Lyapunov Function Constructions for Slowly Time-Varying Systems - PowerPoint PPT Presentation

Lyapunov Function Constructions for Slowly Time-Varying Systems MICHAEL MALISOFF Department of Mathematics Louisiana State University Joint with Fr´ ed´ eric Mazenc, Projet MERE INRIA-INRA Stability Regular Session Paper FrA15.3 45th IEEE Conference on Decision and Control Manchester Grand Hyatt Hotel, San Diego, CA December 13-15, 2006

REVIEW of MODEL and LITERATURE Goals: For large constants α > 0 , prove input-to-state stability (ISS) for x = f ( x, t, t/α ) + g ( x, t, t/α ) u, x ( t 0 ) = x o ˙ ( Σ ) and construct explicit corresponding ISS Lyapunov functions.

REVIEW of MODEL and LITERATURE Goals: For large constants α > 0 , prove input-to-state stability (ISS) for x = f ( x, t, t/α ) + g ( x, t, t/α ) u, x ( t 0 ) = x o ˙ ( Σ ) and construct explicit corresponding ISS Lyapunov functions. Literature: Uses exponential-like stability of ˙ x = f ( x, t, τ ) aka (Σ fro ) for all relevant values of the scalar τ to show stability for u ≡ 0 but does not lead to explicit Lyapunov functions for (Σ) (Peuteman-Aeyels, Solo).

REVIEW of MODEL and LITERATURE Goals: For large constants α > 0 , prove input-to-state stability (ISS) for x = f ( x, t, t/α ) + g ( x, t, t/α ) u, x ( t 0 ) = x o ˙ ( Σ ) and construct explicit corresponding ISS Lyapunov functions. Literature: Uses exponential-like stability of ˙ x = f ( x, t, τ ) aka (Σ fro ) for all relevant values of the scalar τ to show stability for u ≡ 0 but does not lead to explicit Lyapunov functions for (Σ) (Peuteman-Aeyels, Solo). Our Contributions: We explicitly construct Lyapunov functions for (Σ) in terms of given Lyapunov functions for (Σ fro ) without assuming any exponential-like stability of (Σ fro ) and we allow τ to be a vector.

REVIEW of MODEL and LITERATURE Goals: For large constants α > 0 , prove input-to-state stability (ISS) for x = f ( x, t, t/α ) + g ( x, t, t/α ) u, x ( t 0 ) = x o ˙ ( Σ ) and construct explicit corresponding ISS Lyapunov functions. Literature: Uses exponential-like stability of ˙ x = f ( x, t, τ ) aka (Σ fro ) for all relevant values of the scalar τ to show stability for u ≡ 0 but does not lead to explicit Lyapunov functions for (Σ) (Peuteman-Aeyels, Solo). Our Contributions: We explicitly construct Lyapunov functions for (Σ) in terms of given Lyapunov functions for (Σ fro ) without assuming any exponential-like stability of (Σ fro ) and we allow τ to be a vector. Significance: Lyapunov functions for (Σ fro ) are often readily available. Explicit Lyapunov functions and slowly time-varying models are important in control engineering e.g. control of friction, pendulums, etc.

MAIN ASSUMPTION and MAIN THEOREM We first assume our (sufficiently regular) system (Σ) has the form x = f ( x, t, p ( t/α )) ˙ ( Σ p ) where p : R → R d is bounded and ¯ p := sup {| p ′ ( r ) | : r ∈ R } < ∞ .

MAIN ASSUMPTION and MAIN THEOREM We first assume our (sufficiently regular) system (Σ) has the form x = f ( x, t, p ( t/α )) ˙ ( Σ p ) where p : R → R d is bounded and ¯ p := sup {| p ′ ( r ) | : r ∈ R } < ∞ . Assume: ∃ δ 1 , δ 2 ∈ K ∞ ; constants c a , c b , T > 0 ; a continuous function q : R d → R ; and a C 1 V : R n × [0 , ∞ ) × R d → [0 , ∞ ) s.t. ∀ x ∈ R n , t ≥ 0 , and τ ∈ R ( p ) := { p ( t ) : t ∈ R } : (i) | V τ ( x, t, τ ) | ≤ c a V ( x, t, τ ) , � t (ii) δ 1 ( | x | ) ≤ V ( x, t, τ ) ≤ δ 2 ( | x | ) , (iii) t − T q ( p ( s ))d s ≥ c b , and (iv) V t ( x, t, τ ) + V x ( x, t, τ ) f ( x, t, τ ) ≤ − q ( τ ) V ( x, t, τ ) all hold.

