A differential Lyapunov framework for contraction analysis F. Forni, R. Sepulchre University of Liège Reykjavik, July 19th, 2013
Outline Incremental stability d(x 1 (t),x 2 (t)) 0 Differential Lyapunov functions δx Closed systems ẋ = f(x) TM Differential Storage / supply rate u S y w = (u,y) Ṡ(t) s(x(t),w(t),δw(t)) Open systems ẋ = f(x,u) - S 1 S 2 interconnection theory
Why? regulation, observer design, synchronization Regulation: from p(t) p e to p(t) sin(ωt) u v ṗ = v v = u p sin(ωt) Filtering: filters forget initial conditions input signal output of a process filter Kalman filter processed output state estimation Synchronization: ϑ i
Linear systems: incremental stability stability ẋ = Ax ż = Az e := x z ė = Ae if e 0 then d(x,z) 0 Use Lyapunov on the error dynamics e: V = (x z) T P(x z) = e T Pe
But the error in nonlinear spaces/dynamics... ẋ = sin(x), ż = sin(z) e := x z ė = sin(x) + sin(z) f(e) ϕ x ẋ = sin(x) M e = x z?, V(x,z) x,z p x,p z
The variational system replaces the error dynamics δx := x z for z x (infinitesimal variation). δx or δx + = f(x +δx) f(x) f(x) x δx If δx 0 then incremental stability δx(0) δx(t) x 2(t) x 1(t)
Time-varying linear system and tangent vectors (intrinsic!) { ẋ = f(x) δx = f(x) x δx { x + = f(x) δx + = f(x) x δx Time-varying linear dynamics of the error δx along any given solution x(t), since δx = f(x(t)) δx. δx(t) is a tangent vector in the tangent space T x(t) M x δx(0) δx(t)
From δx to the distance between solutions by integration { ẋ = f(x) δx = f(x) x δx { x + = f(x) δx + = f(x) x δx x s (0) = γ(s) x s (t) = time evolution of γ along the flow δx s (t) := xs(t) s length(t) = δx s (t) ds δx(0) δx(t) δx(0) δx(t) x 2(t) x 1(t)
We need two ingredients: { ẋ = f(x) δx = f(x) x δx { x + = f(x) δx + = f(x) x δx A metric for δx A way to infer δx(t) 0
A metric for δx: Finsler structures How to measure δx 0? δx T x M δx x Finsler structure δx T x M, λδx x = λ δx x δx 1,δx 2 T x M δx 1 +δx 2 x δx 1 x + δx 2 x Example: δx T P(x)δx P(x) > 0
A way to infer δx(t) 0: Lyapunov theory in TM { ẋ = f(x) δx = f(x) x δx c 1 δx p x V(x,δx) c 2 δx p x V λv V = V(x,δx) x f(x)+ V(x,δx) f(x) x x δx δx(t) x(t) ke λt δx(0) x(0) γ(s) δγ(s) i.e. from any (x 0,δx 0 )
A way to infer δx(t) 0: Lyapunov theory in TM { x + = f(x) δx + = f(x) x δx c 1 δx p x V(x,δx) c 2 δx p x V + λv V + = V(f(x), f(x) x δx) δx(t) x(t) kλ t δx(0) x(0) γ(s) δγ(s) i.e. from any (x 0,δx 0 )
Example: ẋ = sin(x), contraction in ( π,π) ẋ = sin(x) x { ẋ = sin(x) δx = cos(x)δx V(x,δx) = 1 2 δx2 V(x,δx) = cos(x)δx 2 contraction in ( π 2, π 2 ) same as Lyapunov on e := x z V(x,δx) = δx 2 1+cos(x) V(x,δx) = δx 2 contraction in ( π,π)
What about contraction and inputs?
