analysis, rather than one of synthesis. The

motivation for this statement is our desire to find conditions that are applicable to a

class of feedbacks, those that have the form (3.3). In Chapter 5 we solve PBSSP. The

resulting conditions enable a control synthesis procedure presented in Chapter 6.

The reason that Γ is assumed to be a subset of V −1(0) is that the time derivative of

the storage function V along trajectories of the closed-loop system formed by (3.1) and

the PBF u = −ϕ(x) satisfies

dV (φu(t, x0))

dt= LfV (φu(t, x0))− LgV (φu(t, x0))ϕ(φu(t, x0))

≤ −h(φu(t, x0))⊤ϕ(φu(t, x0)) ≤ 0,

(3.4)

where we have used the KYP property. Thus, a PBF makes V nonincreasing along

closed-loop solutions. It is therefore natural to consider goal sets that are contained in

V −1(0).

3.3 State of the art

In this section we review the state of the art in the literature on passivity-based stabi-

lization. When the storage function V is positive definite, and Γ = V −1(0) = 0 is an

equilibrium, the most general stabilization result is that by Byrnes, Isidori, and Willems

in [14]. It relies on the following notion of detectability to guarantee that V tends to zero

along solutions of the closed-loop system.


Definition 3.3.1 (Zero-state detectability). System (3.1) is locally zero-state de-

tectable if there exists a neighbourhood U of 0 such that, for all x0 ∈ U ,

h(φ(t, x0)) = 0 for all t ∈ R =⇒ φ(t, x0) → 0 as t→ +∞.

If U = X , the system is zero-state detectable.

Note that the definition above involves open-loop solutions φ(t, x0), and thus zero-

state detectability is a property of the open-loop system. The work in [14] provides

sufficient conditions for detectability. Assuming that V is Cr, r ≥ 1, define the distribu-

tion

D = spanadkf gi : 0 ≤ k ≤ n− 1, 1 ≤ i ≤ m, (3.5)

and the set

S = x ∈ X : LjfLτV (x) = 0, for all τ ∈ D, and all 0 ≤ j < r. (3.6)

Proposition 3.3.2 (Proposition 3.4 in [14]). If S ∩ L+(X ) = 0 and V is proper (i.e.,

all its sublevel sets are compact) and positive definite, then system (3.1) is zero-state

detectable.

The result above is a slight improvement of analogous results by Jurdjevic-Quinn

in [44] and Lee-Araposthatis in [52]. The main passivity-based equilibrium stabilizing

result by Byrnes, Isidori and Willems is given as follows.

Theorem 3.3.3 (Theorem 3.2 in [14]). Suppose that the storage function V is positive

definite and (3.1) is locally zero-state detectable. Then any PBF u = −ϕ(x) asymptot-

ically stabilizes the equilibrium x = 0. Moreover, if V is proper and (3.1) is zero-state

detectable, then the passivity-based feedback globally asymptotically stabilizes x = 0.

The theorem above implies that, when Γ is an equilibrium and V is positive definite,

zero-state detectability is a condition solving PBSSP. In a series of papers, [83, 82, 81],

Shiriaev and Fradkov extended Theorem 3.3.3 to the case when Γ is a compact set and

Γ = V −1(0), relying on the following notion of detectability.


Definition 3.3.4 (V -detectability). System (3.1) is locally V -detectable if there

exists a constant c > 0 such that for all x0 ∈ V −1([0, c]),

h(φ(t, x0) = 0 for all t ∈ R =⇒ V (φ(t, x0)) → 0 as t→ +∞.

If c = ∞, the system is V -detectable.

Proposition 3.3.5 (Theorem 10 in [82]). If S ∩ L+(X ) ⊂ V −1(0) and V is proper and

positive semi-definite, then system (3.1) is V -detectable.

We remark that a function can be proper and positive semi-definite at the same

time. The main passivity-based set stabilizing result by Shiriaev and Fradkov is given as

follows.

Theorem 3.3.6 (Theorem 2.3 in [81]). Suppose that V −1(0) is a compact set, and (3.1) is

locally V -detectable. Then, any passivity-based feedback of the form (3.3) asymptotically

stabilizes V −1(0). Moreover, if V is proper and (3.1) is V -detectable, then the passivity-

based feedback globally asymptotically stabilizes V −1(0).

In summary, existing literature on passivity-based stabilization addresses the situation

when the goal set is compact and it coincides with the zero level set of the storage function.

We will see in Chapters 6 and 7 that these restrictions limit flexibility when performing

control design.

As a first step in extending the theory to the general setting of PBSSP, the next

section establishes a link between the PBSSP and the so-called reduction problem.

3.4 Set stabilization and the reduction problem

As shown in (3.4), a PBF guarantees that the storage function is nonincreasing along

solutions of the closed-loop system. One expects that if the system enjoys suitable

detectability-like properties, then the storage function should decrease asymptotically to

zero and the solutions should approach a subset of V −1(0), hopefully the goal set Γ.


Our point of departure in understanding what system properties yield the conver-

gence of closed-loop solutions to Γ is the following observation. By inequality (3.4), if

‖h(φu(t, x0))‖ 6= 0 then V (φu(t, x0)) < 0, from which it can be deduced that L+u (x0) ⊂

h−1(0). By Birkhoff’s theorem (Theorem 2.3.1), the positive limit set of the closed-

loop solution, L+u (x0), is invariant for the closed-loop system. Since L+

u (x0) ⊂ h−1(0),

and since ϕ(x) = 0 on h−1(0), we have that L+u (x0) is also invariant for the open-loop

system. Now denote by O the maximal set contained in h−1(0) which is positively in-

variant for the open-loop system. In light of the property above, if L+u (x0) is non-empty,

then it must be contained in O. Then, Birkhoff’s theorem implies that all bounded

trajectories of the closed-loop system asymptotically approach O. Since V is nonnega-

tive, any point x ∈ V −1(0) is a local minimum of V and hence dV (x) = 0. Therefore,

LgV (x) = 0 on V −1(0). By the KYP property, LgV (x) = h(x)⊤, and so we conclude that

Γ ⊂ V −1(0) ⊂ h−1(0).

By the KYP property we have that LfV ≤ 0, implying that V is nonincreasing along

solutions of the open-loop system. In particular, then, V −1(0) is positively invariant for

the open-loop system. Since V −1(0) is positively invariant and contained in h−1(0), it is

necessarily a subset of O. Putting everything together, we conclude that

Γ ⊂ V −1(0) ⊂ O ⊂ h−1(0). (3.7)

The above implies that if the trajectories of the closed-loop system in a neighbourhood of

Γ are bounded, the least a passivity-based feedback guarantees is the attractivity of O -

but this is not sufficient for our purposes. Since ϕ(·) = 0 on O, the closed-loop dynamics

on O coincide with the open-loop dynamics, and thus O is an invariant set for the closed-

loop system. In order to ensure the property of asymptotic stability of Γ, the open-loop

system must enjoy the same property relative to O. Therefore, a necessary condition

for Γ to be asymptotically stable for the closed-loop system is that Γ be asymptotically

stable relative to O for the open-loop system.

When the system is LTI, O is the unobservable subspace. In this case, if Γ = 0,


the property of asymptotic stability of Γ relative to O for the open-loop system coincides

with the classical notion of detectability.

Let us summarize our findings so far. We have determined that, associated with

the open-loop system, there exists a nonempty invariant set O satisfying the following

properties:

(i) For any PBF, all those closed-loop solutions that are bounded converge to O.

(ii) OnO, the dynamics of the closed-loop system formed by (3.1) and any PBF coincide

with the dynamics of the open-loop system. Moreover, Γ ⊂ O.

(iii) A necessary condition for Γ to be asymptotically stable for the closed-loop system

is that Γ be asymptotically stable for the open-loop system relative to O.

It follows from the above that the key question in PBSSP is this: is the condition in

(iii) sufficient, or are extra-properties needed for Γ to be asymptotically stable? This

question leads to the following problem.

Reduction Problem. Consider a dynamical system Σ : x = f(x), where f : X → Rn is

locally Lipschitz and X ⊂ Rn is a domain. Let Γ1 ⊂ Γ2 be closed subsets of X which are

positively invariant for Σ. Assume that Γ1 is, either stable, attractive, or asymptotically

stable relative to Γ2. Find what additional conditions are needed to guarantee that Γ1

is, respectively, stable, attractive, or asymptotically stable for Σ. We also seek to solve

the global version of each of the problems above.

In the case of LTI systems x = Ax, when X = Rn and Γ1 ⊂ Γ2 ⊂ Rn are A-

invariant subspaces, the reduction problem has an easy solution which follows directly

from the representation theorem of linear algebra. Indeed, one can find an isomorphism

x 7→ (y1, y2, y3) yielding

y1

y1

y3

=

A11 ⋆ ⋆

0 A22 ⋆

0 0 A33

y1

y2

y3

,


and such that, in new coordinates, Γ1 = (y1, y2, y3) : y2 = y3 = 0 and Γ2 = (y1, y2, y3) :

y3 = 0. We see that Γ1 is asymptotically stable if and only if A22 and A33 are Hurwitz

matrices, i.e., if and only if Γ2 is asymptotically stable and Γ1 is asymptotically stable

relative to Γ2.

In the nonlinear setting, the geometric decomposition above is not always available

and, as a matter of fact, finding conditions such that such a decomposition exists for

given Γ1 and Γ2 remains an open problem. For this reason, the reduction problem for

nonlinear systems is much harder than in the LTI setting. The next chapter is devoted

entirely to this topic.

Chapter 4

Reduction Principles

In this chapter we investigate the reduction problem introduced in Section 3.4. Besides

being relevant for passivity-based stabilization, this problem arises in other areas of

nonlinear control theory, including the stability of invariant sets in cascade-connected

systems and the separation principle in output feedback control. This relationship is

explained in Section 4.1.

The reduction problem was first stated by P. Seibert and J.S. Florio in 1969-1970.

Seibert and Florio proved reduction theorems for stability and asymptotic stability (but

not attractivity) of dynamical systems on metric spaces assuming that Γ1 is compact.

Their conditions first appeared in [76] and [77], while the proofs are found in [78]. The

main results are reviewed in Section 4.2. In [45], B.S. Kalitin investigated Seibert-Florio’s

problem in the context of locally compact metric spaces and closed, but not necessarily

compact, Γ1. Kalitin used a different approach than Seibert and Florio which is based on

a property of B-stability1. In [42, 46], Kalitin’s notion of B-stability and his reduction

theorems were applied to extend Lyapunov’s theorems.

In Section 4.3 we illustrate with two examples the difficulties in extending Seibert-

1Kalitin’s reduction results were based on the notion of a set of type B. This property is morerestrictive than the conditions used in this chapter. Furthermore, Kalitin’s proofs in [45] are open toquestion.

46

Chapter 4. Reduction Principles 47

Florio’s theory and then, in Section 4.4 we generalize Seibert-Florio’s reduction theorems

for stability and asymptotic stability to the case when the goal set is closed but not

bounded. Moreover, we present a novel reduction theorem for attractivity. The proofs

of the reduction theorems are in Section 4.5. These results are applied, in Section 4.6, to

the investigation of stability of invariant sets for cascade-connected systems. Finally, in

Section 4.7, we present a conceptual reduction-based procedure for control design.

4.1 Motivation

Let us recall the reduction problem presented in Section 3.4.

Reduction Problem. Consider a dynamical system Σ : x = f(x), where f : X → Rn is

locally Lipschitz and X ⊂ Rn is a domain. Let Γ1 ⊂ Γ2 be closed subsets of X which are

positively invariant for Σ. Assume that Γ1 is either stable, attractive or asymptotically

stable relative to Γ2. Find what additional conditions are needed to guarantee that Γ1

is, respectively, stable, attractive or asymptotically stable for Σ. We also seek to solve

the global version of each of the problems above.

Besides the connection with PBSSP, the reduction problem appears in various areas

of control theory. Consider, for instance, cascade-connected systems of the form

x = f(x, y), (x, y) ∈ Rn1 × Rn2

y = g(y).

(4.1)

Suppose that g(0) = 0 and that a closed set Γ1 ⊂ Rn1 is asymptotically stable for

x = f(x, 0). Under what conditions is the set Γ1 × 0 asymptotically stable for the whole

system (4.1)? Equivalently, if we let Γ1 = Γ1 × 0 and Γ2 = Rn1 × 0, we ask when is it

that the asymptotic stability of Γ1 relative to Γ2 implies that Γ1 is asymptotically stable

relative to Rn1 × Rn2? We will return to this problem in Section 4.6.


The reduction problem also arises in the investigation of the separation principle in

output feedback control [6], [93]. Consider a control system

x = f(x, u), x ∈ Rn, u ∈ Rm

y = h(x), y ∈ Rp,

(4.2)

and suppose that a state feedback controller u(x) is available which asymptotically stabi-

lizes an equilibrium x = x⋆. What properties should be possessed by the feedback u and

by an asymptotic observer

˙x = f(x, u, y), x ∈ Rn, (4.3)

in order that the output feedback controller u(x) asymptotically stabilizes the equilibrium

(x, x) = (x⋆, x⋆) of the closed-loop system? In this case we have Γ1 = (x, x) : x = x =

x⋆ and Γ2 = (x, x) : x = x.

4.2 Seibert-Florio’s reduction theorems

In this section we present Seibert and Florio’s reduction theorems that solve the reduction

problem for stability and asymptotic stability when Γ1 is compact. Throughout this

chapter we consider the dynamical system

Σ : x = f(x) (4.4)

where f : X → Rn is locally Lipschitz and X ⊂ Rn.

Theorem 4.2.1 (Theorem 3.4 in [78]). Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed

positively invariant sets for Σ in (4.4), and assume that Γ1 is compact. Then, Γ1 is stable

for Σ if the following conditions hold:

(i) Γ1 is asymptotically stable relative to Γ2,

(ii) Γ2 is locally stable near Γ1.


As mentioned earlier, when Γ1 is stable then Γ2 is locally stable near Γ1. Thus

condition (ii) is also necessary.

Theorem 4.2.2 (Theorem 4.13 and Corollary 4.11 in [78]). Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X ,

be two closed positively invariant sets for Σ in (4.4), and assume that Γ1 is compact. If,

and only if, the following conditions hold:


(ii) Γ2 is locally stable near Γ1,

(iii) Γ2 is locally attractive near Γ1,

then Γ1 is asymptotically stable for Σ. Furthermore, if

(iv) all trajectories of Σ are bounded,

and conditions (i) and (iii) are replaced by

(i)’ Γ1 is globally asymptotically stable relative to Γ2,

(iii)’ Γ2 is a global attractor for Σ,

then Γ1 is globally asymptotically stable for Σ.

Remark 4.2.3. In [78] Seibert and Florio give a definition for local stability near a

compact set which is slightly different from that of Definition 2.2.5. Their definition goes

as follows. “Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be closed positively invariant sets and Γ1

be compact. The set Γ2 is locally stable near Γ1 if there exists a neighbourhood of Γ1,

N (Γ1), such that for all ε > 0, there exists δ > 0 such that for all x0 ∈ Bδ(Γ1) and all

t > 0, whenever φ([0, t], x0) ⊂ N (Γ1) one has φ([0, t], x0) ⊂ Bε(Γ2).” It is easy to see that

if the conditions in Definition 2.2.5 hold, then Γ2 is locally stable near Γ1 in the sense of

Seibert and Florio. Since local stability of Γ2 near Γ1 in the sense of Definition 2.2.5 is

a necessary condition for stability of Γ, the assumptions in Theorems 4.2.1 and 4.2.2 are

equivalent to the conditions in [78].


4.3 Examples

In this section we illustrate with two examples some of the complications one encounters

in extending Seibert-Florio’s reduction theory. The first example shows that the reduction

theorem for asymptotic stability (Theorem 4.2.2) may no longer hold when the goal set is

not compact, due to the presence of unbounded solutions. The second example illustrates

that the attractivity of Γ1 relative to Γ2 is a fragile property which is not sufficient to

obtain a reduction principle for attractivity.

Example 4.3.1. Consider the dynamical system

x1 = x1x3

x2 = −x2 + x1x3

x3 = −x33.

The sets Γ1 = x : x2 = x3 = 0 and Γ2 = x : x3 = 0 are closed and invariant. The set

Γ2 is globally asymptotically stable. On Γ2, the motion is described by

x1 = 0

x2 = −x2,

so Γ1 is globally exponentially stable relative to Γ2. Yet, Γ1 is neither stable nor attractive

because for all x3(0) > 0, x1(t) → ∞ and x1(t)x3(t) → ∞, implying that x2(t) → ∞ as

well. This is illustrated in Figure 4.3.1. The source of the problem is the unboundedness

of Γ1 and the presence of unbounded trajectories on arbitrarily small neighbourhoods of

Γ1. If Γ1 were compact, the Seibert-Florio reduction principle for asymptotic stability,

Theorem 4.2.2, would imply that unbounded trajectories near Γ1 cannot exist, and Γ1

would be asymptotically stable. This observation suggests that in order to develop re-

duction principles for unbounded sets a suitable boundedness property of trajectories is

needed. Later, we show that the required property is that of local uniform boundedness.


0

2

4

6

8

10

6

2 0 0.20.6

1

0

0.2

0.4

0.6

0.8

1

Γ1

Γ2x1

x2

x3

Figure 4.1: Γ2 is globally asymptotically stable and Γ1 is globally exponentially stable

relative to Γ2. Yet, Γ1 is not asymptotically stable.

Example 4.3.2. Consider the following system

x1 = (x22 + x23)(−x2)

x2 = (x22 + x23)(x1)

x3 = −x33.

Let Γ1 = (x1, x2, x3) : x2 = x3 = 0 and Γ2 = (x1, x2, x3) : x3 = 0, both invariant

sets. Clearly, Γ2 is globally asymptotically stable. The system dynamics on Γ2 take the

form

x1 = −x2(x22)

x2 = x1(x22).

On Γ1 ⊂ Γ2, every point is an equilibrium. Phase curves on Γ2 off of Γ1 are concentric

semicircles x21+x22 = c, and therefore Γ1 is a global, but unstable, attractor relative to

Γ2. As shown in Figure 4.2, for initial conditions not in Γ2 the trajectories are bounded

and their positive limit set is a circle inside Γ2 which intersects Γ1 at equilibrium points.

Thus, the attractivity of Γ1 relative to Γ2 is a fragile property which is lost outside of Γ2,


even though Γ2 is globally asymptotically stable. Our reduction principle for attractivity

will show that this phenomenon of attractivity loss is due to the instability of Γ1 relative

to Γ2.

−2

−1

0

1

2

−2

−1

0

1

2

0

0.2

0.4

0.6

0.8

1

x1x2

x3

Γ1

Γ2

Figure 4.2: Γ1 is globally attractive rel. to Γ2, Γ2 is globally asymptotically stable, and

yet Γ1 is not attractive.

4.4 Reduction theorems

In this section we present three reduction theorems for stability, attractivity, and asymp-

totic stability which do not require Γ1 to be bounded.

Theorem 4.4.1 (Stability). Let Γ1 ⊂ Γ2 be two closed positively invariant subsets for

Σ in (4.4). Then, Γ1 is stable if the following conditions hold:



(iii) If Γ1 is unbounded, then Σ is locally uniformly bounded near Γ1.

Condition (ii) is also necessary.


Remark 4.4.2. Condition (i) cannot be relaxed by just requiring that Γ1 be stable

relative to Γ2. This fact was already pointed out by Seibert and Florio in [78] using the

following simple counter-example.

Example 4.4.3. Consider the linear system

x1 = x2

x2 = 0.

Let Γ1 be the origin and Γ2 be the x1-axis. The dynamics on Γ2 are x1 = 0 and so Γ1

is stable relative to Γ2. Moreover, Γ2 is stable and hence also locally stable near Γ1.

However, Γ1 is unstable.

By noting that if Γ2 is stable for Σ, then it is also locally stable near Γ1, we get the

following useful corollary.

Corollary 4.4.4. Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed positively invariant sets.

Then, Γ1 is stable if conditions (i) and (iii) in Theorem 4.4.1 hold and condition (ii) is

replaced by the following one:

(ii)’ Γ2 is stable.

The stability of Γ2 in condition (ii)’ is not necessary for the stability of Γ, as shown

by Seibert and Florio in [78] by the following counter-example.

Example 4.4.5. Consider the following system

x1 = −2x31 + x2 + x41

x2 = −x21x2(1− x1).

Let Γ1 be the origin and Γ2 be the x1-axis. Γ2 is invariant and locally stable near Γ1.

However, Γ2 is not stable. On Γ2 the dynamics takes the form x1 = −2x31 + x41 and thus

Γ1 is asymptotically stable relative to Γ2. Thus, by Theorem 4.4.1 Γ1 is stable.


Theorem 4.4.6 (Attractivity). Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed positively

invariant sets for Σ in (4.4). Then, Γ1 is attractive if the following conditions hold:

(i) Γ1 is asymptotically stable relative to Γ2

(ii) Γ2 is locally attractive near Γ1,

(iii) there exists a neighbourhood N (Γ1) such that, for all initial conditions in N (Γ1),

the associated solutions are bounded and such that the set cl(φ(R+,N (Γ1))) ∩ Γ2

is contained in the domain of attraction of Γ1 relative to Γ2.

The set Γ1 is globally attractive if:

(i)’ Γ1 is globally asymptotically stable relative to Γ2,

(ii)’ Γ2 is a global attractor,

(iii)’ all trajectories in X are bounded.

Conditions (ii) and (ii)’ are also necessary.

