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.
Chapter 1. Introduction 3
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)
Chapter 1. Introduction 4
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
Chapter 1. Introduction 5
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
Chapter 1. Introduction 6
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.
Chapter 1. Introduction 7
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
Chapter 1. Introduction 8
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 .
Chapter 1. Introduction 9
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.
Chapter 1. Introduction 10
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
Chapter 1. Introduction 11
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.
Chapter 1. Introduction 12
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.
Chapter 1. Introduction 13
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.
Chapter 1. Introduction 14
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
Chapter 1. Introduction 15
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.
Chapter 1. Introduction 16
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
Chapter 1. Introduction 17
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
Chapter 1. Introduction 18
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.
Chapter 1. Introduction 19
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
Chapter 1. Introduction 20
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.
Chapter 1. Introduction 21
• 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).
Chapter 2. Preliminaries 24
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)
Chapter 2. Preliminaries 25
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.
Chapter 2. Preliminaries 26
(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
Chapter 2. Preliminaries 27
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
Chapter 2. Preliminaries 28
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
Chapter 2. Preliminaries 29
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
Chapter 2. Preliminaries 30
-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).
Chapter 2. Preliminaries 31
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.
Chapter 2. Preliminaries 32
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).
Chapter 2. Preliminaries 33
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).
Chapter 2. Preliminaries 34
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.
Consider the control-affine system
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).
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 37
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)
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 38
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.
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 39
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) =
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 40
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.
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 41
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.
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 42
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.
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 43
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,
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 44
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
,
Chapter 3. Passivity-Based Set Stabilization I: Preliminaries 45
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.
Chapter 4. Reduction Principles 48
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.
Chapter 4. Reduction Principles 49
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:
(i) Γ1 is asymptotically stable relative to Γ2,
(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].
Chapter 4. Reduction Principles 50
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.
Chapter 4. Reduction Principles 51
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,
Chapter 4. Reduction Principles 52
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:
(i) Γ1 is asymptotically stable relative to Γ2,
(ii) Γ2 is locally stable near Γ1,
(iii) If Γ1 is unbounded, then Σ is locally uniformly bounded near Γ1.
Condition (ii) is also necessary.
Chapter 4. Reduction Principles 53
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.
Chapter 4. Reduction Principles 54
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.
Chapter 4. Reduction Principles 55
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,
(ii) Γ2 is locally stable near Γ1,
(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.
Chapter 4. Reduction Principles 56
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
Chapter 4. Reduction Principles 57
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.
Chapter 4. Reduction Principles 58
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.
4.5.2 Proof of Theorem 4.4.6
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
Chapter 4. Reduction Principles 59
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].
4.5.3 Proof of Theorem 4.4.8
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
Chapter 4. Reduction Principles 60
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.
Chapter 4. Reduction Principles 61
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.
Chapter 4. Reduction Principles 62
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
Chapter 4. Reduction Principles 63
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].
Chapter 5. Passivity-Based Set Stabilization II: Theory 66
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).
Chapter 5. Passivity-Based Set Stabilization II: Theory 67
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.
Chapter 5. Passivity-Based Set Stabilization II: Theory 68
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)
Chapter 5. Passivity-Based Set Stabilization II: Theory 69
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.
Chapter 5. Passivity-Based Set Stabilization II: Theory 70
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.
Chapter 5. Passivity-Based Set Stabilization II: Theory 71
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
Chapter 5. Passivity-Based Set Stabilization II: Theory 72
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]).
Chapter 5. Passivity-Based Set Stabilization II: Theory 73
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.
Chapter 5. Passivity-Based Set Stabilization II: Theory 74
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).
