Stabilization of persistently excited linear systems
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1 Stabilization of persistently excited linear systems Yacine Chitour Laboratoire des signaux et systèmes & Université Paris-Sud, Orsay Exposé LJLL Paris, 28/9/2012
2 Stabilization & intermittent control Consider a linear control system ẋ = Ax + Bu and a feedback u = Kx stabilizing at 0.
3 Stabilization & intermittent control Consider a linear control system ẋ = Ax + Bu and a feedback u = Kx stabilizing at 0. Let α : [0, ) {0, 1} (or, more generally, α : [0, ) [0, 1]) represent a switching signal which determines whether the feedback u = Kx is active: ẋ = Ax + αbkx. The signal α may model Unfaithful transmission of the control law (α(t) {0, 1}) Approximately periodic or quasi-periodic parameter affecting the control efficiency Allocation of control resources
4 Stabilization & intermittent control Consider a linear control system ẋ = Ax + Bu and a feedback u = Kx stabilizing at 0. Let α : [0, ) {0, 1} (or, more generally, α : [0, ) [0, 1]) represent a switching signal which determines whether the feedback u = Kx is active: ẋ = Ax + αbkx. The signal α may model Unfaithful transmission of the control law (α(t) {0, 1}) Approximately periodic or quasi-periodic parameter affecting the control efficiency Allocation of control resources Under which conditions on A, B, K and on α is the non-autonomous system asymptotically stable at 0?
5 Stabilizable linear control system in R n A linear control system ẋ = Ax + Bu, x R n, u R m is stabilizable at the origin if there exists a feedback u = Kx such that A + BK is Hurwitz.
6 Stabilizable linear control system in R n A linear control system ẋ = Ax + Bu, x R n, u R m is stabilizable at the origin if there exists a feedback u = Kx such that A + BK is Hurwitz. It is well known that this is true if and only if there exists a system of coordinates in which ( A1 A A = 2 0 A 3 ), B = ( B1 0 and A 3 is Hurwitz while (A 1, B 1 ) is controllable. )
7 Stabilizable linear control system in R n A linear control system ẋ = Ax + Bu, x R n, u R m is stabilizable at the origin if there exists a feedback u = Kx such that A + BK is Hurwitz. It is well known that this is true if and only if there exists a system of coordinates in which ( A1 A A = 2 0 A 3 ), B = ( B1 0 and A 3 is Hurwitz while (A 1, B 1 ) is controllable. (Pole shifting theorem:) if (A, B) is controllable, then the system can be stabilized with an arbitrary rate of convergence, i.e., for every λ > 0 there exist K and C > 0 such that x(t) C x(0) e λt for every trajectory x of ẋ = Ax + BKx. )
8 Persistent excitation We always assume that for α 1 the system is stabilizable. Definition ((T, µ)-signal) Let 0 < µ T. A (T, µ)-signal is a function α L (R, [0, 1]) satisfying t+t G(T, µ) = set of all (T, µ)-signals. Definition ((T, µ)-stabilizer) t α(s)ds µ, t R. Let 0 < µ T. The feedback u = Kx is said to be a (T, µ)-stabilizer if there exist C, γ > 0 such that, for every (T, µ)-signal α, and every x 0 R n, the solution x of ẋ = (A + αbk)x, x(0) = x 0, satisfies x(t) Ce γt x 0, t 0.
9 A neutrally stable Lemma Let (A, B) be stabilizable and A neutrally stable (Re(σ(A)) 0 and the eigenvalues with real-part equal to zero have trivial Jordan blocks). Then there exists K which is a (T, µ)-stabilizer for every 0 < µ T.
10 A neutrally stable Lemma Let (A, B) be stabilizable and A neutrally stable (Re(σ(A)) 0 and the eigenvalues with real-part equal to zero have trivial Jordan blocks). Then there exists K which is a (T, µ)-stabilizer for every 0 < µ T. Idea of the proof. Without loss of generality A skew-symmetric and (A, B) controllable. Take K = B T (independent on (T, µ)). V (t) = x(t) 2 Lyapunov function. V = 2α(t) B T x 2 0. Concludes with Lasalle-type argument.