MAIN ASSUMPTION and MAIN THEOREM We first assume our (sufficiently regular) system (Σ) has the form x = f ( x, t, p ( t/α )) ˙ ( Σ p ) where p : R → R d is bounded and ¯ p := sup {| p ′ ( r ) | : r ∈ R } < ∞ . Assume: ∃ δ 1 , δ 2 ∈ K ∞ ; constants c a , c b , T > 0 ; a continuous function q : R d → R ; and a C 1 V : R n × [0 , ∞ ) × R d → [0 , ∞ ) s.t. ∀ x ∈ R n , t ≥ 0 , and τ ∈ R ( p ) := { p ( t ) : t ∈ R } : (i) | V τ ( x, t, τ ) | ≤ c a V ( x, t, τ ) , � t (ii) δ 1 ( | x | ) ≤ V ( x, t, τ ) ≤ δ 2 ( | x | ) , (iii) t − T q ( p ( s ))d s ≥ c b , and (iv) V t ( x, t, τ ) + V x ( x, t, τ ) f ( x, t, τ ) ≤ − q ( τ ) V ( x, t, τ ) all hold. Theorem A: For each constant α > 2 Tc a ¯ p/c b , (Σ p ) is UGAS and t t � � α α q ( p ( l ))d l d s α T t V ♯ α − T s α ( x, t ) := e V ( x, t, p ( t/α )) is a Lyapunov function for (Σ p ) . [UGAS: | φ ( t ; t o , x o ) | ≤ β ( | x o | , t − t o ) ]

SKETCH of PROOF of THEOREM A Step 1: Set ˆ V ( x, t ) := V ( x, t, p ( t/α )) . Along trajectories of (Σ p ) , d − q ( p ( t/α )) + c a ¯ p � � ˆ ˆ V ≤ V ( x, t ) . ( ⋆ ) dt α Important: The term involving α in ( ⋆ ) vanishes if V τ ≡ 0 .

SKETCH of PROOF of THEOREM A Step 1: Set ˆ V ( x, t ) := V ( x, t, p ( t/α )) . Along trajectories of (Σ p ) , d − q ( p ( t/α )) + c a ¯ p � � ˆ ˆ V ≤ V ( x, t ) . ( ⋆ ) dt α Important: The term involving α in ( ⋆ ) vanishes if V τ ≡ 0 . Step 2: Substitute ( ⋆ ) into � � t � � � α ˙ dt ˆ ˆ q ( p ( t/α )) − 1 V ♯ d = E ( t, α ) V + q ( p ( l ))d l V α T t α − T � c a ¯ α − c b p � ˆ ≤ E ( t, α ) V ( x, t ) , T α ( x, t ) = E ( t, α ) ˆ where V ♯ V ( x, t ) and t �� t � � α α q ( p ( l ))d l d s α T t α − T s E ( t, α ) := e .

EXAMPLE 1: STABILITY for ALL PARAMETER VALUES The assumptions of Theorem A hold for 1 − 90 cos 2 � t x = f ( x, t, cos 2 ( t/α )) := x � �� ˙ √ α 1+ x 2 √ 1+ x 2 − e. V ( x, t, τ ) ≡ ¯ V ( x ) := e

EXAMPLE 1: STABILITY for ALL PARAMETER VALUES The assumptions of Theorem A hold for 1 − 90 cos 2 � t x = f ( x, t, cos 2 ( t/α )) := x � �� ˙ √ α 1+ x 2 √ 1+ x 2 − e. V ( x, t, τ ) ≡ ¯ V ( x ) := e This follows from the estimates √ � � 2 ∇ ¯ ¯ 2 e V ( x ) f ( x, t, τ ) ≤ e − 1 − 45 τ V ( x ) � t √ √ � � � � 2 2 45 cos 2 ( s ) − 2 e 45 2 − 2 e d s = π > 0 t − π e − 1 e − 1