Remark 4.4.7. Being of a rather technical nature, Assumption (iii) is difficult to check

and of limited practical use. It has, however, theoretical significance because it is used to

prove the reduction principle for asymptotic stability stated in the sequel. A similar, but

slightly stronger, assumption is found in Theorem 10.3.1 in [43] concerning the attractiv-

ity of equilibria of cascade-connected systems. In fact, the result in [43] is a corollary of

Theorem 4.4.6. If condition (i) is replaced by the stronger (i)’, then one can replace (iii)

by the simpler requirement that trajectories in some neighbourhood of Γ1 be bounded.

Returning to Example 4.3.2, the loss of attractivity of Γ1 is due to the fact that Γ1 is

only attractive relative to Γ2, and not asymptotically stable, and thus condition (i) in

the above theorem is violated.


It is interesting to note that it is not enough to assume, in place of condition (i), that

Γ1 is an attractor relative to Γ2 (or, in place of condition (i)’, that Γ is a global attractor

relative to Γ2) as was shown in Example 4.3.2.

By combining Theorems 4.4.6 and 4.4.1 we obtain a reduction principle for asymptotic

stability.

Theorem 4.4.8 (Asymptotic stability). Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed

positively invariant sets for Σ in (4.4). Then, Γ1 is [globally] asymptotically stable if the

following conditions hold:

(i) Γ1 is [globally] asymptotically stable relative to Γ2,


(iii) Γ2 is locally attractive near Γ1 [Γ2 is globally attractive],

(iv) if Γ1 is unbounded, then Σ is locally uniformly bounded near Γ1,

(v) [all trajectories of Σ are bounded.]

Conditions (i), (ii), and (iii) in the theorem above are necessary.

By combining Theorem 4.4.8 and Corollary 4.4.4 we obtain the following corollary.

Corollary 4.4.9. Let Γ1 and Γ2, Γ1 ⊂ Γ2 ⊂ X , be two closed positively invariant

sets. Then, Γ1 is [globally] asymptotically stable if conditions (i), (iii), (iv) [and (v)] in

Theorem 4.4.8 hold, and condition (ii) is replaced by the following one:

(ii)’ Γ2 is stable.

4.5 Proofs of reduction theorems

4.5.1 Proof of Theorem 4.4.1

To prove the theorem, we need the following lemma.


Lemma 4.5.1. Let Γ1 ⊂ X be a closed set which is positively invariant for Σ. If Γ1 is

unstable, then there exist ε > 0, a bounded sequence xi ⊂ X , a sequence ti ⊂ R+

and x ∈ Γ1 such that xi → x ∈ Γ1, and ‖φ(ti, xi)‖Γ1= ε for all i.

Proof. The instability of Γ1 implies that there exists ε > 0, a sequence xi ⊂ X , and a

sequence ti ⊂ R+, such that ‖xi‖Γ1→ 0, and ‖φ(ti, xi)‖Γ1

= ε. If we show that xi

above can be chosen to be bounded, then, without loss of generality, there exists x ∈ Γ1

such that xi → x and we are done. Let S be defined as follows

S = x ∈ Bε(Γ1) : (∃t > 0) ‖φ(t, x)‖ = ε.

The instability of Γ1 implies that S is not empty. Moreover, since Γ1 is positively invari-

ant, S ∩ Γ1 = ∅. Suppose, by way of contradiction, that there does not exist a bounded

sequence xi and a sequence ti such that ‖xi‖Γ1→ 0 and ‖φ(ti, xi)‖Γ1

= ε. This

implies that, for any x ∈ Γ1, there exists δ(x) > 0 such that Bδ(x)(x) ∩ S = ∅. For, if

this were not true, then there would exist a bounded sequence xi ⊂ S, with xi → Γ1

contradicting the assumption we have made. Let U =⋃

x∈Γ1Bδ(x)(x). By construction,

U is a neighbourhood of Γ1 such that U ∩ S = ∅. In other words, for all x ∈ U , there

does not exist t > 0 such that ‖φ(t, x)‖Γ1= ε, contradicting the assumption that Γ1 is

unstable.

Proof of Theorem 4.4.1. By way of contradiction, suppose that Γ1 is unstable.

Then, by Lemma 4.5.1, there exist ε > 0, a bounded sequence xi ⊂ X , with xi → x ∈

Γ1, and a sequence ti ⊂ R+, such that

‖φ(ti, xi)‖Γ1= ε and φ([0, ti), xi) ∈ Bε(Γ1).

By local uniform boundedness of Σ near Γ1, there exist λ,m > 0 such that φ(R+, Bλ(x)) ⊂

Bm(x). We can assume xi ⊂ Bλ(x). Take a decreasing sequence εi ⊂ R+, εi → 0.

By assumption (ii), Γ2 is locally stable near Γ1. Using the definition of local stability


with c = m and ε = εi, there exists δi > 0 such that for all x0 ∈ Bδi(x) and all t > 0, if

φ([0, t], x0) ⊂ Bm(x), then φ([0, t], x0) ⊂ Bεi(Γ2). By taking δi ≤ λ we have

(∀x0 ∈ Bδi(x)) φ(R+, x0) ⊂ Bεi(Γ2).

By passing, if needed, to a subsequence we can assume without loss of generality that,

for all i, xi ∈ Bδi(x) so that

lim supi→∞

d(φ([0, ti], xi),Γ2) = 0.

Using assumptions (i) and (iii) (if Γ1 is unbounded), by Lemma 2.2.9 it follows that Γ1

is a uniform semi-attractor relative to Γ2. Therefore,

(∀x ∈ Γ1)(∃µ > 0)(∀ε′ > 0)(∃T > 0) s.t. φ([T,+∞), Bµ(x) ∩ Γ2) ⊂ Bε′(Γ1). (4.5)

Consider the set Γ′1 = Γ1 ∩ cl(B2m(x)). Since Γ′

1 is compact, then the infimum of µ(x),

in (4.5), for all x ∈ Γ′1 exists and is greater than zero. Thus, we infer the existence of

µ > 0 such that

(∀x ∈ Γ′1)(∀ε′ > 0)(∃T > 0) φ([T,+∞), Bµ(x) ∩ Γ2) ⊂ Bε′(Γ1). (4.6)

By reducing, if necessary, ε in the instability definition, we may assume that2 ε < µ. Now

choose ε′ < ε/2. Using again a compactness argument, by (4.6) one infers the following

condition

(∃T > 0)(∀x ∈ Γ′1)φ([T,+∞), Bµ(x) ∩ Γ2) ⊂ Bε′(Γ1). (4.7)

We claim that Bµ(Γ1) ∩Bm(x) ⊂ Bµ(Γ′1). For, if µ ≥ m, then

Bµ(Γ1) ∩ Bm(x) = Bm(x) ⊂ Bµ(x) ⊂ Bµ

(

Γ1 ∩ cl(B2m(x)))

.

If µ < m, then x ∈ Bµ(Γ1)∩Bm(x) if and only if ‖x‖Γ1< µ and ‖x−x‖ < m; in particular,

there exists y ∈ Γ1 such that ‖x−y‖ < µ. Since ‖y−x‖ ≤ ‖x−y‖+‖x−x‖ ≤ µ+m < 2m,

we have that y ∈ Γ1 ∩ cl(B2m(x)), and thus x ∈ Bµ(Γ1 ∩ cl(B2m(x))).

2In the contradiction assumption that Γ1 is unstable we employ ε > 0 as in Lemma 4.5.1. Byinstability of Γ1, any ǫ ∈ (0, ε] works in place of ε. Therefore, it is always possible to find ε < µ.


Using (4.7) and the claim we have just proved we obtain

(∀x ∈ Bµ(Γ1) ∩Bm(x) ∩ Γ2) φ([T,+∞), x) ⊂ Bε′(Γ1). (4.8)

Now, since tk is unbounded there exists K1 > 0 such that tk > T for all k ≥ K1. Since

φ([0, tk), xk) ⊂ Bε(Γ1) we have φ(tk − T, xk) ∈ Bε(Γ1) for all k ≥ K1. Let

yk = φ(tk, xk), and zk = φ(tk − T, xk).

Thus, yk = φ(T, zk), ‖yk‖Γ1= ε and zk ∈ Bε(Γ1). By local uniform boundedness, it also

holds that zk ∈ Bm(x). Pick δ ∈ (0, µ− ε). Since zk ∈ φ([0, tk), xk) ⊂ Bm(x), and since

lim supk→∞

d(φ([0, tk], xk),Γ2) = 0,

then there exists K2 ≥ K1 such that, for all k ≥ K2, there exists z′k ∈ Bm(x) ∩ Γ2 such

that ‖zk − z′k‖ < δ. Since zk ∈ Bε(Γ1), then

z′k ∈ Bε+δ(Γ1) ∩ Bm(x) ∩ Γ2 ⊂ Bµ(Γ1) ∩Bm(x) ∩ Γ2

and, by (4.8), φ([T,+∞), z′k) ⊂ Bε′(Γ1). By continuous dependence on initial conditions,

δ can be chosen small enough that

(∀x ∈ Bm(x))(∀x0 ∈ Bδ(x)) ‖φ(T, x)− φ(T, x0)‖ < ε/2.

We have zk ∈ Bm(x) and ‖zk − z′k‖ < δ, hence ‖φ(T, zk)−φ(T, z′k)‖ < ε/2, which implies

yk ∈ Bε/2(φ(T, z′k)) ⊂ Bε/2+ε′(Γ1) ⊂ Bε(Γ1),

contradicting ‖yk‖Γ1= ε.


By assumption (ii), there exists a neighbourhood N1(Γ1) of Γ1 such that all trajectories

originating there asymptotically approach Γ2 in positive time. Let N2(Γ1) be the neigh-

bourhood in assumption (iii), and define N3(Γ1) = N1(Γ1)∩N2(Γ1). Clearly, N3(Γ1) is a


neighbourhood of Γ1. By construction, for all x0 ∈ N3(Γ1), the solution is bounded and

approaches Γ2. Therefore, the positive limit set L+(x0) is non-empty, compact, invariant,

and L+(x0) ⊂ Γ2. Moreover, by definition of positive limit set, and by assumption (iii)

we have the following inclusion,

L+(x0) ⊂ cl(φ(R+, x0)) ∩ Γ2 ⊂ domain of attraction of Γ1 rel. to Γ2. (4.9)

We need to show that L+(x0) ⊂ Γ1. Assume, by way of contradiction, that there exists

ω ∈ L+(x0) and ω /∈ Γ1. By the invariance of L+(x0), φ(R, ω) ⊂ L+(x0), and therefore

L−(ω) ⊂ L+(x0). By the inclusion in (4.9), all trajectories in L−(ω) asymptotically

approach Γ1 in positive time, and so since L−(ω) is closed, L−(ω) ∩ Γ1 6= ∅. Let p ∈

L−(ω) ∩ Γ1. Pick ε > 0 such that ‖ω‖Γ1> ε. By the stability of Γ1 relative to Γ2,

there exists a neighbourhood N4(Γ1) of Γ1 such that φ(R+,N4(Γ1)∩Γ2) ⊂ Bε(Γ1). Since

p ∈ L−(ω), there exists a sequence tk ⊂ R+, with tk → +∞, such that φ(−tk, ω) → p

at k → +∞. Since p ∈ Γ1, we can pick k⋆ large enough that φ(−tk⋆ , ω) ∈ N4(Γ1). Let

T = tk⋆ and z = φ(−tk⋆ , ω). We have thus obtained that z ∈ N4(Γ1), but φ(T, z) = ω

is not in Bε(Γ1). This contradicts the stability of Γ1, and therefore, for all x0 ∈ N3(Γ1),

L+(x0) ⊂ Γ1, proving that Γ1 is an attractor for Σ.

To prove global attractivity of Γ1 it is sufficient to notice that by assumptions (ii)’

and (iii)’, for all x0 ∈ X , L+(x0) is non-empty and L+(x0) ⊂ Γ2. On Γ2, by assumption

(i)’ all trajectories approach Γ1, so by the contradiction argument above we conclude

that L+(x0) ⊂ Γ1.

Remark 4.5.2. Part of the proof above was inspired by the stability results using positive

semidefinite Lyapunov functions presented in [42] and by the proof of Lemma 1 in [21].


If Γ1 is compact, the theorem coincides with Theorem 4.2.2. Suppose that Γ1 is un-

bounded. That the “global” assumptions imply global asymptotic stability is a direct


consequence of Theorems 4.4.6 and 4.4.1. To prove that the “local” assumptions imply

asymptotic stability of Γ1, we need to show that assumption (iii) in Theorem 4.4.6 is

satisfied.

Assumptions (i), (ii), and (iv) in imply that Γ1 is stable. Moreover, by assumption

(i), Γ1 is attractive relative to Γ2. Let N ⊂ Γ2 denote the domain of attraction of Γ1

relative to Γ2. By assumption (iv), for each x ∈ Γ1 there exist two positive numbers λ(x)

and m(x) such that φ(R+, Bλ(x)(x)) ⊂ Bm(x)(x). Fix x ∈ Γ1, and let ε(x) > 0 be small

enough that

cl(

Bε(x)(Γ1) ∩Bm(x)(x))

∩ Γ2 ⊂ N.

The constant ε is guaranteed to exist because the set on left-hand side of the inclu-

sion is compact and can be made arbitrarily small. Since Γ1 is stable, there exists a

neighbourhood Nx(Γ1) such that φ(R+,Nx(Γ1)) ⊂ Bε(x)(Γ1). Now define

U =⋃

x∈Γ1

Bλ(x)(x) ∩Nx(Γ1).

Clearly, U is a neighbourhood of Γ1. By definition, for each y ∈ U , there exists x ∈ Γ1

such that y ∈ Bλ(x)(x) ∩Nx(Γ1), so that the solution originating in y is bounded and

φ(R+, y) ⊂ Bε(x)(Γ1) ∩Bm(x)(x).

Therefore, cl(φ(R+, y)) ∩ Γ2 ⊂ cl(

Bε(x)(Γ1) ∩Bm(x)(x))

∩ Γ2 ⊂ N .

4.6 Cascade-connected systems

We now return to the cascade-connected system in (4.1). When f(0, 0) = 0 and g(0) = 0,

conditions for asymptotic stability and attractivity of the equilibrium (x, y) = (0, 0)

are well-known in the control literature (see [99, Theorem 3.1], [91, Corollary 5.2], [43,

Corollaries 10.3.2, 10.3.3]), and in fact they are consequences of Seibert and Florio’s

reduction theory, specialized to the case when Γ1 is the origin and Γ2 = (x, y) : y = 0.


Motivated by this observation, we present a straightforward application of Theorem 4.4.6

and Corollary 4.4.9.

Corollary 4.6.1. Consider system (4.1) with f and g locally Lipschitz on Rn1 × Rn2 ,

and let Γ1 ⊂ Rn1 be a positively invariant set for system x = f(x, 0). Denote Γ1 := Γ1×0

and suppose that g(0) = 0. Then, Γ1 is an attractor [global attractor] for (4.1) if

(i) Γ1 is globally asymptotically stable for x = f(x, 0),

(ii) y = 0 is a [globally] attractive equilibrium for y = g(y),

(iii) all solutions of (4.1) originating in some neighbourhood of Γ1 [originating in Rn1 ×

Rn2 ] are bounded.

Moreover, Γ1 is [globally] asymptotically stable if

(iv) Γ1 is [globally] asymptotically stable for x = f(x, 0),

(v) y = 0 is a [globally] asymptotically stable equilibrium of y = g(y),

(vi) if Γ1 is unbounded, then (4.1) is locally uniformly bounded near Γ1,

(vii) [all trajectories of (4.1) are bounded.]

4.7 Reduction-based control design

The reduction principles presented in Section 4.4 motivates a reduction-based perspective

for control design and set stabilization. The general idea is to use a feedback transforma-

tion to decompose the control input into two parts. The first part is designed to stabilize

a set Γ2 that contains the goal set Γ1. The second part is designed to stabilize the goal

set Γ1 relative to the larger set Γ2. Such a decomposition may significantly reduce the

complexity control design, as it will be seen later in this thesis.


Consider the control-affine system (3.1) and assume that it is required to design a

feedback u(x) to stabilize a goal set Γ1 ⊂ X . Consider the following feedback transfor-

mation

u = β1(x)u+ β2(x)u,

where u ∈ Rk and u ∈ Rm−k are new control inputs, β1(·) is locally Lipschitz matrix-

valued function β1(x) : X → Rm×k, for some k ∈ 1, . . . , m − 1, such that β1(x) has

full rank k and β2(x) : X → Rm×m−k is another locally Lipschitz function such that

[β1(x) β2(x)] is nonsingular for all x ∈ X .

Suppose that this feedback transformation can be chosen such that, when u = 0, a

set Γ2 that contains the goal set Γ1 is invariant for the system

x = f(x) + g(x)β1(x)u(x),

for any feedback u(x).

In this case, to design feedbacks u and u to stabilize Γ1 it is sufficient, according to

Theorem 4.4.8, to do the following. First, design u to asymptotically stabilize Γ1 relative

to Γ2 for the system

x = [f(x) + g(x)β1(x)u(x)]|Γ2.

Then, design u to asymptotically stabilize Γ2 for the closed loop system

x = f(x) + g(x)β1(x)u(x) + g(x)β2(x)u(x),

and also ensure, if Γ1 is unbounded, that the closed-loop system is LUB near Γ1. By

the reduction principle of Theorem 4.4.8 this approach yields asymptotic stabilization

of the set Γ1. Moreover, if u(x) globally asymptotic stabilizes Γ1 relative to Γ2, if u(x)

globally asymptotically stabilizes Γ2, and if the closed-loop system is LUB near Γ1 and

all solutions are bounded, then Γ1 is globally asymptotically stable for the closed loop

system.

The procedure described above is fairly general and can involve many details if one

wants to work out its steps. However, this general perspective can be quite beneficial in


certain situations. In Chapter 6 we use this general perspective to present a passivity-

based set stabilizing procedure and in Chapter 7 we use this passivity-based procedure

and the general reduction-based perspective discussed above to solve a challenging prob-

lem in the field of multiagent systems.

Chapter 5

Passivity-Based Set Stabilization II:

Theory

In this chapter we return to the control-affine system

x = f(x) +

m∑

i=1

gi(x)ui

y = h(x)

(5.1)

and leverage the reduction theory of Chapter 4 to solve the passivity-based set stabi-

lization problem introduced in Chapter 3. The solution we present extends the existing

passivity-based stabilization theory reviewed in Section 3.3, generalizing it even in the

special case when the goal set is an equilibrium. Our results rely on a notion of detectabil-

ity, presented in Section 5.1, which encompasses both of the existing notions of zero-state

and V -detectability. In Section 5.1 we also give sufficient conditions for detectability to

hold. The solution of PBSSP is presented in Section 5.2. In Section 5.3 we compare

our result to the existing literature and discuss the significance of the new detectability

notion. Finally, in Section 5.4 we return to the point-mass example presented in the

introduction and use the theory developed in this chapter to solve it.

64

Chapter 5. Passivity-Based Set Stabilization II: Theory 65

5.1 Γ-Detectability

As seen in Section 3.3, the passivity-based stabilization theory of Byrnes-Isidori-Willems

and Shiriaev-Fradkov relies on notions of zero-state and V -detectability, respectively.

Recall the definition of the set O in Section 3.4.

Definition 5.1.1 (Set O). Given the control system (5.1), we denote by O the maximal

set contained in h−1(0) which is positively invariant for the open-loop system x = f(x).

In Section 3.4 we determined that a necessary condition for PBSSP to be solvable

is that the goal set Γ be asymptotically stable relative to O. We call this property

Γ-detectability.

Definition 5.1.2 (Γ-detectability). System (5.1) is locally Γ-detectable if Γ is

asymptotically stable relative toO for the open-loop system. The system is Γ-detectable

if Γ is globally asymptotically stable relative to O for the open-loop system.

When system (5.1) is linear time-invariant (LTI), the set O is the unobservable sub-

space and Γ = 0. In this case, the above definition requires that all open-loop tra-

jectories on the unobservable subspace O converge to 0. Therefore, in the LTI setting,

Γ-detectability coincides with the classical notion of detectability.

We now show that the notion of Γ-detectability generalizes that of zero-state de-

tectability. As a matter of fact, when V is positive definite, and thus Γ = 0, the two

detectability notions coincide.

Lemma 5.1.3. If V is positive definite and Γ = V −1(0) = 0, then the following three

conditions are equivalent:

(a) System (5.1) is locally zero-state detectable [zero-state detectable],

(b) the equilibrium x = 0 is [globally] attractive relative to O for the open-loop system,

(c) system (5.1) is locally Γ-detectable [Γ-detectable].


Proof. The set of points x0 ∈ X such that the open-loop solution satisfies h(φ(t, x0)) ≡ 0

is precisely the maximal open-loop invariant subset of h−1(0), i.e., the set O. Thus,

conditions (a) and (b) are equivalent. Since (5.1) is passive, by the KYP property we

have LfV ≤ 0. By the assumption that V is positive definite, it follows that x = 0 is a

stable equilibrium of the open-loop system. Thus, x = 0 is [globally] asymptotically stable

relative to O for the open-loop system if and only if x = 0 is [globally] attractive relative

to O for the open-loop system, proving that conditions (b) and (c) are equivalent.

The next lemma shows that Γ-detectability also encompasses the notion of V -detectability.

Lemma 5.1.4. If Γ = V −1(0) is a compact set, then the following three conditions are

equivalent:

(a) System (5.1) is locally V -detectable,

(b) the set Γ is attractive relative to O for the open-loop system,

(c) system (5.1) is locally Γ-detectable.

Moreover, if V is proper, then the global versions of conditions (a)-(c) are equivalent.

