stabilization of non linear control systems€¦ · stabilization of non linear control systems...

Stabilization of non linear control systems

Jean-Michel Coron

Laboratory J.-L. Lions, University Pierre et Marie Curie (Paris 6)BCAM OPTPDE summer school„ Bilbao, July 4-8 2011

Chapter 1: Stabilization: General resultsChapter 2: Design tools

Chapter I

Stabilization: General results

Outline: Stabilization: General results

1 Motivation and an example

2 Some abstract results in finite dimension

The cart inverted pendulum control system

FC

The cart inverted pendulum: the equilibrium

C

Instability of the equilibrium

C

Stabilization of the equilibrium

C F : fonction de l’état

F est un feedback

Outline: Stabilization: General results

1 Motivation and an example

2 Some abstract results in finite dimension

Aleksandr Lyapunov (1857-1918)

In the following we assume that the vector fields X : Rn → Rn arecontinuous. Then the Cauchy problem ẋ = X(x) x(0) = x0 have alwaysclassical solutions. However these solutions may not be unique.

Example

We take n = 1, X(x) = |x|1/2 and x0 = 0. One has infinitely manysolutions: for every a ∈ [0,+∞), x : R → R defined by

x(t) = 0 for t in ]−∞, a],(1)

x(t) =(t− a)2

4for t in (a,+∞),(2)

is solution to the Cauchy problem.

After important works by many mathematicians (e.g. J.-L. Lagrange, P.-S.Laplace, G. Dirichlet), A. Lyapunov gave in 1892 the “good” definition of“asymptotic stability”.

Stability

Attractor

Asymptotically stable

Definition (Asymptotically stable)

Let xe be an equilibrium point of X ∈ C0(Rn;Rn), i.e. a point of Rn suchthat X(xe) = 0. One says that xe is (locally) asymptotically stable forẋ = X(x) if it is stable and attractor for ẋ = X(x). One says that xe isunstable for ẋ = X(x) if it not stable stable.

Second Lyapunov’s theorem

Theorem

The point xe is asymptotically stable for ẋ = X(x) if and only if thereexists η > 0 and V : B(xe, η) : {x ∈ Rn; |x− xe| < η} → R of class C∞such that

V (x) > V (xe), ∀x ∈ B(xe, η) \ {xe}(1)

∇V (x) ·X(x) =n∑

i=1

∂V

∂xiXi(x) < 0, ∀x ∈ B(xe, η) \ {xe}.(2)

Remark

• For the only if part: J. Massera and J. Kurzweil.• This theorem explains why “asymptotic stability” is the good notion:

asymptotic stabilization gives robustness with respect to smallperturbation.

The stabilizability problem

We consider the control system ẋ = f(x, u) where the state is x in Rn andthe control is u in Rm. We assume that f(0, 0) = 0.

Problem

Does there exists u : Rn → Rm vanishing at 0 such that 0 ∈ Rn is (locally)asymptotically stable for ẋ = f(x, u(x))? (If the answer is yes, one saysthat the control system is locally asymptotically stabilizable.)

Remark

The map u : x ∈ Rn 7→ Rm is called a feedback (or feedback law). Thedynamical system ẋ = f(x, u(x)) is called the closed loop system.

Regularity of feedback laws

The regularity of x 7→ u(x) is an important point. With u continuous,asymptotic stability implies the existence of a smooth strict Lyapunovfunction and one has robustness with respect to small actuator errors aswell as small measurement errors. See above.If u is discontinuous, one needs to define the notion of solution of theclosed loop system ẋ = f(x, u(x)) and study carefully the robustness ofthe closed loop system.

Discontinuous feedback laws

Filippov solution: H. Hermes (1967); JMC and L. Rosier (1994), E.Ryan (1994).

Euler solutions: F. Clarke, Y. Ledyaev, E. Sontag and A. Subbotin(1997); E. Sontag (1999); L. Rifford (2002, 2005).

Carathéodory solutions: F. Ancona and A. Bressan (1999, 2004).

Add extra variables in order to have more robustness (hybrid feedbacklaws): Y. Ledyaev and E. Sontag (1997); C. Prieur (2005); R. Goebel,C. Prieur and A. Teel (2007, 2009).

Unless otherwise specified, from now on we assume that the feedback lawsare continuous.

Controllability

Given two states x0 and x1, does there exist a control t ∈ [0, T ] 7→ u(t)which steers the control system from x0 to x1, i.e. such that

(

ẋ = f(x, u(t)), x(0) = x0)

⇒(

x(T ) = x1)

?(1)

Controllability of linear control systems

The control system is

ẋ = Ax+Bu, x ∈ Rn, u ∈ Rm,

where A ∈ Rn×n and B ∈ Rn×m.Theorem (Kalman’s rank condition)

The linear control system ẋ = Ax+Bu is controllable on [0, T ] if and onlyif

Span {AiBu;u ∈ Rm, i ∈ {0, 1, . . . , n− 1}} = Rn.

Remark

This condition does not depend on T . This is no longer true for nonlinearsystems and for systems modeled by linear partial differential equations.

Small time local controllability

We assume that (xe, ue) is an equilibrium, i.e., f(xe, ue) = 0. Manypossible choices for natural definitions of local controllability. The mostpopular one is Small-Time Local Controllability (STLC): the stateremains close to xe, the control remains to ue and the time is small.

xe

ε

xe

ε

η

xe

ε

η

x0

x1

xe

ε

η

x0t = 0

x1

t = ε

|u(t)− ue| 6 ε

The linear test

We consider the control system ẋ = f(x, u) where the state is x ∈ Rn andthe control is u ∈ Rm. Let us assume that f(xe, ue) = 0. The linearizedcontrol system at (xe, ue) is the linear control system ẋ = Ax+Bu with

A :=∂f

∂y(xe, ue), B :=

∂f

∂u(xe, ue).(1)

If the linearized control system ẋ = Ax+Bu is controllable, thenẋ = f(x, u) is small-time locally controllable at (xe, ue).

Stabilizability of linear controllable systems

Notations. For a matrix M ∈ Rn×n, PM denotes the characteristicpolynomial of M : PM (z) := det (zI −M).Let us denote by Pn the set of polynomials of degree n in z such that thecoefficients are all real numbers and such that the coefficient of zn is 1.One has the following theorem

Theorem (Pole shifting theorem, M. Wonham (1967))

Let us assume that the linear control system ẋ = Ax+Bu is controllable.Then

{

PA+BK ; K ∈ Rm×n}

= Pn.

Stabilizability of linear controllable systems

Notations. For a matrix M ∈ Rn×n, PM denotes the characteristicpolynomial of M : PM (z) := det (zI −M).Let us denote by Pn the set of polynomials of degree n in z such that thecoefficients are all real numbers and such that the coefficient of zn is 1.One has the following theorem

Theorem (Pole shifting theorem, M. Wonham (1967))

Let us assume that the linear control system ẋ = Ax+Bu is controllable.Then

{

PA+BK ; K ∈ Rm×n}

= Pn.

Corollary

If the linear control system ẋ = Ax+Bu is controllable, there exists alinear feedback x 7→ u(x) = Kx such that 0 ∈ Rn is (globally)asymptotically stable for the closed loop system ẋ = Ax+Bu(x).

Application to nonlinear controllable systems

We assume that f(0, O) = 0. Let us consider the linearized control systemẋ = Ax+Bu of ẋ = f(x, u) at (0, 0) ∈ Rn ×Rm:

A :=∂f

∂x(0, 0), B :=

∂f

∂u(0, 0).

Let us assume that the linearized control system ẋ = Ax+Bu iscontrollable. Then, by the pole-shifting theorem, there exists K ∈ Rm×nsuch that σ(A+BK) = {−1}. Let us consider the feedback u(x) = Kx.Then, if X(x) := f(x, u(x)), X ′(0) = A+BK. Hence 0 ∈ Rn is locallyasymptotically stable for the closed loop system ẋ = f(x, u(x)).In conclusion, if the the linearized control system is controllable, then

The control system ẋ = f(x, u) is small-time locally controllable at(0, 0).

The control system ẋ = f(x, u) is locally asymptotically stabilizable(at the equilibrium (0, 0)).

An example: The cart-inverted pendulum

ξ M

mθ

l

F

C

The Cart-inverted pendulum: The equations

Let

x1 := ξ, x2 := θ, x3 := ξ̇, x4 := θ̇, u := F,(1)

The dynamics of the cart-inverted pendulum system is ẋ = f(x, u), withx = (x1, x2, x3, x4)

tr and

f :=

x3x4

mlx24 sinx2 −mg sinx2 cos x2M +m sin2 x2

+u

M +m sin2 x2−mlx24 sinx2 cos x2 + (M +m)g sinx2

(M +m sin2 x2)l− u cos x2

(M +m sin2 x2)l

.

