Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear - PowerPoint PPT Presentation

Analysis of Discrete and Hybrid Stochastic Systems by Nonlinear Contraction Theory Phạm Quang Cường Laboratoire de Physiologie de la Perception et de l’Action (LPPA) Collège de France – CNRS, Paris, France Work funded by the BACS (Bayesian Approach to Complex Systems) European Project Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 1 / 21

Outline Nonlinear contraction theory 1 Stochastic nonlinear contraction theory 2 Hybrid-resetting stochastic contraction 3 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 2 / 21

Outline Nonlinear contraction theory 1 Stochastic nonlinear contraction theory 2 Hybrid-resetting stochastic contraction 3 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 3 / 21

Motivations Real systems are typically complex with many feedback loops, so there is no a priori reason for stability Evolution, engineering: accumulation of stable subparts How stability can be preserved through combinations ? Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 4 / 21

A new tool for stability analysis Consider a nonlinear time-varying system ˙ x = f ( x , t ) If there exists a (nonlinear) coordinate transform Θ ( x , t ) such that M = Θ ( x , t ) ⊤ Θ ( x , t ) > 0 uniformly , and �� Θ + Θ ∂ f � � ˙ Θ − 1 ∀ x , t λ max < − λ ∂ x s then all system trajectories converge exponentially to a single trajectory, independently of the initial conditions (incremental stability) � � Θ − 1 is called the generalized Jacobian of f in the ˙ Θ + Θ ∂ f Note : J = ∂ x metric M Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 5 / 21

Sketch of proof (case of identity metric) Consider a smooth path between any pair of trajectories, parameterized by u system trajectories Lewis, Am. J. of Mathematics , 1951 Lohmiller & Slotine, Automatica , 1998 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 6 / 21

Sketch of proof (case of identity metric) Consider a smooth path between any pair of trajectories, parameterized by u Consider an element of distance system trajectories ⊤ ∂ x ∂ x ∂ u on that path. ∂ u Lewis, Am. J. of Mathematics , 1951 Lohmiller & Slotine, Automatica , 1998 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 6 / 21

Sketch of proof (case of identity metric) Consider a smooth path between any pair of trajectories, parameterized by u Consider an element of distance system trajectories ⊤ ∂ x ∂ x ∂ u on that path. ∂ u We have „ ∂ x « „ d x « d = ∂ = ∂ u ∂ u dt dt ∂ u ( f ( x , t )) = ∂ f ∂ ∂ x ∂ x ∂ u Lewis, Am. J. of Mathematics , 1951 Lohmiller & Slotine, Automatica , 1998 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 6 / 21

Sketch of proof (case of identity metric) Consider a smooth path between any pair of trajectories, parameterized by u Consider an element of distance system trajectories ⊤ ∂ x ∂ x ∂ u on that path. ∂ u We have „ ∂ x « „ d x « d = ∂ = ∂ u ∂ u dt dt ∂ u ( f ( x , t )) = ∂ f ∂ ∂ x ∂ x ∂ u Thus „ ∂ x ⊤ ∂ x ⊤ ∂ f « d = 2 ∂ x ∂ x Lewis, Am. J. of Mathematics , 1951 dt ∂ u ∂ u ∂ u ∂ x ∂ u Lohmiller & Slotine, Automatica , 1998 ⊤ ∂ x ≤ − 2 λ ∂ x ∂ u ∂ u Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 6 / 21

Interesting properties Exact and global analysis, as opposed to linearization techniques “Separation principle” for nonlinear systems Converse theorem (global exponential stability ⇒ contraction in some metric) Combination properties: Parallel Hierarchy Negative feedback Small gains Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 7 / 21

Example: negative feedback Consider the combination � d x 1 = f 1 ( x 1 , x 2 , t ) dt d x 2 = f 2 ( x 1 , x 2 , t ) dt with system x i contracting with rate λ i in the metric M i = Θ ⊤ i Θ i Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 8 / 21

Example: negative feedback Consider the combination � d x 1 = f 1 ( x 1 , x 2 , t ) dt d x 2 = f 2 ( x 1 , x 2 , t ) dt with system x i contracting with rate λ i in the metric M i = Θ ⊤ i Θ i Assume that the connection is negative feedback, i.e. � ∂ f 1 � ∂ f 2 � T � Θ − 1 Θ − 1 Θ 1 = − k Θ 2 2 1 ∂ x 2 ∂ x 1 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 8 / 21