Proof. Suppose that (5.1) is locally V -detectable. Then, for all x0 ∈ V −1([0, c]) ∩ O,

V (x(t)) → 0. Since V −1(0) is compact, in a sufficiently small neighbourhood of Γ,

V (φ(t, x0)) → 0 implies φ(t, x0) → V −1(0), and thus Γ = V −1(0) is attractive relative to

O for the open-loop system, showing that condition (a) implies (b). Since LfV ≤ 0, Γ

is also stable for the open-loop system. Thus, condition (b) implies (c). Now suppose

that (5.1) is locally Γ-detectable. Then, there exists a neighbourhood S of Γ such that,

for all x0 ∈ S ∩O, φ(t, x0) → Γ. Since Γ = V −1(0) is compact and V is continuous, there

exists c > 0 such that V −1([0, c]) ⊂ S. Hence, for all x0 ∈ V −1([0, c])∩O or, equivalently

for all x0 ∈ V −1([0, c]) such that h(φ(t, x0)) ≡ 0, we have φ(t, x0) → V −1(0). By the

continuity of V and the compactness of V −1(0) the latter fact implies that V (φ(t, x0)) →

0. This proves that condition (c) implies (a).


The proof of equivalence of the global notions of detectability follows directly from

the fact that if V is proper, then V (φ(t, x0)) → 0 if and only if φ(t, x0) → V −1(0).

Despite their equivalence when Γ = V −1(0) is compact, the two notions of Γ- and V -

detectability have a different flavor, in that the latter notion utilizes the storage function

V (·) to define a property of the open-loop system, detectability, which is independent of

V . On the other hand, the definition of Γ-detectability, being independent of V , is closer

in spirit to the original definition of zero-state detectability. Finally, the notion of V -

detectability cannot be generalized to the case when Γ is unbounded, even if Γ = V −1(0),

because in this case V (φ(t, x0)) → 0 no longer implies φ(t, x0) → V −1(0). For instance,

suppose that for a second order system, φ(t, x0) = [t 1]⊤ and that V (x1, x2) = x22/(1+x21).

Then, V (φ(t, x0)) → 0 but it is not true that φ(t, x0) → V −1(0).

We now give sufficient conditions for (5.1) to be Γ-detectable that extend the results

in Propositions 3.3.2 and 3.3.5. Recall the definition of the set S in (3.5)-(3.6), repeated

here for convenience:

S = x ∈ X : LjfLτV (x) = 0, for all τ ∈ D, and all 0 ≤ j < r, where

D = spanadkf gi : 0 ≤ k ≤ n− 1, 1 ≤ i ≤ m.

Let

S ′ = x ∈ X : Lmf h(x) = 0, 0 ≤ m ≤ r + n− 2.

Notice that the definition of S ′, unlike that of S, does not directly involve the storage

function (but recall that h⊤ = LgV , so it does indirectly depend on V ). The next result

clarifies the relationship between S and S ′.

Lemma 5.1.5. Given any subset X ⊂ X , S ′ ∩ L+(X) = S ∩ L+(X).

This result is interesting in its own right because it implies that the conditions in

Propositions 3.3.2 and 3.3.5 can be equivalently stated as S ′∩L+(X ) ⊂ Γ. This condition

can be checked without directly knowing the storage function.


Proof. We show that (S ′ ∩ L+(X)) ⊂ (S ∩ L+(X)). Let x be an arbitrary point in

S ′ ∩ L+(X). Since x is a positive limit point of an open-loop trajectory of (5.1), and

since LfV ≤ 0, then V (φ(t, x)) is constant and hence

dV (φ(t, x))

dt= LfV (φ(t, x)) ≡ 0.

The identity LfV (φ(t, x)) ≡ 0 and the fact that LfV ≤ 0 imply that LfV (φ(t, x)) is

maximal. Therefore, dLfV (φ(t, x)) ≡ 0, yielding LgiLfV (φ(t, x)) ≡ 0. This and the fact

that x ∈ S ′ give

L[f,gi]V (x) = LfLgiV (x)− LgiLfV (x)

= LfLgiV (x) = Lfh(x) = 0.

Next, notice that since LgiLfV (φ(t, x)) ≡ 0, we have

0 ≡ dm

dtmLgiLfV (φ(t, x)) = Lm

f LgiLfV (φ(t, x)), 0 ≤ m < r.

Thus, for 0 ≤ m < r,

Lmf L[f,gi]V (x) = Lm+1

f LgiV (x)− Lmf LgiLfV (x)

= Lm+1f hi(x) = 0.

A simple extension of this argument leads to

Lmf LτV (x) = 0, for all τ ∈ D, 0 ≤ m < r,

and thus x ∈ S ∩ L+(X). The proof that S ∩ L+(X) ⊂ S ′ ∩ L+(X) is almost identical

and is therefore omitted.

Proposition 5.1.6. Suppose that all open-loop trajectories that originate and remain

on S ′ are bounded and that the open-loop system in (5.1) is locally uniformly bounded

near Γ. If

S ′ ∩ J+(S ′, S ′) ⊂ Γ, (5.2)

then system (5.1) is Γ-detectable. Moreover, if Γ = V −1(0), then condition (5.2) may be

replaced by the following one:

S ′ ∩ L+(S ′) ⊂ V −1(0). (5.3)


Proof. By definition of Γ-detectability, we need to show that Γ is globally asymptotically

stable relative to O. To this end, it is sufficient to show that J+(O,O) ⊂ Γ. For,

L+(O) ⊂ J+(O,O) ⊂ Γ implies that Γ is a global attractor. Moreover, J+(O,O) ⊂ Γ

implies, by Proposition 2.3.5, that Γ is a uniform semi-attractor relative to O and so, by

Lemma 2.2.9, it is stable.

Since O ⊂ h−1(0) is open-loop invariant, we have h(φ(t, x)) ≡ 0 for all x ∈ O, and

thus also Lmf h(φ(t, x)) ≡ 0, for m = 0, 1, . . ., showing that O ⊂ S ′.

It can be shown (see the proof of Proposition 3.4 in [14], which is Proposition 3.3.2

in this chapter) that L+(O) ⊂ S, and so L+(O) ⊂ S ∩ L+(S ′). By Lemma 5.1.5,

L+(O) ⊂ S ∩ L+(S ′) = S ′ ∩ L+(S ′). Using condition (5.2) or, when Γ = V −1(0),

condition (5.3), we obtain

L+(O) ⊂ S ′ ∩ L+(S ′) ⊂ Γ.

Since all open-loop trajectories contained in S ′, and hence in O, are bounded, the above

inclusion implies that Γ is a global attractor relative to O. Let p ∈ O be arbitrary. We

next show that J+(p,O) is compact. Let ω ∈ L+(p) ⊂ Γ. By local uniform bound-

edness of the open-loop system near Γ, there exist two positive scalars λ and m such

that φ(R+, Bλ(ω)) ⊂ Bm(ω). By definition of prolongational limit set, for any δ > 0,

J+(ω,O) ⊂ cl (φ(R+, Bδ(ω))). Taking δ = λ, we have that J+(ω,O) ⊂ cl(Bm(ω)). Thus,

J+(ω,O) is a compact set. By Proposition 2.3.4, J+(p,O) ⊂ J+(ω,O), and so J+(p,O)

is a compact set as well.

We claim that, for all p ∈ O, J+(p,O) ⊂ V −1(0). Suppose that the claim is false.

Then, by the compactness of J+(p,O), there exists y ∈ J+(p,O) such that V (y) > 0. Put

µ = V (y). Since y ∈ J+(p,O), there exist two sequences xk ⊂ O and tk such that

xk → p, tk → +∞, and φ(tk, xk) → y. By the continuity of V , one can find K > 0 such

that, for all k > K, V (φ(tk, xk)) > 3µ/4. Since p ∈ O and Γ is a global attractor relative

to O, φ(t, p) → Γ ⊂ V −1(0). Since all solutions on O are bounded and V is continuous,

V (φ(t, p)) → 0 and hence there exists T > 0 such that, for all t ≥ T , V (φ(t, p)) < µ/4.


Using again the continuity of V , there exists ε > 0 such that, for all x ∈ Bε(φ(T, p)),

V (x) < µ/2. Now, by continuous dependence on initial conditions, there exists δ > 0

such that, for all x ∈ Bδ(p), ‖φ(t, x)−φ(t, p)‖ < ε for all t ∈ [0, T ]. Since, for sufficiently

large k > K, xk ∈ Bδ(p) and tk > T , we have V (φ(tk, xk)) > 3µ/4 > µ/2 > V (φ(T, xk))

which contradicts the fact that LfV ≤ 0, proving the claim.

So far we have established that J+(O,O) ⊂ V −1(0). If Γ = V −1(0), we are done. If

Γ ( V −1(0), we reach the desired conclusion by means of condition (5.2) as follows. Note

that J+(O,O) ⊂ V −1(0) ⊂ O ⊂ S ′, and, further, J+(O,O) ⊂ J+(S ′, S ′). In conclusion,

J+(O,O) ⊂ S ′ ∩ J+(S ′, S ′) ⊂ Γ, as required.

Remark 5.1.7. The natural way to check Γ-detectability is to compute the set O in

Definition 5.1.1, and then assess the asymptotic stability of Γ relative to O. Should the

computation of the set O be too difficult, Proposition 5.1.6 above provides an alterna-

tive, but conservative, criterion for Γ-detectability that may prove useful in some cases.

Example 5.1.8 illustrates this result. It is important to notice that condition (5.2) may

be hard to check in practice because it involves the computation of the prolongational

limit set J+(S ′, S ′). The conditions used in Propositions 3.3.2 and 3.3.5 suffer from the

same limitation because they too involve the computation of limit sets.

Propositions 3.3.2 and 3.3.5 are corollaries of Proposition 5.1.6 above. As a matter

of fact, Proposition 5.1.6 relaxes the sufficient conditions for detectability found in [14]

and [81]. To see this fact, note that, when V is proper and Γ = V −1(0), all trajectories

of the open-loop system are bounded and system (5.1) is locally uniformly bounded

with respect to Γ. Therefore, in this setting Proposition 5.1.6 states that a sufficient

condition for Γ-detectability is the inclusion S ′ ∩ L+(S ′) ⊂ V −1(0). Since S ′ ∩ L+(S ′) =

S∩L+(S ′) ⊂ S∩L+(X ), this condition is weaker than the condition S∩L+(X ) ⊂ V −1(0)

used in Propositions 3.3.2 and 3.3.5.


Example 5.1.8. Consider the control system on X = R5

x1 = −x1 − x1x4

x2 = −x2 + x1 − x24

x3 = x25 + u1

x4 = x21 + e−1/x24u2

x5 = −x3x5

(we set e−1/x24 |x4=0 := 0) with output y = col

(

x3, x4e−1/x2

4

)

. This system is passive with

storage V (x) = 1/2(x21 + x23 + x24 + x25). The goal set is Γ = 0. It is not hard to see

that S ′ = x : x3 = x4 = x5 = 0. Let (x1(t), x2(t), 0, 0, 0) be any solution of the open-

loop system lying in S ′ for all time. Since x1(t) = −x1(t) and x2(t) = −x2(t) + x1(t),

any such solution is bounded. Next, we check condition (5.2). Pick any x0 ∈ S ′, i.e.,

x0 = (x10, x20, 0, 0, 0), and consider the corresponding open-loop solution x(t). If x10 6= 0,

then x4(t) → ∞, and so J+(x0, S′) = ∅. On the other hand, if x10 = 0, then we claim that

J+(x0, S′) = 0. For, the equilibrium x = 0 is globally asymptotically stable relative to

the set x1 = x3 = x4 = x5 = 0, and hence a uniform attractor relative to the same set.

By Proposition V.1.2 in [11], J+(x0, S′) = 0. In conclusion, S ′ ∩ J+(S ′, S ′) = 0, and

the system is Γ-detectable.

In this example, Γ-detectability can be checked without using Proposition 5.1.6, since

it is easily seen that the maximal open-loop invariant subset of h−1(0) is O = x1 = x3 =

x4 = x5 = 0. As noted above, 0 is globally asymptotically stable relative to this set.

5.2 Solution of PBSSP

In this section we solve the set stabilization problem by presenting conditions that guar-

antee that a passivity-based controller of the form (3.3) makes Γ stable, attractive, or

asymptotically stable for the closed-loop system. All results are straightforward conse-

quences of the reduction principles presented in Section 4.4, and they rely on the next


fundamental observation.

Proposition 5.2.1. Consider the passive system (5.1) with a passivity-based feedback

of the form (3.3), and the set O in Definition 5.1.1. Then, the set O is locally stable near

Γ for the closed-loop system.

Proof. Given arbitrary x in Γ and c > 0, we need to show that

(∀ε > 0)(∃δ > 0) s.t. (∀x0 ∈ Bδ(Γ))(∀t ≥ 0) φu([0, t], x0) ⊂ Bc(x) =⇒ φu([0, t], x0) ⊂ Bε(O).

Let U = cl(Bc(x)) and pick any ε > 0. Define

v := minV (x) : x ∈ U ∩ x : ‖x‖V −1(0) = ε,

and notice that v > 0 because U ∩ x : ‖x‖V −1(0) = ε is compact and disjoint from

V −1(0). Using v, we define

δ := min‖x‖V −1(0) : x ∈ U ∩ V −1(v).

Since U ∩ V −1(v) is compact and disjoint from V −1(0), then δ > 0. Note that δ ≤ ε for,

if not, then we would have that

(∀x ∈ U ∩ V −1(v)) ‖x‖V −1(0) > ε,

and this would contradict the definition of v. By the definitions of v and δ it follows that

U ∩Bδ(V−1(0)) ⊂ U ∩ V −1([0, v]) ⊂ U ∩ Bε(V

−1(0)).

Since Γ ⊂ V −1(0) ⊂ O, for any x ∈ X we have ‖x‖O ≤ ‖x‖V −1(0) ≤ ‖x‖Γ, and so

Bδ(Γ) ⊂ Bδ(V−1(0)) and Bε(V

−1(0)) ⊂ Bε(O). By inequality (3.4) we have that all level

sets of V are positively invariant for the closed-loop system. Putting everything together

we have

x0 ∈ U ∩ Bδ(Γ) =⇒ x0 ∈ U ∩ Bδ(V−1(0)) =⇒ x0 ∈ U ∩ V −1([0, v])

=⇒ φ(R+, x0) ⊂ V −1([0, v]).


From the above, for any t ≥ 0, the condition φ([0, t], x0) ⊂ U implies

φ([0, t], x0) ⊂ U ∩ V −1([0, v]) ⊂ Bε(V−1(0)) ⊂ Bε(O),

and thus O is locally stable near Γ for the closed-loop system.

Theorem 5.2.2 (Solution of PBSSP). Consider system (5.1) with a passivity-based

feedback of the form (3.3). If Γ is compact, then

• Γ is asymptotically stable for the closed-loop system if, and only if, system (5.1) is

locally Γ-detectable,

• if all trajectories of the closed-loop system are bounded, then Γ is globally asymptot-

ically stable for the closed-loop system if, and only if, system (5.1) is Γ-detectable.

If Γ is unbounded and the closed-loop system is locally uniformly bounded near Γ, then

• Γ is asymptotically stable for the closed-loop system if, and only if, system (5.1) is

locally Γ-detectable.

• if all trajectories of the closed-loop system are bounded, then Γ is globally asymptot-

ically stable for the closed-loop system if, and only if, system (5.1) is Γ-detectable.

Remark 5.2.3. The theorem above enables a procedure, presented in the next chapter,

to synthesize feedbacks that stabilize the goal set Γ.

Proof. The sufficiency part of the theorem follows from the following considerations. By

Proposition 5.2.1, O is locally stable near Γ. If Γ is compact, by Theorem 4.4.1 local

Γ-detectability implies stability of Γ. The stability of Γ and its compactness in turn

imply that all closed-loop trajectories in some neighbourhood of Γ are bounded. Since

all bounded trajectories asymptotically approach O, O is locally attractive near Γ. If

all trajectories of the closed-loop system are bounded, then O is globally attractive.

Theorem 4.4.8 yields the required result.


Now suppose that Γ is unbounded. By local uniform boundedness near Γ we have

that all closed-loop solutions in some neighbourhood of Γ are bounded and hence O is

locally attractive near Γ. Once again, if all closed-loop trajectories are bounded, then O

is globally attractive. The required result now follows from Theorem 4.4.8.

The various necessity statements follow from the following basic observation. Any

passivity-based feedback of the form (3.3) makes O an invariant set for the closed-loop

system (see Section 3.2). Therefore, if Γ is [globally] asymptotically stable for the closed-

loop system, necessarily Γ is [globally] asymptotically stable relative to O for the closed-

loop system. In other words, (5.1) is necessarily locally Γ-detectable [Γ-detectable].

The following result gives conditions that are alternatives to the Γ-detectability as-

sumption.

Proposition 5.2.4. Theorem 5.2.2 still holds if the local Γ-detectability [Γ-detectability]

assumption is replaced by the following condition:

(i’) Γ is stable relative to V −1(0) and Γ is [globally] attractive relative to O.

The proof of this proposition relies on essentially identical arguments as those used to

prove the reduction principles in Theorems 4.4.6 and 4.4.8 and therefore omitted. If the

sufficient conditions for Γ-detectability in Proposition 5.1.6 fail, rather than checking for

Γ-detectability one may find it easier to check condition (i’) in Proposition 5.2.4. This

is because verifying whether Γ is stable relative to V −1(0) does not require finding the

maximal open-loop invariant subset O of h−1(0); moreover, checking that Γ is attractive

relative to O amounts to checking the familiar condition

h(φ(t, x0)) ≡ 0 =⇒ φ(t, x0) → Γ as t→ +∞.

Note that, in the framework of [14] and [81], the requirement that Γ be stable relative to

V −1(0) is trivially satisfied because in these references it is assumed that Γ = V −1(0).


5.3 Discussion

Theorems 3.3.3 and 3.3.6, dealing with the special case when Γ = V −1(0) (= 0) and

Γ is compact, become corollaries of our main result, Theorem 5.2.2. We have already

shown (see Lemmas 5.1.3 and 5.1.4) that in this special case the properties of zero-

state detectability (when Γ = 0), and V -detectability coincide with our notion of

Γ-detectability. Therefore, Theorems 3.3.3 and 3.3.6 state that local Γ-detectability is a

sufficient condition for the asymptotic stabilization of the origin using a passivity-based

feedback. We have shown that actually this condition is also necessary. When the storage

function is proper, Theorems 3.3.3 and 3.3.6 assert that Γ-detectability is a sufficient

condition for the global stabilization of Γ by means of a passivity-based feedback of the

form (3.3). If V is proper, then all trajectories of the closed-loop system are bounded,

and so Theorem 5.2.2 gives the same result. Moreover, once again, the theorem states

that Γ-detectability is necessary for the stabilizability of Γ by means of a passivity-based

feedback.

The theory in [14] and [81] does not handle the special case when Γ is compact and

Γ ( V −1(0), while our theory does. This case includes the important situation when

one wants to stabilize an equilibrium (Γ = 0) but the storage is only positive semi-

definite. Based on the results in [14] and [81], it may be tempting to conjecture that

Theorems 3.3.3 and 3.3.6 still hold if one employs the following notion of detectability:

(∀x0 ∈ N (Γ)) h(φ(t, x0)) = 0 for all t ∈ R =⇒ φ(t, x0) → Γ, (5.4)

which corresponds to requiring that on the set O, in Definition 5.1.1, Γ is an attractor

for the open-loop system (i.e., Γ is attractive relative to O). This conjecture is false:

we have shown that (local) Γ-detectability (i.e., the asymptotic stability of Γ relative to

O for the open-loop system) is a necessary condition for the stabilization of Γ. Even

if one relaxes the asymptotic stability requirement and just asks for attractivity of Γ,

the above conjecture is still false. As a matter of fact, Theorem 4.4.6 suggests that


even in this case (local) Γ-detectability is a key property. A counter-example illustrating

this loss of attractivity is the pendulum. The upright equilibrium is globally attractive,

but unstable, relative to the homoclinic orbit of the pendulum. Despite the fact that

a passivity-based feedback can be used to asymptotically stabilize the homoclinic orbit

(see, e.g., [30], [71], and the related work in [26]), the upright equilibrium is unstable for

the closed-loop system. This well-known phenomenon finds explanation in the theory

developed in Chapter 4: the cause of the problem is the instability of the upright equilib-

rium relative to the homoclinic orbit. We next present another explicit counter-example

illustrating our point.

Example 5.3.1. Consider the control system with state (x1, x2, x3),

r = −r(r − 1)

θ = sin2(θ/2) + x3

x3 = u

y = x33,

(5.5)

where (r, θ) ∈ (0,+∞)× S1 represent polar coordinates for (x1, x2). The control system

is passive with storage V (x) = x43/4. Let Γ be the equilibrium point (x1, x2, x3) : x1 =

1, x2 = x3 = 0 and note that O = (x1, x2, x3) : x3 = 0. On O, the open-loop dynamics

read as

r = −r(r − 1)

θ = sin2(θ/2),

(5.6)

and it is easily seen that the equilibrium Γ attracts every point in O except the origin.

Hence, Γ is attractive relative to O, but unstable (indeed, the unit circle is a homoclinic

orbit of the equilibrium); see Figure 5.1. Therefore, condition (5.4) holds but the system

is not locally Γ-detectable. Consider the passivity-based feedback u = −y, which ren-

ders O globally asymptotically stable. Now for any initial condition off of O such that


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

1

−1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

x1

x1

x2

x2

x3Γ

ΓO

Figure 5.1: On the left-hand side, phase portrait on O for the open-loop system (5.6).

On the right-hand side, closed-loop system (5.5) with feedback u = −y. Note that Γ is

not attractive.

