Stabilization of Nonlinear Systems by Oscillating Controls with - PowerPoint PPT Presentation

Stabilization of Nonlinear Systems by Oscillating Controls with Application to Nonholonomic and Fluid Dynamics Alexander Zuyev School and Workshop on Mixing and Control, ICTP, Trieste 16–20 September 2019

Outline 1. Motivation: Systems with Uncontrollable Linearization Controllability ⇒ Stabilizability ? Controllability ⇒ Stabilizability ! 2. Stabilization by Fast Oscillating Controls Nonholonomic Systems Control Design Scheme Exponential Stability Results Examples 3. Systems with Drift: Small-Time Local Controllability (STLC) Conditions Control Design Scheme Euler’s Equations in Rigid Body Dynamics 4. Hydrodynamical Models The Navier–Stokes and Euler Equations Lie Brackets and Energy Cascades Stabilization of Finite-Dimensional Systems Zuyev Stabilization by oscillating controls 2/41

Systems with Uncontrollable Linearization Unicycle Euler’s equations in rigid body dynamics x 1 = u 1 cos x 3 , ˙ J 1 ˙ x 1 = ( J 2 − J 3 ) x 2 x 3 + µ 11 u 1 + µ 21 u 2 , x 2 = u 1 sin x 3 , ˙ J 2 ˙ x 2 = ( J 3 − J 1 ) x 1 x 3 + µ 12 u 1 + µ 22 u 2 , x 3 = u 2 . ˙ J 3 ˙ x 3 = ( J 1 − J 2 ) x 1 x 2 + µ 13 u 1 + µ 23 u 2 . The Navier–Stokes and Euler equations on T 2 (incompressible case) m ∂ v ∂ t + ( v · ∇ ) v + 1 � ρ ∇ p − ν ∆ v = u j F j ( y ) , j =1 ∇ · v = 0 , y = ( y 1 , y 2 ) ∈ T 2 , v = ( v 1 ( t , y ) , v 2 ( t , y )) – velocity, p = p ( t , y ) – pressure. The Euler equations: ν = 0. Zuyev Stabilization by oscillating controls 3/41

Motivation: Controllability ⇒ Stabilizability ? Consider m � x ∈ D ⊂ R n , u ∈ R m , 0 ∈ D , x = f 0 ( x ) + ˙ u j f j ( x ) ≡ f ( x , u ) , (Σ) j =1 where f 0 , f 1 , ..., f m are smooth, f 0 (0) = 0, and m < n . Zuyev Stabilization by oscillating controls 4/41

Motivation: Controllability ⇒ Stabilizability ? Consider m � x ∈ D ⊂ R n , u ∈ R m , 0 ∈ D , x = f 0 ( x ) + ˙ u j f j ( x ) ≡ f ( x , u ) , (Σ) j =1 where f 0 , f 1 , ..., f m are smooth, f 0 (0) = 0, and m < n . Stabilization by a time-invariant feedback Find a continuous u = k ( x ), k (0) = 0 s.t. the solution x = 0 of x = f ( x , k ( x )) ≡ F ( x ) is asymptotically stable in the sense of Lyapunov. ˙ References R.E. Kalman (1961), N.N. Krasovskii (1966), G.V. Kamenkov (1972), V.I. Korobov (1973), Z. Artstein (1983), R.W. Brockett (1983), V.G. Veretennikov (1984), M. Kawski (1989), J.-M. Coron, L. Praly, A. Teel (1995), F.H. Clarke, Yu. S. Ledyaev, E.D. Sontag, A.I. Subbotin (1997), S. Celikovsky, E. Aranda-Bricaire (1999), P. Morin, J.-B. Pomet, C. Samson (1999), ... , F. Gao, Y. Wu, H. Li, Y. Liu (2018), ... Zuyev Stabilization by oscillating controls 4/41

Motivation: Obstacles for asymptotic stability Krasnoselskii–Zabreiko theorem (1974) x = f ( x , k ( x )) ≡ F ( x ) , x ∈ R n , then If x = 0 is asymptotically stable for ˙ γ [ F , S ε ] = ( − 1) n for any small enough ε > 0. Rotation (degree) of a continuous vector field F : S ε → R n If F ( x ) � = 0 on a sphere S ε = ε S n − 1 R n then γ [ F , S ε ] ∈ Z is well-defined. Zuyev Stabilization by oscillating controls 5/41