11 Persistent excitation in the infinite-dimensional case: new phenomena The stabilizability in the neutrally stable case does not generalize to infinite-dimensional systems.
12 Persistent excitation in the infinite-dimensional case: new phenomena The stabilizability in the neutrally stable case does not generalize to infinite-dimensional systems. Consider the wave equation on a string of finite length L, fixed at both ends and damped on a subset (a, b) (0, L), v tt (t, y) = v yy (t, y) α(t)1 (a,b) (y)v t (t, y) v(t, 0) = v(t, L) = 0 x(t) = (v(t, ), v t (t, )) H = H 1 0 (0, 1) L2 (0, 1) A(x 1, x 2 ) = (x 2, yy x 1 ), B(x 1, x 2 ) = (0, 1 (a,b) x 2 ) A preserves the norm (x 1, x 2 ) H = y x 1 L 2 (0,1) + x 2 L 2 (0,1) A BB makes the norm (weakly) decreasing
13 Persistent excitation in the infinite-dimensional case: new phenomena Given T µ > 0, it suffices to take a traveling wave with sufficiently small support in order to design α that satisfies the persistent excitation condition and switches off the actuator when the wave arrives. The stabilizability in the neutrally stable case does not generalize to infinite-dimensional systems. Consider the wave equation on a string of finite length L, fixed at both ends and damped on a subset (a, b) (0, L), v tt (t, y) = v yy (t, y) α(t)1 (a,b) (y)v t (t, y) v(t, 0) = v(t, L) = 0 x(t) = (v(t, ), v t (t, )) H = H 1 0 (0, 1) L2 (0, 1) A(x 1, x 2 ) = (x 2, yy x 1 ), B(x 1, x 2 ) = (0, 1 (a,b) x 2 ) A preserves the norm (x 1, x 2 ) H = y x 1 L 2 (0,1) + x 2 L 2 (0,1) A BB makes the norm (weakly) decreasing
14 Persistent excitation in the infinite-dimensional case [F. Hante, M. Sigalotti, M. Tucsnak, JDE, 2012] Theorem Exponential stability. Let ϑ, c > 0 be such that ϑ 0 α(t) B e ta z 0 2 H dt c z 0 2 H, for each (T, µ)-signal α( ). Then B is a (T, µ)-stabilizer. Weak stability. Let ϑ > 0 be such that ϑ 0 α(s) B e sa z 0 2 H ds = 0 z 0 = 0 for every (T, µ)-signal α( ). Then each solution t z(t) of ż = Az αbb z converges weakly to 0 in H as t for any initial data z 0 H and any (T, µ)-signal α( ).
15 Examples Exponential stability: wave equation Ω bounded domain of R N v tt (t, x) = v(t, x) α(t)d(x) 2 v t (t, x), (t, x) (0, ) Ω, v(0, x) = y 0 (x), v t (0, x) = y 1 (x), x Ω, v(t, x) = 0, (t, x) (0, ) Ω, with d L (Ω), d(x) d 0 > 0. The generalized observability inequality is satisfied with ϑ = T, H = H 1 0 (Ω) L2 (Ω), (z 1, z 2 ) = z 1 L 2 (Ω) + z 2 L 2 (Ω).
16 Examples Exponential stability: wave equation Ω bounded domain of R N v tt (t, x) = v(t, x) α(t)d(x) 2 v t (t, x), (t, x) (0, ) Ω, v(0, x) = y 0 (x), v t (0, x) = y 1 (x), x Ω, v(t, x) = 0, (t, x) (0, ) Ω, with d L (Ω), d(x) d 0 > 0. The generalized observability inequality is satisfied with ϑ = T, H = H 1 0 (Ω) L2 (Ω), (z 1, z 2 ) = z 1 L 2 (Ω) + z 2 L 2 (Ω). Weak stability: Schrödinger equation y t (t, x) = i y(t, x) α(t)1 ω (x)y(t, x), (t, x) (0, ) Ω, y(t, x) = 0, t (0, ) Ω, y(0, x) = y 0 (x), t Ω, with ω Ω open nonempty. Generalized unique continuation principle with θ = T, H = L 2 (Ω).