(x1(0), x2(0)) 6= (0, 0), x3(0) > 0, the corresponding trajectory is bounded, but its posi-

tive limit set is the unit circle on O, and therefore it is not a subset of Γ; see Figure 5.1.

In conclusion, Γ is not attractive for the closed-loop system (and neither is it stable).

This example illustrates the fact that, when Γ ( V −1(0) is compact, simply requiring

condition (5.4) in place of Γ-detectability may not be enough for attractivity of Γ.

In the light of Theorem 5.2.2 and the example above, it is clear that the addition of

the stability requirement on Γ, relative to O, is a crucial enhancement to the notions of

detectability in [14] and [81].

5.4 Path following for the point-mass system

In this section we provide a passivity-based set stabilizing solution to the circular motion

problem presented in Section 1.1 for the point-mass system

Mx = f. (5.7)


Define the state of this system as X ∈ R6 where

X = [x x]⊤ = [x1 x2 x3 x4 x5 x6]⊤,

so that system (5.7) in state space form can be written as

X = AX +Bu (5.8)

where u = (u1, u2, u3) =1Mf and

A =

03×3 I3×3

03×3 03×3

, B =

03×3

I3×3

.

We aim to design a feedback u(X) solving the following problem.

Circular path following for the point-mass system.

1. Make the point-mass approach and follow a circular path with radius r, as in

Figure 5.2. The circle lies on a plane perpendicular to a vector c. The orientation

of the path is counter-clockwise relative to c. We will assume without loss of

generality that the circular path is centred at the origin.

2. On the circle, make the point move with a constant forward speed, v, counter-

clockwise relative to c.

ab

c

x1

x2

x3

Figure 5.2: Circular path


As pointed out in Section 1.1, this problem can be posed as the asymptotic stabiliza-

tion of the goal set

Γ = X = (x, x) ∈ R6 : c · x = 0, ‖x‖ = r, x = v1(x) =v

rc× x. (5.9)

Notice that when the point-mass moves along the circle, i.e., when the state is on the set

X = (x, x) ∈ R6 : c · x = 0, ‖x‖ = r, v1(x) restricts to the desired velocity: tangent to

the circle in the counter-clockwise direction with linear speed v.

To design a passivity-based controller stabilizing the goal set Γ, one would start by

finding a candidate storage function V (X). A straightforward choice would be

V (X) =1

2(c · x)2 + 1

4

(

‖x‖2 − r2)2

+1

2‖x− v1(x)‖2,

so that Γ = V −1(0). By taking the derivative of V along the dynamics (5.8) we get

V (X) = (c · x)(c · x) + (‖x‖2 − r2)(x · x) + (x− v1(x))⊤(u− v1(x)),

where v1(x) is the Lie derivative of v1(x) along the dynamics (5.8). From this we conclude

that

LfV = (c · x)(c · x) + (‖x‖2 − r2)(x · x)− (x− v1(x))⊤v1(x)

LgV = (x− v1(x))⊤

where f = AX and g = B. From the above it is clear that (5.8) is not passive with

storage (5.4) since the condition LfV ≤ 0 does not hold.

In this situation one might still try to render the system passive by finding a feedback

transformation u = up(X)+u, where up(X) is a passifying feedback such that Lf+gupV ≤

0. If such a feedback were found, the system with input u would be passive. However, this

is not possible here because it is not possible to find up(X) such that LfV + LgupV ≤ 0.

Consider instead the following candidate storage function

V (X) =1

2‖x− v(x)‖2 (5.10)

where

v(x) = v1(x) + v2(x) (5.11)


with v1(x) =vrc× x and v2(x) is to be defined later. By taking the derivative of (5.10)

along the dynamics (5.8) we get V (X) = (x− v(x))⊤(u− ˙v(x)), and so

LfV = −(x− v(x))⊤ ˙v(x)

LgV = (x− v(x))⊤.

By setting

u = up(X) + u, up(X) = ˙v(x) = dv(x)x,

we get Lf+gupV = 0. Thus, system

X = AX +Bup(X) +Bu (5.12)

with input u and output

y = h(X) = LgV⊤ = x− v(x) (5.13)

is passive. Notice that h−1(0) = V −1(0), thus the maximal subset O of h−1(0) with the

property of being invariant for the open loop system X = AX +Bup(X) is

O = V −1(0) = X ∈ R6 : x = v(x). (5.14)

We set u to be the passivity-based feedback

u = −ky = −k(x − v(x)),

with k > 0. From Theorem 5.2.2, this feedback asymptotically stabilizes Γ if v2(x)

in (5.11) is chosen so that

(1) Γ ⊂ V −1(0),

(2) Γ is asymptotically stable relative to O = (x, x) : x = v1(x) + v2(x), i.e., sys-

tem (5.12) is locally Γ-detectable.

The set Γ can be expressed as Γ = (x, x) ∈ O : v2(x) = 0, c · x = 0, ‖x‖ = r and the

system dynamics on O read as

x = (v/r)c× x+ v2(x). (5.15)


Therefore, letting

Γ′ = x : c · x = 0, ‖x‖ = r,

requirement (1) above is met provided that v2(x) = 0 on Γ′, and requirement (2) is met

provided that Γ′ is asymptotically stable for (5.15). We thus have the following reduced

control problem: Design v2 to stabilize the set Γ′ for (5.15), and such that v2(x) = 0 on

Γ′. In terms of the orthonormal body coordinate system shown in Figure 5.2, Γ′ can be

written as

Γ′ = x : c · x = 0, (x · a)2 + (x · b)2 = r2.

Consider the following candidate Lyapunov function

W (x) =1

4

[

(x · a)2 + (x · b)2 − r2]2

+1

2(c · x)2,

so that Γ′ = W−1(0). Let v2(x) = au′1 + bu′2 + cu′3 where u′1, u′2, u

′3 ∈ R, and so

x =v

r(c× x) + au′1 + bu′2 + cu′3. (5.16)

We have

W =[

(x · a)2 + (x · b)2 − r2] [

(x · a)a⊤ + (x · b)b⊤]

x+ (c · x)c⊤x

=[

(x · a)2 + (x · b)2 − r2] [

(x · a)a⊤ + (x · b)b⊤] v

r(c× x) + (c · x)c⊤ v

r(c× x)

+[

(x · a)2 + (x · b)2 − r2] [

(x · a)a⊤ + (x · b)b⊤]

(au′1 + bu′2 + cu′3)

+ (c · x)c⊤(au′1 + bu′2 + cu′3)

Since (c · x)c⊤ vr(c× x) = 0 and (c× x) = −(x · b)a + (x · a)b we get

W =[

(x · a)2 + (x · b)2 − r2] [

(x · a)u′1 + (x · b)u′2]

+ (c · x)u′3

Selecting

u′1 = −(x · a)[

(x · a)2 + (x · b)2 − r2]

u′2 = −(x · b)[

(x · a)2 + (x · b)2 − r2]

u′3 = −(c · x).


we get W = −[

(x · a)2 + (x · b)2 − r2]2 (

(x · a)2 + (x · b)2)

− (c · x)2. Therefore, Γ′ is

rendered almost globally asymptotically stable for (5.15), with domain of attraction x :

x · a 6= 0 or x · b 6= 0 = R3\(c axis). Moreover, v2(x) = 0 on Γ′. We have thus proved

the following result.

Proposition 5.4.1. The feedback

f =M (dv(x)x− k(x− v(x)))

with k > 0 and

v(x) = (v/r)c× x+ au′1 + bu′2 + cu′3

u′1 = −(x · a)[

(x · a)2 + (x · b)2 − r2]

u′2 = −(x · b)[

(x · a)2 + (x · b)2 − r2]

u′3 = −(c · x)

asymptotically stabilizes the set Γ in (5.9) and solves the circular path-following problem

for the point-mass system Mx = f .

−2

−1

0

1

2 −1

−0.5

0

0.5

1

1.5

2

−2

0

2

x1x2

x3

Figure 5.3: Set stabilization for point-mass system


Figure 5.3 shows simulation results for control (5.4.1) with the following parameters.

M = r = v = k = 1 and [a b c] =

1/√2 1/

√6 1/

√3

−1/√2 1/

√6 1/

√3

0 −2/√6 1/

√3

.

Remark 5.4.2. Another approach to solving the path following problem for the point-

mass system would be to express the goal set Γ, in (5.9), as the zero level set of some

output function y = h(X), and apply input-output feedback linearization. From (5.9),

we see that y ∈ R5. Since the control force f ∈ R3, the system has three inputs and five

outputs, and it is therefore impossible to solve the problem using input-output feedback

linearization.

Chapter 6

Passivity-Based Set Stabilization III:

Control Design

In this chapter we leverage the theory in Chapter 5 and present a control design pro-

cedure for passivity-based stabilization of closed sets. The procedure is an adaptation

to the setting of passive systems of the reduction-based design philosophy presented in

Section 4.7. The idea behind the procedure is to use part of the control freedom to

enforce detectability, while the remaining part is used for passivity-based stabilization.

Whenever feasible, this methodology has the advantage of simplifying the control design,

because stabilizing the goal set Γ amounts to designing a stabilizer for a system of smaller

dimension, so the dimensionality of the problem is effectively reduced. The control design

procedure is presented in Section 6.1. To illustrate the procedure, in Sections 6.2 to 6.4

we present three case studies concerning the path following problem for one kinematic

unicycle, maneuvering of one unicycle and coordination for two unicycles. Our examples

have independent interest, but their primary objective is to elucidate different aspects

of the theory in Chapter 5, and demonstrate the design flexibility gained by eliminating

the requirement that the goal set coincides with the zero level set of the storage func-

tion. The design procedure presented here is used in Chapter 7 where a more substantial

84

Chapter 6. Passivity-Based Set Stabilization III: Control Design 85

problem is considered. In that chapter we solve a circular formation control problem for

n-unicycles.

6.1 Set stabilization procedure


x = f(x) +m∑

i=1

gi(x)ui := f(x) + g(x)u

y = h(x)

(6.1)

with state space X ⊂ Rn, set of input values U = Rm and set of output values Y = Rm.

We assume that X is either an open subset of Rn or a smooth submanifold therein.

Further, f and gi, i = 1, . . . , m, are smooth vector fields on X , and h is a smooth

mapping.

Let Γ be a closed goal set that is controlled invariant (see Definition 3.1.1), for (6.1).

1. Candidate storage function and feedback transformation.

(a) Find a candidate C1 storage function V : X → R+ such that Γ ⊂ V −1(0) and

LfV (x) ≤ 0 for all x ∈ X .

(b) Find, if possible, a locally Lipschitz matrix-valued function β1(x) : X → Rm×k,

for some k ∈ 1, . . . , m− 1, such that β1(x) has full rank k and LgV (x)β1(x) =

01×k for all x ∈ X . Therefore, the columns of β1(x) are in the kernel of LgV (x).

(c) Let β2(x) : X → Rm×m−k be any locally Lipschitz function such that the square

matrix [β1(x) β2(x)] is nonsingular for all x ∈ X . Define the feedback transfor-

mation

u = β1(x)u+ β2(x)u, (6.2)

where u ∈ Rk and u ∈ Rm−k are new control inputs. Define an output function

h : X → Rm−k as h(x) := Lgβ2V (x)⊤.


2. Γ-detectability enforcement. Find, if possible, a feedback u(x) such that Γ is (globally)

asymptotically stable relative to O for the system

x = [f(x) + g(x)β1(x)u(x)]|O,

where O is the maximal subset of h−1(0) invariant under the vector field f + gβ1u.

3. Passivity-based stabilization. Pick any PBF u(x) with respect to h(x), and let u(x) =

β1(x)u(x) + β2(x)u(x), where u(x) is the feedback chosen in step 2.

Remark 6.1.1. (a) The procedure above is most effective when the system has more

than one input. For single-input systems the feedback transformation becomes trivial:

u = u. Therefore, in this case, there is no hope to enforce Γ-detectability if the open-

loop system is not already Γ-detectable.

(b) The inputs u and u after feedback transformation (6.2) represent control directions

tangential and transversal to V −1(0), so that when u = 0 and u(x) is any smooth

feedback, the set V −1(0) is invariant. Moreover, since for any smooth u(x) it holds

that Lf+gβ1uV (x) = LfV (x) + LgV (x)β1(x)u(x) = LfV (x) ≤ 0, the system with

input u and output h(x) = Lgβ2V (x)⊤ is passive. The idea then is to use u(x) to

enforce Γ-detectability (step 2), while u is chosen to be any passivity-based feedback

(step 3).

(c) The procedure is greatly simplified when f = 0, because in this case any function

V ≥ 0 gives LfV = 0, and hence it qualifies as a viable storage function.

(d) In step 1a, it may be possible to ensure that LfV ≤ 0 through a preliminary feedback

up(x), so that Lf+gupV ≤ 0. In this case, we would let u = up(x) + u, define

f := f + gup, and continue the procedure with f and u in place of f and u. The

preliminary controller up(x) has the role of passifying feedback.


(e) Suppose that the set Γ is expressed as the level set of C1 functions, Γ = x ∈ X :

ψ1(x) = 0, . . . , ψl(x) = 0. In this case, the functions ψi(x) can be used to produce

guesses for the storage V by setting, for instance, V = ψ2i (x) or V = ψ2

i (x) + ψ2j (x),

and so on. Since we only require Γ to be a subset of V −1(0), there is some freedom

in which of the functions to use, and how to combine them. The storage functions

in the three case studies presented in this chapter are chosen using this method.

(f) A feature of the set stabilization procedure is that, whenever it is feasible, it allows

one to reduce the control design to the design of a controller u(x) that asymptotically

stabilizes Γ for the system x = [f(x) + g(x)β1(x)u(x)]∣

∣

O, with state space O. Typi-

cally, O is a submanifold of the state space, and hence the restriction of f + gβ1u to

O is a system of dimension smaller than the original system (6.1).

(g) A challenge in applying the previous passivity-based set stabilization procedure is

the problem of finding the set O (the maximal subset of h−1(0) invariant under the

vector field f+gβ1u). In the language of viability theory [7], O is the viability kernel

of the set h−1(0). An analogous problem is encountered in the LaSalle invariance

principle [49], where one seeks to find the largest invariant subset of a given compact

set. Although finding such maximal invariant subsets is generally a very difficult

problem, the approach commonly used of imposing that repeated Lie derivatives of

h along the vector field f be zero works well in practice. This is the approach to

finding O used in all examples presented in this thesis.

(h) As shown in Proposition 6.1.2, in order to be able to stabilize an unbounded Γ using

the previous procedure, one has to verify that the closed-loop system is LUB near Γ.

Showing that this property applies with a certain choice of feedbacks u(x) and u(x)

can be challenging, as will be seen in case study 3 and in Chapter 7.

(i) As shown in Proposition 6.1.4 below, the outcome of the control design procedure is

independent of the choice of β2(x) in step 1c.


Proposition 6.1.2. The feedback u(x) designed according to the procedure above has

the following properties:

(a) If Γ is compact, then u(x) asymptotically stabilizes it.

(b) If Γ is closed and unbounded, then u(x) asymptotically stabilizes it provided that

the closed-loop system is LUB near Γ.

(c) In both cases above, if all trajectories of the closed-loop system are bounded, and

the Γ-detectability property enforced in step 2 of the procedure is global, then the

stabilization of Γ is global as well.

Remark 6.1.3. When Γ is unbounded, a suitable choice of PBF u(x) may help achieve

the LUB property. This fact is illustrated in case study 3 below.

Proof. Let u = β1(x)u(x) + β2(x)u, where u(x) is as in step 2 of the procedure, and

consider the system

x = [f(x) + g(x)β1(x)u(x)] + g(x)β2(x)u

y = Lgβ2V (x)⊤.

Since Lf+gβ1uV (x) = LfV (x) + LgV (x)β1(x)u(x) = LfV (x) ≤ 0, the system above is

passive. By the construction in step 2, Γ is [globally] asymptotically stable relative to O,

and hence the system above is locally Γ-detectable [Γ-detectable]. Now the proposition

follows directly from Theorem 5.2.2.

Proposition 6.1.4. Steps 2 and 3 of the set stabilization procedure are independent of

the choice of β2(x).

Proof. We need to show that the set O is independent of the choice of β2(x). Let

β1(x), β2(x) be as in step 1 of the procedure and let β2(x) be another locally Lipschitz

function X → Rm×m−k such that [β1(x) β2(x)] is nonsingular. Denote h(x) = Lgβ2V (x)⊤


the corresponding output. Since [β1 β2] is nonsingular, there exist continuous matrix-

valued functions K1(x) and K2(x) such that β2(x) = β1(x)K1(x) + β2(x)K2(x) and

therefore

h(x)⊤ = Lgβ2V (x) = Lgβ1

V (x)K1(x) + Lgβ2V (x)K2(x) = Lgβ2

V (x)K2(x) = h(x)⊤K2(x).

Since the matrix [β1 β2] is nonsingular, the matrix-valued functionK2 : X → Rm−k×(m−k)

must be nonsingular. Therefore, the sets h−1(0) and h−1(0) coincide, proving that the

set O is the same for both outputs h(x) and h(x).

The rest of this chapter is dedicated to the application of the procedure above to three

case studies illustrating different aspects of the approach :

CS1. Path following control design for the kinematic unicycle and strictly convex paths.

In this case, we will have Γ = V −1(0) and Γ compact.

CS2. Stabilizing the kinematic unicycle to the unit circle with a constant heading re-

quirement on the circle. Here, Γ ( V −1(0) and Γ is compact.

CS3. Coordination of two unicycles: make two unicycles meet at a fixed distance facing

each other. Here, Γ ( V −1(0) and Γ is unbounded.

6.2 Case study 1: path following for the kinematic

unicycle

We consider the path following problem for the kinematic unicycle model with state

(x1, x2, x3) ∈ R2 × S1,

x1 = u1 cosx3

x2 = u1 sin x3

x3 = u2,

(6.3)


and a smooth regular path C ⊂ R2 which is closed and does not have self-intersections

(i.e., it is a Jordan curve). The path following problem for kinematic unicycles and, more

generally, for wheeled vehicles was the subject of considerable research in the 1990s. The

seminal work in [75] (see also the review paper [22]) proposed a smooth time-varying con-

trol law based on the conversion of the path following problem to equilibrium stabilization

by using Frenet-Serret frames moving along the path. The idea of using Frenet-Serret

frames for path following is also found in [1], where a virtual target is used to make a uni-

cycle converge to the path. Virtual targets are further explored in the recent work [51]. A

global discontinuous path following controller for a circle is proposed in the work of [15].

To the best of our knowledge, no global solution to the unicycle path following prob-

lem has been found by means of a smooth, static, and time-invariant feedback. In this

section, we present such a solution for the class of strictly convex paths, i.e., paths with

strictly positive signed curvature. The next lemma provides a useful characterization of

strictly convex paths.

Lemma 6.2.1. If C is a smooth Jordan curve, then the following statements are equiv-

alent:

i. C is strictly convex.

ii. There exists a regular parameterization σ : S1 → R2 of C such that, for each θ ∈ S1,

the angle of the tangent vector σ′(θ) is precisely θ mod 2π. In other words, σ′(θ) =

‖σ′(θ)‖ col(cos θ, sin θ).

Proof. Let σ : R → R2 be a unit speed parameterization of C, and for each t denote

by ϕ(t) the angle of the vector σ′(t) modulo 2π. If L is the length of C, then σ is L-

periodic, and we change the domain of σ from R to R mod L, so that σ maps R mod L

diffeomorphically onto C.

(i)⇒ (ii). The curvature of C at a point σ(t) is ϕ′(t), and it is a smooth function. Since

ϕ′(t) > 0 for all t ∈ R, the function t 7→ ϕ(t) is invertible, its inverse ϕ−1 : θ ∈ S1 → t ∈ R


mod L is smooth, and the derivative of ϕ−1(·) is positive. The function σ(θ) := σϕ−1(θ)

has the required properties: its derivative σ′(θ) = σ′(ϕ−1(θ))(ϕ−1)′(θ) is never zero, and

so it is a regular parametrization. Moreover, the angle of σ′(θ) is the same as that of

σ′(ϕ−1(θ)), which is precisely θ.

(ii) ⇒ (i). Let σ : S1 → R2 be a regular parameterization of C such that σ′(θ) =

‖σ′(θ)‖ col(cos(θ), sin(θ)). The signed curvature k(θ) is given by the formula k(θ) =

[σ′1(θ)σ

′′2(θ)− σ′

2(θ)σ′′1 (θ)]/‖σ′(θ)‖3 = 1/‖σ′(θ)‖, which is everywhere positive.

Example 6.2.2. Suppose that C is a circle of radius r centred at the origin, and consider

the regular parameterization σ(θ) = r col(sin θ,− cos θ). The tangent vector at σ(θ) is

σ′(θ) = r col(cos θ, sin θ), whose angle is θ. Next, suppose that C is an ellipse with major

semi-axis a and minor semi-axis b, centred at the origin. The regular parameterization

σ(θ) =

a2 sin θ√a2 sin2 θ+b2 cos2 θ

−b2 cos θ√a2 sin2 θ+b2 cos2 θ

satisfies σ′(θ) = ‖σ′(θ)‖ col(cos θ, sin θ), where ‖σ′(θ)‖ = a2b2/(a2 sin2 θ + b2 cos2 θ)3/2.

We now return to the path following problem for the unicycle. Suppose that C is

a strictly convex curve with parameterization σ : S1 → R2, as in Lemma 6.2.1. We

will design a global path following controller making the unicycle follow the curve in

the counter-clockwise direction. In order to make the unicycle follow C in the clockwise

direction, it suffices to replace θ by −θ in the definition of σ.