Chapter 5. Passivity-Based Set Stabilization II: Theory 75
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
Chapter 5. Passivity-Based Set Stabilization II: Theory 76
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
Chapter 5. Passivity-Based Set Stabilization II: Theory 77
−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)
Chapter 5. Passivity-Based Set Stabilization II: Theory 78
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
Chapter 5. Passivity-Based Set Stabilization II: Theory 79
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)
Chapter 5. Passivity-Based Set Stabilization II: Theory 80
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)
Chapter 5. Passivity-Based Set Stabilization II: Theory 81
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).
Chapter 5. Passivity-Based Set Stabilization II: Theory 82
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
Chapter 5. Passivity-Based Set Stabilization II: Theory 83
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
Consider the control-affine system
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)⊤.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 86
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.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 87
(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.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 88
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)⊤
Chapter 6. Passivity-Based Set Stabilization III: Control Design 89
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)
Chapter 6. Passivity-Based Set Stabilization III: Control Design 90
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
Chapter 6. Passivity-Based Set Stabilization III: Control Design 91
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)
Chapter 6. Passivity-Based Set Stabilization III: Control Design 92
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].
Chapter 6. Passivity-Based Set Stabilization III: Control Design 93
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.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 94
-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.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 95
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
Chapter 6. Passivity-Based Set Stabilization III: Control Design 96
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,
Chapter 6. Passivity-Based Set Stabilization III: Control Design 97
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)
Step 2: Γ-detectability enforcement.
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
Chapter 6. Passivity-Based Set Stabilization III: Control Design 98
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,
Chapter 6. Passivity-Based Set Stabilization III: Control Design 99
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.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 100
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)
Chapter 6. Passivity-Based Set Stabilization III: Control Design 101
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)
Step 2: Γ-detectability enforcement.
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
Chapter 6. Passivity-Based Set Stabilization III: Control Design 102
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).
Step 4: Passivity-based stabilization.
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)
Chapter 6. Passivity-Based Set Stabilization III: Control Design 103
−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)
Chapter 6. Passivity-Based Set Stabilization III: Control Design 104
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.
Chapter 6. Passivity-Based Set Stabilization III: Control Design 105
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-
Chapter 7. Circular Formation Control of Unicycles 108
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
Chapter 7. Circular Formation Control of Unicycles 109
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.
Chapter 7. Circular Formation Control of Unicycles 110
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.
Chapter 7. Circular Formation Control of Unicycles 111
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
Chapter 7. Circular Formation Control of Unicycles 112
(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
Chapter 7. Circular Formation Control of Unicycles 113
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
Chapter 7. Circular Formation Control of Unicycles 114
(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)
Chapter 7. Circular Formation Control of Unicycles 115
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
Chapter 7. Circular Formation Control of Unicycles 116
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
]⊤
.
Chapter 7. Circular Formation Control of Unicycles 117
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
Chapter 7. Circular Formation Control of Unicycles 118
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)
Chapter 7. Circular Formation Control of Unicycles 119
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)
Step 2: Γ-detectability enforcement.
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)
Chapter 7. Circular Formation Control of Unicycles 120
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)
Chapter 7. Circular Formation Control of Unicycles 121
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
Chapter 7. Circular Formation Control of Unicycles 122
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.
Step 3: Passivity-based stabilization.
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.
Chapter 7. Circular Formation Control of Unicycles 123
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)
Chapter 7. Circular Formation Control of Unicycles 124
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
Chapter 7. Circular Formation Control of Unicycles 125
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
Chapter 7. Circular Formation Control of Unicycles 126
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
Chapter 7. Circular Formation Control of Unicycles 127
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.
Chapter 7. Circular Formation Control of Unicycles 128
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
Chapter 7. Circular Formation Control of Unicycles 129
−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.
Chapter 7. Circular Formation Control of Unicycles 130
−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
Chapter 7. Circular Formation Control of Unicycles 131
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.
Chapter 7. Circular Formation Control of Unicycles 132
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
.
Chapter 7. Circular Formation Control of Unicycles 133
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.
