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1/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Output stabilization at unobservable points: analysis via an example

J.P. Gauthiera M.A. Lagacheab

• U. Serresb

aUniversit´

e de Toulon, France

bUniversit´

e de Lyon, France

60th birthday of Jean-Michel Coron IHP, June 2016

2/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Table of contents

1

Introduction

2

Practical stability

3

Numerical simulations

4

Ongoing work: exact stabilization

5

Conclusion

3/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

1

Introduction

2

Practical stability

3

Numerical simulations

4

Ongoing work: exact stabilization

5

Conclusion

4/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

System under consideration

Consider the closed quantum system 1

          

˙ x = A(u)x =

  

e u1 −e u2 −u1 −u2

   x

y = Cx = x3 (Σ) where

◮ x = (x1, x2, x3) ∈ S2 is the state variable ◮ y ∈ R is the measured output ◮ u = (u1, u2) ∈ R2 is the control variable

Aim: Stabilize (Σ) to the target point xt = (0, 0, −1) by mean of a smooth dynamic time invariant output feedback

1see e.g. [Boscain et al., 2015]

5/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Problem

Problem: The equilibrium point xt corresponds to the null input, which makes the system unobservable

time(s) 5 10 15 20 25 30 35 40 45 50

• 1
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 x1(t) x2(t) x3(t)

Figure: State variables of system (Σ) using a ”naive” approach

Some results about output feedback:

◮ [Teel and Praly, 1994] ◮ [Coron, 1994] ◮ [Besancon and Hammouri, 2000]

6/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

1

Introduction

2

Practical stability

3

Numerical simulations

4

Ongoing work: exact stabilization

5

Conclusion

7/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Stabilizing state feedback

Consider the state feedback λs(x) = r1(x1, x2), where r1 is an arbitrary positive constant. Proposition The target point xt is an asymptotically stable equilibrium for the closed-loop system resulting from applying the feedback control u = λs(x) to system (Σ). Moreover, its basin of attraction is S2 \ {−xt}. Sketch of proof. A direct application of LaSalle’s principle (see e.g. [LaSalle, 1968]) using V (x) = x3, as a candidate Lyapunov function gives the desired result.

8/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

The observer

The equations of the controller-observer system are

˙

ˆ x = A (λs

δ(ˆ

x)) ˆ x − r2C′Cε ˙ ε =

A (λs

δ(ˆ

x)) − r2C′C

ε,

(ˆ x, ε) ∈ R3 × R3, (CLO) where λs

δ(ˆ

x) = λs(ˆ x) + (δ, δ), and δ and r2 are positive constants.

9/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

The observer

Lemma 1 All the inputs of the form u = λs

δ(ˆ

x) applied to the full coupled system (CLO) make system (Σ) observable on any time interval [0, T], T > 0. Sketch of proof.

◮ By contradiction: there exist a positive T and an input

λs

δ(ˆ

x(·)) that renders system (Σ) unobservable on [0, T]

◮ There exists a ω(·) = (ω1, ω2, ω3) ≡ 0 solution of (Σ) such

that ω3(·) ≡ 0

◮ Differentiating with respect to t and solving with respect to

ω(·), we get that ω(·) vanishes identically

10/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Step 1 : Estimation errors goes to zero

Definition (from [Celle et al., 1989]) A persistent input (for bilinear systems) is a measurable bounded input u for which there exists a time interval T > 0, such that lim sup

θ→+∞

ind(u(· + θ), T) > 0, where ind(u(·), T) is the index of universality of u on [0, T], i.e. the smallest eigenvalue of the Gram-observability matrix. Corollary All the inputs λs

δ

ˆ

x(·)

are persistent.

11/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Step 1 : Estimation error goes to zero

Sketch of proof.

◮ x ∈ S2 and ε is decreasing (since 1 2 d dt ε2 = −r2(Cε)2)

− → {(ˆ x(t), ε(t)) | t 0} lies in a compact K

◮ The mapping

F : R3 × R3 → R+ (ˆ x0, ε0) → ind(λs

δ (ˆ

x(·)) , T) is continuous and nonnegative for all T > 0

◮ Since

inf

K F lim sup θ→+∞

F(ˆ x(θ), ε(θ)) = lim sup

θ→+∞

ind(λs

δ

ˆ

x(· + θ)

, T)

◮ By continuity, the infimum of F over K is reached, and is

positive by the crucial Lemma 1

12/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Step 1 : Estimation error goes to zero

Theorem ([Celle et al., 1989]) If u ∈ L∞(R+, R2) is a persistent input, then the observation error tends to zero, i.e. lim

t→+∞ ε(t) = 0.

Corollary 2 For any trajectory of the coupled system (CLO), we have limt→+∞ ε(t) = 0.

13/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Step 2 : Asymptotic stability of an equilibrium point

Lemma 3 If δ is small enough, system (CLO) admits an asymptotically stable equilibrium point (xs

δ, 0) arbitrarily close to (xt, 0) and an unstable

equilibrium point (xu

δ , 0) arbitrarily close to (−xt, 0).

Sketch of proof.

◮ Compute the two equilibrium points ◮ Rewrite the system using the constraint x = ˆ

x − ε = 1

◮ Linearize around the equilibrium points ◮ Perform the stability analysis on the linearized system

14/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Step 3 : The main result

Theorem If δ is small enough, the point (xs

δ, 0) ∈ R6 is asymptotically stable

for system (CLO), and its region of attraction is R6 \ {(xu

δ , 0)}.