17 A stability result in a similar spirit [Martinez-Vancostenoble, 2002] and [Haraux-Martinez-Vancostenoble, 2005] studied (in particular) the damped wave equation v tt (t, x) = v xx (t, x) α(t)v t (t, x) v(t, 0) = v(t, L) = 0. They proved that if {t α(t) = 1} = n N (a n, b n ) with b n a n+1 and (b n a n ) 3 = n N then the solution converges exponentially to zero.
18 Stability results are then obtained by estimating the asymptotics of c(t ) for T small. Example: Wave with damping everywhere: c(t ) T 3, as proved in [Haraux-Martinez-Vancostenoble, 2005] for the case α 1. Strong stability in infinite dimension Theorem Assume that there exists ρ > 0 and a continuous function c : (0, ) (0, ) such that for all T, T 0 α(t) dt ρt ϑ 0 α(t) B e ta z 0 2 H dt c(t ) z 0 2 H, z 0. Assume moreover that I n = (a n, b n ) is a sequence of disjoint intervals in [0, ) with b n a n α(t) dt ρ b n a n and n=1 min(c(b n a n ), 1) =. Then each solution of ż = Az αbb z satisfies z(t) H 0 as t.
19 Example: finite-dimensional control systems Proposition Let H = R n. Let A be skew-symmetric and (A, B) controllable. Let r be the minimal non-negative integer such that rank[b, AB,..., A r B] = n. Then for every ρ > 0 there exists κ > 0 such that, for every T (0, 1] and every α L ([0, T ], [0, 1]), if T 0 α(s)ds ρt then T 0 α(s) B e sa z 0 2 ds κt 2r+1 z 0 2. Then c(t ) T 2r+1 as proved in [Seidman, 1988] for the case α 1.
20 Spectra with non-positive real part [A. Chaillet, Y. C., A. Lorìa, M. Sigalotti, MCSS, 2008] & [Y. C., M. Sigalotti, SICON, 2010] From now on we restrict our attention to systems of the type ẋ = Ax + αbu, x R n, u R, α [0, 1] with (A, b) controllable. Theorem Let (A, b) M n (R) R n be a controllable pair and assume that Re(σ(A)) 0. Then, for every 0 < µ T there exists a (T, µ)-stabilizer. The uncontrolled system ẋ = Ax can have trajectories such that x(t) as t +. The proof is based on a compactness argument and a time-contraction procedure, transforming the integral constraint in a pointwise one.
21 First case: the n-integrator Proposition For every 0 < µ T the system ẋ 1 = x 2. ẋ n 1 = x n 2 ẋ n = αu admits a (T, µ)-stabilizer.
22 First case: the n-integrator Proposition For every 0 < µ T the system ẋ 1 = x 2. ẋ n 1 = x n 2 ẋ n = αu admits a (T, µ)-stabilizer. First step of the proof: Lemma (homogeneity and time-rescaling) Let λ > 0. Then (k 1,..., k n ) is a (T, µ)-stabilizer if and only if (λ n k 1,..., λk n ) is a (T/λ, µ/λ)-stabilizer. x λ (t) = diag(1, λ,..., λ n 1 )x(λt) solution with (k 1,..., k n ) replaced by (λ n k 1,..., λk n ) and α( ) by α(λ ).
23 (T, µ)-stabilizability of the n-integrator: limit switched systems We have a one-parameter family of equivalent stabilization problems: we fix K and we assume by contradiction that, for every λ > 0, K is not a (T/λ, µ/λ)-stabilizer. By a compactness argument the proof of the theorem is reduced to the stability of the limit switched system ẋ 1 = x 2. ẋ n 1 = x n 2 ẋ n = βk T x, β = β(t) [µ/t, 1]. The existence of a common quadratic Lyapunov function follows from a uniform observability result by Dayawansa, Gauthier and Kupka.