Stabilization of the cart-inverted pendulum

For the cart-inverted pendulum, the linearized control system around(0, 0) ∈ R4 × R is ẋ = Ax+Bu with

A =

0 0 1 00 0 0 1

0 −mgM

0 0

0(M +m)g

Ml0 0

, B =1

Ml

00l−1

.(1)

One easily checks that this linearized control system satisfies the Kalmanrank condition and therefore is controllable. Hence the cart-invertedpendulum is small-time locally controllable at (0, 0) ∈ R4 ×R and is locallyasymptotically stabilizable (at the equilibrium (0, 0)).

Obstruction to the stabilizability

Theorem (R. Brockett (1983))

If the control system ẋ = f(x, u) can be locally asymptotically stabilizedthen

(N) the image by f of every neighborhood of (0, 0) ∈ Rn × Rm is aneighborhood of 0 ∈ Rn.

An example: The baby stroller

The baby stroller: The model

x1

x2

x3

ẋ1 = u1 cosx3, ẋ2 = u1 sinx3, ẋ3 = u2, n = 3, m = 2.

The baby stroller and the Brockett condition

The baby stroller control system

ẋ1 = u1 cosx3, ẋ2 = u1 sinx3, ẋ3 = u2

is small-time locally controllable at (0, 0). (This follows from theChow-Rashevski theorem.) However (N) does not hold for the baby strollercontrol system. Hence the baby stroller control system cannot be locallyasymptotically stabilized.

Another example: The under-actuated satellite

ω̇ = J−1S(ω)Jω +

m∑

i=1

uibi, η̇ = A(η)ω,(1)

We consider again the case where m = 2. Using the fact that A(0) = Id,one easily sees that (N) never holds. However

Span {b1, b2, S(ω)J−1ω; ω ∈ Span {b1, b2}} = R3.

then the control system (1) is small-time locally controllable at(0, 0) ∈ R6 × R2. (This follows from a sufficient condition for localcontrollability proved by H. Sussmann in 1987.)

A solution: Time-varying feedback laws

Instead of u(x), use u(t, x). Note that asymptotic stability for time-varyingfeedback laws is also robust (there exists again a strict Lyapunov function).

Time-varying feedback laws for driftless control systems

Theorem (JMC (1992))

Assume that

{g(x); g ∈ Lie{f1, . . . , fm}} = Rn, ∀x ∈ Rn \ {0}.(1)

Then, for every T > 0, there exists u in C∞(R× Rn;Rm) such that

u(t, 0) = 0, ∀t ∈ R,u(t+ T, x) = u(t, x), ∀x ∈ Rn, ∀t ∈ R,

0 is globally asymptotically stable for ẋ =

m∑

i=1

ui(t, x)fi(x).

Sketch of proof

Sketch of the proof of the theorem. Let T > 0. Assume that thereexists ū in C∞(R× Rn;Rm) T -periodic with time, vanishing for x = 0,and such that, if ẋ = f(x, ū(t, x)), then

(i) x(T ) = x(0),

(ii) If x(0) 6= 0, then the linearized control system around the trajectoryt ∈ [0, T ] 7→ (x(t), ū(t, x(t))) is controllable on [0, T ].

Using (i) and (ii) one easily sees that one can construct a “small" feedbackv in C∞(R × Rn;Rm) T -periodic with time and vanishing for x = 0 suchthat, if ẋ = f(x, (ū+ v)(t, x)) and x(0) 6= 0, then |x(T )| < |x(0)|, whichimplies that 0 is globally asymptotically stable for ẋ = f(x, (ū+ v)(t, x)).

Construction of ū

In order to get (i), one just imposes on ū the condition

ū(t, x) = −ū(T − t, x), ∀(t, x) ∈ R× Rn,

which implies that x(t) = x(T − t), ∀t ∈ [0, T ], for every solution ofẋ = f(x, u(t, x)), and therefore gives x(0) = x(T ). Finally, one provesthat (ii) holds for “many" ū’s (this is the difficult part of the proof).

T t

x

˙̄x = f(x̄, ū(t, x̄))

T t

x

˙̄x = f(x̄, ū(t, x̄))

ẋ = f(x, ū(t, x) + v(t, x))

T t

x

˙̄x = f(x̄, ū(t, x̄))

ẋ = f(x, ū(t, x) + v(t, x))

bc

bc

General control systems

Definition

The origin (of Rn) is locally continuously reachable in small time for thecontrol system ẋ = f(x, u) if, for every positive real number T , there exista positive real number ε and an element u in C0

(

B̄ε;L1 ((0, T );Rm)

)

such that

Sup{|u(a)(t)|; t ∈ (0, T )} → 0 as a→ 0,((ẋ = f(x, u(a)(t)), x(0) = a) ⇒ (x(T ) = 0)),∀a ∈ B̄ε.

General control systems

Definition

The origin (of Rn) is locally continuously reachable in small time for thecontrol system ẋ = f(x, u) if, for every positive real number T , there exista positive real number ε and an element u in C0

(

B̄ε;L1 ((0, T );Rm)

)

such that

Sup{|u(a)(t)|; t ∈ (0, T )} → 0 as a→ 0,((ẋ = f(x, u(a)(t)), x(0) = a) ⇒ (x(T ) = 0)),∀a ∈ B̄ε.

Open problem

Assume that f is analytic and that ẋ = f(x, u) is small-time locallycontrollable at (0, 0) ∈ Rn × Rm. Is the origin (of Rn) locally continuouslyreachable in small time for the control system ẋ = f(x, u)?

Theorem

Let us assume that 0 is locally continuously reachable in small-time for thecontrol system ẋ = f(x, u). Then the control system ẋ = f(x, u) issmall-time locally controllable at (0, 0) ∈ Rn ×Rm.

Theorem

Let us assume that 0 is locally continuously reachable in small-time for thecontrol system ẋ = f(x, u). Then the control system ẋ = f(x, u) issmall-time locally controllable at (0, 0) ∈ Rn ×Rm.


Assume that 0 ∈ Rn is locally continuously reachable in small time for thecontrol system ẋ = f(x, u), that f is analytic and that n 6∈ {2, 3}. Then,for every positive real number T , there exist ε in (0,+∞) and u inC0(R× Rn;Rm), of class C∞ on R× (Rn \ {0}), T -periodic with respectto time, vanishing on R× {0} and such that, for every s ∈ R,

((ẋ = f(x, u(t, x)) and x(s) = 0) ⇒ (x(τ) = 0, ∀τ > s)) ,(ẋ = f(x, u(t, x)) and |x(s)| 6 ε) ⇒ (x(τ) = 0, ∀τ > s+ T )) .

Stabilization of the under-actuated satellite

ω̇ = J−1S(ω)Jω +

m∑

i=1


We consider again the case where m = 2 and assume that


Then 0 ∈ R6 is locally continuously reachable in small-time for the controlsystem the control system (1) and therefore can be locally asymptoticallystabilized by means of periodic time-varying feedback laws.

Stabilization of the under-actuated satellite

ω̇ = J−1S(ω)Jω +

m∑

i=1


We consider again the case where m = 2 and assume that


Then 0 ∈ R6 is locally continuously reachable in small-time for the controlsystem the control system (1) and therefore can be locally asymptoticallystabilized by means of periodic time-varying feedback laws. Construction ofexplicit time-varying stabilizing feedback laws:

Special cases: G. Walsh, R. Montgomery and S. Sastry (1994); P.Morin, C. Samson, J.-B. Pomet and Z.-P. Jiang (1995).

General case: JMC and E.-Y. Keraï (1996); P. Morin and C. Samson(1997).

Output feedback laws

In most practical situations, only part of of the state (called theobservation y = h(x) ∈ Rp) is measured. Hence, one cannot use u(x) oru(t, x). At a first glance, one would like to use u(h(x)) or u(t, h(x)).

Output feedback laws

In most practical situations, only part of of the state (called theobservation y = h(x) ∈ Rp) is measured. Hence, one cannot use u(x) oru(t, x). At a first glance, one would like to use u(h(x)) or u(t, h(x)).Question (separation principle): Do controllability and a goodobservability condition imply stabilizability by means of output feedbacklaws?