Sketch of proof.

◮ From Corollary 2 the ω-limit points of system (CLO) are of

the form (ˆ x, 0).

◮ Set C− = {x ∈ S2 | x3 0} and consider the function

L : C− → R+ defined by L(ˆ x) = 1

2ˆ

x − xs

δ2. ◮ Using LaSalle’s principle and Lemma 3 we get the desired

result.

15/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

1

Introduction

2

Practical stability

3

Numerical simulations

4

Ongoing work: exact stabilization

5

Conclusion

16/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Output feedback with perturbation

• 1
• 0.5

0.5 1 δ = 0.1 x1(t) x2(t) x3(t) time(s) 2000 4000 6000 8000

• 1
• 0.5

0.5 1 δ = 0.05 x1(t) x2(t) x3(t) time(s) 2000 4000 6000 8000

Figure: State variables of system (Σ) with u = λs

δ(ˆ

x) for δ = 0.1 and δ = 0.05.

17/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Output feedback with perturbation

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 δ = 0.1 ε1(t) ε2(t) ε3(t) time(s) 2000 4000 6000 8000

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 δ = 0.05 ε1(t) ε2(t) ε3(t) time(s) 2000 4000 6000 8000

Figure: Observation errors of system (Σ) with u = λs

δ(ˆ

x) for δ = 0.1 and δ = 0.05.

18/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

1

Introduction

2

Practical stability

3

Numerical simulations

4

Ongoing work: exact stabilization

5

Conclusion

19/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Decreasing perturbation

Consider the feedback λs

δ,α(ˆ

x) = r1(ˆ x1, ˆ x2) + (|ˆ x3|α − 1)(δ, δ). where r1, δ and α are arbitrary positive constants. The equations of the controller-observer system are

    

˙ ˆ x = A

• λs

δ,α(ˆ

x)

• ˆ

x − r2C′Cε ˙ ε =

• A
• λs

δ,α(ˆ

x)

• − r2C′C
• ε,

(ˆ x, ε) ∈ R6. (CLO2)

20/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Main result

Lemma The point (xt, 0) is asymptotically stable for system (CLO2) and its basin of attraction is R3 × R3 \ {−xt, 0}. Sketch of proof.

◮ Prove that λs δ,α is a stabilizing state feedback for system (Σ) ◮ Write system (CLO2) in R5 using ˆ

x − ε = 1 and prove that 0 ∈ R5 is (locally) stable

◮ Prove that any trajectory of system (CLO2) converges to

(xt, 0) using ω-limit arguments

21/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Numerical simulations

• 1
• 0.5

0.5 1 δ = 0.1 α = 2 x1(t) x2(t) x3(t) time(s) ×105 2 4 6 8 10

• 1
• 0.5

0.5 1 δ = 0.1 α = 6 x1(t) x2(t) x3(t) time(s) ×105 2 4 6 8 10

Figure: State variables of system (Σ) with u = λs

δ(ˆ

x) for α = 2 and α = 6 (δ = 0.1 in both cases).

22/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Numerical simulations

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 δ = 0.1 α = 2 ε1(t) ε2(t) ε3(t) time(s) ×105 2 4 6 8 10

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 δ = 0.1 α = 6 ε1(t) ε2(t) ε3(t) time(s) ×105 2 4 6 8 10

Figure: Observation errors of system (Σ) with u = λs

δ(ˆ

x) for α = 2 and α = 6 (δ = 0.1 in both cases).

23/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

1

Introduction

2

Practical stability

3

Numerical simulations

4

Ongoing work: exact stabilization

5

Conclusion

24/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Conclusion

For a particular example we saw:

◮ a method to stabilize the system arbitrarily close to the target

point

◮ a method to stabilize the system to the target point ◮ in both cases the feedbacks are smooths and time invariant ◮ the second method is a lot slower than the first one

Perspectives:

◮ Find more qualitative proofs and extend to more general

systems (of the same form)

◮ Extend to bilinear systems

25/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

Joyeux anniversaire Jean-Michel !

26/26 Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion

References

Besancon, G. and Hammouri, H. (2000). Some remarks on dynamic output feedback control of non uniformly observable systems. In Decision and Control, 2000. Proceedings of the 39th IEEE Conference on, volume 3, pages 2462–2465 vol.3. Boscain, U., Gauthier, J.-P., Rossi, F., and Sigalotti, M. (2015). Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems.

• Comm. Math. Phys., 333(3):1225–1239.

Celle, F., Gauthier, J.-P., Kazakos, D., and Sallet, G. (1989). Synthesis of nonlinear observers: a harmonic-analysis approach.

• Math. Systems Theory, 22(4):291–322.

Coron, J.-M. (1994). On the stabilization of controllable and observable systems by an output feedback law.

• Math. Control Signals Systems, 7(3):187–216.

Gauthier, J.-P. and Kupka, I. (2001). Deterministic observation theory and applications. Cambridge University Press, Cambridge. LaSalle, J. P. (1968). Stability theory for ordinary differential equations.

• J. Differential Equations, 4:57–65.

Teel, A. and Praly, L. (1994). Global stabilizability and observability imply semi-global stabilizability by output feedback. Systems Control Lett., 22(5):313–325.