If (x1(t), x2(t), x3(t)) is a solution of (6.3), then x3(t) is the angle of the tangent vector

to the curve (x1(t), x2(t)). This fact, and the property, due to strict convexity, that the

angle of σ′(θ) is θ, together imply that solving the path following problem is equivalent

to stabilizing the controlled invariant set

Γ = (x1, x2, x3) ∈ R2 × S1 : x1 = σ1(x3), x2 = σ2(x3). (6.4)


Remark 6.2.3. Note that the set Γ = (x1, x2, x3) : (x1, x2) ∈ C is not controlled

invariant unless u1(t) ≡ 0. For, if (x1, x2) ∈ C and the unicycle’s heading is not tangent

to C, then the unicycle will leave C. The set Γ in (6.4) is the largest controlled invariant

subset of Γ subject to the requirement that u1 is bounded away from zero.

Step 1: Candidate storage function. We make the obvious choice

V (x) =1

2

[

(x1 − σ1(x3))2 + (x2 − σ2(x3))

2]

.

Note that Γ = V −1(0) and Γ is a compact set because x3 ∈ S1, which is compact1. For

the unicycle (6.3), f = col(0, 0, 0) and g = [g1 g2], with g1 = col(cos(x3), sin(x3), 0),

g2 = col(0, 0, 1). Since LfV = 0, V satisfies the requirements of step 1a of the procedure.

Next, we find a feedback transformation of the form (6.2). We have

LgV = [(x1 − σ1) cosx3 + (x2 − σ2) sin x3 − (x1 − σ1)σ′1 − (x2 − σ2)σ

′2],

and since, by strict convexity, σ′(x3) = ‖σ′(x3)‖ col(cosx3, sin x3), setting β1(x) = col(1, 1/‖σ′(x3)‖)

we have LgV (x)β1(x) = 0. Next, we need to pick a vector β2 that is linearly independent

of β1. We choose β2 = col(0, 1). The feedback transformation

u = β1(x)u+ β2(x)u =

1

1/‖σ′(x3)‖

u+

0

1

u

guarantees that, for any smooth u(x), the system with input u and output y = h(x) :=

Lgβ2V (x)⊤,

h(x) := −[x1 − σ1(x3)]σ′1(x3)− [x2 − σ2(x3)]σ

′2(x3) (6.5)

is passive.

Step 2: Γ-detectability enforcement.

Lemma 6.2.4. Let u(x) be any smooth positive feedback bounded away from 0, i.e.,

infx u(x) > 0. Then, the maximal subset O of h−1(0) invariant under f + gβ1u is Γ.

1S1 is a compact topological space, i.e., any open cover of S1 has a finite subcover, [102].


Proof. If u = β1(x)u(x), we have

x1 = u(x) cosx3

x2 = u(x) sin x3

x3 =u(x)

‖σ′(x3)‖.

Using the fact that σ′(θ) = ‖σ′(θ)‖ col(cos θ, sin θ), we have

d

dt

x1 − σ1(x3(t))

x2 − σ2(x3(t))

= 0,

and so the vector col(x1(t), x2(t)) − σ(x3(t)) is constant. Therefore, if infx u(x) > 0,

the curve t 7→ col(x1(t), x2(t)) coincides with C modulo a translation. Consider any

solution x(t) such that the corresponding output signal y(t) is identically zero. Then,

the vectors col(x1(t) − σ1(x3(t)), x2(t) − σ2(x3(t))) and σ′(x3(t)) are orthogonal for all

t ≥ 0. Therefore, either col(x1(t)− σ1(x3(t)), x2(t)− σ2(x3(t))) is zero (i.e., x(t) ∈ Γ), or

σ′(x3(t)) has a constant angle. However, the angle of σ′(x3(t)) is x3(t), whose derivative

is positive. Thus, x(t) ∈ Γ, proving that O = Γ.

Letting u = β1(x)u + β2(x)u, Lemma 6.2.4 guarantees that the system with input u

and output h(x) is Γ-detectable.

Step 3: Passivity-based stabilization. The next result is a direct consequence of The-

orem 5.2.2.

Proposition 6.2.5. For any smooth u : R2 × S1 → R bounded away from zero, i.e.,

u(x) > ǫ > 0, and any PBF u(x) = −ϕ(x) with respect to the output h(x) in (6.5), the

feedback

u1 = u(x)

u2 =u(x)

‖σ′(x3)‖− ϕ(x)

(6.6)

globally asymptotically stabilizes the set Γ in (6.4), and thus solves the path following

problem for C globally.


-4 -3 -2 -1 0 1 2 3 4-2

-1. 5

-1

-0. 5

0

0.5

1

1.5

2

x1

x2

Figure 6.1: Simulation results for the global path following controller in (6.6), where C

is an ellipse with major semi-axis length 2 and minor semi-axis length 1.

Example 6.2.6. If C is a circle of radius r centred at the origin, then a global solution

to the path following problem in the counter-clockwise direction is given by the feedback

u1 = v

u2 =v

r+ r(x1 cosx3 + x2 sin x3).

If C is an ellipse centred at the origin with major semi-axis a and minor semi-axis b, then

a global solution to the path following problem in the counter-clockwise direction is given

by the feedback

u1 = v

u2 = vµ(x)3/2

a2b2+ a2b2

[

(b2 − a2) sin x3 cosx3µ(x)2

+x1 cosx3 + x2 sin x3

µ(x)3/2

]

,

where µ(x) = a2 sin2 x3+ b2 cos2 x3. Simulation results for this controller, with a = 2 and

b = 1, are displayed in Figure 6.1.


Remark 6.2.7. An important advantage of the feedback (6.6) is that it can be made to

be compatible with any input saturation constraint. For, if the controller is subject to

saturation constraints |u1| ≤ U1, |u2| ≤ U2, one can choose u(·) > 0 small enough that

u(·) ≤ U1 and u/‖σ′(x3)‖ < U2. Then, choose ϕ in the PBF u(x) = −ϕ(x) to be any

odd function of h(x) such that u/‖σ′(x3)‖+ supR |ϕ(·)| ≤ U2.

6.3 Case study 2: stabilizing the unicycle to a circle

with heading angle requirement

Consider again the kinematic unicycle model in (6.3), and the problem of stabilizing the

unicycle to a unit circle centred at the origin, with a constant desired heading on the

circle. This problem can be stated equivalently as the stabilization of the set

Γ = (x1, x2, x3) : x21 + x22 = 1, x3 = a mod 2π, (6.7)

where a is the desired reference heading.

Step 1: Candidate storage function. If one chooses a storage function V such that

V −1(0) = Γ, then a passivity-based feedback does not stabilize Γ. In order to illustrate

this fact, consider the storage function V = (x21 + x22 − 1)2/2 + (x3 − a)2/2. The unique

value of u rendering V −1(0) invariant is u = 0, so the feedback transformation (6.2)

becomes trivial, u = u. Since f = 0 for the kinematic unicycle, the system is passive

with any storage function V (x1, x2, x3), and output

LgV⊤ =

2(x21 + x22 − 1)(x1 cosx3 + x2 sin x3)

x3 − a

.

We now show that the system with input u and the output above is not Γ-detectable and

hence, since Γ-detectability is a necessary condition for passivity-based stabilization, no

PBF can stabilize Γ with the above choice of V . In order to check Γ-detectability, we

need to find O. Suppose that u(t) ≡ 0 and LgV (t) ≡ 0. Then, the unicycle dynamics are


x1

x2

x3 = a

u ≡ 0 and y ≡ 0

Figure 6.2: Failure of Γ-detectability in case study 2 when Γ = V −1(0).

stationary and O = x : (x21 + x22 − 1)(x1 cos a+ x2 sin a) = 0, x3 = a, and all points on

O are equilibria. Figure 6.2 illustrates the set of configurations of the unicycle on O. It

is clear that O contains and is not equal to Γ in (6.7). Therefore, Γ is not asymptotically

stable relative to O, and the system is not Γ-detectable. More generally, if we choose for

the system a storage function V (e1, e2), where e1 = (x21 + x22 − 1)/2, e2 = x3 − a, and

(e1, e2) 7→ V (e1, e2) is positive definite, then

Lg1V =∂V

∂e1(x1 cosx3 + x2 sin x3),

gives the same obstruction to Γ-detectability.

The above suggests that if one wants to stabilize Γ in (6.7) using a passivity-based

approach, one should not attempt to find a storage V with the property that V −1(0) = Γ.

Guided by this principle, we choose the simplest storage V such that Γ ( V −1(0), namely

V (x) =1

4(x21 + x22 − 1)2.

Next, we define a feedback transformation according to step 2 of the procedure. Since

LgV = (x21 + x22 − 1)[x1 cosx3 + x2 sin x3 0], we choose β1 = col(0, 1), so LgV (x)β1 = 0,


and β2 = col(1, 0), so the matrix [β1 β2] is nonsingular. The feedback transformation

u = β1(x)u+ β2(x)u =

0

1

u+

1

0

u.

guarantees that, for any feedback u(x), the system with input u and output y = h(x) :=

Lgβ2V (x)⊤ below is passive,

h(x) := (x21 + x22 − 1)(x1 cosx3 + x2 sin x3). (6.8)


Lemma 6.3.1. Let u(·) be any feedback such that, for any solution x(t) of x = f+gβ1u,

u(x(t)) ≡ 0 implies V (x(t)) ≡ 0. Then, the maximal subset of h−1(0) invariant under

the vector field f + gβ1u is

O = V −1(0) ∪ x : x1 = x2 = 0.

Proof. We have f+gβ1u = col(0, 0, u), and so x1(t) and x2(t) are constant. If h(x(t)) ≡ 0,

then either x(t) ∈ V −1(0), or x1(t) cosx3(t) + x2(t) sin x3(t) ≡ 0. If x(t) 6∈ V −1(0), then

the latter identity can only be satisfied if x1(t) ≡ x2(t) ≡ 0, because otherwise we would

have x3(t) =constant, implying that x3(t) ≡ u(x(t)) ≡ 0 and this, by assumption, can

only hold on V −1(0).

Under the assumption of the above lemma, O is the union of two disconnected com-

ponents, V −1(0) and x : x1 = x2 = 0. On V −1(0), f + gβ1u = col(0, 0, u). To

enforce Γ-detectability, choose u = −ϕ1(x3 − a) − ϕ2(x21 + x22 − 1) sin t, where ϕ1(·) is

2π-periodic and such that ϕ1(y) sin y > 0 for all y 6= 0, π mod 2π, and ϕ2 is positive

definite. If u(x(t)) ≡ 0, then x3(t) is constant. Thus, ϕ1(x3(t) − a) is constant and so

−ϕ1(x3(t)−a)−ϕ2(x21(t)+x

22(t)−1) sin t can only be zero if x21(t)+x

22(t) ≡ 1. Therefore,

this choice of u satisfies the assumption of Lemma 6.3.1. Moreover, on V −1(0) we have


x3 = −ϕ1(x3−a). By the choice of ϕ1, x3 = a mod 2π is almost globally asymptotically

stable for this differential equation, with domain of attraction x3 6= a+π mod 2π. Thus,

Γ is almost globally asymptotically stable relative to V −1(0), and hence almost globally

asymptotically stable relative to O (because the set x1 = x2 = 0 has measure zero).

Step 3: Passivity-based stabilization.

Proposition 6.3.2. Let ϕ1(y) be a locally Lipschitz and 2π-periodic function such that

ϕ1(y) sin y > 0 for all y 6= 0, π mod 2π, and let ϕ2 : R → R+ be positive definite. Then,

for any PBF −ϕ(x) with respect to the output h(x) in (6.8), the feedback

u1 = −ϕ(x)

u2 = −ϕ1(x3 − a) + ϕ2(x21 + x22 − 1) sin t

(6.9)

almost globally stabilizes the set

V −1(0) = (x1, x2, x3) : x21 + x22 = 1

with domain of attraction D = R2 × S1\(x1, x2, x3) : x1 = x2 = 0, and asymptotically

stabilizes the set

Γ = (x1, x2, x3) : x21 + x22 = 1, x3 = a mod 2π.

Simulation results for the controller (6.9) solving case study 2 are found in Figure 6.3,

in which we have chosen ϕ1(·) = sin(·), ϕ(·) = arctan(·), and ϕ2 = ‖ · ‖.

Proof. In order to handle the presence of the term sin t in the control input, consider the

augmented system

x1 = u cosx3

x2 = u sin x3

x3 = −ϕ1(x3 − a) + ϕ2(x21 + x22 − 1) sin θ

θ = 1,


2 1 0 1 2

2

1. 5

1

0. 5

0

0.5

1

1.5

2

x1

x2

Figure 6.3: Simulation results for the controller in (6.9).

with (x1, x2, x3, θ) ∈ R2 × S1 × S1. For notational simplicity, we will still denote by Γ,

O, V −1(0), and h−1(0) the lift of these sets to the augmented state space. Thus, for

instance, we will denote by Γ the set (x1, x2, x3, θ) : (x1, x2, x3) ∈ Γ. We have shown

in step 2 of the procedure that the system above with input u and output h(x) in (6.8)

is passive and locally Γ-detectable. Let u = −ϕ(x) be any PBF with respect to the

output h(x). By Theorem 5.2.2, since Γ is compact, Γ is asymptotically stable for the

closed-loop system. Moreover, V is proper. To see why this is the case, note that for

any c ≥ 0, the set V −1(c) = (x1, x2, x3, t) : (x21 + x22 − 1)2 = 4c. This set is compact

because (x3, t) ∈ S1 × S1. Since V is proper, all trajectories of the closed-loop system

are bounded. On x1 = x2 = 0, V has a local maximum. Therefore, for any initial

condition in D, the corresponding solution of the closed-loop system remains in D and

converges to the maximal invariant subset of V = 0 = h−1(0), i.e., it converges to

O ∩ D = V −1(0). This fact, together with the properness of V , implies that V −1(0) is

almost globally asymptotically stable with domain of attraction D.


6.4 Case study 3: coordination of two unicycles

Consider two kinematic unicycles

x1 = ux1 cos x3

x2 = ux1 sin x3

x3 = ux2

z1 = uz1 cos z3

z2 = uz1 sin z3

z3 = uz2,

(6.10)

and let χ = col(x, z). For this system, we have f = 0 and

g = blockdiag

cos x3 0

sin x3 0

0 1

,

cos z3 0

sin z3 0

0 1

.

Consider the problem of making the unicycles meet at a distance ∆ > 0 facing each

other. Solving this problem corresponds to stabilizing the set

Γ =

χ ∈ X :√

d1(χ)2 + d2(χ)2 = ∆, z3 = θ(χ), x3 = θ(χ) + π

. (6.11)

where d1(χ) = x1 − z1, d2(χ) = x2 − z2, and θ(χ) = arg(d1(χ) + i d2(χ)), with θ ∈ S1.

Step 1: Candidate storage function. Once again, choosing a candidate storage func-

tion V with the property that Γ = V −1(0) does not lead to a solution of the problem,

because such a choice would lead to an obstruction to Γ-detectability similar to the one

described in case study 2. Instead, we choose

V (χ) =1

4

[

d1(χ)2 + d2(χ)

2 −∆2]2, (6.12)

which has the property that Γ ( V −1(0). We choose the feedback transformation

ux1

ux2

uz1

uz2

= β1u+ β2u =

0 0

1 0

0 0

0 1

ux

uz

+

1 0

0 0

0 1

0 0

ux

uz

. (6.13)


For any feedback u(χ), when u = 0, we have that the unicycles rotate without translating,

and therefore the distance between them, (d1(χ), d2(χ)), remains constant. In other

words, LgV β1 = 0. Therefore, for any feedback u(χ), the system with input u and

output y = h(χ) := Lgβ2V (χ)⊤ below is passive,

y = h(χ) = (d21 + d22 −∆2)

d1 cos x3 + d2 sin x3

−(d1 cos z3 + d2 sin z3)

. (6.14)


Lemma 6.4.1. Let u(·) be any feedback which does not vanish on the set χ : (d1(χ), d2(χ)) 6=

0, d1(χ) cosx3+ d2(χ) sin x3 = 0, d1(χ) cos z3+ d2(χ) sin z3 = 0. Then, the maximal sub-

set O of h−1(0) invariant under the vector field f + gβ1u is O = V −1(0) ∪ χ : d1(χ) =

d2(χ) = 0.

Proof. The solutions of χ = f + gβ1u correspond to the two unicycles rotating and not

translating. Therefore, d1, d2 are constant along solutions. The set χ : d1(χ) = d2(χ) =

0, being invariant under the vector field f + gβ1u and contained in h−1(0), is contained

in O. Now suppose that (d1(χ(0)), d2(χ(0))) 6= 0, so that (d1(χ(t)), d2(χ(t))) 6= 0 for all

t ∈ R, and that h(χ(t)) ≡ 0. Then, dh(χ(t))/dt ≡ 0 so either d21(χ(t)) + d22(χ(t)) ≡ ∆2

(i.e., χ(t) ∈ V −1(0)), or

d1 cosx3(t) + d2 sin x3(t) ≡ 0

d1 cos z3(t) + d2 sin z3(t) ≡ 0

(−d1 sin x3(t) + d2 cosx3(t))ux ≡ 0

(−d1 sin z3(t) + d2 cos z3(t))uz ≡ 0.

By assumption, ux, uz are not zero on the set where the first two equations are satisfied.

Therefore, the equations can only be satisfied if d1(χ(t)) = d2(χ(t)) ≡ 0, which is not the

case.

The sets V −1(0) and d1(χ) = d2(χ) = 0 are disjoint and, for any u, they are

invariant under f+gβ1u. In order to enforce Γ-detectability, we need to design u such that


Γ is asymptotically stable relative to V −1(0) and u 6= 0 on the set χ : (d1(χ), d2(χ)) 6=

0, d1(χ) cosx3+d2(χ) sin x3 = 0, d1(χ) cos z3+d2(χ) sin z3 = 0. The restriction of f+gβ1u

to V −1(0) is

x1 = 0, x2 = 0, x3 = ux

z1 = 0, z2 = 0, z3 = uz.

The function θ(χ) is constant along solutions of the above differential equation, so sta-

bilizing Γ corresponds to stabilizing the equilibria x3 = θ(χ) + π, z3 = θ(χ) modulo 2π.

There are many ways to achieve this goal. We choose

ux = −K1

√

d21 + d22 sin(x3 − θ(χ)− π)

= K1 [d1 sin x3 − d2 cosx3]

uz = −K1

√

d21 + d22 sin(z3 − θ(χ))

= K1 [−d1 sin z3 + d2 cos z3] ,

with K1 > 0, which almost globally stabilizes Γ relative to V −1(0), with domain of

attraction χ ∈ V −1(0) : x3 6= θ(χ), z3 6= θ(χ)+π. Our choice of u is not zero on the set

χ : (d1(χ), d2(χ)) 6= 0, d1(χ) cosx3+ d2(χ) sinx3 = 0, d1(χ) cos z3+ d2(χ) sin z3 = 0 and

therefore, by Lemma 6.4.1, the feedback above almost globally stabilizes Γ relative to O

with domain of attraction χ ∈ O : d1(χ) 6= 0, d2(χ) 6= 0, x3 6= θ(χ), z3 6= θ(χ)+π, and

thus ensures local Γ-detectability for system (6.10) with feedback transformation (6.13),

input u and output y defined in (6.14).


Proposition 6.4.2. For any positive scalars K1, K2, the feedback

ux1 = −K2h1(χ)

ux2 = K1 [d1(χ) sin x3 − d2(χ) cosx3]

uz1 = −K2h2(χ)

uz2 = K1 [−d1(χ) sin z3 + d2(χ) cos z3] ,

(6.15)


−5 −4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

z1

x1

z2x2

x3

z3

x1, z1

x2,z

2

Figure 6.4: Simulation results for the coordination controller in (6.15).

where h(χ) is defined in (6.14), renders Γ asymptotically stable for the closed-loop system

and solves the coordination problem.

Remark 6.4.3. To compute the output function h(χ) in (6.14) each unicycle needs

to sense its orientation and relative displacement with respect to the other unicycle.

Therefore, feedback (6.15) only uses local information.

Simulation results for the controller in (6.15), with K1 = K2 = 1, are found in Fig-

ure 6.4.

Proof. In order to prove that Γ is asymptotically stable, by Theorem 5.2.2 it suffices to

show that the closed-loop system is LUB near Γ. Solutions of the closed-loop system are

defined for all t ≥ 0 because x3 and z3 are variables in S1, a compact set, and ux1(χ(t)),

uz1(χ(t)) are uniformly bounded,

|ux1|, |uz1| ≤ 2K2

√

V (2√V +∆2), (6.16)


and V is nonincreasing along solutions. The derivative of the storage function along

closed-loop solutions is

V = −4V[

(d1 cosx3 + d2 sin x3)2 + (d1 cos z3 + d2 sin z3)

2]

. (6.17)

We will show that for all initial conditions in some neighborhood N (Γ), the term in

square brackets in (6.17) is bounded away from zero. This fact then implies the LUB

property. For, the claim implies that on N (Γ), V converges to zero exponentially and

thus, by (6.14), h(χ(t)) tends to zero exponentially. Therefore, ux1, uz1 tend to zero

exponentially and so x1, x2, z1, z2 are bounded. Moreover, their bound is uniform on Γ,

proving the LUB property.

It is easy to see that (d1 cosx3 + d2 sin x3)2 = (d21 + d22) cos

2(x3 − θ). Since V is

nonincreasing, if d1(χ0)2 + d2(χ0)

2 6= 0 and V (χ0) < (1/4)∆4, then the solution χ(t) is

such that, for all t ≥ 0, d21(χ(t)) + d22(χ(t)) ≥ ∆2 − 2√

V (χ0) > 0. Therefore, for the

purpose of showing that the term in square brackets in (6.17) is bounded away from zero,

it is enough to show that there exists a neighborhood N (Γ) ⊂ V < (1/4)∆4 such that

all closed-loop solutions originating in N (Γ) yield, for all t ≥ 0, cos2(x3(t)− θ(t)) ≥ 1/2.

