Stochastic contraction BACS Workshop Chamonix, January 14, 2008

Transcription

1 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

2 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

3 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

4 Modelling the random perturbations In physics, engineering, neuroscience, finance,... random perturbations are traditionnally modelled with Itô stochastic differential equations (SDE) dx = 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

5 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 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 5 / 19

6 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 Formally: W (t) W (0) = t 0 ξ tdt or dw /dt = ξ t or dw = ξ t dt Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 6 / 19

7 Some notions of stochastic modelling : Iô SDE Stochastic differential equation : dx/dt = f(x, t) + σ(x, t)ξ t or by mutiplying by dt : dx = f(x, t)dt + σ(x, t)dw The last equation was made rigourous by K. Itô in 1951 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 7 / 19

8 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

9 The stochastic contraction theorem If the noise-free system is contracting λ max (J s ) λ and the noise variance is upper-bounded ( ) tr σ(x, t) T σ(x, t) C Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 8 / 19

10 The stochastic contraction theorem If the noise-free system is contracting λ max (J s ) λ and the noise variance is upper-bounded ( ) tr σ(x, t) T σ(x, t) C Then after exponential transients, the mean square distance between any two trajectories is upper-bounded by C/λ t 0 E a(t) b(t) 2 C» λ + a 0 b 0 2 C + e 2λt λ Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 8 / 19

11 Practical meaning After exponential transients, we have E ( a(t) b(t) ) C λ a 0 a 0 C/λ b 0 1 b 0 1 Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 9 / 19

12 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

13 Optimality of the theorem The mean square bound in the theorem is optimal (consider an Ornstein-Uhlenbeck process dx = λxdt + σdw 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

14 Noisy and noise-free trajectories The theorem can be used to compare noisy and noise-free versions of a (stochastically) contracting system a 0 da = f(a, t)dt + σ(a, t)dw db = f(b, t)dt C/2λ 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 1

15 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

16 Example: Negative feedback combination Two systems coupled by negative feedback gain k ( J1 kj J = T ) 21 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

17 Application: contracting observers and noisy measurements Consider the system ẋ = f(x, t) with the measurements y = H(t)x Typically dim(y) < dim(x) Recall the deterministic contracting observer ˆx = f(ˆx, t) + K(t)(ŷ y) i.e. ˆx = f(ˆx, t) + K(t)(H(t)ˆx H(t)x) ( ) If K is chosen such that f(x,t) x K(t)H(t) is negative definite, 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 ˆx will exponentially converge to x Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 15 / 19

18 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 dˆx = (f(ˆx, t) + K(t)(H(t)x H(t)ˆx))dt + K(t)Σ(t)dW Using the same K as earlier, the system is stochastically contracting with rate λ and bound C where λ = inf f(x, t) x,t λmax x K(t)H(t)«C = sup tr(σ(t) T K(t) T K(t)Σ(t)) t 0 After exponential transients, ˆx x C/2λ Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 16 / 19

19 Application: stochastic synchronization See next talk by Nicolas Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 17 / 19

20 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

21 The end Thank you for your attention! Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 19 / 19