Chapter 7. Circular Formation Control of Unicycles 134
(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
Chapter 7. Circular Formation Control of Unicycles 135
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)
Chapter 7. Circular Formation Control of Unicycles 136
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)
Chapter 7. Circular Formation Control of Unicycles 137
−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
Chapter 7. Circular Formation Control of Unicycles 138
−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,
Chapter 7. Circular Formation Control of Unicycles 139
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
,
Chapter 7. Circular Formation Control of Unicycles 140
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.
Chapter 7. Circular Formation Control of Unicycles 141
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.
Chapter 7. Circular Formation Control of Unicycles 142
−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
Chapter 7. Circular Formation Control of Unicycles 143
−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
Chapter 8. Conclusion 146
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
Chapter 8. Conclusion 148
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
Chapter 8. Conclusion 149
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.
Bibliography
[1] M. Aicardi, G. Casalino, A. Bicchi, and A. Balestrino. Closed loop steering of
unicycle-like vehicles via Lyapunov techniques. IEEE Robotics & Automation Mag-
azine, 2:27–35, 1995.
[2] F. Albertini and E.D. Sontag. Continuous control-Lyapunov functions for asymp-
totically controllable time-varying systems. International Journal of Control,
72:1630–1641, 1999.
[3] B. D. O. Anderson and S. Vongoanitlerd. Network Analysis and Synthesis. Prentice-
Hall, Engelwood Cliffs, 1973.
[4] Z. Artstein. Stabilization with relaxed controls. Nonlinear Analysis, Theory Meth-
ods and Applications, 7:1163–1173, 1983.
[5] K. J. Astrom and B. Wittenmark. Adaptive Control. Dover Publications, 2008.
[6] A. N. Atassi and H. Khalil. A separation principle for the stabilization of a class
of nonlinear systems. IEEE Transactions on Automatic Control, 44(9):1672–1687,
September 1999.
[7] J. P. Aubin. Viability Theory. Birkhauser, 1991.
[8] A. Banaszuk and J. Hauser. Feedback linearization of transverse dynamics for
periodic orbits. Systems and Control Letters, 29:95–105, 1995.
150
Bibliography 151
[9] J. Bang-Jensen and G. Gutin. Digraphs: Theory, Algorithms and Applications.
Springer-Verlag, 2002.
[10] N. P. Bathia and G. P. Szego. Dynamical Systems: Stability Theory and Applica-
tions. Springer-Verlag, Berlin, 1967.
[11] N. P. Bathia and G. P. Szego. Stability Theory of Dynamical Systems. Springer-
Verlag, Berlin, 1970.
[12] R. Bhatia. Matrix Analysis. Springer-Verlag, 1996.
[13] G. D. Birkhoff. Dynamical Systems. American Mathematical Society Colloquium
Publications, 1927.
[14] C. Byrnes, A. Isidori, and J. C. Willems. Passivity, feedback equivalence, and
the global stabilization of nonlinear systems. IEEE Transactions on Automatic
Control, 36:1228–1240, 1991.
[15] N. Ceccarelli, M. Di Marco, A. Garulli, and A. Giannitrapani. Collective circular
motion of multi-vehicle systems with sensory limitations. In Proc. 44th IEEE Conf.
on Decision and Control, and the European Control Conference 2005, pages 740 –
745, Seville, Spain, 2005.
[16] N. Ceccarelli, M. D. Marco, A. Garulli, and A. Giannitrapani. Collective circular
motion of multi-vehicle systems. Automatica, 44(12):3025–3035, 2008.
[17] F. H. Clarke, Y. S. Ledyaev, and E. D. Sontag. Asymptotic controllability implies
feedback stabilization. IEEE Transaction on Automatic Control, 42:1394 – 140,
1997.
[18] L. Consolini, M. Maggiore, C. Nielsen, and M. Tosques. Path following for the
PVTOL aircraft. Automatica, 46:1284–1296, 2010.