Let

W (χ) =1

2[x3 − θ(χ)− π]2 .

The time derivative of W along closed-loop solutions is

W =− (x3 − θ − π) sin(x3 − θ − π)√

d21 + d22

(

1 +ux1

d21 + d22

)

+ (x3 − θ − π) sin(z3 − θ)uz1

√

d21 + d22

≤−√2W sin(

√2W )

√

d21 + d22

(

1− |ux1|d21 + d22

)

+√2W

|uz1|√

d21 + d22.

Note that, when ux1 = uz1 = 0, if W (χ0) < π2/2, then the solution asymptotically

converges to W = 0. Moreover, given any c, with 0 < c < π2/2, there exists U > 0

such that, for |ux1|, |uz1| < U the set χ ∈ X : W (χ) ≤ c is positively invariant. Pick

c = 1/2(π/4)2, and let U be as above.


Given any V0 > 0, by the inequalities in (6.16) and the fact that V is nonincreasing along

solutions of the closed-loop system, for any initial condition χ0 ∈ χ ∈ X : V (χ) ≤ V0,

we have

|ux1(t)|, |uz1(t)| ≤ 2K2

√

V0(

2√

V0 +∆2)

.

Let V0 be small enough that 2K2

√

V0(

2√V0 +∆2

)

< U and V0 < (1/4)∆4. Consider the

set

N (Γ) = χ : V < V0 ∩ χ : W < 1/2(π/4)2.

On Γ, V = 0 andW = 0, soN (Γ) is a neighborhood of Γ. By construction, the setN (Γ) is

positively invariant. In particular, for all χ0 ∈ N (Γ) and all t ≥ 0, |x3(t)−θ(t)−π| < π/4,

and hence

cos2(x3(t)− θ(t)− π) > 1/2,

as required.

Chapter 7

Circular Formation Control of

Unicycles

Using the theoretical results of Chapters 4 and 5 and the control design approach pre-

sented in Chapter 6, in this chapter we present an application in the field of multi-agent

systems control. This field has been the subject of vigorous research in the past ten years,

and different instances of the problem we investigate have been previously addressed in

the control literature. In our problem, a group of n kinematic unicycles with a certain

information flow graph is required to follow a circle, with specified radius, and also to

acquire a certain desired formation on the circle. We show that this problem has an

intrinsic reduction aspect in that it can be broken down into two tasks: circular path

following and formation stabilization. Using this insight, we leverage our reduction and

set stabilization theory, and apply the passivity-based control design approach presented

in Chapter 6. Three different cases of increasing difficulty and generality are addressed,

based on the structure of the information flow graph. The first case is when the infor-

mation flow graph is undirected. The second case addresses circulant information flow

graph, and finally the third case addresses general static information flow graphs.

106

Chapter 7. Circular Formation Control of Unicycles 107

7.1 Introduction

In this chapter we show how the reduction-based and passivity-based control design

perspectives, introduced in Chapters 4 and 5, can be applied in the field of multi-agent

systems. In this field, the main objective is to obtain a global behaviour through the use

of distributed control. One of the main problems in this field is the formation control

of unmanned aerial, land, underwater vehicles, and spacecrafts. This problem is also

relevant in the areas of sensor networks and cooperative robotics.

In this chapter we address a circular formation control problem for a network of planar

vehicles under communication constraints. Different instances of this problem have been

previously addressed and different solutions have been proposed. One of the main results

is the work presented in [57], [58], which addresses the cyclic pursuit control problem

where agent i has communication link with agent i+ 1. The authors study an intuitive

cyclic pursuit control for unicycles. This law is linear in the difference between the i-th

unicycle orientation and the i + 1 unicycle relative position direction. Using this law,

the authors obtain circular formations and show that the resulting relative equilibria are

generalized regular polygons. The possible equilibrium formations depends on certain

control parameters. The authors study the local stability properties of the equilibrium

polygons and show which equilibrium formations are asymptotically stable.

The cyclic pursuit law in [57] has been studied in many other works, including [86],

where nonlinear cyclic pursuit were studied, [69], which improved traditional cyclic pur-

suit by rotating the pursuit direction, and [24], which studied hierarchical pursuit strate-

gies.

Another important research direction on formation stabilization is found in [79], where

the authors investigate problems of synchronization for systems of particles modeled

as unicycles. Potential functions are defined for various tasks and used to generate

gradient control laws. Among other things, the authors stabilize the unicycles to a

circle and study symmetric formations on the circle. The results are based on an all-


to-all communication assumption whereby each unicycle can sense the state of all other

unicycles. In [80] the authors extend the results in [79] to address different communication

topologies. First, they provide a direct extension to the case of undirected time invariant

communications topologies. To address the case where the communication topology is

time varying and directed, the authors design a dynamic feedback by using consensus

filters that asymptotically reconstruct the averaged quantities required by the all-to-

all stabilizing control law. This scheme requires extra communication where particles

exchange relative estimated variables in addition to relative communication variables.

The ideas used in [79] and [80] are incorporated in several other works. In [65], particles in

a uniform flow field are considered and symmetry breaking to stabilize a particular circle

is studied. This work is generalized in [68], where the flow field is arbitrary but known.

In [66], symmetric formation on closed curves, not necessarily circular, are addressed.

The communication topology is undirected. A super-ellipse function is incorporated to

express and stabilize different curves. In [67], the result in [79] and [80] is extended to

formations around convex loops.

In the above results the unicycles are stabilized to a circle with unspecified centre.

Other results in the literature address the problem from a different perspective. In [16],

circular formation around a beacon is addressed whereby all unicycles have access to the

relative distance and relative orientation to the direction of the beacon. In this case the

communication topology has no role and the problem is conceptually different from what

we address. In [50] stabilizing regular polygon formations is addressed, without circular

motion. The gradient control is derived from a potential function based on undirected

infinitesimally rigid graphs.

In this chapter we provide a control design that stabilizes n-unicycles to a circle, with

unspecified centre, and at the same time provide desired formations on the circle. We

address different communication topologies beginning with undirected, then circulant,

and finally general directed topological schemes. We approach this problem from a set


stabilization perspective. The desired formation is expressed as an unbounded set in the

composite state space of the unicycles, and then control design is performed to stabilize

that set. Thus, unlike some previous results, our control design is objective oriented.

Our design takes into account the 2π modularity in the unicycles headings, and in some

instances the result is global or possibly almost global. Unlike most of the previous results

where only symmetric formations were addressed, we provide stabilization for arbitrary

formations on the circle. This is accomplished through applying reduction principles and

decomposition of control design.

Some of the results cited earlier implicitly involve the stabilization of closed un-

bounded sets. Proofs are often carried out in relative coordinates, where the sets in

question become compact. The problem with this approach, however, is that the trans-

formation from absolute to relative coordinates is not a diffeomorphism, and therefore

stability claims made in relative coordinates cannot be extrapolated to deduce analo-

gous claims in absolute coordinates. As an illustration, it may happen that the relative

heading and distance between two unicycles is predicted to converge to zero, but in con-

verging to each other the two unicycles may diverge to infinity in finite time, invalidating

the analysis. Although the problem just described is of purely theoretical interest, it is

nonetheless important to provide a rigorous stability analysis. The controllers presented

in this chapter are derived and analyzed in absolute coordinates, and leverage the full

power of the reduction principles for closed unbounded sets presented in Chapter 4.

The chapter is organized as follows. In Section 7.2 we review basic notations of

multi-agent systems concerning information flow and directed graphs. In Section 7.3 we

formalize the circular formation control problem addressed in this chapter. Section 7.4

presents the solution for the circular formation control problem when the information

flow graph is undirected. In Section 7.5 the solution is extended to the case where the

information flow graph is circulant. Finally, in Section 7.6 we address the general case

where the information flow graph is an arbitrary static directed graph.


7.2 Information flow and digraphs

In the formation control problem we present in this chapter, the information flow through-

out the unicycles formation is crucial for control design. We model this information flow

by a directed graph (digraph). In this section, we present the definition of a directed graph

and other related concepts and tools, from algebraic graph theory [9], [28]. An excellent

overview of key results is presented in [55].

A directed graph (digraph) G consists of a non-empty finite set V of elements called

nodes and a finite set E of ordered pairs of nodes called arcs, see Figure 7.1. The digraph

is written as G = (V, E) where V and E are the node set and the arc set of G.

For an arc (i, j) the first node is the tail and the second node is the head. One says arc

1 2

34

Figure 7.1: Digraph

(i, j) leaves i and enters j. The head and tail of an arc are its end-nodes.

In this chapter we model the information flow throughout the unicycles formation by

a simple digraph with n nodes, each node representing a unicycle, and each arc from node

i to node j indicating that unicycle i has access to the relative position and orientation

of unicycle j. We also assume that each unicycle has access to its own orientation.

Reachability. For a digraph G, if there is a path from one node i to another node

j then j is said to be reachable from i, written i → j. If not, then j is said to be not

reachable from i, written i 6→ j. If a node j is reachable from every other node in G,

then it is called globally reachable.


The Laplacian of a Digraph. An important matrix associated with a digraph G is the

graph Laplacian defined as L = D − E, where D, the degree matrix of G, is a diagonal

matrix with the out-degree of each node along the diagonal, i.e., the number of arcs with

tail at the node. E is a nonnegative matrix associated with G and called its adjacency

matrix. It is defined as E = (eij) ∈ Rn×n where eij = 1 if there is an arc from node i to

node j in G, and eij = 0 otherwise. The Laplacian L has the property that its row sums

are all zero. For example, consider the digraph in Figure 7.1, the out-degree matrix, the

adjacency matrix and the Laplacian are given by

D =

2 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, E =

0 1 0 1

0 0 1 0

1 0 0 0

0 0 1 0

, L =

2 −1 0 −1

0 1 −1 0

−1 0 1 0

0 0 −1 1

Next, we present two results from [54] that will be used in the chapter. Consider a digraph

G = (V, E) with Laplacian L. The first result is a useful algebraic characterization of the

property that G has a globally reachable node.

Lemma 7.2.1 (Lemma 2, [54]). The digraph G has a globally reachable node if and only

if 0 is a simple eigenvalue of L.

By this Lemma, if a digraph with Laplacian L has a globally reachable node then

kerL = span 1 where 1 = col(1, . . . , 1) ∈ Rn. Note also that, by the Gershgorin circle

Theorem [12], the eigenvalues of any Laplacian are either zero or have positive real part.

Thus if the digraph has a globally reachable node then all the eigenvalues of L have

positive real part except for one which is zero.

The second results is given in terms of the definition of H(α,m) stability introduced

in [54]. Let α = α1, α2, · · · , αp be a partition of 1, 2, · · · , n. A block diagonal matrix

with diagonal blocks indexed by α1, α2, · · · , αp is said to be α-diagonal.

Definition 7.2.2 (Definition 2, [54]). Let α = α1, α2, · · · , αp be a partition of 1, 2, · · · , n

and m ≥ 0 an integer. An n× n matrix A is said to be H(α,m)-stable if


(a) 0 is an eigenvalue of A of algebraic and geometric multiplicity m while all other

eigenvalues have negative real part,

(b) For every α-diagonal positive definite symmetric matrix R, 0 is an eigenvalue of RA

of algebraic and geometric multiplicity m,while all other eigenvalues have negative

real part.

Letting L(2) = L⊗I2, where ⊗ denotes the Kronecker product, we have the following:

Lemma 7.2.3 (Lemma 4, [54]). Let α = 1, 2, 3, 4, · · · , 2n− 1, 2n. the matrix

−L(2) is H(α, 2) stable if and only if the digraph G has a globally reachable node.

7.3 Problem statement

Consider a system of n kinematic unicycles, n ≥ 2, where the i’s unicycle model is given

by

xi1 = ui1 cosxi3

xi2 = ui1 sin xi3

xi3 = ui2

(7.1)

for i = 1, . . . , n, with state xi = (xi1, xi2, x

i3) ∈ R2 × S1. The state space of the system is

X = (R2 × S1)n. Let χ = col(x1, · · · , xn) and x3 = col(x13, · · · , xn3 ). For this system, we

have f = 0 and

g = blockdiag

cosx13 0

sin x13 0

0 1

, · · · ,

cosxn3 0

sin xn3 0

0 1

.

As pointed out earlier, we shall model the information flow among the n-unicycles with

a digraph G, where the individual unicycles are the nodes of G and the arcs represent

the information flows. An arc from unicycle i to unicycles j means that unicycle i has

access to the relative displacement and heading of unicycle j. We also assume that each


unicycle has access to its own absolute orientation. In practice, this can be achieved if

the vehicle has an on-board compass. As before, we will let L be the Laplacian of the

digraph G of the n-unicycles. We will use the notation Li for the i-th row of L, and we

denote L(2) = L⊗ I2.

Circular Formation Control Problem (CFCP). Consider the n-unicycles in (7.1).

For a given information flow digraph G with a globally reachable node, design a dis-

tributed control law achieving the following objectives:

(i) Circular path following. For a suitable set of initial conditions, the unicycles should

converge to a common circle of radius r > 0, whose centre is stationary but depen-

dent on the initial condition, and traverse the circle in a desired direction (clockwise

or counter-clockwise). The unicycles’ forward speed should be bounded away from

zero, so that the unicycles travel around the circle without stopping.

(ii) Formation stabilization. On the circle in part (i) of the problem, the n-unicycles are

required to converge to a formation expressed by desired separations and ordering

of the unicycles.

Our control design for CFCP provides circular path following in the counter-clockwise di-

rection, but can be easily modified to achieve clockwise path following. We now make the

problem statement above precise by formulating it within the set stabilization framework.

Define the function ci(xi) as

ci(xi) = (xi1 − r sin xi3, xi2 + r cosxi3) (7.2)

for i = 1, · · · , n. For unicycle i, the point ci(xi) lies at a distance r from (xi1, xi2), and the

vector col(xi1, xi2) − ci(xi) is orthogonal to the normalized velocity vector (cosxi3, sin x

i3)

of unicycle i, see Figure 7.2. Therefore, the point ci(xi) is the centre of the circle that

the unicycle would follow in the counter-clockwise direction if its controls were chosen as

ui1 = v and ui2 = v/r. Using (7.2), part (i) of the CFCP can be stated as the stabilization


(xi1, xi2)

xi3

r

ci(xi)

Figure 7.2: The centre ci(xi)

of the set

Γ1 = χ : ci+1(xi+1) = ci(xi), i = 1, · · · , n (7.3)

with the additional requirements that the linear velocities of the unicycles be bounded

away from zero and that ci(xi(t)), i = 1, . . . , n, tend to constant values. In the above,

and in what follows, the indices i ∈ 1, . . . , n are evaluated modulo n. For instance,

n+ 1 is identified with 1.

Remark 7.3.1. The function ci(xi) gives a smooth map R2×S1 → R2×S1, (xi1, xi2, x

i3) 7→

(ci(xi), xi3) which is is a diffeomorphism. Using this, instead of the dynamics (7.1), one

can express the unicycle model as

ci1 = (ui1 − rui2) cosxi3

ci2 = (ui1 − rui2) sin xi3

xi3 = ui2.

(7.4)

We now turn our attention to part (ii) of CFCP. Consider a formation where unicycle

j travels on the circle at distance d from unicycle i, as shown in Figure 7.3. This formation

constraint can be equivalently expressed as xi3−xj3 = 2 sin−1(

d2r

)

mod 2π. In light of this

observation, part (ii) of CFCP can be restated as the stabilization of the configuration

on the circle where the unicycles headings differ by prespecified constant angles, or

xi3(t) = α(t) + αi mod 2π, i = 1, · · · , n (7.5)


j

i

xi3 − xj3

r

r

d

2 sin−1 d2r

Figure 7.3: Formation on the circle

for some differentiable function α(t) and desired angles αi. The angles αi ∈ [0, 2π)

determine the ordering of the unicycles on the circle and their inter-distances. Part (ii)

of CFCP can be restated as the stabilization of the set Γ2 defined as

Γ2 = χ : L(x3 − α) = 0 mod 2π (7.6)

where α = col(α1, · · · , αn) is the vector of desired angles specifying the formation. Notice,

indeed, that since we assume that G has a globally reachable node, kerL = span 1 and

χ(t) ∈ Γ2 if and only if (7.5) holds.

Using the sets Γ1 in (7.3) and Γ2 in (7.6), CFCP can be restated as follows.

CFCP (equivalent statement). Consider the n-unicycles in (7.1). For a given

information flow digraph G with a globally reachable node, design a distributed control

law which asymptotically stabilizes the set

Γ = Γ1 ∩ Γ2 = χ : L(x3 − α) = 0, ci+1(xi+1) = ci(xi), i = 1, · · · , n, (7.7)

where α = col(α1, · · · , αn) is a vector of angles specifying the formation on the circle and

ci(xi) is defined in (7.2). Additionally, the linear velocities ui1 and angular velocities ui2

of the unicycles should be bounded away from zero on Γ and they should have a common


asymptotic centre of rotation, by which it is meant that for all χ(0) ∈ X there exists

c ∈ R2 such that c(xi(t)) → c as t→ ∞, i = 1, . . . , n.

Note that Γ is closed but not compact since there are no restrictions on the centres

of rotation ci(xi).

Example 7.3.2. Consider as example of the circular formation control problem the cyclic

pursuit with uniform spacing problem, see Figure 7.4. In this case there is an information

12

3

. . . n− 1

n

Figure 7.4: Cyclic pursuit with uniform spacing

link from unicycle i to unicycle i + 1 and the the Laplacian of the information digraph

takes the form

L =

1 −1 0 · · · · · · 0

0 1 −1 0 · · · 0

.... . .

...

0 · · · · · · 0 1 −1

−1 0 · · · · · · 0 1

For uniform spacing between the unicycles, the vector α is chosen as

α =

[

02π

n

4π

n· · · (n− 1)

2π

n

]⊤

.


In this chapter, we solve CFCP in three steps of increasing complexity:

Case I. The Laplacian L is symmetric, which corresponds to the situation of undi-

rected information flow graph.

Case II. The information flow graph has a circulant Laplacian L.

Case III. The information flow graph is an arbitrary static directed graph.

7.4 Case I: Undirected information flow graph

In this section we present a passivity-based control design to solve CFCP when the infor-

mation flow digraph is undirected, i.e, its Laplacian matrix symmetric. This corresponds

to the situation where if unicycle i sees unicycle j, then necessarily j sees i.

Step 1: Candidate storage function.

Let c(χ) = col(c1(x1), · · · , cn(xn)) ∈ R2n with ci(xi) defined in (7.2), and let L(2) =

L ⊗ I2, where L is the Laplacian of the n-unicycles information digraph G and I2 =

1 0

0 1

. Consider the following candidate storage function

V (χ) =1

2c(χ)⊤L(2) c(χ) (7.8)

Since L is symmetric, L(2) is positive semidefinite. Also, since the information digraph

has a globally reachable node, from Lemma 7.2.1 we have that L(2) has 2 eigenvalues at

0 with geometric multiplicity 2, and thus

kerL(2) = Image col(I2, · · · , I2) , (7.9)

from which it follows that V −1(0) is the set where all the centres of rotation coincide,

i.e., V −1(0) = Γ1. Based on the observation that any feedback of the form

(ui1, ui2) = (ui(χ), ui(χ)/r), i = 1, · · · , n


keeps the centres of rotation, and hence V , constant along solutions of the closed-loop

system, we choose the feedback transformation

ui =

ui1

ui2

= βi

1 ui + βi

2 ui =

1

1/r

ui +

0

1

ui, i = 1, . . . , n.

Setting

u =

[

u11 u12 · · · ui1 u

i2 · · · un1 u

n2

]⊤

u =

[

u1 · · · ui · · · un]⊤

u =

[

u1 · · · ui · · · un]⊤

β1 = blockdiag

1

1/r

, · · · ,

1

1/r

β2 = blockdiag

0

1

, · · · ,

0

1

we have

u = β1 u+ β2 u. (7.10)

The above feedback transformation has the property that LgV (x)β1 = 01×n. Moreover,

LfV = 0 because f = 0. Therefore, for any feedback u(χ), the system with input u and

output y = h(χ) := Lgβ2V (χ)⊤ is passive. The output y is given as follows

y = h(χ) =

[

∂V∂x1

3

· · · ∂V∂xi

3

· · · ∂V∂xn

3

]⊤

, (7.11)

with

yi =∂V

∂xi3=∂c⊤

∂xi3L(2) c(χ)

=

[

0 0 · · · ∂ci1(xi)

∂xi3

∂ci2(xi)

∂xi3

· · · 0 0

]

L(2) c(χ)

=

[

0 0 · · · −r cos xi3 −r sin xi3 · · · 0 0

]

L(2) c(χ).

(7.12)


Letting

R(x3) = blockdiag

[

cosx13 sin x13

]

, · · · ,[

cosxn3 sin xn3

]

, (7.13)

we get

y = h(χ) = −rR(x3)L(2) c(χ) (7.14)


Lemma 7.4.1. Let u(χ) be any feedback which is bounded away from zero component-

wise, i.e., for some ε > 0, infχ ui(χ) ≥ ε > 0 for i = 1, · · · , n. Then, the maximal subset

of h−1(0) invariant under the vector field f + gβ1u is Γ1, i.e., O = Γ1.