EXAMPLE 1: STABILITY for ALL PARAMETER VALUES The assumptions of Theorem A hold for 1 − 90 cos 2 � t x = f ( x, t, cos 2 ( t/α )) := x � �� ˙ √ α 1+ x 2 √ 1+ x 2 − e. V ( x, t, τ ) ≡ ¯ V ( x ) := e This follows from the estimates √ � � 2 ∇ ¯ ¯ 2 e V ( x ) f ( x, t, τ ) ≤ e − 1 − 45 τ V ( x ) � t √ √ � � � � 2 2 45 cos 2 ( s ) − 2 e 45 2 − 2 e d s = π > 0 e − 1 e − 1 t − π so for all α > 0 we get UGAS and the Lyapunov function √ t �� t � � � 2 45 cos 2 ( l ) − 2 e � α α d l d s α π e − 1 t ¯ α − π s V ♯ α ( x, t ) := e V ( x ) � √ � 2 sin( 2 t α )+ π − 4 πe 45 α √ 1+ x 2 − e ] 4 45( e − 1) = e [ e

EXAMPLE 2: MECHANICAL SYSTEM with FRICTION Model: Dynamics for x 1 =mass position and x 2 =velocity: x 1 ˙ = x 2 x 2 ˙ = − σ 1 ( t/α ) x 2 − k ( t ) x 1 + u (MSF) σ 2 ( t/α ) + σ 3 ( t/α ) e − β 1 µ ( x 2 ) � � − sat( x 2 ) σ i are positive friction-related coefficients; β 1 is a positive constant corresponding to Stribeck effect; µ ∈ PD is related to Stribeck effect; k is a positive time-varying spring stiffness-related coefficient; and sat( x 2 ) = tanh( β 2 x 2 ) , where β 2 is a large positive constant. α > 1 .

EXAMPLE 2: MECHANICAL SYSTEM with FRICTION Model: Dynamics for x 1 =mass position and x 2 =velocity: x 1 ˙ = x 2 x 2 ˙ = − σ 1 ( t/α ) x 2 − k ( t ) x 1 + u (MSF) σ 2 ( t/α ) + σ 3 ( t/α ) e − β 1 µ ( x 2 ) � � − sat( x 2 ) σ i are positive friction-related coefficients; β 1 is a positive constant corresponding to Stribeck effect; µ ∈ PD is related to Stribeck effect; k is a positive time-varying spring stiffness-related coefficient; and sat( x 2 ) = tanh( β 2 x 2 ) , where β 2 is a large positive constant. α > 1 . Assumptions: (a) σ i ∈ C 1 ,valued in (0 , 1] , σ ′ i bounded; (b) ∃ constants � t t − T σ 1 ( r ) dr ≥ c b ∀ t ≥ 0 ; (c) k ∈ C 1 , k ′ bounded, c b , T > 0 such that ∃ k o , ¯ k > 0 s . t . k o ≤ k ( t ) ≤ ¯ k and k ′ ( t ) ≤ 0 ∀ t ≥ 0 .

EXAMPLE 2: MECHANICAL SYSTEM with FRICTION Model: Dynamics for x 1 =mass position and x 2 =velocity: x 1 ˙ = x 2 x 2 ˙ = − σ 1 ( t/α ) x 2 − k ( t ) x 1 + u (MSF) � σ 2 ( t/α ) + σ 3 ( t/α ) e − β 1 µ ( x 2 ) � − sat( x 2 ) σ i are positive friction-related coefficients; β 1 is a positive constant corresponding to Stribeck effect; µ ∈ PD is related to Stribeck effect; k is a positive time-varying spring stiffness-related coefficient; and sat( x 2 ) = tanh( β 2 x 2 ) , where β 2 is a large positive constant. α > 1 . Assumptions: (a) σ i ∈ C 1 ,valued in (0 , 1] , σ ′ i bounded; (b) ∃ constants � t t − T σ 1 ( r ) dr ≥ c b ∀ t ≥ 0 ; (c) k ∈ C 1 , k ′ bounded, c b , T > 0 such that ∃ k o , ¯ k > 0 s . t . k o ≤ k ( t ) ≤ ¯ k and k ′ ( t ) ≤ 0 ∀ t ≥ 0 . We apply our theorem to (MSF) with p ( t ) = ( σ 1 ( t ) , σ 2 ( t ) , σ 3 ( t )) .