Bibliography 152
[19] P. J. Davis. Circulant Matrices. Chelsea Publishing, New York, second edition,
1994.
[20] E. J. Davison. The robust control of a servomechanism problem for linear time-
invariant multivariable systems. IEEE Transactions on Automatic Control, 21:25–
34, 1976.
[21] P. de Leenheer and D. Aeyels. Stabilization of positive systems with first integrals.
Automatica, 38:1583 – 1589, 2002.
[22] A. De Luca, G. Oriolo, and C. Samson. Feedback control of a nonholonomic car-
like robot. In J.-P. Laumond, editor, Robot Motion Planning and Control, pages
173–253. Springer-Verlag, London, 1998.
[23] C. A. Desoer and M. Vidyasagar. Feedback Systems: Input-Output Properties.
Academic Press, New York, 1975.
[24] W. Ding, G. Yan, and Z. Lin. Formation on two-layer pursuit systems. In
Proc. IEEE international conference on robotics and automation, pages 3496–3501,
Koobe, Japan, 2009.
[25] P. Encarnacao and A. Pascoal. Combined trajectory tracking and path following:
an application to the coordinated control of autonomous marine craft. In Proc.
40th IEEE Conference on Decision and Control, pages 964–969, Orlando, FL, USA,
2001.
[26] I. Fantoni and R. Lozano. Stabilization of the Furuta pendulum around its homo-
clinic orbit. International Journal of Control, 75(6):390–398, 2002.
[27] J. Ferber. Multi-Agent System: An Introduction to Distributed Artificial Intelli-
gence. Harlow: Addison Wesley Longman, 1999.
[28] L. R. Foulds. Graph Theory Applications. Springer-Verlag, 1992.
Bibliography 153
[29] A. L. Fradkov. Speed-gradient scheme and its application in adaptive control.
Automation and Remote Control, 9:1333–1342, 1980.
[30] A. L. Fradkov. Swinging control of nonlinear oscillations. International Journal of
Control, 64(6):1189–1202, 1996.
[31] A. L. Fradkov. Introduction to Control of Oscillations and Chaos, volume 35 of
World Scientific Series on Nonlinear Science. Series a, Monographs and Treatises.
World Scientific Publishing Company, 1999.
[32] A. L. Fradkov, M. V. Miroshnik, and V. O. Nikiforov. Nonlinear and Adaptive
Control of Complex Systems. Klumer Academic publishers, 1999.
[33] B. A. Francis. The linear multivariable regulator problem. Siam Journal on Control
and Optimization, 14:486–505, 1977.
[34] B. A. Francis and W. M. Wonham. The internal model principle of control theory.
Automatica, 12:457–465, 1976.
[35] R. A. Freeman and P. V. Kokotovic. Robust Nonlinear Control Design: State-Space
and Lyapunov Techniques. Modern Birkhauser Classics. Birkhauser, 2008.
[36] C. Nielsen C. Fulford and M. Maggiore. Path following using transverse feedback
linearization: Application to a maglev positioning system. Automatica, 46:585–590,
2010.
[37] D. Hill and P. Moylan. The stability of nonlinear dissipative systems. IEEE Trans-
actions on Automatic Control, 21:708–711, 1976.
[38] D. Hill and P. Moylan. Stability results for nonlinear feedback systems. Automatica,
13:377–382, 1977.
[39] D. Hill and P. Moylan. Connections between finite gain and asymptotic stability.
IEEE Transactions on Automatic Control, 25:931–936, 1980.
Bibliography 154
[40] D. Hill and P. Moylan. Dissipative dynamical systems: Basic input-output and
state properties. Journal of the Franklin Institute, 309:327–357, 1980.
[41] R. Hindman and J. Hauser. Maneuver modified trajectory tracking. In Proc. of
the International Symposium on Mathematical Theory of Networks and Systems
(MTNS), St. Louis, MO, USA, 1996.
