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

1

Nonlinear contraction theory

2

Stochastic nonlinear contraction theory

3

Hybrid-resetting stochastic contraction

Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 2 / 21

Outline

1

Nonlinear contraction theory

2

Stochastic nonlinear contraction theory

3

Hybrid-resetting stochastic contraction

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 ∀x, t λmax

• ˙

Θ + Θ ∂f ∂x

• Θ−1
• s

< −λ then all system trajectories converge exponentially to a single trajectory, independently of the initial conditions (incremental stability) Note : J =

• ˙

Θ + Θ ∂f

∂x

• Θ−1 is called the generalized Jacobian of f in the

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

∂x ∂u ⊤ ∂x ∂u on that path.

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

∂x ∂u ⊤ ∂x ∂u on that path.

We have

d dt „ ∂x ∂u « = ∂ ∂u „dx dt « = ∂ ∂u (f(x, t)) = ∂f ∂x ∂x ∂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

∂x ∂u ⊤ ∂x ∂u on that path.

We have

d dt „ ∂x ∂u « = ∂ ∂u „dx dt « = ∂ ∂u (f(x, t)) = ∂f ∂x ∂x ∂u

Thus

d dt „ ∂x ∂u

⊤ ∂x

∂u « = 2 ∂x ∂u

⊤ ∂f

∂x ∂x ∂u ≤ −2λ ∂x ∂u

⊤ ∂x

∂u

system trajectories

Lewis, Am. J. of Mathematics, 1951 Lohmiller & Slotine, Automatica, 1998

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 dx1 = f1(x1, x2, t)dt dx2 = f2(x1, x2, t)dt with system xi contracting with rate λi in the metric Mi = Θ⊤

i Θi

Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 8 / 21

Example: negative feedback

Consider the combination dx1 = f1(x1, x2, t)dt dx2 = f2(x1, x2, t)dt with system xi contracting with rate λi in the metric Mi = Θ⊤

i Θi

Assume that the connection is negative feedback, i.e. Θ1 ∂f1 ∂x2

• Θ−1

2

= −kΘ2 ∂f2 ∂x1 T Θ−1

1

Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 8 / 21

Example: negative feedback

Consider the combination dx1 = f1(x1, x2, t)dt dx2 = f2(x1, x2, t)dt with system xi contracting with rate λi in the metric Mi = Θ⊤

i Θi

Assume that the connection is negative feedback, i.e. Θ1 ∂f1 ∂x2

• Θ−1

2

= −kΘ2 ∂f2 ∂x1 T Θ−1

1

Then the global system is contracting with rate min(λ1, λ2) in the metric M = ΘTΘ where Θ = Θ1 √ kΘ2

• 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

1

Nonlinear contraction theory

2

Stochastic nonlinear contraction theory

3

Hybrid-resetting stochastic contraction

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) 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”)

Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 11 / 21

The stochastic contraction theorem

If the noise-free system is contracting in some metric λmax(Js) ≤ −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 λmax(Js) ≤ −λ and the noise variance is upper-bounded in the same metric tr

• σ(x, t)⊤M(t)σ(x, t)
• ≤ C

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 λmax(Js) ≤ −λ and the noise variance is upper-bounded in the same metric tr

• σ(x, t)⊤M(t)σ(x, t)
• ≤ C

Then, for any pair of system trajectories (a, b) ∀t ≥ 0 E

• a(t) − b(t)2

≤ 1 β C λ + E

• a0 − b02

e−2λt

• where β is the lower-bound of the metric : ∀x, x⊤Mx ≥ βx2

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 E (a(t) − b(t)) ≤

• C

βλ

a0 b0

1

b0 a0 C/λ

1

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 − b2 And differentiate using Ito’s rule LV ≤ −2λV + 2C Then use Dynkin’s formula and a Gronwall-type lemma (see paper) to conclude that EV (a(t), b(t)) ≤ C λ +

• V (a0, b0) − C

λ + e−2λt

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

1

Nonlinear contraction theory

2

Stochastic nonlinear contraction theory

3

Hybrid-resetting stochastic contraction

Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 17 / 21

An example: synchronization of continuous noisy nonlinear

• scillators through discrete noisy couplings

Consider three Andronov-Hopf oscillators Discrete noisy couplings at instants t = kτ, k ∈ N xi(kτ +) = xi(kτ −) + γ

• R
• xi+1(kτ −) + σd

√ 2 wk

• − xi(kτ −)
• with 0 < γ < 1 the coupling strength, x4 = x1 and

R =

• −1

2

−

√ 3 2 √ 3 2

−1

2

• Between two interaction instants: each oscillator follows

dxi = f(xi)dt + σc √ 2 dW where f(xi) = f xi yi

• =

xi − yi − x3

i − xiy2 i

xi + yi − y3

i − yix2 i

• Quang-Cuong Pham (LPPA)

Stochastic nonlinear contraction theory 18 / 21

Result

Using hybrid stochastic contraction, we can show that: if 3γ2 − 3γ + 1 < e−2τ then after exponential transients, E

• sync error2

≤ 2γ2σ2

d + (1 − α)(1 + α − αe2τ)e2τσ2 c

2(1 − α)(1 − αe2τ) where α = 3γ2 − 3γ + 1.

Quang-Cuong Pham (LPPA) Stochastic nonlinear contraction theory 19 / 21

Simulations

γ = 0.01 γ = 0.2 x against y 2π/3-shifted x against t squared sync error against t

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

0.5 1 1.5 (a)

• 1.5
• 1
• 0.5

0.5 1 1.5 1 2 3 4 5 (b) 1 2 3 4 5 1 2 3 4 5 (c)

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1.5
• 1
• 0.5

0.5 1 1.5 (d)

• 1.5
• 1
• 0.5

0.5 1 1.5 1 2 3 4 5 (e) 1 2 3 4 5 1 2 3 4 5 (f)

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Thanks

Many thanks to J.-J Slotine (MIT, USA) and N. Tabareau (Collège de France Thank you for your attention !

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