Observability

As least two possible definitions are possible (even for small time)(i) For every T > 0 and for every u : [0, T ] → Rm,(ẋ1 = f(x, u(t)), ẋ2 = f(x, u(t)), h(x1) = h(x2) on [0, T ])

⇒ (x1(0) = x2(0)).(ii) For every T > 0, for every (a1, a2) ∈ Rn × Rn, there existsu : [0, T ] → Rm such that(ẋ1 = f(x, u(t)), ẋ2 = f(x, u(t)), x1(0) = a1, x2(0) = a2)

⇒ (∃τ ∈ [0, T ] such that h(x1(τ)) 6= h(x1(τ))).Of course (ii) is weaker than (i). We shall use (ii). Moreover, since wewant to use a control which vanishes if the output is 0, it is natural torequire that, for every T > 0(iii) (ẋ = f(x, 0), |x(0)| small and h(x(t)) = 0, ∀t ∈ [0, T ]) ⇒

(x(0) = 0).From now on, by “observable” we mean (ii) and (iii).

u(h(x)) and u(t, h(x)) give poor results

Example (A 1-D car)

Let us consider the control system

ẋ1 = x2, ẋ2 = u, y = x1.(C)

Note that (C) is controllable and observable. However there is noy 7→ u(y) or more generally no (t, y) 7→ u(t, y) such that (0, 0) isasymptotically stable for the closed loop system ẋ1 = x2, ẋ2 = u(t, x1).(Proof: Take the divergence of (x2, u(t, x1))

tr...)

Dynamic output feedback laws

Definition

The control system

ẋ = f(x, u), y = h(x), x ∈ Rn, u ∈ Rm, y ∈ Rp

is locally stabilizable by means of dynamic output feedback laws if thereexists k ∈ N such that the control system

ẋ = f(x, u), ż = v, ỹ = (h(x); z), x ∈ Rn, z ∈ Rk,

where the state is (x; z) ∈ Rn+k, the control is (u; v) ∈ Rn+m and theobservation is ỹ ∈ Rp+k.

Dynamic output feedback laws and the 1-D car


ẋ1 = x2, ẋ2 = u, y = x1,(C)

where the state is (x1, x2)tr ∈ R2, the control is u ∈ R and the observation

is y ∈ R. Let us take k = 1. Let us choose the following general dynamiclinear output feedback law

ẋ = x2, ẋ2 = αx1 + βz, ż = γx1 + δz.(*)

The point 0 ∈ R3 is asymptotically stable if and only if the zeroes of thepolynomial

P (λ) := λ3 − δλ2 − αλ+ αδ − γβ.One can choose (α, β, γ, δ)tr ∈ R4 such that P (λ) = (λ+ 1)3.

Interest of time-varying output feedback laws

We go back to the control system

ẋ = u, y = x2.(*)

This control system is controllable and observable. However one has thefollowing proposition.

Proposition

Let k ∈ N. There are no u ∈ C0(R1+k;R), (y; z) 7→ u(y; z), andv ∈ C0(R1+k;Rk), (y; z) 7→ v(y; z), such that (0; 0) ∈ R1+k is locallyasymptotically stable for the closed loop system

ẋ = u(x2; z), ż = v(x2; z), x ∈ R, z ∈ Rk.

(One uses the convention that, if k = 0, the closed system is justẋ = u(x2) and (0; 0) ∈ R1+k is just 0 ∈ R.)

Sketch of the proof of the proposition

Let X ∈ C0(R1+k;R1+k) be defined by

X(x; z) := (u(x2; z); v(x2; z)), x ∈ R, z ∈ Rk.(2)

By a theorem due to Krasnosel′skĭı, the fact that 0 is locally asymptoticallystable for ẋ = X(x) implies de existence of ε > 0 such that, withBε :=

{

(x; z) ∈ R1+k; x2 + |z|2 < ε2}

,

X(x; z) 6= 0, ∀(x, z) ∈ R× Rk such that x2 + |z|2 = ε2,(3)degree (X,Bε, 0) = (−1)k+1.(4)

Note that, by (2),

X(x; z) = X(−x; z), ∀(x; z) ∈ Bε,(5)

from which we get that degree (X,Bε, 0) = 0, a contradiction with (4).

Output stabilization of ẋ = u, y = x2 by means of

time-varying feedback laws

t

t = 0 t = T ′ t = 2T ′

x+

x−

Separation principle and time-varying feedback laws


Assume that f and h are analytic. Assume that 0 ∈ Rn is locallycontinuously reachable in small time for ẋ = f(x, u). Assume that theobservability properties (ii) and (iii) hold. Then, there exist k ∈ N∗ suchthat, for every T > 0, there exist ε > 0, u ∈ C0(R × Rp × Rk;Rm) andv ∈ C0(R× Rp × Rk;Rk), of class C∞ on R× (Rp × Rk \ {0}),T -periodic with respect to time, vanishing on R× {0} such that, ∀s ∈ R,(

ẋ = f(x, u(t, (h(x), z))), ż = v(t, (h(x), z)), (x(s), z(s)) = 0)

⇒(

(x(τ), z(τ)) = 0, ∀τ > s)

,(

ẋ = f(x, u(t, (h(x), z))), ż = v(t, (h(x), z)),

|x(s)|+ |z(s)| 6 ε)

⇒(

(x(τ), z(τ)) = 0, ∀τ > s+ T)

.

Time-varying feedback laws and measurement: An

experiment

Material:A jigsawA viseTwo Meccano c© strips(length ≃ 30 and 4 cm)A nut and 3 boltsA plastic tube

pendule-unstable-sans-son.mp4Media File (video/mp4)

pendule-stable.mp4Media File (video/mp4)

Chapter II

Design tools

Outline: Design tools

3 Control Lyapunov function

4 Damping

5 Phantom tracking

6 Averaging

7 Backstepping

Design tools: Commercial break

JMC, Control and nonlinearity,Mathematical Surveys andMonographs, 136, 2007, 427 p. Pdffile freely available from my webpage.



4 Damping

5 Phantom tracking

6 Averaging

7 Backstepping

Control Lyapunov function

A basic tool to study the asymptotic stability of an equilibrium point is theLyapunov function. In the case of a control system, the control is at ourdisposal, so there are more “chances” that a given function could be aLyapunov function for a suitable choice of feedback laws. Hence Lyapunovfunctions are even more useful for the stabilization of control systems thanfor dynamical systems without control.

Definition

A function V ∈ C1(Rn;R) is a control Lyapunov function for the controlsystem (C) if

V (x) → +∞, as |x| → +∞,V (x) > 0, ∀x ∈ Rn \ {0},

∀x ∈ Rn \ {0},∃u ∈ Rm s.t. f(x, u) · ∇V (x) < 0.

Moreover, V satisfies the small control property if, for every strictlypositive real number ε, there exists a strictly positive real number η suchthat, for every x ∈ Rn with 0 < |x| < η, there exists u ∈ Rm satisfying|u| < ε and f(x, u) · ∇V (x) < 0.

Theorem

If the control system (C) is globally asymptotically stabilizable by means ofcontinuous stationary feedback laws, then it admits a control Lyapunovfunction satisfying the small control property. If the control system (C)admits a control Lyapunov function satisfying the small control property,then it can be globally asymptotically stabilized by means of

1 Continuous stationary feedback laws if the control system (C) iscontrol affine (f(x, u) = f0(x) +

∑mi=1 uifi(x)) (Z. Artstein (1983)),

2 Time-varying feedback laws for general f (JMC-L. Rosier (1994)).



4 Damping

5 Phantom tracking

6 Averaging

7 Backstepping

Damping

For mechanical systems at least, a natural candidate for a control Lyapunovfunction is given by the total energy, i.e., the sum of potential and kineticenergies.

Damping

For mechanical systems at least, a natural candidate for a control Lyapunovfunction is given by the total energy, i.e., the sum of potential and kineticenergies. Consider the classical spring-mass control system.

bc um

bc

x1

u

m


ẋ1 = x2, ẋ2 = −k

mx1 +

u

m,(Spring-mass)

where m is the mass of the point attached to the spring, x1 is thedisplacement of the mass (on a line), x2 is the speed of the mass, k is thespring constant, and u is the external force applied to the mass. The stateis (x1, x2)

tr ∈ R2 and the control is u ∈ R.

The total energy E of the system is

E =1

2(kx21 +mx

22).

One has

Ė = ux2.

Hence if x2 = 0, one cannot have Ė < 0. However it tempting to considerthe following feedback laws

u := −νx2,

where ν > 0. Using the LaSalle invariance principle, one gets that thesefeedback laws globally asymptotically stabilize the spring-mass controlsystem.

Application: Orbit transfer with low-thrust systems (JMC

and L. Praly (1996))

Electric propulsion is characterized by a low-thrust acceleration level but ahigh specific impulse. They can be used for large amplitude orbit transfersif one is not in a hurry.The state of the control system is the position of the satellite (hereidentified to a point: we are not considering the attitude of the satellite)and the speed of the satellite. Instead of using Cartesian coordinates, oneprefers to use the “orbital” coordinates. The advantage of this set ofcoordinates is that, in this set, the first five coordinates remain unchangedif the thrust vanishes: these coordinates characterize the Keplerian ellipticorbit. When thrust is applied, they characterize the Keplerian ellipticosculating orbit of the satellite. The last component is an angle whichgives the position of the satellite on the Keplerian elliptic osculating orbitof the satellite.