[42] A. Iggidr, B. Kalitin, and R. Outbib. Semidefinite Lyapunov functions stability
and stabilization. Mathematics of Control, Signals and Systems, 9:95–106, 1996.
[43] A. Isidori. Nonlinear Control Systems II. Springer-Verlag, London, 1999.
[44] V. Jurdjevic and J. P. Quinn. Controllability and stability. Journal of Differential
equations, 28:381–389, 1978.
[45] B. S. Kalitin. B-stability and the Florio-Seibert problem. Differential Equations,
35:453–463, 1999.
[46] B. S. Kalitin. Stability of closed invariant sets of semidynamical systems. the
method of sign definite lyapunov functions. Differential Equations, 38(11):1662–
1664, 2002.
[47] C. M. Kellett and A. R. Teel. Asymptotic controllability to a set implies locally
Lipschitz control-Lyapunov function. In Proceedings of the 39th IEEE Conference
on Decision and Control, Sydney, Australia, 2000.
[48] C. M. Kellett and A. R. Teel. Weak converse Lyapunov theorems and control-
Lyapunov functions. SIAM Journal on Control and Optimization, 42(6):1934–1959,
2004.
[49] H. K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, New Jersey,
3rd edition, 2002.
Bibliography 155
[50] L. Krick, M. E. Broucke, and B. A. Francis. Stabilization of infinitesimally rigid
formations of multi-robot networks. International Journal of Control, 82(3):423–
439, 2009.
[51] L. Lapierre, D. Soetanto, and A. Pascoal. Nonsingular path following control of
a unicycle in the presence of parametric modelling uncertainties. International
Journal of Robust and Nonlinear Control, 16(10):485–503, 2006.
[52] K. K. Lee and A. Araposthatis. Remarks on smooth feedback stabilization of
nonlinear systems. Systems & Control Letters, 10:41–44, 1988.
[53] Y. Lin, E.D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for
robust stability. SIAM Journal on Control and Optimization, 34:124–160, 1996.
[54] Z. Lin, M. Maggiore, and B. A. Francis. Necessary and sufficient graphical condi-
tons for formation control of unicycles. IEEE Transactions on Automatic Control,
20(1):121–127, 2005.
[55] Zhiyun Lin. Coupled Dynamic Systems: From Structure Towards Stability And
Stabilizability. PhD thesis, University of Toronto, 2006.
[56] L. Marconi, L. Praly, and A. Isidori. Output stabilization via nonlinear Luenberger
observers. SIAM Journal on Control and Optimization, 45(6):2277–2298, 2007.
[57] J. A. Marshall, M. E. Broucke, and B. A. Francis. Formations of vehicles in cyclic
pursuit. IEEE Transactions on Automatic Control, 49:1963–1974, 2004.
[58] J. A. Marshall, M. E. Broucke, and B. A. Francis. Pursuit formations of unicycles.
Automatica, 42(1):3–12, 2006.
[59] A. Meystel. Autonomous Mobile Robots: Vehicles With Cognitive Control. World
Scientific Series in Automation. World Scientific Publishing Company, 1993.
Bibliography 156
[60] C. Nielsen and M. Maggiore. Output stabilization and maneuver regulation: A
geometric approach. Systems and Control Letters, 55:418–427, 2006.
[61] C. Nielsen and M. Maggiore. On local transverse feedback linearization. SIAM
Journal on Control and Optimization, 47(5):2227–2250, 2008.
[62] R. Ortega, A. Loria, P.J. Nicklasson, and H. Sira-Ramirez. Passivity-based control
of Euler-Lagrange Systems. Springer-Verlag, London, 1998.
[63] R. Ortega, A. J. van der Schaft, I. Mareels, and B. Maschke. Putting energy back
in control. Control Systems Magazine, 21(2):18–33, April 2001.
[64] R. Ortega, A. J. van der Schaft, B.M. Maschke, and G. Escobar. Interconnection
and damping assignment passivity-based control of port-controlled Hamiltonian
systems. Automatica, 38:585–596, 2002.