Nonlinear filters: uniform contraction w.r.t. input signal { ẋ = f(x) δx = f(x) x δx { ẋ = f(x,t) δx = f(x,t) x δx { ẋ =f(x,u(t)) δx = f(x,u(t)) δx x c 1 δx p x V(x,δx) c 2 δx p x c 1 δx p x V(t,x,δx) c 2 δx p x V λv δx(t) x(t) ke λt δx(0) x(0) γ(s) δγ(s) i.e. from any (x 0,δx 0 )
Example: ẋ = sin(x)+u ẋ = sin(x) x { ẋ = sin(x)+u δx = cos(x)δx V(x,δx) = 1 2 δx2 V(x,δx) = cos(x)δx 2 uniform contraction in ( π 2, π 2 ) Solutions converge towards each other provided that u(t) preserves the invariance of the contraction region.
Example: ẋ = sin(x)+u ẋ = sin(x) x { ẋ = sin(x)+u δx = cos(x)δx V(x,δx) = δx 2 1+cos(x) V(x,δx) = δx 2 + δx2 sin(x)u (1+cos(x)) 2 no uniform contraction in ( π, π) Solutions converge towards each other only for suitable inputs u(t)
Example: ẋ = sin(x)+cos( x 2 )u ẋ = sin(x) x { ẋ = sin(x)+cos( x 2 )u δx = [ cos(x) 1 2 sin(x 2 )u] δx V(x,δx) = ( V(x,δx) = δx 2 +δx 2 u δx 2 1+cos(x) cos(x 2 ) (1+cos(x)) 2 sin( x 2 ) ) 2(1+cos(x)) }{{} =0! uniform contraction in ( π, π)
Example: ẋ = sin(x)+cos( x 2 )u ẋ = sin(x) x { ẋ = sin(x)+cos( x 2 )u δx = [ cos(x) 1 2 sin(x 2 )u] δx x 3 2 1 0-1 -2 x 3 2 1 0-1 -2-3 0 2 4 6 8 10 t -3 0 2 4 6 8 10 t
What we gain from a differential Lyapunov theory?
Lyapunov (stability) vs Finsler-Lyapunov (incr. stability) ẋ = f(x) Find Lyapunov V(x) T V(x) f(x) λv(x) x Ẋ = F(X) X := [ x δx], F(X) := [ f(x) ] f(x) x δx Find Finsler-Lyapunov V(x, δx) T V(X) F(X) λv(x) X Then, x(t) x e. Then, δx(t) 0. f(x) V x δx(0) x 2(t) V(x) = l > 0 δx(t) x 1(t)
Finsler-Lyapunov unifies many contraction approaches V = δx A := sup x=1 Ax, µ(a) := lim I+hA 1 h 0 + h, c > 0, x µ( f(x) x ) < c V = δx T Pδx P = P T > 0, Q = Q T > 0, x V = δx T M(x)δx, M(x) = M(x) T > 0, x T f(x) P +P f(x) < Q x x T f(x) M(x)+M(x) f(x) +[Ṁ(x)] < cm(x) x x
IS, IAS, IES [ X := [ δx] x f(x), F(X) := J(x)δx ], Ẋ = F(X) Let V : TM R 0 be a Finsler-Lyapunov function s.t. X T F(X) 0 Incremental Stability. V(X) X T F(X) α(v(x)) α K Incremental Asymptotic Stability. 1 lim t d(ϕ(t,x 0 ),ϕ(t,x 1 )) lim t 0 V(...)ds = 0 V(X) X T F(X) cv(x) c > 0 Incremental Exponential Stability. d(ϕ(t,x 0 ),ϕ(t,x 1 )) 1 0 e ct...ds = e ct 1 0...ds V(X) X
Advantages: relaxations, design, I/O Relaxations (LaSalle): from V α(v) to V W(x,δx) 0 Design: u = k(x) such that V α(v) open systems...
What about systems that contract only partially?
Horizontal contraction - symmetries or first integrals α α α = 3 ϕ(t, ) α = 2 ϕ(t, ) V x Hx α = 1 V(x,δx) = V(x,δx h ) δx = δx h +δx v (Cooperative irreducibile systems of ordinary differential equations with first integral, Mierczynski)
Horizontal contraction - attractors - Bendixon s criterion stable limit cycle H x contraction V x no contraction no limit cycles if the system contracts in the direction of the vector field f(x) - connections with Bendixon-like criteria.