A usual set of orbital coordinates is

p := a(1− e2),ex := e cos ω̃, with ω̃ = ω +Ω,

ey := e sin ω̃,

hx := tani

2cos Ω,

hy := tani

2sinΩ,

L := ω̃ + v,

where a, e, ω, Ω, i characterize the Keplerian osculating orbit:

1 a is the semi-major axis,

2 e is the eccentricity,

3 i is the inclination with respect to the equator,

4 Ω is the right ascension of the ascending node,

5 ω is the angle between the ascending node and the perigee,

and where v is the true anomaly.

ṗ = 2

√

p3

µ

1

ZS,

ėx =

√

p

µ

1

Z[Z(sinL)Q+AS − ey(hx sinL− hy cosL)W ] ,

ėy =

√

p

µ

1

Z[−Z(cosL)Q+BS − ex(hx sinL− hy cosL)W ] ,

ḣx =1

2

√

p

µ

X

Z(cosL)W, ḣy =

1

2

√

p

µ

X

Z(sinL)W,

L̇ =

√

µ

p3Z2 +

√

p

µ

1

Z(hx sinL− hy cosL)W,

where µ > 0 is a gravitational coefficient depending on the centralgravitational field, Q, S, W, are the radial, orthoradial, and normalcomponents of the thrust and where

Z := 1 + ex cosL+ ey sinL, A := ex + (1 + Z) cosL,

B := ey + (1 + Z) sinL, X := 1 + h2x + h

2y.

We study the case, useful in applications, where

Q = 0,

and, for some ε > 0,

|S| 6 ε and |W | 6 ε.

Note that ε is small, since the thrust acceleration level is low.The goal: give feedback laws, which (globally) asymptotically stabilize agiven Keplerian elliptic orbit characterized by the coordinatesp̄, ēx, ēy, h̄x, h̄y.In order to simplify the notations (this is not essential for the method), werestrict our attention to the case where the desired final orbit isgeostationary, that is,

ēx = ēy = h̄x = h̄y = 0.

We start with a change of “time”. One describes the evolution of(p, ex, ey, hx, hy) as a function of L instead of t. Then our system reads

dp

dL= 2KpS,

dexdL

= K[AS − ey(hx sinL− hy cosL)W ],deydL

= K[BS − ex(hx sinL− hy cosL)W ],dhxdL

=K

2X(cosL)W,

dhydL

=K

2X(sinL)W,

dt

dL= K

√

µ

pZ,

with

(6) K =

[

µ

p2Z3 + (hx sinL− hy cosL)W

]−1

.

Typically, one consider the following control Lyapunov function

V (p, ex, ey, hx, hy) =1

2

(

(p− p̄)2p

+e2

1− e2 + h2

)

,

with e2 = e2x + e2y < 1 and h

2 = h2x + h2y. The time derivative of V along a

trajectory of our control system is is given by

V̇ = K(αS + βW ),

with

α := 2p∂V

∂p+A

∂V

∂ex+B

∂V

∂ey,

β := (hy cosL− hx sinL)(

ey∂V

∂ex+ ex

∂V

∂ey

)

+1

2X

(

(cosL)∂V

∂hx+ (sinL)

∂V

∂hy

)

.

Following the damping method, one defines

S := −σ1(α),W := −σ2(β)σ3(p, ex, ey, hx, hy),

where σ1 : R → R, σ2 : R → R and σ3 : (0,+∞)× B1 × R2 → (0, 1] aresuch that

σ1(s)s > 0, σ2(s)s > 0, ∀s ∈ R \ {0},‖ σ1 ‖L∞(R)< ε, ‖ σ2 ‖L∞(R)< ε,

σ3(p, ex, ey, hx, hy) 61

1 + ε

µ

p2(1− |e|)3

|h| .

Following the damping method, one defines

S := −σ1(α),W := −σ2(β)σ3(p, ex, ey, hx, hy),

where σ1 : R → R, σ2 : R → R and σ3 : (0,+∞)× B1 × R2 → (0, 1] aresuch that

σ1(s)s > 0, σ2(s)s > 0, ∀s ∈ R \ {0},‖ σ1 ‖L∞(R)< ε, ‖ σ2 ‖L∞(R)< ε,

σ3(p, ex, ey, hx, hy) 61

1 + ε

µ

p2(1− |e|)3

|h| .

It works!

Comparison with optimal control

It is interesting to compare the feedback constructed here to the open-loopoptimal control for the minimal time problem (reach (p̄, 0, 0, 0, 0) in aminimal time with the constraint |u(t)| 6M). Numerical experimentsshow that the use of the previous feedback laws (with suitable saturationsσi, i ∈ {1, 2, 3}) gives trajectories which are nearly optimal if the state isnot too close to (p̄, 0, 0, 0, 0). Note that our feedback laws are quite easyto compute compared to the optimal trajectory and provide already goodrobustness properties compared to the open-loop optimal trajectory (theoptimal trajectory in a closed-loop form being, at least for the moment, outof reach numerically).

Comparison with optimal control

It is interesting to compare the feedback constructed here to the open-loopoptimal control for the minimal time problem (reach (p̄, 0, 0, 0, 0) in aminimal time with the constraint |u(t)| 6M). Numerical experimentsshow that the use of the previous feedback laws (with suitable saturationsσi, i ∈ {1, 2, 3}) gives trajectories which are nearly optimal if the state isnot too close to (p̄, 0, 0, 0, 0). Note that our feedback laws are quite easyto compute compared to the optimal trajectory and provide already goodrobustness properties compared to the open-loop optimal trajectory (theoptimal trajectory in a closed-loop form being, at least for the moment, outof reach numerically). However, when one is close to the desired target,our feedback laws are very far from being optimal. When one is close tothe desired target, it is much better to linearize around the desired targetand apply a standard Linear-Quadratic strategy.

ẋ1 = x2, ẋ2 = −x1 + u, |u| 6 2

u = −2

u = 2

1 2 3 4 5 x1

−1−2−3−4−5

x2

ẋ1 = x2, ẋ2 = −x1 + u, |u| 6 1

u = −1

u = 1

1 2 3 4 5 x1

−1−2−3−4−5

x2

ẋ1 = x2, ẋ2 = −x1 + u, |u| 6 1/2

u = −1/2

u = 1/2

1 2 3 4 5 x1

−1−2−3−4−5

x2

ẋ1 = x2, ẋ2 = −x1 + u, |u| 6 1/4

u = −1/4

u = 1/4

1 2 3 4 5 x1

−1−2−3−4−5

x2

An important limitation of the damping method

Let us come back to the spring-mass control system (with normalizedphysical constants)

ẋ1 = x2, ẋ2 = −x1 + u.

With the Lyapunov strategy used above, let V R2 → R) be defined by

V (x) = x21 + x22,∀x = (x1, x2)tr ∈ R2.

As we have sen above V̇ = 2x2u. and it is tempting to take, at least if weremain in the class of linear feedback laws, u := −νx2, where ν is somefixed positive real number. An a priori guess would be that, if we let ν bequite large, then we get a quite good convergence, as fast as we want.

An important limitation of the damping method

Let us come back to the spring-mass control system (with normalizedphysical constants)

ẋ1 = x2, ẋ2 = −x1 + u.

With the Lyapunov strategy used above, let V R2 → R) be defined by

V (x) = x21 + x22,∀x = (x1, x2)tr ∈ R2.

As we have sen above V̇ = 2x2u. and it is tempting to take, at least if weremain in the class of linear feedback laws, u := −νx2, where ν is somefixed positive real number. An a priori guess would be that, if we let ν bequite large, then we get a quite good convergence, as fast as we want.But this is completely wrong. On a given [0, T ] time-interval, as ν → +∞,x2 goes very quickly to 0 and x1 does not change.

ẋ1 = x2, ẋ2 = −x1 − (1/10)x2

ẋ1 = x2, ẋ2 = −x1 − (1/2)x2

ẋ1 = x2, ẋ2 = −x1 − x2

ẋ1 = x2, ẋ2 = −x1 − 2x2

ẋ1 = x2, ẋ2 = −x1 − 3x2

ẋ1 = x2, ẋ2 = −x1 − 4x2

ẋ1 = x2, ẋ2 = −x1 − 5x2

ẋ1 = x2, ẋ2 = −x1 − 6x2

ẋ1 = x2, ẋ2 = −x1 − 10x2

ẋ1 = x2, ẋ2 = −x1 − 20x2

Damping and quantum control systems (M. Mirrahimi, P.

