Output stabilization at unobservable points: analysis via an example - PowerPoint PPT Presentation

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Output stabilization at unobservable points: analysis via an example J.P. Gauthier a M.A. Lagache ab U. Serres b a Universit´ e de Toulon, France b Universit´ e de Lyon, France 60th birthday of Jean-Michel Coron IHP, June 2016 1/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Table of contents Introduction 1 Practical stability 2 Numerical simulations 3 Ongoing work: exact stabilization 4 Conclusion 5 2/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Introduction 1 Practical stability 2 Numerical simulations 3 Ongoing work: exact stabilization 4 Conclusion 5 3/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion System under consideration Consider the closed quantum system 1    0 e u 1    x = A ( u ) x = ˙ − e 0 u 2  x      (Σ) − u 1 − u 2 0     y = Cx = x 3  where ◮ x = ( x 1 , x 2 , x 3 ) ∈ S 2 is the state variable ◮ y ∈ R is the measured output ◮ u = ( u 1 , u 2 ) ∈ R 2 is the control variable Aim: Stabilize (Σ) to the target point x t = (0 , 0 , − 1) by mean of a smooth dynamic time invariant output feedback 1 see e.g. [Boscain et al., 2015] 4/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Problem Problem: The equilibrium point x t corresponds to the null input, which makes the system unobservable x 1 ( t ) x 2 ( t ) x 3 ( t ) 1 0.8 0.6 Some results about output feedback: 0.4 0.2 ◮ [Teel and Praly, 1994] 0 -0.2 ◮ [Coron, 1994] -0.4 -0.6 ◮ [Besancon and Hammouri, 2000] -0.8 -1 0 5 10 15 20 25 30 35 40 45 50 time(s) Figure: State variables of system (Σ) using a ”naive” approach 5/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Introduction 1 Practical stability 2 Numerical simulations 3 Ongoing work: exact stabilization 4 Conclusion 5 6/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Stabilizing state feedback Consider the state feedback λ s ( x ) = r 1 ( x 1 , x 2 ) , where r 1 is an arbitrary positive constant. Proposition The target point x t is an asymptotically stable equilibrium for the closed-loop system resulting from applying the feedback control u = λ s ( x ) to system (Σ) . Moreover, its basin of attraction is S 2 \ {− x t } . Sketch of proof. A direct application of LaSalle’s principle (see e.g. [LaSalle, 1968]) using V ( x ) = x 3 , as a candidate Lyapunov function gives the desired result. 7/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion The observer The equations of the controller-observer system are � ˙ x = A ( λ s ˆ δ (ˆ x )) ˆ x − r 2 C ′ C ε (CLO) � ε, x , ε ) ∈ R 3 × R 3 , � A ( λ s x )) − r 2 C ′ C ε = ˙ δ (ˆ (ˆ where λ s x ) = λ s (ˆ δ (ˆ x ) + ( δ, δ ) , and δ and r 2 are positive constants. 8/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion The observer Lemma 1 All the inputs of the form u = λ s 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 λ s δ (ˆ x ( · )) that renders system (Σ) unobservable on [0 , T ] ◮ There exists a ω ( · ) = ( ω 1 , ω 2 , ω 3 ) �≡ 0 solution of (Σ) such that ω 3 ( · ) ≡ 0 ◮ Differentiating with respect to t and solving with respect to ω ( · ), we get that ω ( · ) vanishes identically 9/26