Motivation: Obstacles for asymptotic stability Krasnoselskii–Zabreiko theorem (1974) x = f ( x , k ( x )) ≡ F ( x ) , x ∈ R n , then If x = 0 is asymptotically stable for ˙ γ [ F , S ε ] = ( − 1) n for any small enough ε > 0. Rotation (degree) of a continuous vector field F : S ε → R n If F ( x ) � = 0 on a sphere S ε = ε S n − 1 R n then γ [ F , S ε ] ∈ Z is well-defined. Topological constraints for asymptotic stability γ [ F , S ε ] = 0 γ [ F , S ε ] = 1 γ [ F , S ε ] = 2 Zuyev Stabilization by oscillating controls 5/41

Motivation: Obstacles for asymptotic stability Topological constraints for asymptotic stability γ [ F , S ε ] = 0 γ [ F , S ε ] = 1 γ [ F , S ε ] = 2 Principle of nonzero rotation If F ∈ C ( ¯ B ), ¯ B - closed ball, γ [ F , ∂ B ] � = 0 ⇒ ∃ ˜ x ∈ B : F (˜ x ) = 0. Zuyev Stabilization by oscillating controls 5/41

Motivation: Obstacles for asymptotic stability Topological constraints for asymptotic stability γ [ F , S ε ] = 0 γ [ F , S ε ] = 1 γ [ F , S ε ] = 2 Brockett’s necessary stabilizability condition (1983) If x = 0 is stabilizable for ˙ x = f ( x , u ) by a continuous feedback law u = k ( x ), k (0) = 0, then ∀ ε > 0 ∃ δ > 0 s.t. B ε ( x ∗ ) := { x : � x − x ∗ � < ε } . � � B δ (0) ⊂ f B ε (0) , B ε (0) , Zuyev Stabilization by oscillating controls 5/41

Motivation: Obstacles for asymptotic stability Examples of non-stabilizable systems ( R . W . Brockett ′ 83) x 1 = u 1 , ˙ ˙ x 2 = u 2 , ˙ x 3 = x 2 u 1 − x 1 u 2 . x 2 = x 2 1 − 2 x 1 x 2 ( J . − M . Coron & L . Rosier ′ 92) x 1 = x 3 , ˙ ˙ 3 , ˙ x 3 = u . z = f 0 z s + ug 0 z q , z = x 1 + ix 2 , 2 q − 1 > s > 1 . ( B . Jakubczyk & A . Z . ′ 05) ˙ An academic example (Brockett’s example) x 1 = u 1 , ˙ ˙ x 2 = u 2 , ˙ x 3 = x 2 u 1 − x 1 u 2 . Brockett’s condition fails: the system of algebraic equations     u 1 0  = u 2 0 has no solutions if p 3 � = 0 .    x 2 u 1 − x 1 u 2 p 3 Zuyev Stabilization by oscillating controls 6/41

Motivation: Obstacles for asymptotic stability Examples of non-stabilizable systems ( R . W . Brockett ′ 83) x 1 = u 1 , ˙ ˙ x 2 = u 2 , ˙ x 3 = x 2 u 1 − x 1 u 2 . x 2 = x 2 1 − 2 x 1 x 2 ( J . − M . Coron & L . Rosier ′ 92) x 1 = x 3 , ˙ ˙ 3 , ˙ x 3 = u . z = f 0 z s + ug 0 z q , z = x 1 + ix 2 , 2 q − 1 > s > 1 . ( B . Jakubczyk & A . Z . ′ 05) ˙ A practical motivation: stabilization of nonholonomic systems x 1 = u 1 cos x 3 , ˙ x 2 = u 1 sin x 3 , ˙ x ∈ R 3 , u ∈ R 2 . x 3 = u 2 , ˙ Control Lyapunov functions do not exist for underactuated ( m < n ) driftless ( f 0 ( x ) ≡ 0) Unicycle systems! Zuyev Stabilization by oscillating controls 6/41