Proof. As observed earlier, if u = 0 and infχ u > ε > 0 component-wise, then each

unicycle moves along a circle of radius r, and so the vector L(2) c, in the output function

(7.12), is constant. Suppose that, for some solution χ(t) of the system with u = 0,

h(χ(t)) ≡ 0. Then, either L(2) c(χ(t)) = 0 which, because of (7.9), is only possible when

all the centres coincide, i.e., when χ(t) ∈ Γ1, or, for some i, the constant vector L(2)c

is perpendicular to the vector [0 0 · · · cosxi3(t) sin xi3(t) · · · 0 0]⊤, for i = 1, · · · , n

and t ∈ R, implying that xi3(t) is constant. However, by assumption the unicycles move

along n circles with nonzero linear velocity vectors, and therefore the angle xi3(t) is not

constant.

As mentioned earlier, the functions ci(xi) in (7.2) remain constant along the solutions

of (7.1) with feedback transformation (7.10) and u = 0. When u = 0, the restriction of

the vector field f + gβ1u to O = Γ1 is

xi1 = ui cos xi3

xi2 = ui sin xi3

xi3 =1

rui

(7.15)


Using the model (7.4), the dynamics above takes the form

ci1 = 0

ci2 = 0

xi3 =1

rui,

(7.16)

i.e., the dynamics of the unicycles are entirely described by those of their angular velocities

xi3. Under the assumption of Lemma 7.4.1, the goal set Γ can be expressed as

Γ = χ ∈ O : L(x3 − α) = 0, (7.17)

so we need to design u to stabilize the set x3 : L(x3−α) = 0. In designing the stabilizer,

we must take into account the fact that xi3 ∈ S1, so the stabilization must be performed

modulo 2π. To fulfill the assumption of Lemma 7.4.1, we also need ui to be bounded

away from zero. There are many ways to obtain these objectives. We base our design on

the following candidate Lyapunov function

W (x3) =

n∑

i=1

[1− cos(Li(x3 − α))] (7.18)

where Li is the i-th row of the Laplacian L. Note that W ≥ 0 and W = 0 if and only if

Li(x3 − α) = 0 mod 2π, for i = 1, · · · , n. Thus W−1(0) is precisely the set we wish to

stabilize. The derivative of W along (7.15) is

W =

n∑

i=1

sin(Li(x3 − α))Li u/r =1

rS(x3)

⊤L u, (7.19)

where

S(x3) =

sin(L1(x3 − α))

...

sin(Ln(x3 − α))

.

Lemma 7.4.2. The feedback

ui = v − v1 sin(Li(x3 − α)), i = 1, . . . , n, (7.20)


where and v > v1 > 0 are design constants, is bounded away from zero component-wise

and makes the set Γ asymptotically stable relative to Γ1 for system (7.1) after feedback

transformation (7.10) and u = 0, thus enforcing local Γ-detectability of the system with

input u and output y = h(χ) in (7.14).

Proof. By Lemma 7.4.1, the maximal subset of h−1(0) invariant under the vector field

f + gβ1u is O = Γ1. Referring to the system restriction on O in (7.16), to prove the

Lemma it suffices to show that the set W−1(0) is asymptotically stable for the system

xi3 = ui/r, i = 1, · · · , n, with ui given in (7.20). By substituting the control (7.20) into

the derivative (7.19) we get

W = −v1S(x3)⊤LS(x3).

The matrix L is positive semidefinite with one eigenvalue at zero and so W is nonin-

creasing along solutions, proving that W−1(0) is stable. As for its attractivity, since

(x13, . . . , xn3 ) ∈ Sn is compact, we can apply the LaSalle invariance principle and conclude

that, for all initial conditions,

S(x13(t), . . . , xn3 (t)) → kerL = span 1.

Therefore, there exists a C1 real-valued function s(t) such that sin(Li(x3 − α)) → s(t)

for all i. Let

Ω = x3 : W (x3) < 1−mincos(2π/n), 0.

The set Ω is positively invariant. Moreover, since for each x3 ∈ Ω and each i ∈ 1, . . . , n,

1− cos(Li(x3 − α)) ≤W (x3) < 1−min

cos2π

n, 0

,

we have cos(Li(x3 − α)) > mincos(2π/n), 0, so that

|Li(x3 − α)| < min2π/n, π/2 modulo 2π. (7.21)

Now let x3(0) be an arbitrary initial condition in Ω. Since for all i ∈ 1, . . . , n, |Li(x3−

α)| < π/2 we can invert the sin function and deduce that

(∀i ∈ 1, . . . , n) Li(x3 − α) → arcsin s(t) mod 2π


or

L(x3 − α) → 1 arcsin s(t) mod 2π.

Since kerL = kerL⊤ = span 1, we have 1⊤L(x3 − α) = 0, and therefore it must be that

1⊤1 arcsin s(t) = 0 mod 2π, or n arcsin s(t) = 0 mod 2π. In other words,

arcsin s(t) ∈ 2πk/n+ 2πl, k, l ∈ N.

But since |Li(x3 − α)| < min2π/n, π/2 mod 2π, it must be that arcsin s(t) = 0

mod 2π, proving that W−1(0) is attractive, and hence asymptotically stable.


We are now ready to solve CFCP in the case of undirected information flow graph.

Proposition 7.4.3 (Solution of CFCP for undirected information flow graph). Assume

that the information flow graph is undirected and has a globally reachable node. For any

v > v1 > 0, there exists K⋆ > 0 such that for all K ∈ (0, K⋆) the feedback

ui1 = v − v1 sin(Li(x3 − α))

ui2 =ui1r

−Khi(χ), i = 1, · · · , n(7.22)

where h(χ) is given in (7.14), solves CFCP and renders the goal set Γ in (7.7) asymptot-

ically stable, and Γ1 in (7.3) globally asymptotically stable for the closed-loop system.

Remark 7.4.4. Note that unicycle i needs to compute

hi(χ) = [0 0 · · · − r cosxi3 − r sin xi3 · · · 0 0]L(2)c(χ)

and Li(x3−α). In order to perform this computation, the unicycle needs to sense its rel-

ative displacement and orientation with respect to its neighbours in the information flow

graph, as well as its absolute orientation xi3. Therefore, feedback (7.22) is a distributed

control law respecting the information flow graph.


Proof. Consider the system with input u and output y = h(χ) in (7.14), obtained after

applying the feedback transformation (7.10) to (7.1). This system is passive by construc-

tion, and, as shown previously, choosing u as in (7.20) yield local Γ-detectability. The

feedback in (7.22) results from using the PBF u = −Kh(χ). Therefore, V is nonincreas-

ing along solutions of the closed-loop system, and in light of Theorem 5.2.2, in order to

prove that Γ is asymptotically stable for the closed-loop system it suffices to show that

the closed-loop system is LUB near Γ. Since xi3 ∈ S1, a compact set, we need to prove

the LUB property for the displacements xi1, xi2 or, equivalently, for the centres of rotation

ci(xi).

All trajectories of the closed-loop system are defined for all t ≥ 0 because |xi1|, |xi2| ≤

|ui1| and 0 < ui1 ≤ 2v. Using (7.4), along solutions of the closed-loop system we have

ci(xi) =

r cosxi3

r sin xi3

Khi(χ)

= −r2K

r cosxi3

r sin xi3

[

0 0 · · · cosxi3 sin xi3 · · · 0 0

]

L(2) c(χ)

Letting

R(x3) = R⊤(x3)R(x3) = blockdiagR1(x13), · · · , Rn(x

n3 )

= blockdiag

cos2 x13 sin x13 cos x13

sin x13 cos x13 sin2 x13

, · · · ,

cos2 xn3 sin xn3 cosxn3

sin xn3 cosxn3 sin2 xn3

(7.23)

we rewrite the c dynamics as

c = −r2KR(x3)L(2) c (7.24)


where

R(x3) = R⊤(x3)R(x3) = blockdiagR1(x13), · · · , Rn(x

n3 )

= blockdiag

cos2 x13 sin x13 cos x13

sin x13 cos x13 sin2 x13

, · · · ,

cos2 xn3 sin xn3 cosxn3

sin xn3 cosxn3 sin2 xn3

(7.25)

Since V −1(0) ⊂ h−1(0), there exists V0 > 0 such that V (χ) ≤ V0 implies that ‖h(χ)‖ is

small enough that (∃µ > 0)(∀i ∈ 1, · · · , n) xi3 > µ.

We next show that for any initial condition χ(0) ∈ X , the solution χ(t) reaches

the positively invariant set V (χ) ≤ V0 in finite time t, and so xi3(t) > µ > 0 for all

i ∈ 1, · · · , n and t ≥ t. First, note that the feedback (7.22) is bounded on any sublevel

set of V , and thus, from passivity, all closed-loop solutions are globally defined. Along

solutions of the closed-loop system we have,

V = −K‖h(χ)‖2.

By this, the continuity of V , and its nonnegativity, limt→∞ V (χ(t)) exists and is finite.

Also, from (7.24), it is easy to see that V (χ(t)) is bounded. Thus, by invoking Barbalat’s

Lemma [49] we get V (χ(t)) → 0, and so

y(t) = h(χ(t)) → 0.

Assume, by way of contradiction, that there exists an initial condition χ(0) ∈ X such

that the solution χ(t) does not reach the positively invariant set V (χ) ≤ V0. From

this we have V (χ(t)) → V1, for some constant V1 > V0 > 0. Using this, there exists a

constant ǫ > 0 and a time t1 > 0 such that, for all t ≥ t1, |L(2)c(χ(t))i| ≥ ǫ for some

i ∈ 1, · · · , 2n, where L(2)c(χ(t))i is the i-th component of the vector L(2)c(χ(t)).

Since y(t) → 0 we have, from (7.24), c → 0. From this, t1 can be picked large

enough that if |L(2)c(χ(t0))i| ≥ ǫ for some t0 ≥ t1, then |L(2)c(χ(t0))i| ≥ ǫ/2 for all

t ∈ [t0, t0+2π/µ]. Also, since y(t) → 0 there exists a time t2 > 0 such that xi3(t) > µ > 0


for all i ∈ 1, · · · , n and t ≥ t2. Using this, and (7.14), we get that for all t > max (t1, t2)

there there exists T > t such that ‖h(χ(T ))‖ ≥ ǫ/2 contradicting the fact that y(t) → 0.

Consider the linear time varying system

c = −r2KR(x3(t))L(2) c (7.26)

and the corresponding averaged system

cavg = −r2KRL(2) cavg (7.27)

where R = blockdiagR1, · · · , Rn and

Ri = limT→∞

1

T

∫ t+T

tcos2 xi3(τ)dτ

∫ t+T

tsin xi3(τ) cosx

i3(τ)dτ

∫ t+T

tsin xi3(τ) cosx

i3(τ)dτ

∫ t+T

tsin2 xi3(τ)dτ

.

The limits above exist and are finite.

Obviously,∫ t+T

tcos2 xi3(τ)dτ > 0. Moreover, by the Cauchy-Schwarz inequality, for

each T > 0(

∫ t+T

t

sin xi3(τ) cosxi3(τ)dτ

)2

≤(

∫ t+T

t

sin2 xi3(τ)dτ

)(

∫ t+T

t

cos2 xi3(τ)dτ

)

and the inequality is strict because xi3(t) is not constant. Therefore, det R > 0 and R

is positive definite. Since x3 ≥ µ on the positively invariant set V (χ) ≤ V0, det R is

bounded away from zero uniformly over V (χ) ≤ V0, and thus the eigenvalues of Ri,

i = 1, . . . , n, are bounded away from zero uniformly on V (χ) ≤ V0. Recall that the

matrix −L(2) has two eigenvalues at 0 with geometric multiplicity 2, and the remain-

ing eigenvalues have negative real part. By Lemma 7.2.3, the matrix −RL(2) has the

same properties. In conclusion, −RL(2) has two eigenvalues at zero and the remaining

eigenvalues are negative and bounded away from zero uniformly over V (χ) ≤ V0.

We now apply the representation theorem of Linear Algebra to isolate the asymptot-

ically stable subsystem of the averaged system. Consider the coordinate transformations

z = P−1c, zavg = P−1cavg


where

P =

1 0 · · · 0 0

1 1 · · · 0 0

.... . .

. . .

1 0 · · · 0 1

⊗ I2. (7.28)

System (7.26) after this coordinate transformation becomes

z = −r2KP−1R(x3(t))L(2)P z,

while its average (7.27) becomes

zavg = −Kr2P−1R L(2)P zavg,

Partitioning z and zavg as z = col(z, z), zavg = col(zavg, zavg), with z, zavg ∈ R2 and

z, zavg ∈ R2n−2, we have

˙z = KA12(t)z

˙z = KA22(t)z

˙zavg = KA12zavg

˙zavg = KA22zavg,

where, in light of the discussion above, A22 is Hurwitz. Therefore, the origin of the zavg

subsystem is globally exponentially stable. By the general averaging theorem (Theorem

10.5, [49]), there exists K⋆ > 0 such that for all K ∈ (0, K⋆), the origin of the z

subsystem is globally exponentially stable as well. The exponential convergence of z(t)

to zero implies that z(t) has a finite limit as t → ∞. Going back to χ coordinates

and summarizing our discussion so far, we have that for all initial conditions χ(0) ∈ X

the solution χ(t) is defined for all t ≥ 0 and there exists c ∈ R2 such that ci(xi(t)) →

c as t → ∞ exponentially, i = 1, . . . , n, proving that the unicycles have a common

asymptotic centre of rotation. Moreover, the exponential rate of convergence is uniform

over initial conditions in χ(0) ∈ V (χ) ≤ V0. Therefore, referring to (7.26), there exists

M > 0 such that for all χ(0) ∈ V (χ) ≤ V0, ‖c(χ(t))‖ ≤ M‖L(2)c(χ(0))‖, and so the

bound on ‖c(χ(t))‖ is uniform over initial conditions on a neighbourhood of Γ. The


LUB property is therefore proven, and so Γ is asymptotically stable for the closed-loop

system. Finally, concerning the global asymptotic stability of Γ1, the fact that for all

χ(0) ∈ X there exists c ∈ R2 such that ci(xi(t)) → c as t → ∞ implies that Γ1 is

globally attractive. Since V (χ) = (1/2)c(χ)⊤L(2)c(χ) is nonincreasing along solutions of

the closed-loop system, and since Γ1 = χ : L(2)c(χ) = 0, it follows that Γ1 is stable for

the closed-loop system.

7.4.1 Simulations

We present simulation results for the following two cases, for 6 unicycles.

A. The unicycles are uniformly distributed on the circle in a cyclic order as shown in

Figure 7.5.

1

2

34

5

6

Figure 7.5: CFCP Simulation - A

To achieve this formation the vector α is set as:

α =

[

02π

6

4π

6

6π

6

8π

6

10π

6

]⊤

.

B. The unicycles are uniformly distributed on half the circle in a cyclic order as shown

in Figure 7.6.


1

2

3

4

56

Figure 7.6: CFCP Simulation - B

To achieve this formation the vector α is set as:

α =

[

02π

10

4π

10

6π

10

8π

10

10π

10

]⊤

.

Figures 7.7 and 7.8 show the simulations results for A. and B. using feedback 7.22

with the following parameters: r = 1, v = 1, v1 = 0.9, K = 1 and

L =

2 −1 0 0 0 −1

−1 2 −1 0 0 0

0 −1 2 −1 0 0

0 0 −1 2 −1 0

0 0 0 −1 2 −1

−1 0 0 0 −1 2

.

7.4.2 Global solution of CFCP

The passivity-based design in Section 7.4 took into account the fact that xi3 ∈ S1 and

so the stabilization was performed modulo 2π. This was accomplished by using the

function W in (7.18), which is 2π-periodic with respect to xi3, i = 1, . . . , n. Moreover,

the centres ci(xi), upon which the output (7.14) depends, are 2π-periodic with respect


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

1

1

2

2

3

3

4

4

5

5

6

6

x1

x2

Figure 7.7: CFCP Simulation - A: Case I

to xj3, j = 1, . . . , n. Motivated by the fact that several results in literature, including

the work in [57], do not account for the fact that x3 ∈ S1, in this section we present a

variation of the controller solving CFCP in Proposition 7.4.3 which assumes that x3i ∈ R,

rather than S1, but globally asymptotically stabilizes the goal set Γ, hence solving CFCP

globally.

Proposition 7.4.5 (Global solution of CFCP for undirected information flow graph).

Assume that the information flow graph is undirected and has a globally reachable node.

Let v > 0, and let1 ϕ : Rn → Rn be defined as ϕ(y) = φ(y)y, where c : Rn → (0,+∞) is

a locally Lipschitz map such that supRn ‖ϕ‖ < v. Then, there exists K⋆ > 0 such that,

1One possible choice of function ϕ is ϕ(y) = v1 min1, 1/‖y‖y, with 0 < v1 < v.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

1

1

2

2

3

3

4

45

5

6

6

x1

x2

Figure 7.8: CFCP Simulation - B: Case I

for all K ∈ (0, K⋆), the feedback

ui1 = v − ϕi(L(x3 − α))

ui2 =ui1r

−Khi(χ), i = 1, · · · , n,(7.29)

where h(χ) is defined in (7.14) and ϕi denotes the i-th component of ϕ, globally asymp-

totically stabilizes Γ in (7.7) and solves CFCP globally when the state space is taken to

be X = R3n.

Proof. Feedback (7.29) results from the feedback transformation (7.10), setting u = 1v−

ϕ(L(x3−α)), and u = −Kh(χ). By Theorem 5.2.2 , to show that this feedback solves the

circular formation control problem globally we need to show: (i) Feedback (7.29) provides

Γ-detectability; (ii) the closed-loop system (7.1), (7.29) is locally uniformly bounded near


Γ; and, (iii) all trajectories of the closed-loop system are bounded and the unicycles have

a common asymptotic centre of rotation.

The proof of properties (ii) and (iii) is identical to the argument presented in the

proof of Proposition 7.4.3. We will now show that the system with input u and output

y = h(χ) is Γ-detectable. The definition of ϕ guarantees that ui = v − ϕi(L(x3 − α)),

i = 1, . . . , n, are bounded away from zero. By Lemma 7.4.1, we have O = Γ1. The

dynamics on O, setting u = 0, are given by

ci1 = 0

ci2 = 0

xi3 =v

r− ϕi(L(x3 − α))/r,

(7.30)

for i = 1, · · · , n. Solutions onO are defined for all t ≥ 0 because |xi3| < 2v/r, i = 1, . . . , n.

Consider the following candidate Lyapunov function

W (x3) =1

2

n∑

i=1

(

Li(x3 − α)))2.

Since W ≥ 0 and W = 0 if and only if Li(x3−α) = 0, i = 1, · · · , n, we need to show that

the set W−1(0) is asymptotically stable relative to O. Using the fact that kerL = span 1,

the derivative of W along (7.30) is

W = −1

r(L(x3 − α))⊤Lϕ(L(x3 − α)).

Since L is positive semidefinite and the vectors L(x3 − α) and ϕ(L(x3 − α)) are parallel

and have positive inner product, W ≤ 0 and so W is nonincreasing along solutions on O.

Moreover, W = 0 if L(x3 − α) ∈ kerL. Since L is symmetric, kerL = (ImL)⊥, and thus

W = 0 if and only if x3 − α ∈ kerL. In other words, x3 : W = 0 =W−1(0). Since, for

all ǫ > 0, infχ:W (χ)≥ǫ W < 0, it follows that the set W−1(0) is globally asymptotically

stable, which proves Γ-detectability.


7.5 Case II: Circulant information flow graph

In this section we present a reduction-based control design to solve the circular formation

control problem when the information flow digraph Laplacian is no longer undirected, but

is circulant, i.e., it takes the form (see [19])

L =

l1 l2 · · · ln

ln l1 · · · ln−1

......

...

l2 l3 · · · l1

In the development that follows we will need the next Lemma.

Lemma 7.5.1. If the Laplacian L of a digraph G with a globally reachable node is

circulant, then the matrix L + L⊤ is positive semidefinite with a simple eigenvalue at 0

with geometric multiplicity 1.

Proof. If L is circulant, then L⊤ is also the Laplacian of a graph with the same node set,

which we denote G⊤. Therefore, L+L⊤ is the Laplacian of a graph with the same nodes

as those of G, and whose arcs are the arcs of G and those of G⊤. Such graph, therefore,

has a globally reachable node, and its Laplacian L+ L⊤ has one eigenvalue at zero with

geometric multiplicity 1 and n− 1 positive eigenvalues.

Remark 7.5.2. Lemma 7.5.1 is not applicable to digraphs with non-circulant Laplacians

because if L is the Laplacian of G, in general it is not true that L⊤ is the Laplacian of a

digraph. As an example, consider the digraph in Figure 7.1. In this case, we have

L⊤ =

2 0 −1 0

−1 1 0 0

0 −1 1 −1

−1 0 0 1

.


Since the row sums of L⊤ are not all zero, L⊤ is not the Laplacian of a digraph, and in

fact L+ L⊤ has one negative eigenvalue.

The passivity-based approach presented in Section 7.4 cannot be used to derive dis-

tributed control laws solving CFCP when the graph is directed, and hence its Laplacian

L is not symmetric. To see why this is the case, we recall that the storage function is

given by

V = c⊤L(2) c =1

2c⊤(L(2) + L⊤

(2)) c.

By Lemma 7.5.1, this storage is positive semidefinite and V −1(0) = Γ1, as before. Using

the feedback transformation (7.10), the passive output is now given by

y = −rR(x3)(L(2) + L⊤(2)) c.

Choosing a PBF u based on this new output would result in a feedback that violates

the information structure constraints. For example, using the feedback u = −y, we have

that ui depends on the rows 2i− 1 and 2i of (L(2) + L⊤(2)). These are different from the

corresponding rows of Li(2), unless L is symmetric.