Rouchon, G. Turinici (2005))

The quantum control system considered is

ιψ̇ = H0ψ + uH1ψ + ωψ,

where H0 and H1 are N ×N Hermitian matrices. The state is ψ ∈ S2N−1,the unit sphere of R2N ≃ CN , the control is (u, ω)tr ∈ R2. The control ωis a fictitious phase control which allows to take care of the fact that globalphase of the state is physically meaning-less. Let ψe ∈ S2N−1 and λe ∈ Rbe such that H0ψe = λψe. Replacing ψ by ψe

−ιλe and ω by ω − λe, wemay assume that λe = 0. Then (ψ, (u, ω)) := (ψe, (0, 0)

tr) is anequilibrium of our quantum control system. The goal is to stabilizeasymptotically this equilibrium.

Remark

Since S2N−1 is not contractible, one cannot a global stabilizability.However one can try to get global stabilizability on S2N−1 \ {−ψe}.

A natural control Lyapunov function to consider is

V := |ψ − ψe|2 .

Indeed, the time-derivative of V along the trajectory of our control systemis

V̇ = 2uℑ(〈H1ψ,ψe〉) + 2ωℑ(〈ψ,ψe〉).This leads to choose the following feedback laws

u := −ν1ℑ(〈H1ψ,ψe〉), ω := −ν2ℑ(〈ψ,ψe〉),

where ν1 and ν2 are two strictly positive real numbers. With thesefeedback laws one has

V̇ = −2ν1ℑ(〈H1ψ,ψe〉)2 − 2ν2ωℑ(〈ψ,ψe〉)2 6 0.

Theorem (M. Mirrahimi, P. Rouchon, G. Turinici (2005))

The above feedback law insures global asymptotic stabilization onS2N−1 \ {−ψe} if and only the two following properties hold(i) If (α, β) ∈ σ(H0)2 and |α| = |β|, then α = β,(ii) If φ in an eigenvector of H0 which is not colinear to Ψe, then

〈φ,H1ψe〉 6= 0.

Remark

The properties (i) and (ii) hold together if and only if the linearized controlsystem around (ψe, 0) ∈ S2N−1 is controllable.

Theorem (M. Mirrahimi, P. Rouchon, G. Turinici (2005))

The above feedback law insures global asymptotic stabilization onS2N−1 \ {−ψe} if and only the two following properties hold(i) If (α, β) ∈ σ(H0)2 and |α| = |β|, then α = β,(ii) If φ in an eigenvector of H0 which is not colinear to Ψe, then

〈φ,H1ψe〉 6= 0.

Remark

The properties (i) and (ii) hold together if and only if the linearized controlsystem around (ψe, 0) ∈ S2N−1 is controllable.

Question: What to do if (i) or (ii) do not hold but the quantum controlsystem is controllable? Partial solution: use the “phantom tracking”method (see below).

Schrödinger pde control systems

We are now in infinite dimension. In order to apply LaSalle invarianceprinciple one needs to have the precompactness of the trajectories. This isan open problem. However with clever and important modifications of themethod one can get global approximate controllability results. Let usmention in particular

M. Mirrahimi (2006),

V. Nersesyan (2009,2010),

K. Beauchard and M. Mirrahimi (2009).

LaSalle invariance principle/Strict Lyapunov function

Another possibility to overcome the problem of the precompactness of thetrajectories is to try to modify the control Lyapunov function in order toget a strict Lyapunov function. Some recent examples of this possibility:

1 Partially dissipative hyperbolic systems (K. Beauchard, E. Zuazua,2011):

yt +

n∑

j=1

(F j(y))xj = (0, B(y))tr, x ∈ Rn.

2 1D quasilinear hyperbolic equations on a finite interval (dissipativeboundary conditions): JMC, G. Bastin, B. d’Andréa-Novel (2007,2008):

yt +A(y)yx = 0, x ∈ (0, L),incoming Riemann invariants = F (outgoing Riemann invariants).

The hyperbolic control system considered

Our hyperbolic control system is

(1) yt +A(y)yx = 0, (t, x) ∈ [0, T ]× [0, L],

where, at time t ∈ [0, T ], the state is x ∈ [0, L] 7→ y(t, x) ∈ Rn. Lety∗ ∈ Rn be fixed. Assume that A(y∗) has n distinct real distincteigenvalues: λ1(y

∗) < . . . < λk(y∗) < . . . < λn(y

∗). We assume that

λ1(y∗) < . . . < λm−1(y

∗) < 0 < λm+1(y∗) < . . . < λn(y

∗),(2)

for some m ∈ {1, · · · , n}.

Stabilization: Dissipative boundary conditions

Motivated by 1994 Li Tatsien’s book

(1) yt +A(y)yx = 0, y ∈ Rn, x ∈ [0, 1], t ∈ [0,+∞),

• Assumptions on A

A(0) = diag (λ1, λ2, . . . , λn),(2)

λi > 0, ∀i ∈ {1, . . . ,m}, λi < 0, ∀i ∈ {m+ 1, . . . , n},

λi 6= λj, ∀(i, j) ∈ {1, . . . , n}2 such that i 6= j.

• Boundary conditions on y:

(1)

(

y+(t, 0)y−(t, 1)

)

= G

(

y+(t, 1)y−(t, 0)

)

, t ∈ [0,+∞),

where

(i) y+ ∈ Rm and y− ∈ Rn−m are defined by

y =

(

y+y−

)

,

(ii) the map G : Rn → Rn vanishes at 0.

Notations

For K ∈ Mn,m(R),

‖K‖ := max{|Kx|;x ∈ Rn, |x| = 1}.

If n = m,

ρ1(K) := Inf {‖∆K∆−1‖; ∆ ∈ Dn,+},

where Dn,+ denotes the set of n× n real diagonal matrices with strictlypositive diagonal elements.

Theorem (JMC-G. Bastin-B. d’Andréa-Novel (2008))

If ρ1(G′(0)) < 1, then the equilibrium ȳ ≡ 0 of the quasi-linear hyperbolic

systemyt +A(y)yx = 0,

with the above boundary conditions, is exponentially stable for the SobolevH2-norm.

Complements:

yt +Ayx +By = 0: G. Bastin and JMC (2010), A. Diagne, G. Bastinand JMC (2010), R. Vazquez, M. Krstic and JMC (2011),

yt +A(x, y)yx +B(x, y)y = 0: A. Diagne and A. Drici (2011), R.Vazquez, JMC, M. Krstic and G. Bastin (2011),

Integral action: V. Dos Santos, G. Bastin, JMC and B.d’Andréa-Novel (2008), A. Drici (2010),

BV -solutions: V. Perrollaz (2011).

Estimate on the exponential decay rate

For every ν ∈ (0,−min{|λ1|, . . . , |λn|} ln(ρ1(G′(0)))), there exist ε > 0and C > 0 such that, for every y0 ∈ H2((0, 1),Rn) satisfying|y0|H2((0,1),Rn) < ε (and the usual compatibility conditions) the classicalsolution y to the Cauchy problem

yt +A(y)yx = 0, y(0, x) = y0(x) + boundary conditions

is defined on [0,+∞) and satisfies

|y(t, ·)|H2((0,1),Rn) 6 Ce−νt|y0|H2((0,1),Rn), ∀t ∈ [0,+∞).

The Li Tatsien condition

R2(K) := Max {n∑

j=1

|Kij |; i ∈ {1, . . . , n}},

ρ2(K) := Inf {R2(∆K∆−1); ∆ ∈ Dn,+}.

Theorem (Li Tatsien, 1994)

If ρ2(G′(0)) < 1, then the equilibrium ȳ ≡ 0 of the quasi-linear hyperbolic

systemyt +A(y)yx = 0,

with the above boundary conditions, is exponentially stable for theC1-norm.

The Li Tatsien proof relies mainly on the use of direct estimates of thesolutions and their derivatives along the characteristic curves.

C1/H2-exponential stability

1 Open problem: Does there exists K such that one has exponentialstability for the C1-norm but not for the H2-norm?

2 Open problem: Does there exists K such that one has exponentialstability for the H2-norm but not for the C1-norm?

Comparison of ρ2 and ρ1

Proposition

For every K ∈ Mn,n(R),

ρ1(K) 6 ρ2(K).(1)

Example where (1) is strict: for a > 0, let

Ka :=

(

a a−a a

)

∈ M2,2(R).

Then

ρ1(Ka) =√2a < 2a = ρ2(Ka).

Open problem: Does ρ1(K) < 1 implies the exponential stability for theC1-norm?

