Stochastic contraction BACS Workshop Chamonix, January 14, 2008 Q.-C. Pham N. Tabareau J.-J. Slotine Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 1 / 19
Why stochastic contraction? Nice features of deterministic contraction theory: Powerful stability analysis tool for nonlinear systems Combination properties (modularity and stability) Hybrid, switching systems (Concurrent) synchronization in large-scale systems Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 2 / 19
Why stochastic contraction? Goal: extend contraction theory to the stochastic case Analyze real-life systems, which are typically subject to random perturbations Benefit from the nice features of contraction theory Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 3 / 19
Modelling the random perturbations In physics, engineering, neuroscience, finance,. . . random perturbations are traditionnally modelled with Itˆ o stochastic differential equations (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”) Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 4 / 19
Some notions of stochastic modelling : Random walk and Wiener process Random walk (discrete-time): x t +∆ t = x t + ξ t ∆ t where ( ξ t ) t ∈ N are Gaussian and mutually independent If one is interested in very rapidly varying perturbations, ∆ t has to be very small Wiener process (or Brownian motion) (continuous-time): limit of the random walk when ∆ t → 0 15 10 5 0 −5 −10 0 2 4 6 8 10 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 5 / 19
Some notions of stochastic modelling : Wiener process and “white noise” Problem: a Wiener process is not differentiable (why?), thus it is not the solution of any ordinary differential equation Define formally ξ t (“white noise”) = “derivative” of the Wiener process � t Formally: W ( t ) − W (0) = 0 ξ t dt or dW / dt = ξ t or dW = ξ t dt Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 6 / 19
Some notions of stochastic modelling : Iˆ o SDE Stochastic differential equation : d x / dt = f ( x , t ) + σ ( x , t ) ξ t or by mutiplying by dt : d x = f ( x , t ) dt + σ ( x , t ) dW The last equation was made rigourous by K. Itˆ o in 1951 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 7 / 19
The stochastic contraction theorem If the noise-free system is contracting λ max ( J s ) ≤ − λ Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 8 / 19
The stochastic contraction theorem If the noise-free system is contracting λ max ( J s ) ≤ − λ and the noise variance is upper-bounded � � σ ( x , t ) T σ ( x , t ) ≤ C tr Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 8 / 19
The stochastic contraction theorem If the noise-free system is contracting λ max ( J s ) ≤ − λ and the noise variance is upper-bounded � � σ ( x , t ) T σ ( x , t ) ≤ C tr Then after exponential transients, the mean square distance between any two trajectories is upper-bounded by C /λ – + » ≤ C � a 0 − b 0 � 2 − C “ � a ( t ) − b ( t ) � 2 ” e − 2 λ t ∀ t ≥ 0 E λ + λ Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 8 / 19
Practical meaning After exponential transients, we have � C E ( � a ( t ) − b ( t ) � ) ≤ λ a 0 a 0 C/ λ b 0 1 1 b 0 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 9 / 19
Vocabulary and remarks We say that a system that verifies the conditions of the stochastic contraction theorem is stochastically contracting with rate λ and bound C Discrete and continuous-discrete versions of the theorem are available The theorem can be easily generalized to time-varying metrics Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 10 / 19
“Optimality” of the theorem The mean square bound in the theorem is optimal (consider an Ornstein-Uhlenbeck process d x = − λ x dt + σ dW where the bound is attained) In general, one cannot obtain asymptotic almost-sure stability (consider again the Ornstein-Uhlenbeck process) Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 11 / 19
Noisy and noise-free trajectories The theorem can be used to compare noisy and noise-free versions of a (stochastically) contracting system d a = f ( a , t ) dt + σ ( a , t ) dW d b = f ( b , t ) dt a 0 C/2 λ 1 b 0 Any contracting system is automatically protected against white noise (robustness) Very useful in applications (see later) Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 12 / 19
Combinations of stochastically contracting systems Combinations results in deterministic contraction can be adapted very naturally for stochastic contraction Parallel combinations Hierarchical combinations Negative feedback combinations Small gains Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 13 / 19
Example: Negative feedback combination Two systems coupled by negative feedback gain k � J 1 − k J T � 21 J = J 21 J 2 System 1 stochastically contracting with rate λ 1 and bound C 1 System 2 stochastically contracting with rate λ 2 and bound C 2 Then the coupled system is stochastically contracting with rate min( λ 1 , λ 2 ) and bound C 1 + kC 2 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 14 / 19
Application: contracting observers and noisy measurements Consider the system ˙ x = f ( x , t ) with the measurements y = H ( t ) x Typically dim( y ) < dim( x ) Recall the deterministic contracting observer ˙ ˆ x = f (ˆ x , t ) + K ( t )(ˆ y − y ) ˙ ˆ x = f (ˆ x , t ) + K ( t )( H ( t )ˆ i . e . x − H ( t ) x ) � � ∂ f ( x , t ) If K is chosen such that − K ( t ) H ( t ) is negative definite, ∂ x then the observer system is contracting Since actual state of the system x is a particular solution of the observer system, the state of the observer ˆ x will exponentially converge to x Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 15 / 19
Application: contracting observers and noisy measurements Now, the measurements are corrupted by “white noise” y = H ( t ) x + Σ( t ) ξ ( t ) Using the formal rule ξ ( t ) dt = dW , the observer equation becomes d ˆ x = ( f (ˆ x , t ) + K ( t )( H ( t ) x − H ( t )ˆ x )) dt + K ( t )Σ( t ) dW Using the same K as earlier, the system is stochastically contracting with rate λ and bound C where ˛ „ ∂ f ( x , t ) «˛ ˛ ˛ λ = inf − K ( t ) H ( t ) ˛ λ max ˛ ˛ ∂ x x , t ˛ tr (Σ( t ) T K ( t ) T K ( t )Σ( t )) C = sup t ≥ 0 � After exponential transients, � ˆ x − x � ≤ C / 2 λ Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 16 / 19
Application: stochastic synchronization See next talk by Nicolas Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 17 / 19
Current directions of research Extension to space-dependent metrics Stochastic contraction analysis of Kalman filters (which are a Bayesian filters) and other Bayesian algorithms Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 18 / 19
The end Thank you for your attention! Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 19 / 19
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