The considerations above suggest that in order to generate distributed control laws

solving CFCP, we should replace the PBF u = −Kh(χ) in (7.22) by a suitable distributed

feedback that asymptotically stabilizes the set Γ1, and then invoke the reduction principle

for asymptotic stability of Section 4.4 to deduce asymptotic stability of the goal set Γ

in (7.7). Our design strategy is as follows:

(i) Using the feedback transformation (7.10) and letting u be defined as in (7.20),

we show that the result of Lemma 7.4.2 holds when L is circulant, i.e, that Γ is

asymptotically stable relative to Γ1 when u = 0.

(ii) We design a distributed feedback u(χ) which asymptotically stabilizes Γ1 and guar-

antees that the closed-loop system is LUB near Γ. Moreover, we show that with

this feedback the unicycles have a common asymptotic centre of rotation.


(iii) By invoking the reduction theorem for asymptotic stability in Section 4.4, we con-

clude that the feedback in question solved CFCP.

Step 1: Asymptotic stabilization of Γ relative to Γ1

Consider system (7.1) with feedback transformation (7.10). Let u be defined as

in (7.20), and let u = 0. In the proof of Lemma 7.4.2 it was shown that the deriva-

tive of the function W in (7.18) along solutions of the closed-loop system is given by

W = −v1S(x3)⊤LS(x3) = −v12S(x3)

⊤(L+ L⊤)S(x3).

Since, by Lemma 7.5.1, L + L⊤ is positive semidefinite and has one eigenvalue at zero

with geometric multiplicity one, the proof of Lemma 7.4.2 shows that the set Γ is asymp-

totically stable relative to Γ1.

Step 2: Stabilization of Γ1 and LUB property

Consider system (7.1) with feedback transformation (7.10). Let u be defined as

in (7.20), and let

u(χ) = KR(x3)ϕ(L(2)c(χ)),

where K > 0 and ϕ(y) = φ(y)y, with φ : R2n → (0,+∞) a locally Lipschitz function

such that supy∈R2n ‖φ(y)y‖ < (v − v1)/(2Kr). Since

xi3 =ui

r+ ui

=v

r− v1

rsin(Li(x3 − α)) +K[0 0 · · · cosxi3 sin xi3 · · · 0 0]φ(L(2)c(χ))L(2)c(χ),

our choice of φ guarantees that ‖x3‖ ≥ µ > 0 for some µ > 0. Next, the dynamics of the

centres of rotation are given by

c = −rKR(x3(t))⊤u(χ) = −rKφ(L(2)c(χ))R(x3)

⊤R(x3)L(2)c(χ)

= −rKφ(L(2)c(χ))R(x3(t))L(2)c(χ).

The above can be viewed as a time-varying system whose time-dependency is brought

about by the signal x3(t). Our discussion in the rest of this section will follow closely the


arguments of the proof of Proposition 7.4.3. The averaged system is

cavg = −rKφ(L(2)cavg)RL(2)cavg,

where R is, as before, positive definite. Using the coordinate transformations z = P−1c,

zavg = P−1cavg, and partitioning z = (z, z), zavg = (zavg, zavg) as before, we obtain

˙z = Kφ(L(2)Pz)A12(t)z

˙z = Kφ(L(2)Pz)A22(t)z

˙zavg = Kφ(L(2)Pzavg)A12zavg

˙zavg = Kφ(L(2)Pzavg)A22zavg,

where the matrix A22 is Hurwitz. Note that, by the definition of P in (7.28), the terms

L(2)Pz and L(2)Pzavg are linear functions of only z and zavg, respectively. Since the real-

valued function φ(·) is bounded away from zero on any compact set, the origin of the zavg

subsystem is exponentially stable and globally asymptotically stable. By the averaging

theorem, for small enough K the linear time-varying system with matrix KA22(t) is

globally exponentially stable. This fact implies that for small enough K the origin of the

z subsystem is exponentially stable and globally uniformly asymptotically stable. We

thus have, as before, that the unicycles have a common asymptotic centre of rotation

and there exists M > 0 such that for all χ(0) ∈ X , ‖c(χ(t))‖ ≤ M‖L(2)c(χ(0))‖, thus

proving that the closed-loop system is LUB near Γ.

Step 3: Solution of CFCP

The arguments presented in the previous two steps and the reduction principle for

asymptotic stability of Section 4.4 yield the following result.

Proposition 7.5.3. Assume that the information flow graph has a circulant Laplacian

with a globally reachable node. Let v > v1 > 0 and φ : R2n → (0,+∞) be a locally

Lipschitz function such that supy∈R2n ‖φ(y)y‖ <∞. Then, there exists K⋆ > 0 satisfying

supy∈R2n ‖φ(y)y‖ < (v − v1)/(2K⋆r) such that for all K ∈ (0, K⋆) the feedback

ui1 = v − v1 sin(Li(x3 − α))

ui2 =ui1r

+Kφ(L(2)c(χ))[

cosxi3L2i−1(2) c(χ) + sin xi3L

2i(2)c(χ)

]

, i = 1, . . . , n

(7.31)


solves CFCP and renders the goal set Γ in (7.7) asymptotically stable, and Γ1 in (7.3)

globally asymptotically stable for the closed-loop system.

Remark 7.5.4. If we replace the expression for ui1 in (7.31) by that in (7.29) and we take

the state space to be X = R3n, then the set Γ becomes globally asymptotically stable

relative to Γ1, and the feedback above solves CFCP globally.

7.5.1 Simulations

Figures 7.9 and 7.10 show the simulations results for A. and B., given in Section 7.4.1,

using feedback 7.31 with the following parameters: r = 1, v = 1, v1 = 0.2, K = 0.7 and

L =

1 −1 0 0 0 0

0 1 −1 0 0 0

0 0 1 −1 0 0

0 0 0 1 −1 0

0 0 0 0 1 −1

−1 0 0 0 0 1

.

The function φ : R2n → (0,+∞) is chosen as

φ(y) =

c ‖y‖ ≤ c

c2/‖y‖ ‖y‖ ≤ c

where c =√

0.99(v − v1)/Kr.

7.6 Case III: General information flow graph

The solution of CFCP in the case of circulant information flow digraph relies on the feed-

back transformation (7.10) and the design of two feedbacks u(χ) and u(χ). The feedback

u(χ) asymptotically stabilizes Γ relative to Γ1, while the feedback u(χ) asymptotically

stabilizes Γ1 and yields the LUB property. The stability analysis for the feedback u(χ)


−3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3

4

1

12

23

3

4

4

5

5

6

6

x1

x2

Figure 7.9: CFCP Simulation - A: Case II

does not rely on the fact that the graph Laplacian L is circulant, and is therefore appli-

cable to general information flow graphs which have a globally reachable node. On the

other hand, the analysis for feedback u is based on Lemma 7.4.2, whose proof relies on

the circulant property. In this section we develop a different analysis proving that the

feedback u(χ) in Lemma 7.4.2 stabilizes Γ relative to Γ1 even when L is not circulant,

and thus the distributed feedback (7.31) solves CFCP in the general case of information

flow graphs with a globally reachable node.

Proposition 7.6.1. Assume that the information flow graph has a globally reachable


−3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3

4

1

1

2

2

3

3

3

4

4

5

5

6

6

x1

x2

Figure 7.10: CFCP Simulation - B: Case II

node. Let v > v1 > 0 and φ : R2n → (0,+∞) be a locally Lipschitz function such

that supy∈R2n ‖φ(y)y‖ < ∞. Then, there exists K⋆ > 0 satisfying supy∈R2n ‖φ(y)y‖ <

(v−v1)/(2K⋆r) such that for allK ∈ (0, K⋆) the feedback (7.31) solves CFCP and renders

the goal set Γ in (7.7) asymptotically stable, and Γ1 in (7.3) globally asymptotically stable

for the closed-loop system.

Proof. We only need to show that the set Γ is asymptotically stable relative to Γ1. The

dynamics on Γ1 are described by x3 = u(χ)/r, with u(χ) defined in (7.20). Letting

S = x3 : L(x3 − α) = 0,


we need to show that S is asymptotically stable for the x3 dynamics. Recall the definition

of S(x3),

S(x3) =

sin(L1(x3 − α))

...

sin(Ln(x3 − α))

.

We have

S(x3) =

cos(L1(x3 − α))L1u/r

...

cos(Ln(x3 − α))Lnu/r

=1

r

cos(L1(x3 − α)) 0 · · · 0

.... . .

...

0 · · · 0 cos(Ln(x3 − α))

L u.

Since kerL = 1, substituting u from (7.20) in the above we get

S = −v1r

cos(L1(x3 − α)) 0 · · · 0

.... . .

...

0 · · · 0 cos(Ln(x3 − α))

LS

= −v1rLS − v1

r∆(x3)LS,

(7.32)

where

∆(x3) =

cos(L1(x3 − α))− 1 0 · · · 0

.... . .

...

0 · · · 0 cos(Ln(x3 − α))− 1

,

is equal to zero on Γ. Recall that the Laplacian matrix L has one eigenvalue at zero

with eigenvector 1 and all its other eigenvalues have positive real part. Consider the

coordinate transformation Rn → R× Rn−1,

S 7→ col(S, S) = P−1S, P =

1

∣

∣

∣

∣

∣

01×n−1

In−1

,


which gives

˙S = A21S +∆1(x3)S

˙S = A22S +∆2(x3)S,

where col(∆1(x3),∆2(x3)) = −v1rP−1∆(x3)LP and the matrix A22 is Hurwitz. Since

∆1, ∆2 are globally bounded functions, all solutions of the system above are defined for

all t ≥ 0. The S subsystem is composed of two terms: an asymptotically stable LTI

nominal part, A22S, and a globally bounded perturbation, ∆2(x3)S, with the property

that ∆2(x3) = 0 when L(x3 − α) = 0. Letting

N = x3 : cos(Li(x3 − α)) > min0, cos(2π/n), i = 1, . . . , n,

we claim that

S := x3 : L(x3 − α) = 0 = x3 ∈ N : S(x3) = 0,

from which it follows that, on S, ∆2(x3) = 0. Obviously, S ⊂ x3 ∈ N : S(x3) = 0.

Suppose that x3 ∈ N is such that S(x3) = 0. Then, S(x3) = s1, for some s ∈ R or,

since x3 ∈ N , L(x3 − α) = 1 arcsin s. Since 1⊤L = 0, it must hold that n arcsin s = 0

mod 2π, and therefore, arcsin s ∈ 2πk/n : k ∈ N mod 2π. Since x3 ∈ N , the above

can only be true if arcsin s = 0, proving that L(x3 − α) = 0.

Next, let R be the positive definite solution of Lyapunov’s equation A22⊤R+RA22 =

−In−1, and define W (x3) = S(x3)⊤RS(x3). The derivative of W along solutions on Γ1 is

W = −‖S‖22 + 2S⊤R∆2(x3)S

≤ −[

1−M‖∆2(x3)‖∞]

‖S‖22

for some positive scalarM . Since ∆2(x3) = 0 on S, and since N is a neighbourhood of S,

there exists a neighbourhood W of S, with S ⊂ W ⊂ N , such that ‖∆2(x3)‖∞ < 1/M

on W, and therefore the set x3 ∈ N : S(x3) = 0 is exponentially stable for the x3

dynamics or, what is the same, Γ is exponentially stable relative to Γ1.


7.6.1 Simulations

Figures 7.11 and 7.12 show the simulations results for A. and B., given in Section 7.4.1,

using feedback 7.31 with the following parameters: r = 1, v = 1, v1 = 0.14, K = 1.9 and

L =

1 0 −1 0 0 0

0 1 −1 0 0 0

0 −1 1 0 0 0

0 0 0 1 −1 0

0 −1 0 0 2 −1

0 −1 0 0 0 1

.

The function φ : R2n → (0,+∞) is set as in Section 7.5.1.


−3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3

4

1

1

2

2

3

3

4

4

5

5

6

6

x1

x2

Figure 7.11: CFCP Simulation - A: Case III


−3 −2 −1 0 1 2 3−4

−3

−2

−1

0

1

2

3

4

1

1 2

2

3

3

4

4

5

5

6

6

x1

x2

Figure 7.12: CFCP Simulation - B: Case III

Chapter 8

Conclusion

Set stabilization is a central problem in control science and engineering which finds

application in areas like robotics and aerospace engineering. Despite its importance,

set stabilization has received less attention in the control literature as compared to other

classical problems such as equilibrium stabilization and reference tracking. The main

objective of this thesis is the investigation of set stabilization from the perspective of

passivity theory. The reason for doing so is that the passivity paradigm represents one of

the most successful control design approaches in nonlinear control, and can be elegantly

tied to the modelling framework of port-Hamiltonian systems.

This thesis makes three main contributions to the set stabilization problem for passive

systems. The first contribution is the development, in Chapter 4, of three reduction

principles for stability, attractivity, and asymptotic stability, which we envision will be

useful general-purpose tools in nonlinear control theory. Our second main contribution,

stemming from the reduction principles, is the definition, in Chapter 5, of a new notion of

detectability which is the main system property characterising whether or not a passivity-

based feedback stabilizes a given goal set Γ contained in the zero level set of the storage

function. When the goal set is compact, this detectability property is both necessary

and sufficient for the stabilizability of the goal set by passivity-based feedback. The key

144

Chapter 8. Conclusion 145

result in Theorem 5.2.2 generalises previous results on passivity-based stabilization even

when the goal set is an equilibrium, and enables a control design procedure, presented

in Chapter 6, which separates the control freedom into two parts, one used to enforce

detectability, and the other used for passivity-based control. The third main contribution

of this thesis is the application of the control design procedure to solve a challenging

distributed coordination problem for a network of unicycles. This result is presented in

Chapter 7.

A number of research directions stem out of this work.

Our reduction theorems address the cases of local and global stability of the goal

set, but not the case of almost-global asymptotic stability (i.e., when the domain of

attraction of the goal set is the entire state space minus a set of measure zero). Consider

the reduction principle provided in Theorem 4.4.8. Assume that all the conditions in the

theorem apply except for (i) which is replaced by

(i)’ Γ1 is almost-globally asymptotically stable relative to Γ2.

Then the problem is to find any extra conditions needed to insure that Γ1 is almost-

globally asymptotically stable. We conjecture that no extra conditions are needed to

extend the reduction principle in Theorem 4.4.8 to obtain a reduction theorem for al-

most global stability. If that were the case, then we would have conditions to solve the

passivity-based set stabilization problem. In other words, we conjecture the following:

Conjecture: Consider system (5.1) with a passivity-based feedback of the form (3.3).

If Γ is compact and all trajectories of the closed-loop system are bounded, then Γ is

almost-globally asymptotically stable for the closed-loop system if system (5.1) is almost

Γ-detectable. If Γ is unbounded and the closed-loop system is locally uniformly bounded

near Γ, and all trajectories of the closed-loop system are bounded, then Γ is almost-

globally asymptotically stable for the closed-loop system if system (5.1) is almost Γ-

detectable.

This problem has practical implications, because it often happens that the obstacle


to global stability is the topology of the state space, but in such a situation almost

global stability may still be viable. Take for instance a simple pendulum with angle θ,

whose state space is the cylinder X = (θ, θ) ∈ S1 × R. It is impossible to globally

asymptotically stabilize an equilibrium (θ, 0) by continuous feedback, but it is possible

to almost globally stabilize it.

Another extension to our theory that would be interesting to pursue is the develop-

ment of stability and stabilization results that do not require boundedness of trajectories.

This would be particularly useful in addressing stabilization of unbounded system be-

haviors like in path following of unbounded paths. In the case of unbounded trajectories,

one can no longer use the notions of limit set and prolongational limit set to characterize

convergence and uniform convergence. This provides interesting challenges.

Passivity-based stabilization is central in the area of control of Euler-Lagrange and

port-Hamiltonian systems. Important research endeavours in this field include what is

known as interconnection and damping assignment passivity-based control schemes. An

interesting research direction would be to address the set stabilization problem for port-

Hamiltonian systems. We conjecture that useful results could arise from investigating the

connection between our work and the interconnection and damping assignment passivity-

based schemes.

In Chapter 7 we have presented an application of the passivity-based set stabilization

to the field of coordination of multiagent systems. The problem had a particular reduction

aspect that proved useful in applying the passivity-based set stabilizing paradigm. We

think that reduction and passivity-based set stabilization may be very helpful in solving

many problems in this area.

Appendix

Proof of Lemma 2.2.9

We first show that if Γ is a uniform semi-attractor, then it is asymptotically stable.

Suppose, by way of contradiction, that Γ is unstable. This implies that there exists ε > 0

and sequences xi ⊂ X and ti ⊂ R+, with ‖xi‖Γ → 0 such that ‖φ(ti, xi)‖Γ = ε. By

Lemma 4.5.1, we can assume, without loss of generality, that xi is bounded and has a

limit x ∈ Γ. Using x and ε in the definition of uniform semi-attractivity, we get λ > 0

and T > 0 such that φ([T,+∞), Bλ(x)) ⊂ Bε(Γ). For sufficiently large i, xi ∈ Bλ(x) and

therefore, necessarily, 0 < ti < T . Having established that ti is a bounded sequence,

we can assume that ti has a limit τ < ∞. Since Γ is positively invariant, φ(τ, x) ∈ Γ.

This gives a contradiction since φ(ti, xi) → φ(τ, x) and, for all i, ‖φ(ti, xi)‖Γ = ε.

Next we show that if Σ is locally uniformly bounded near Γ and Γ is asymptotically

stable, then Γ is a uniform semi-attractor for Σ. By Proposition 2.3.5, we need to show

that there exists a neighbourhood N (Γ) such that J+(N (Γ)) ⊂ Γ. By local uniform

boundedness, for all x in a neighbourhood of Γ, J+(x) 6= ∅. Moreover, since Γ is an

attractor, by Proposition 2.3.4 we have J+(x) ⊂ J+(L+(x)) ⊂ J+(Γ). Therefore, to

prove uniform semi-attractivity it is enough to show that J+(Γ) ⊂ Γ. Consider an

arbitrary point x ∈ Γ, and let p ∈ J+(x). By local uniform boundedness, there exist

positive constants λ and m such that φ(R+, Bλ(x)) ⊂ Bm(x). By the definition of

prolongational limit set, there exist sequences xn ⊂ X and tn ⊂ R+, with xn → x

147


and tn → +∞, such that φ(tn, xn) → p. Without loss of generality, we can assume

that xn ⊂ Bλ(x). Take a decreasing sequence εn ⊂ R+, with εn → 0. By the

stability of Γ, there exists a nested sequence of neighborhoods Nn+1(Γ) ⊂ Nn(Γ) such

that φ(R+,Nn(Γ)) ⊂ Bεn(Γ). SinceNn(Γ)∩Bλ(x) is a bounded set, for each n there exists

δn > 0 such that Bδn(Γ)∩Bλ(x) ⊂ Nn(Γ)∩Bλ(x). We thus obtain a decreasing sequence

δn, δn → 0, such that φ(R+, Bδn(x)) ⊂ Bm(x) ∩Bεn(Γ). Take subsequences xnk and

Bδnk(x) such that, for each k, xnk

∈ Bδnk(x). Since xn → x ∈ Γ, for each n there are

infinitely many xn’s in Bδn(x), and therefore the subsequences just defined have infinite

elements. We have that φ(tnk, xnk

) → p and, by construction, φ(tnk, xnk

) ∈ Bεnk(Γ).

This implies that p ∈ Γ, and so J+(x) ⊂ Γ.

The proofs of the statements involving relative stability concepts are identical.

Proof of Proposition 2.3.5

We only prove sufficiency. Assume that there exists a neighbourhood N (Γ) such that

J+(N (Γ)) ⊂ Γ. By local uniform boundedness, we can assume that all trajectories

on N (Γ) are bounded, and hence for each x ∈ N (Γ), L+(x) 6= ∅. Since L+(N (Γ)) ⊂

J+(N (Γ), U) ⊂ J+(N (Γ)), we have that for each x ∈ N (Γ) [and x ∈ U ], J+(x) [J+(x, U)]

is not empty. To prove that Γ is a uniform semi-attractor, we need to show that, for all

x ∈ Γ,

(∃δ > 0)(∀ε > 0)(∃T > 0) s.t. φ([T,+∞), Bδ(x)) ⊂ Bε(Γ).

Suppose, by way of contradiction, that there exists x ∈ Γ such that

(∀δ > 0)(∃ε > 0) s.t. (∀T > 0)(∃x ∈ Bδ(x), ∃t ≥ T ) s.t. ‖φ(t, x)‖Γ ≥ ε. (8.1)

By the local uniform boundedness assumption, there exist positive λ and m such that

φ(R+, Bλ(x)) ⊂ Bm(x). We can take small enough δ that δ ≤ λ and cl(Bδ(x)) ⊂ N (Γ).

Let ε > 0 be as in (8.1). Take a sequence Ti ⊂ R+, with Ti → ∞. By (8.1), there exist


sequences xi ⊂ Bδ(x) and ti ⊂ R+, with ti → ∞, such that ‖φ(ti, xi)‖Γ ≥ ε. Since

xi ∈ Bδ(x) ⊂ Bλ(x), then φ(xi, ti) ∈ Bm(x). By boundedness of xi and φ(ti, xi),

we can assume that xi → x⋆ ∈ cl(Bδ(x)), and φ(ti, xi) → p, with ‖p‖Γ ≥ ε. We

have thus obtained that there exists x⋆ ∈ cl(Bδ(x)) such that J+(x⋆) 6⊂ Γ. However,

cl(Bδ(x)) ⊂ N (Γ), and so J+(cl(Bδ(x))) ⊂ Γ, a contradiction.

The proof that Γ is a uniform semi-attractor relative to U if there exists N (Γ) such

that J+(N (Γ), U) ⊂ Γ is identical.

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