Comparison with stability conditions for linear hyperbolic

systems

For simplicity we assume that λi are all positive and consider we considerthe special case of linear hyperbolic systems

yt + Λyx = 0, y(t, 0) = Ky(t, 1),

where

Λ := diag (λ1, . . . , λn), with λi > 0, ∀i ∈ {1, . . . , n}.

Theorem

Exponential stability for the C1-norm is equivalent to the exponentialstability in the H2-norm.

A Necessary and sufficient condition for exponential stability

Notation:

ri =1

λi, ∀i ∈ {1, . . . , n}.

Theorem

ȳ ≡ is exponentially stable for the system

yt + Λyx = 0, y(t, 0) = Ky(t, 1)

if and only if there exists δ > 0 such that

(

det (Idn − (diag (e−r1z, . . . , e−rnz))K) = 0, z ∈ C)

⇒ (ℜ(z) 6 −δ).

An example

This example is borrowed from the book Hale-Lunel (1993). Let us chooseλ1 := 1, λ2 := 2 (hence r1 = 1 and r2 = 1/2) and

Ka :=

(

a aa a

)

, a ∈ R.

Then ρ1(K) = 2|a|. Hence ρ1(Ka) < 1 is equivalent to a ∈ (−1/2, 1/2).However exponential stability is equivalent to a ∈ (−1, 1/2).

Robustness issues

For a positive integer n, let

λ1 :=4n

4n + 1, λ2 =

4n

2n+ 1.

Then(

y1y2

)

:=

(

sin(

4nπ(t− (x/λ1)))

sin(

4nπ(t− (x/λ2)))

)

is a solution of yt + Λyx, y(t, 0) = K−1/2y(t, 1) which does not tends to 0as t→ +∞. Hence one does not have exponential stability. Howeverlimn→+∞ λ1 = 1 and limn→+∞ λ2 = 2. The exponential stability is notrobust with respect to Λ: small perturbations of Λ can destroy theexponential stability.

Robust exponential stability

Notations:

ρ0(K) := max{ρ(diag (eιθ1 , . . . , eιθn)K); (θ1, . . . , θn)tr ∈ Rn},

Theorem (R. Silkowski, 1993)

If the (r1, . . . , rn) are rationally independent, the linear systemyt + Λyx = 0, y(t, 0) = Ky(t, 1) is exponentially stable if and only ifρ0(K) < 1.

Note that ρ0(K) depends continuously on K and that “(r1, . . . , rn) arerationally independent” is a generic condition. Therefore, if one wants tohave a natural robustness property with respect to the ri’s, the conditionfor exponential stability is

ρ0(K) < 1.

This condition does not depend on the λi’s!

Comparison of ρ0 and ρ1

Proposition (JMC-G. Bastin-B. d’Andréa-Novel, 2008)

For every n ∈ N and for every K ∈ Mn,n(R),

ρ0(K) 6 ρ1(K).

For every n ∈ {1, 2, 3, 4, 5} and for every K ∈ Mn,n(R),

ρ0(K) = ρ1(K).

For every n ∈ N \ {1, 2, 3, 4, 5}, there exists K ∈ Mn,n(R) such thatρ0(K) < ρ1(K).

Open problem: Is ρ0(G′(0)) < 1 a sufficient condition for exponential

stability (for the H2-norm) in the nonlinear case ?

Proof of the exponential stability if A is constant and G is

linear

Main tool: a Lyapunov approach. A(y) = Λ, G(y) = Ky. For simplicity,all the λi’s are positive. Lyapunov function candidate:

V (y) :=

∫ 1

0ytrQye−µxdx, Q is positive symmetric.

V̇ = −∫ 1

0(ytrxΛQy + y

trQyx)Λe−µxdx

= −µ∫ 1

0ytrΛQy e−µxdx−B,

with

B := [ytrΛQye−µx]x=1x=0 = y(1)tr(ΛQe−µ −KtrΛQK)y(1)

Let D ∈ Dn,+ be such that ‖DKD−1‖ < 1 and let ξ := Dy(1). We takeQ = D2Λ−1. Then

B = e−µ|ξ|2 − |DKD−1ξ|2.

Therefore it suffices to take µ > 0 small enough.

Remark

Introduction of µ:

• JMC (1998) for the stabilization of the Euler equations.• Cheng-Zhong Xu and Gauthier Sallet (2002) for symmetric linear

hyperbolic systems.

New difficulties if A(y) depends on y

We try with the same V :

V̇ = −∫ 1

0

(

ytrxA(y)trQy + ytrQA(y)yx

)

e−µxdx

= −µ∫ 1

0ytrA(y)Qye−µxdx−B +N1 +N2

with

N1 :=

∫ 1

0ytr(QA(y)−A(y)Q)yxe−µxdx,

N2 :=

∫ 1

0ytr

(

A′(y)yx)trQye−µxdx

Solution for N1

Take Q depending on y such that A(y)Q(y) = Q(y)A(y),Q(0) = D2F (0)−1. (This is possible since the eigenvalues of F(0) aredistinct.) Now

V̇ = −µ∫ 1

0ytrA(y)Q(y)ye−µxdx−B +N2

with

N2 :=

∫ 1

0ytr

(

A′(y)yxQ(y) +A(y)Q′(y)yx

)trye−µxdx

What to do with N2?

Solution for N2

New Lyapunov function:

V (y) = V1(y) + V2(y) + V3(y)

with

V1(y) =

∫ 1

0ytrQ(y)y e−µxdx,

V2(y) =

∫ 1

0ytrxR(y)yx e

−µxdx,

V3(y) =

∫ 1

0ytrxxS(y)yxx e

−µxdx,

where µ > 0, Q(y), R(y) and S(y) are symmetric positive definitematrices.

Choice of Q, R and S

• Commutations:

A(y)Q(y)−Q(y)A(y) = 0,A(y)R(y)−R(y)A(y) = 0,A(y)S(y)− S(y)A(y) = 0,

•Q(0) = D2A(0)−1, R(0) = D2A(0), S(0) = D2A(0)3.

Estimates on V̇

Lemma

If µ > 0 is small enough, there exist positive real constants α, β, δ suchthat, for every y : [0, 1] → Rn such that |y|C0([0,1]) + |yx|C0([0,1]) 6 δ, wehave

1

β

∫ 1

0(|y|2 + |yx|2 + |yxx|2)dx 6 V (y) 6 β

∫ 1

0(|y|2 + |yx|2 + |yxx|2)dx,

V̇ 6 −αV.

...

La Sambre (The same + Luc Moens)

xL0

H(t,x)

V(t,x)

u0

uL

Z0

The Saint-Venant equations

The index i is for the i-th reach.Conservation of mass:

Hit + (HiVi)x = 0,

Conservation of momentum:

Vit +

(

gHi +V 2i2

)

x

= 0.

Flow rate: Qi = HiVi.

Barré de Saint-Venant(Adhémar-Jean-Claude)1797-1886Théorie du mouvement non perma-nent des eaux, avec applications auxcrues des rivières et à l’introductiondes marées dans leur lit, C. R. Acad.Sci. Paris Sér. I Math., vol. 53(1871), pp.147–154.

Boundary conditions

Underflow (sluice)

u

u

Overflow (spillway)

34

u

u

Work in progress: La Meuse

Closed loop versus open loop

Closed loop versus open loop

movie1bief.mp4Media File (video/mp4)



4 Damping

5 Phantom tracking

6 Averaging

7 Backstepping

bc

bc

Phantom tracking method: An example

Let us consider the following control system.

ẋ1 = x2, ẋ2 = −x1 + u, ẋ3 = x4, ẋ4 = −x3 + 2x1x2,

where the state is (x1, x2, x3, x4)tr ∈ R4 and the control is u ∈ R.

Phantom tracking method: An example

Let us consider the following control system.

ẋ1 = x2, ẋ2 = −x1 + u, ẋ3 = x4, ẋ4 = −x3 + 2x1x2,

where the state is (x1, x2, x3, x4)tr ∈ R4 and the control is u ∈ R.Roughly,

we have two oscillators which are coupled by means of a quadratic term.The control is acting only on the first oscillator. The point(xγ , uγ) := ((γ, 0, 0, 0)

tr , γ) is an equilibrium of the control system. Thelinearized control system at this equilibrium is the linear control system

ẋ1 = x2, ẋ2 = −x1 + u, ẋ3 = x4, ẋ4 = −x3 + 2γx2,

where the state is (x1, x2, x3, x4)tr ∈ R4 and the control is u ∈ R. This

linear control system is controllable if (and only if ) γ 6= 0. Therefore, ifγ 6= 0 the equilibrium can be asymptotically stabilized for the nonlinearcontrol system.