Example: negative feedback Consider the combination � d x 1 = f 1 ( x 1 , x 2 , t ) dt d x 2 = f 2 ( x 1 , x 2 , t ) dt with system x i contracting with rate λ i in the metric M i = Θ ⊤ i Θ i Assume that the connection is negative feedback, i.e. � ∂ f 1 � ∂ f 2 � T � Θ − 1 Θ − 1 Θ 1 = − k Θ 2 2 1 ∂ x 2 ∂ x 1 Then the global system is contracting with rate min( λ 1 , λ 2 ) in the metric M = Θ T Θ where � Θ 1 � 0 √ Θ = 0 k Θ 2 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 8 / 21

Recent developments and applications of Contraction theory Controllers and observers design Distributed systems driven by nonlinear Partial Derivative Equations Hybrid systems (Elrifai & Slotine, IEEE TAC , 2006) Synchronization (Wang & Slotine, Biological Cybernetics , 2005; Pham & Slotine, Neural Networks , 2007) Analysis of the complex neural systems (Girard et al, Neural Networks , 2008) Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 9 / 21

Outline Nonlinear contraction theory 1 Stochastic nonlinear contraction theory 2 Hybrid-resetting stochastic contraction 3 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 10 / 21

Modelling the random perturbations In physics, engineering, finance, neuroscience,. . . random perturbations are traditionnally modelled with Itô stochastic differential equations (Itô SDE) d x = f ( x , t ) dt + σ ( x , t ) dW f is the dynamics of the noise-free version of the system σ is the noise variance matrix (noise intensity) W is a Wiener process ( dW / dt = “white noise”) Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 11 / 21

The stochastic contraction theorem If the noise-free system is contracting in some metric λ max ( J s ) ≤ − λ Pham, Tabareau & Slotine, to appear in IEEE Transactions on Automatic Control Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 12 / 21

The stochastic contraction theorem If the noise-free system is contracting in some metric λ max ( J s ) ≤ − λ and the noise variance is upper-bounded in the same metric � � σ ( x , t ) ⊤ M ( t ) σ ( x , t ) ≤ C tr Pham, Tabareau & Slotine, to appear in IEEE Transactions on Automatic Control Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 12 / 21

The stochastic contraction theorem If the noise-free system is contracting in some metric λ max ( J s ) ≤ − λ and the noise variance is upper-bounded in the same metric � � σ ( x , t ) ⊤ M ( t ) σ ( x , t ) ≤ C tr Then, for any pair of system trajectories ( a , b ) ≤ 1 � C � � a ( t ) − b ( t ) � 2 � � a 0 − b 0 � 2 � e − 2 λ t � � ∀ t ≥ 0 E λ + E β where β is the lower-bound of the metric : ∀ x , x ⊤ Mx ≥ β � x � 2 Pham, Tabareau & Slotine, to appear in IEEE Transactions on Automatic Control Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 12 / 21

Practical meaning After exponential transients of rate λ , we have � C E ( � a ( t ) − b ( t ) � ) ≤ βλ a 0 a 0 C/ λ b 0 1 1 b 0 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 13 / 21

Sketch of proof (case of identity metric) Consider the Lyapunov-like function V ( a , b ) = � a − b � 2 And differentiate using Ito’s rule LV ≤ − 2 λ V + 2 C Then use Dynkin’s formula and a Gronwall-type lemma (see paper) to conclude that � + � E V ( a ( t ) , b ( t )) ≤ C V ( a 0 , b 0 ) − C e − 2 λ t λ + λ Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 14 / 21

Interesting features This bound is optimal, in the sense that it can be attained (consider an Ornstein-Uhlenbeck process) Combinations results in deterministic contraction can be adapted very naturally for stochastic contraction Parallel combinations Hierarchical combinations Negative feedback combinations Small gains More generally, all results in deterministic contraction theory can be easily translated into the stochastic framework (e.g. observers design under measurement noise, stochastic synchronization, stochastic hybrid systems: see next) Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 15 / 21

Extension to state-dependent metrics Currently, we can prove the theorem only for time-varying, state-independent metrics For state-dependent metrics, the proof is a lot more difficult, but doable using the discretization procedure (Euler-Maruyama method) based on discrete stochastic contraction. Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 16 / 21

Outline Nonlinear contraction theory 1 Stochastic nonlinear contraction theory 2 Hybrid-resetting stochastic contraction 3 Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 17 / 21