Brockett’s stabilizability condition Dynamic extension of Euler’s equations with dim ( u ) = 2 ω = A ω × ω + µ 1 u 1 + µ 2 u 2 , ˙ ˙ φ = ω 1 cos θ + ω 3 sin θ, ˙ θ = ω 1 sin θ tan φ + ω 2 − ω 3 cos θ tan φ, ˙ ψ = − ω 1 sin θ sec φ + ω 3 cos θ sec φ. C. Byrnes (2008): Brockett’s condition is violated! The algebraic equation f ( x , φ, θ, ψ, u 1 , u 2 ) = ( y 1 , y 2 , y 3 , 0 , 0 , 0) T has no solutions generically for small | y | . Zuyev Stabilization by oscillating controls 7/41

Motivation: Controllability ⇒ Stabilizability ? m � x ∈ D ⊂ R n , u ∈ R m , m < n . x = f 0 ( x ) + ˙ u j f j ( x ) ≡ f ( x , u ) , (Σ) j =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General question: Controllability ⇒ Stabilizability ? ? ⇒ ∀ x 0 , x 1 ∈ D ∃ u x 0 x 1 ∈ L ∞ [0 , T ] ∃ k ∈ C ( D ) : k (0) = 0 Zuyev Stabilization by oscillating controls 8/41

Motivation: Controllability ⇒ Stabilizability ? m � x ∈ D ⊂ R n , u ∈ R m , m < n . x = f 0 ( x ) + ˙ u j f j ( x ) ≡ f ( x , u ) , (Σ) j =1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General question: Controllability ⇒ Stabilizability ? Linear and Linearizable Systems x = Ax + Bu , ˙ ∃ u = Kx : ⇒ rank ( B , AB , ..., A n − 1 B ) = n x = 0 - exponentially stable General Systems of the Form (Σ) Lie x =0 { f 0 , f 1 , ..., f m } = R n ∃ u = k ( x ) : k ∈ C ( D ) , k (0) = 0 : �⇒ (Lie algebra rank condition) x = 0 - asymptotically stable Zuyev Stabilization by oscillating controls 8/41

Motivation: Controllability ⇒ Stabilizability ! Existence Results J.-M. Coron (1995) Assume that x = 0 is locally continuously reachable in small time for the control system x = f ( x , u ) , ( x , u ) ∈ O ⊂ R n × R m , (0 , 0) ∈ O , f (0 , 0) = 0 , ˙ (Σ) that (Σ) satisfies the Lie algebra rank condition at (0 , 0) ∈ R n × R m and that n / ∈ { 2 , 3 } . Then (Σ) is locally stabilizable in small time by means of almost smooth periodic time-varying feedback laws u = k ( x , t ). F.H. Clarke, Yu.S. Ledyaev, E.D. Sontag, A.I. Subbotin (1997) System (Σ) is asymptotically controllable if and only if it admits an s -stabilizing feedback u = k ( x ). (Solutions are defined in the sense of sampling – “ π -trajectories” or “ π ε -solutions”). Zuyev Stabilization by oscillating controls 9/41

Sampling and π ε -solutions Partition of t ∈ [0 , + ∞ ) For a given ε > 0, we denote by π ε the partition of [0 , + ∞ ) into intervals I j = [ t j , t j +1 ) , t j = ε j , j = 0 , 1 , 2 , . . . . π ε -solutions Assume given a feedback u = h ( t , x ), h : [0 , + ∞ ) × D → R m , ε > 0, and x 0 ∈ R n . A π ε -solution of system (Σ) corresponding to x 0 ∈ D and h ( t , x ) is an absolutely continuous function x ( t ) ∈ D , defined for t ∈ [0 , + ∞ ), which satisfies the initial condition x (0) = x 0 and the following differential equations x ( t ) = f ( x ( t ) , h ( t , x ( t j ))) , ˙ t ∈ I j = [ t j , t j +1 ) , for each j = 0 , 1 , 2 , . . . . Zuyev Stabilization by oscillating controls 10/41