Stabilization of xγ

One considers the following control Lyapunov function V γ : R4 → Rdefined by

V γ(x) := (x1 − γ)2 + x22 + x23 + x24, ∀x = (x1, x2, x3, x4)tr ∈ R4.

The time derivative of V γ along the trajectory of our control system is

V̇ γ = 2x2(u− γ + 2x1x4).

Hence, in order to asymptotically stabilize xγ for our control system, it isnatural to consider the feedback law uγ : R4 → R defined by

uγ := γ − 2x1x4 − x2.

One gets V̇ γ = −2x22. Using the LaSalle invariance principle, one gets thatthis feedback law globally asymptotically stabilizes xγ .

Let us now follow the phantom tracking strategy. In fact, instead of usinguγ̃ with a suitable γ̃ : R4 → R it is better to use directly a controlLyapunov of the type V γ̃ . Theoretically, the best way to choose γ̃ is todefine it implicitly by proceeding in the following way. There exits an openneighborhood Ω of 0 ∈ R4 and V ∈ C∞(Ω;R) such that

V (0) = 0, ∀x ∈ Ω \ {0},V (x) = (x1 − V (x))2 + x22 + x23 + x24, ∀x = (x1, x2, x3, x4)tr ∈ Ω.

Therefore our choice of γ̃ = V (x), i.e. is such that γ̃(x) = V γ̃(x),γ̃(0) = 0. For the existence of V : use the implicit function theorem. In thissimple case, V can be computed explicitly. One hasV̇ = 2(x1 − V )(x2 − V̇ ) + 2x2(−x1 + u) + x3x4 + x4(−x3 + 2x1x2),i.e.,(1 + 2x1 − 2V )V̇ = 2x2(u− V + x1x4). We define a feedback lawu : Ω → R by u := V − x1x4 − x2, which leads to(1 + 2x1 − 2V )V̇ = −2x22 6 0. One concludes that the feedback law ulocally asymptotically our control system.

Two possible improvements:

(i) One can get global asymptotic stability. It suffices to modify V byrequiring V = V (x) = (x1 − θ(V (x)))2 + x22 + x23 + x24, with a wellchosen function θ : [0,+∞) → [0,+∞).

(ii) One can get explicit feedback laws by using a dynamic extension:Replace the initial control system by the following one

ẋ1 = x2, ẋ2 = −x1 + u, ẋ3 = x4, ẋ4 = −x3 + 2x1x2, γ̇ = v

where the state is (x1, x2, x3, x4, γ)tr ∈ R5 and the control is

(u, v)tr ∈ R2. For z = (x1, x2, x3, x4, γ)tr ∈ R5, one defines

ϕ(z) := (x1 − γ)2 + x22 + x23 + x24,W (z) := ϕ(z) + (γ − ϕ(z))2.

Compute Ẇ etc.

Phantom tracking and stabilization of the Euler equations of

incompressible fluids

bc

x

y(t, x)

Ω

∂Ω

Γ0

R2

Ω is assumed to be connected and simply connected.

The Euler control system

We denote by ν : ∂Ω → R2 the outward unit normal vector field to Ω.Euler equations :

yt + (y · ∇)y +∇p = 0, div y = 0,y · ν = 0 on [0, T ]× (∂Ω \ Γ0).

This system is under-determined. In order to have a determined system,one has to specify what is the control. There are at least two naturalpossibilities:

(a) The control is y(t, x) · n(x) on Γ0 and the time derivative ∂ω/∂t(t, x)of the vorticity at the points x of Γ0 where y(t, x) · n(x) < 0, i.e., atthe points where the fluid enters into the domain Ω.

(b) The control is y(t, x) · n(x) on Γ0 and the vorticity ω at the points xof Γ0 where y(t, x) · n(x) < 0.

To fix ideas, we deal only with the first case.

Failure of the linearization technique

The linearized control is

yt +∇p = 0, div y = 0, y · ν = 0 on [0, T ]× (∂Ω \ Γ0).

Taking the curl of the first equation, on gets, with ω := curl y

ωt = 0.

One cannot change ω! This linear control system is not stabilizable.

Definition of yγ

Take θ : Ω → R such that

∆θ = 0 in Ω,∂θ

∂ν= 0 on ∂Ω \ Γ0.

Let us define define (yγ , pγ) : [0, T ] ×Ω → R2 × R by

yγ(x) = γ∇θ(x), pγ(x) = −γ2

2|∇θ(x)|2.

Then (yγ , pγ) is an equilibrium point of our Euler control system. Thecorresponding control is γ∂θ/∂n on Γ0 and 0 for the vorticity at the pointsx of Γ0 where ∂θ/∂n < 0.

Stabilization of the linearized control system around yγ

The linearized control system around yγ is

{

yt + (yγ · ∇)y + (y · ∇)yγ +∇p = 0, div y = 0 in [0, T ]× Ω,

y · ν = 0 on [0, T ]× (∂Ω \ Γ0).

Taking once more the curl of the first equation, one gets

ωt + (yγ · ∇)ω = 0.(*)

This is a simple transport equation on ω. If there exists a ∈ Ω such that∇θ(a) = 0, then yγ(a) = 0 and ωt(t, a) = 0 showing that (*) is notstabilizable. This is the only obstruction: If ∇θ does not vanish in Ω, onecan easily stabilize (*): just take C > 0 and use the controlωt(t, x) = −Cω(t, x) on the set {x ∈ Γ0; ∂θ/∂ν(x) < 0}.

Construction of a good θ

Ω

∂Ω

Rn


Ω

∂Ω

Rn

Γ0


Ω

∂Ω

Rn

Γ0

Γ−

Γ+


Ω

∂Ω

Rn

Γ0

Γ−

Γ+

g : ∂Ω → R∫

∂Ω gds = 0,

{g > 0} = Γ+, {g < 0} = Γ−


Ω

∂Ω

Rn

Γ0

Γ−

Γ+

g : ∂Ω → R∫

∂Ω gds = 0,

{g > 0} = Γ+, {g < 0} = Γ−

∆θ = 0,∂θ∂ν = g on ∂Ω


Ω

∂Ω

Rn

Γ0

Γ−

Γ+

g : ∂Ω → R∫

∂Ω gds = 0,

{g > 0} = Γ+, {g < 0} = Γ−

∆θ = 0,∂θ∂ν = g on ∂Ω

∇θ

Asymptotic stabilization of the Euler equations

Our stabilizing feedback law is

y · ν :=M |ω|0∂θ

∂νon Γ−,

∂ω

∂t:= −M |ω|C0(Ω) ω on Γ−.


There exists a positive constant M0 such that, for every ε ∈ (0, 1], everyM >M0/ε and every solution ω of the closed loop system,

|ω(t)|0 6 Min{

|ω(0)|C0(Ω),ε

t

}

, ∀t > 0.

Remark

When Ω is not simply connected: O. Glass (2005). For the Camassa-Holmequation (one-dimensional surface waves): O. Glass (2008), V. Perrollaz(2010).

Applications to quantum control systems

1 K. Beauchard, JMC, M. Mirrahimi and P. Rouchon (2007) for

ιψ̇ = H0ψ + uH1ψ, ψ(t, ·) ∈ S2N−1.

2 K. Beauchard and M. Mirrahimi (2009) for a quantum particle in aone-dimensional infinite square potential well:

{

ιψt = −ψxx + u(t)xψ, x ∈ (0, 1), ψ ∈ L2((0, 1);C)∫ 10 |ψ(t, x)|

2 dx = 1.

This control is in infinite dimension and there is a problem to use theLaSalle invariance principle. This leads to important difficulties.

3 JMC, A. Grigoriu (in progress) for

ιψ̇ = H0ψ + uH1ψ + u2H2ψ, ψ(t, ·) ∈ S2N−1.



4 Damping

5 Phantom tracking

6 Averaging

7 Backstepping

Averaging (JMC, A. Grigoriu, C. Lefter and G. Turinici

(2009))


ιψ̇ = H0ψ + uH1ψ + u2H2 + ωψ,

where H0, H1 and H2 are N ×N Hermitian matrices. The state isψ ∈ S2N−1, the unit sphere of R2N ≃ CN , the control is (u, ω)tr ∈ R2.Again we assume that 0 is an eigenvalue of H0 and consider acorresponding eigenvector ψe ∈ S2N−1 We consider the following timedependent feedback:

u(t, ψ) = α(ψ) + β(ψ) sin(t/ε).

The closed loop system is

(Cε)

ιψ̇ =(

H0 + α(ψ)H1 + β(ψ) sin(t/ε)H1

+α2(ψ)H2 + 2α(ψ)β(ψ) sin(t/ε)H2

+β2(ψ) sin2(t/ε)H2 + ω(t))

ψ(t).