[65] D. A. Paley. Stabilization of collective motion in a uniform and constant flow field.
In Proc. of the AIAA Guidance, Navigation and Control Conference and Exhibit,
no. AIAA-2008-7173, Honolulu, Hawaii, 2008.
[66] D. A. Paley, N. Leonard, and R. Sepulchre. Collective motion of self propelled
particles: Stabilizing symmetric formations on closed curves. In Proc. IEEE 45th
Conference on Decision and Control, pages 5067–5072, San Diego-CA, U.S., 2006.
[67] D. A. Paley, N. E. Leonard, and R. Sepulchre. Stabilization of symmetric formations
to motion around convex loops. Systems and Control Letters, 57(3):209–215, 2008.
[68] D. A. Paley and C. Peterson. Stabilization of collective motion in a time-invariant
flowfield. Journal of Guidance, Control, and Dynamics, 32(3):771–779, 2009.
[69] M. Pavone and E. Frazzoli. Decentralized policies for geometric pattern forma-
tion and path coverage. Journal of Dynamic Systems, measurement and control,
129(5):633–643, 2007.
Bibliography 157
[70] F. Plestan, J. W. Grizzle, E. R. Westervelt, and G. Abba. Stable walking of a 7-
DOF biped robot. IEEE Transactions on Robotics and Automation, 19(4):653–668,
2003.
[71] K. J. Astrom and K. Furuta. Swinging up a pendulum by energy control. Auto-
matica, 36:287–295, 2000.
[72] L. Rifford. Existence of Lipschitz and semi-concave control Lyapunov functions.
SIAM Journal on Control and Optimization, 39:1043–1064, 2000.
[73] L. Rifford. Semiconcave control-Lyapunov functions and stabilizing feedbacks.
SIAM Journal on Control and Optimization, 41(3):659–681, 2002.
[74] R. O. Saber, J. A. Fax, and R. M. Murray. Consensus and cooperation in
networkedmulti-agent systems. Proceedings of the IEEE, 95(1):215–233, 2007.
[75] C. Samson. Control of chained systems: Application to path following and time-
varying point-stabilization of mobile robots. IEEE Transactions on Automatic
Control, 40(1):64–77, 1995.
[76] P. Seibert. On stability relative to a set and to the whole space. In Papers presented
at the 5th Int. Conf. on Nonlinear Oscillations (Izdat. Inst. Mat. Akad. Nauk.
USSR, 1970), volume 2, pages 448–457, Kiev, 1969.
[77] P. Seibert. Relative stability and stability of closed sets. In Sem. Diff. Equations
and Dynam. Systs. II; Lect. Notes Math., volume 144, pages 185–189. Springer-
Verlag, Berlin-Heidelberg-New York, 1970.
[78] P. Seibert and J. S. Florio. On the reduction to a subspace of stability properties of
systems in metric spaces. Annali di Matematica pura ed applicata, CLXIX:291–320,
1995.
Bibliography 158
[79] R. Sepulchre, D. A. Paley, and N. E. Leonard. Stabilization of planar collec-
tive motion: All-to-all communication. IEEE Transactions on Automatic Control,
52(5):1–14, 2007.
[80] R. Sepulchre, D. A. Paley, and N. E. Leonard. Stabilization of planar collective
motion with limited communication. IEEE Transactions on Automatic Control,
53(3):706–719, 2008.
[81] A. S. Shiriaev. The notion of V -detectability and stabilization of invariant sets of
nonlinear systems. Systems and Control Letters, 39:327–338, 2000.
[82] A. S. Shiriaev. Stabilization of compact sets for passive affine nonlinear systems.
Automatica, 36:1373–1379, 2000.
[83] A. S. Shiriaev and A. L. Fradkov. Stabilization of invariant sets for nonlinear
non-affine systems. Automatica, 36:1709–1715, 2000.