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 ind( u ( · + θ ) , T ) > 0 , θ → + ∞ 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 � are persistent. All the inputs λ s � ˆ x ( · ) δ 10/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Step 1 : Estimation error goes to zero Sketch of proof. ◮ x ∈ S 2 and ε is decreasing (since 1 dt � ε � 2 = − r 2 ( C ε ) 2 ) d 2 − → { (ˆ x ( t ) , ε ( t )) | t � 0 } lies in a compact K ◮ The mapping R 3 × R 3 F : → R + ind( λ s (ˆ x 0 , ε 0 ) �→ δ (ˆ x ( · )) , T ) is continuous and nonnegative for all T > 0 ◮ Since ind( λ s inf K F � lim sup F (ˆ x ( θ ) , ε ( θ )) = lim sup � ˆ x ( · + θ ) � , T ) δ θ → + ∞ θ → + ∞ ◮ By continuity, the infimum of F over K is reached, and is positive by the crucial Lemma 1 11/26

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 + , R 2 ) is a persistent input, then the observation error tends to zero, i.e. t → + ∞ ε ( t ) = 0 . lim Corollary 2 For any trajectory of the coupled system (CLO) , we have lim t → + ∞ ε ( t ) = 0 . 12/26

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 ( x s δ , 0) arbitrarily close to ( x t , 0) and an unstable equilibrium point ( x u δ , 0) arbitrarily close to ( − x t , 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 13/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Step 3 : The main result Theorem δ , 0) ∈ R 6 is asymptotically stable If δ is small enough, the point ( x s for system (CLO) , and its region of attraction is R 6 \ { ( x u 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 ∈ S 2 | x 3 � 0 } and consider the function L : C − → R + defined by L (ˆ x ) = 1 δ � 2 . x − x s 2 � ˆ ◮ Using LaSalle’s principle and Lemma 3 we get the desired result. 14/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Introduction 1 Practical stability 2 Numerical simulations 3 Ongoing work: exact stabilization 4 Conclusion 5 15/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Output feedback with perturbation δ = 0 . 1 δ = 0 . 05 1 1 x 1 ( t ) x 1 ( t ) x 2 ( t ) x 2 ( t ) x 3 ( t ) x 3 ( t ) 0.5 0.5 0 0 -0.5 -0.5 -1 -1 0 2000 4000 6000 8000 0 2000 4000 6000 8000 time(s) time(s) Figure: State variables of system (Σ) with u = λ s δ (ˆ x ) for δ = 0 . 1 and δ = 0 . 05. 16/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Output feedback with perturbation δ = 0 . 1 δ = 0 . 05 0.8 0.8 ε 1 ( t ) ε 1 ( t ) 0.6 0.6 ε 2 ( t ) ε 2 ( t ) ε 3 ( t ) ε 3 ( t ) 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 0 2000 4000 6000 8000 0 2000 4000 6000 8000 time(s) time(s) Figure: Observation errors of system (Σ) with u = λ s δ (ˆ x ) for δ = 0 . 1 and δ = 0 . 05. 17/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Introduction 1 Practical stability 2 Numerical simulations 3 Ongoing work: exact stabilization 4 Conclusion 5 18/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Decreasing perturbation Consider the feedback x 3 | α − 1)( δ, δ ) . λ s δ,α (ˆ x ) = r 1 (ˆ x 1 , ˆ x 2 ) + ( | ˆ where r 1 , δ and α are arbitrary positive constants. The equations of the controller-observer system are  � � ˙ λ s x − r 2 C ′ C ε x = A ˆ δ,α (ˆ x ) ˆ   (CLO2) � � � � x , ε ) ∈ R 6 . λ s − r 2 C ′ C ε = ˙ A δ,α (ˆ x ) ε, (ˆ   19/26

Introduction Practical stability Numerical simulations Ongoing work: exact stabilization Conclusion Main result Lemma The point ( x t , 0) is asymptotically stable for system (CLO2) and its basin of attraction is R 3 × R 3 \ {− x t , 0 } . Sketch of proof. ◮ Prove that λ s δ,α is a stabilizing state feedback for system (Σ) ◮ Write system (CLO2) in R 5 using � ˆ x − ε � = 1 and prove that 0 ∈ R 5 is (locally) stable ◮ Prove that any trajectory of system (CLO2) converges to ( x t , 0) using ω -limit arguments 20/26