Example: phase synchronization (Stabilization of planar collective motion: All-to-all communication, Sepulchre, Paley, and Leonard) θ k = 1 n n sin(θ j θ k ) j=1 0 s 21 s n1 s 12 0 s n2 θ = 1. n.... 1 S(θ) = S(θ+α1). s n1 s n2 0 }{{} =:S(θ) Contraction in any forward invariant compact set C not containing unstable and saddle points ϑ i
A glimpse into open systems
Open systems - differential dissipativity Incremental stability d(x 1 (t),x 2 (t)) 0 Differential Lyapunov functions δx Closed systems ẋ = f(x) TM Differential Storage / supply rate u S y w = (u,y) Ṡ(t) s(x(t),w(t),δw(t)) Open systems ẋ = f(x,u) - S 1 S 2 interconnection theory
Variational open systems { ẋ = f(x,u) ẏ = h(x,u) δx = A(x,u) δx +B(x,u) δu }{{}}{{} f(x,u) f(x,u) x u δy = C(x,u) δx +D(x,u) δu }{{}}{{} h(x,u) h(x,u) x u
Storage and supply rate lifted to TM S : TM R differential storage To make the connection to incremental stability we need c 1 δx p x S(x,δx) c 2 δx p x Ṡ(x,δx) s(x,w,δw) differential supply rate w := (u,y) δw = (δu,δy) Differential passivity: s(x,w,δw) = δy T δu Quadratic differential supply: s(x,w,δw) = δw T Q(x)δw
Linear systems: differential passivity passivity { ẋ = Ax +Bu y = Cx { δx = Aδx +Bδu δy = Cδx S = 1 2 δxt Pδx Ṡ δy T δu if δu = 0 A T P +PA 0 δy T δu PB = C T
Differential passivity if uniform contraction w.r.t. u. { ẋ = f(x)+g(x)u y = h(x) { δx = (f(x)+g(x)u) x δx +g(x)δu δy = h(x) x δx For differential passivity, let S = δx T P(x)δx. Then, Ṡ δy T δu if δu = 0 P(x) (f(x)+g(x)u) x δy T δu P(x)g(x) = h(x) x +Ṗ(x) 0 0
Differential passivity, incremental stability, interconnections Thm1: differential passivity incremental stability. Proof: Ṡ δy T δu Ṡ 0 (δu = 0) δx(t) x(t) K δx(0) x(0) Thm2: differentially passivity + feedback differentially passivity ] T [ ] Proof: S = S 1 +S 2 Ṡ = δy1 Tδy 2 +δy2 Tδy 1 +[ δy1 δv1 δy 2 δv 2 v 1 u 1 y 2 Σ 1 Σ 2 y 1 u 2 v 2 δv 1 δu 1 δy 2 Σ 1 Σ 2 S 1 S 2 δy 1 δu 2 δv 2
Contraction and differential dissipativity Incremental stability d(x 1 (t),x 2 (t)) 0 Differential Lyapunov functions δx Closed systems ẋ = f(x) TM Differential Storage / supply rate u S y w = (u,y) Ṡ(t) s(x(t),w(t),δw(t)) Open systems ẋ = f(x,u) - S 1 S 2 interconnection theory
Selected Bibliography A differential Lyapunov framework for contraction analysis. F. Forni, R. Sepulchre, submitted to IEEE TAC. Available on arxiv. On differentially dissipative dynamical systems. F. Forni, R. Sepulchre, Nolcos 2013. On contraction analysis for non-linear systems. W. Lohmiller and J. E. Slotine, 1998. An intrinsic observer for a class of Lagrangian systems. N. Aghannan and P. Rouchon, 2003. Uniform output regulation of nonlinear systems: A convergent dynamics approach. A. Pavlov, N. van de Wouw, and H. Nijmeijer, 2005. Global entrainment of transcriptional systems to periodic inputs G. Russo, M. Di Bernardo, and E.D. Sontag, 2010. Metric Properties of Differential Equations D. C. Lewis, 1951. A Lyapunov approach to incremental stability properties. D. Angeli. 2000