[84] A. S. Shiriaev and A. L. Fradkov. Stabilization of invariant sets for nonlinear
systems with application to control of oscillations. International Journal of Robust
and Nonlinear Control, 11:215–240, 2001.
[85] A. S. Shiriaev, A. Pogromsky, H. Ludvigsen, and O. Egeland. On global properties
of passivity based control of an inverted pendulum. International Journal of Robust
and Nonlinear Control, 10:283–300, 2000.
[86] A. Sinha and D. Ghose. Generalization of nonlinear cyclic pursuit. Automatica,
43(11):1954–1960, 2007.
[87] R. Skjetne, A. R. Teel, and P. V. Kokotovic. Stabilization of sets parametrized by a
single variable: Application to ship maneuvering. In Proc. of the International Sym-
posium on Mathematical Theory of Networks and Systems (MTNS), Notre Dame,
IN, USA, 2002.
Bibliography 159
[88] J. J. Slotin. Applied Nonlinear Control. Prentice Hall, Upper Saddle River, NJ,
1991.
[89] E. D. Sontag. A Lyapunov-like characterization of asymptotic controllability. SIAM
Journal on Control and Optimization, 21:462–471, 1983.
[90] E. D. Sontag. A “universal” construction of Artstein theorem on nonlinear stabi-
lization. Systems and Control Letters, 13:542 – 550, 1989.
[91] E. D. Sontag. Further facts about input to state stabilization. IEEE Transactions
on Automatic Control, 35:473–476, 1990.
[92] R. Sutton and G. Roberts. Advances in Unmanned Marine Vehicles. IEE Control
Series. Institution of Engineering and Technology, 2006.
[93] A. Teel and L. Praly. Global stabilizability and observability imply semi-global
stabilizability by output feedback. Systems & Control Letters, 22:313–325, 1994.
[94] K. S. Tsakalis and P. A. Ioannou. Linear Time-Varying Systems: Control and
Adaptation. Prentice Hall, 1992.
[95] T. Ura. Sur le courant exterieur a une region invariante. Funkc. Ekvac., pages
143–200, 1959.
[96] V. I. Utkin. Sliding Modes in Control and Optimizaion. Springer-Verlag, New York,
1992.
[97] K. P. Valavanis. Advances in Unmanned Aerial Vehicles: State of the Art and the
Road to Autonomy. Intelligent Systems, Control and Automation: Science and
Engineering. Springer, 2007.
[98] A. J. van der Schaft. L2-Gain and Passivity Techniques in Nonlinear Control.
Springer Communications and Control Engineering. Springer-Verlag, London, sec-
ond edition, 2000.
Bibliography 160
[99] M. Vidyasagar. Decomposition techniques for large-scale systems with nonaddi-
tive interactions: Stability and stabilizability. IEEE Transactions on Automatic
Control, 25(4):773–779, 1980.
[100] E. R. Westervelt, J. W. Grizzle, and D. E. Koditschek. Hybrid zero dynamics of
planar biped robots. IEEE Transactions on Automatic Control, 48(1):42–56, 2003.
[101] B. Wie. Space Vehicle Dynamics and Control. Aiaa Education Series. American
Institute of Aeronautics & Astronautics, second edition, 2008.
[102] S. Willard. General Topology. Dover Publications, 2004.
[103] J. C. Willems. Dissipative dynamical systems - Part I: General theory. Arch. of
Rational Mechanics and Analysis, 45:321–351, 1972.
[104] J. C. Willems. Dissipative dynamical systems - Part II: Linear systems with
quadratic supply rates. Arch. of Rational Mechanics and Analysis, 45:352–393,
1972.
[105] D. C. Youla, L. J. Castriota, and H.J. Carlin. Bounded real scattering matrices
and the foundations of linear passive network theory. IRE Transactions on Circuit
Theory, pages 102–124, March 1959.