Averaged system

For a differential system ẋ = f(t, x), with f a T -periodic function:f(T + t, x) = f(t, x), the averaged system is defined by ẋav = fav(x)

where fav(x) =1T

∫ T0 f(t, x)dt.

In our case the averaged system corresponding to the closed loop system is:

(Cav) iψ̇ =

(

H0 + αH1 +

(

α2 +1

2β2

)

H2 + ω

)

ψ.

In some sense we have now three independent controls, namely α, β and ω,instead of two, namely u and ω. Moreover, one knows that, if ε > 0 issmall enough, the trajectories of (Cε) are close to the trajectory of (Cav).The strategy is now simple. Using the damping method, one gets feedbacklaws ψ 7→ (α(ψ);β(ψ);ω(ψ)) leading to global stabilizability of ψe onS2N−1 \ {−ψe} for the averaged system. Then, taking ε > 0 small enough,

one gets a “practical” global stabilizability of ψe on S2N−1 \ V for the

closed loop system (Cε), where V is a given neighborhood of −ψe.



4 Damping

5 Phantom tracking

6 Averaging

7 Backstepping

Backstepping

For the backstepping method, we are interested in a control system (C)having the following structure:

ẋ1 = f1(x1, x2), ẋ2 = u,Σ

where the state is x = (x1;x2) := (xtr1 , x

tr2 )

tr ∈ Rn1+m = Rn with(x1, x2) ∈ Rn1 × Rm and the control is u ∈ Rm. The key theorem forbackstepping is the following one.

Theorem

Assume that f1 ∈ C1(Rn1 × Rm;Rn1) and that the control system

ẋ1 = f1(x1, v),Σ1

where the state is x1 ∈ Rn1 and the control v ∈ Rm, can be globallyasymptotically stabilized by means of a feedback law of class C1. Then Σcan be globally asymptotically stabilized by means of a continuous feedbacklaw.

References for the backstepping theorem

Local version: very old, precise father(s)/mother(s) unknown,

Global version:

D. Koditschek (1987),C. Byrnes and A. Isidori (1989),J. Tsinias (1989),L. Praly, B. d’Andréa-Novel and JMC (1991) for low regularity for thefeedback stabilizing ẋ1 = f1(x1, v).

Proof of the theorem

Let v ∈ C1(Rn1 ;Rm) be a feedback law which globally asymptoticallystabilizes 0 ∈ Rn1 for the control system Σ1. Then, by the converse of thesecond Lyapunov theorem, there exists a Lyapunov function of class C∞

for the closed-loop system ẋ1 = f1(x1, v(x1)), that is, there exists afunction V ∈ C∞(Rn1 ;R) such that

f1(x1, v(x1)) · ∇V (x1) < 0, ∀x1 ∈ Rn1 \ {0},V (x1) → +∞ as |x1| → +∞,V (x1) > V (0), ∀x1 ∈ Rn1 \ {0}.

A natural candidate for a control Lyapunov function for Σ is

W (x1;x2) := V (x1) +1

2|x2 − v(x1)|2, ∀(x1, x2) ∈ Rn1 × Rm.

Indeed, one has, for such a W ,

W (x1;x2) → +∞ as |x1|+ |x2| → +∞,W (x1;x2) > W (0, 0), ∀(x1, x2) ∈ (Rn1 ×Rm) \ {(0, 0)}.

The time-derivative Ẇ of W along the trajectories of Σ is

Ẇ = f1(x1, x2) · ∇V (x1)− (x2 − v(x1)) · (v′(x1)f1(x1, x2)− u).

There exists G ∈ C0(Rn1 × Rm × Rm;L(Rm,Rn1)) such thatf1(x1, x2)− f1(x1, y) = G(x1, x2, y)(x2 − y). Therefore

Ẇ = f1(x1, v(x1)) · ∇V (x1)+[

utr − (v′(x1)f1(x1, x2))tr +(∇V (x1))trG(x1, x2, v(x1))]

(x2− v(x1)).

Hence, if one takes as a feedback law for the control system Σ

u := v′(x1)f1(x1, x2)−G(x1, x2, v(x1))tr∇V (x1)− (x2 − v(x1)),

one gets Ẇ = f1(x1, v(x1)) · ∇V (x1)− |x2 − v(x1)|2. Hence

Ẇ (x1;x2) < 0, ∀(x1, x2) ∈ (Rn1 × Rm) \ {(0, 0)}.

In conclusion, u globally asymptotically stabilizes Σ.

Various generalizations

Various (straightforward) adaptations are possible. For example

1 Adaptation to the framework of time-varying feedback laws.

2 It is possible to add an “integrator” to only part of the components ofv.

3 In the above construction, instead of using a strict Lyapunov functionone can use a Lyapunov function satisfying the assumptions of theLaSalle invariance principle.

In the next slides we show how to use these three adaptations together inorder to asymptotically stabilize the baby stroller control system.

Stabilization of the baby stroller

The baby stroller control system is

ẋ1 = u1 cos x3, ẋ2 = u1 sinx3, ẋ3 = u2,Σ

where the state is x := (x1;x2;x3) ∈ R3 and the control is (u1;u2) ∈ R2.Following the backstepping approach, we first deal with the stabilization ofthe control system

ẋ1 = u1 cos x3, ẋ2 = u1 sinx3,Σ1

where the state is (x1;x2) ∈ R2 and the control is (u1;x3) ∈ R2 and thenwe add an “integration x3”. As a potential control Lyapunov for Σ1, weconsider

V (x1;x2) :=1

2

(

x21 + x22

)

, ∀(x1;x2) ∈ R2.

Note that

V (x′) > V (0), ∀x′ ∈ R2, lim|x′|→+∞

V (x′) = +∞.

Along the trajectory of Σ1, we have

V̇ = (x1 cos x3 + x2 sinx3)u1.

Let T > 0. Let f ∈ C1(R) be a T -periodic function which is not constant.We define x̄3 : R× R2 → R and ū1 : R× R2 → R by

x̄3(t, x′) := f(t)|x′|2,

ū1(t, x′) := −(x1 cos(x̄3(t, x′)) + x2 sin(x̄3(t, x′)).

The functions x̄3 and ū1 are T -periodic to time and vanish on R× {0}.One has V̇ = −ū21 6 0. Using the LaSalle invariance principle one checksthat 0 ∈ R2 is globally asymptotically stable for the closed loop system

ẋ1 = ū1 cos x̄3, ẋ2 = ū1 sin x̄3.

Following the backstepping strategy, we consider the potential time-varyingcontrol Lyapunov function W for the baby stroller control system Σ (whichis obtained by “adding an integration” on the variable x3 to the controlsystem Σ1)

W (t, x) := V (x′) +1

2(x3 − x̄3(t, x′))2, ∀t ∈ R, ∀x = (x1;x2;x3) ∈ R3,

with x′ := (x1;x2). Note that W is T-periodic with respect to time andthat

W (t, x) > W (t, 0) = 0, ∀t ∈ R, ∀x ∈ R3 \ {0},lim

|x|→+∞min{W (t, x); t ∈ [0, T ]} = +∞.

The time derivative of W along the trajectory of Σ is

Ẇ = (x1 cos x3 + x2 sinx3)u1 + (x3 − x̄3)(u2 − ξ),

with ξ(t, (x;u1)) := f′(t)|x′|2 + 2f(t)u1(x1 cos x3 + x2 sinx3).

We define our time-varying stabilizing feedback law by

u1(t, (x1;x2;x3)) := −(x1 cos x3 + x2 sinx3),u2(t, (x1;x2;x3)) := ξ(t, (x;−(x1 cos x3 + x2 sinx3)))

−(x3 − x̄3(t, x′)),

so thatẆ = −(x1 cos x3 + x2 sinx3)2 + (x3 − x̄3)2 6 0.

The functions u1 and u2 are T -periodic to time and vanish on R× {0}.Using once more the LaSalle invariance principle, one checks that 0 ∈ R3 isglobally asymptotically stable for the closed loop system

ẋ1 = u1 cos x3, ẋ2 = u1 sinx3, ẋ3 = u2.

Stabilization: General resultsMotivation and an exampleInverted pendulum

Some abstract results in finite dimensionAsymptotic stabilityThe stabilizability problemLinear systems and applications to nonlinear systemsObstruction to the stabilizabilityTime-varying feedback laws

Design toolsControl Lyapunov functionDampingApplication to the stabilization of quasilinear hyperbolic systemsMain resultComparaison with prior resultsProof of the exponential stabilityApplication to the control of open channels

Phantom trackingAveragingBackstepping

stabilization of non linear control systems€¦ · stabilization of non linear control systems...

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