24 Spectra with non-positive real part: the general case Theorem Let (A, b) M n (R) R n be a controllable pair and assume that Re(σ(A)) 0. Then, for every 0 < µ T there exists a (T, µ)-stabilizer. Assume that Re(σ(A)) = 0. Up to a linear change of coordinates { ẋ0 = J r0 x 0 + αb 0 u, ẋ j = (ω j A (j) + Jr C j )x j + αb j u, for j = 1,..., h, where b 0 and b j have all coordinates equal to zero except the last one that is equal to one and ( ) A (j) 0 1 = Id rj, Jr C 1 0 j = J rj Id 2
25 Spectra with non-positive real part Thanks to the dilation, time-rescaling and rotation y 0 (t) = D r0,νx 0 (νt), y j (t) = (D rj,ν Id 2 )e νta(j) x j (νt), for 1 j h, we end up with a limit switched system with a non-scalar switching law { ẏ0 = J r0 y 0 b 0( C 00 K 0 y 0 + h l=1 C ) ( 0l(K l Id 2 )y l, ẏ j = Jr C j y j (b j Id 2 ) C0j T K 0y 0 + ) h l=1 C jl(k l Id 2 )y l with a pointwise (in time) excitation condition of the type (C jl ) j0 j,l h ξid 2h+1 j0 with ξ > 0. An extension of the Gauthier-Kupka result allows to conclude.
26 Arbitrary rate of convergence/divergence λ + log( x(t;0,x (α, K) = sup x0 =1 lim sup 0,K,α) ) t + t λ log( x(t;0,x (α, K) = inf x0 =1 lim inf 0,K,α) ) t + t. Rate of convergence/divergence defined as rc(a, b, T, µ, K) := sup α G(T,µ) λ + (α, K) rd(a, b, T, µ, K) := inf α G(T,µ) λ (α, K).
27 Arbitrary rate of convergence/divergence λ + log( x(t;0,x (α, K) = sup x0 =1 lim sup 0,K,α) ) t + t λ log( x(t;0,x (α, K) = inf x0 =1 lim inf 0,K,α) ) t + t. Rate of convergence/divergence defined as rc(a, b, T, µ, K) := sup α G(T,µ) λ + (α, K) rd(a, b, T, µ, K) := inf α G(T,µ) λ (α, K). Notice that rc(a, b, T, µ, K) minᾱ [µ/t,1] min{ R(σ(A ᾱbk T ))} rc(a, b, T, µ, K) = rc(p AP 1, P b, T, µ, (P 1 ) T K)
28 Arbitrary rate of convergence/divergence λ + log( x(t;0,x (α, K) = sup x0 =1 lim sup 0,K,α) ) t + t λ log( x(t;0,x (α, K) = inf x0 =1 lim inf 0,K,α) ) t + t. Rate of convergence/divergence defined as rc(a, b, T, µ, K) := sup α G(T,µ) λ + (α, K) rd(a, b, T, µ, K) := inf α G(T,µ) λ (α, K). Notice that rc(a, b, T, µ, K) minᾱ [µ/t,1] min{ R(σ(A ᾱbk T ))} rc(a, b, T, µ, K) = rc(p AP 1, P b, T, µ, (P 1 ) T K) Define the maximal rate of convergence/divergence as RC(A, T, µ) = sup K R n rc(a, b, T, µ, K), RD(A, T, µ) = sup K R n rd(a, b, T, µ, K). RC(A + λid n, T, µ) = RC(A, T, µ) λ RC(J n, T, ρt ) = RC(J n, 1, ρ)/t RC and RD monotone with respect to µ
29 Arbitrary rate of convergence/divergence Proposition Let n = 2 and (A, b) controllable. Then RC(A, T, µ) = + if and only if RD(A, T, µ) = +. Open question: is this still true for n > 2?
30 Arbitrary rate of convergence/divergence Proposition Let n = 2 and (A, b) controllable. Then RC(A, T, µ) = + if and only if RD(A, T, µ) = +. Open question: is this still true for n > 2? Proposition There exists ρ (0, 1) (only depending on n) such that for every controllable pair (A, b) M n (R) R n and every T > 0, RC(A, T, ρt ) = + for ρ > ρ (i.e., the system ẋ = Ax + αbu can be (T, µ)-stabilized with an arbitrarily large rate of convergence if µ/t > ρ ).
31 Can the rate of convergence be made arbitrary large by suitably choosing K? Proposition There exists ρ (0, 1) such that for every controllable pair (A, b) M 2 (R) R 2 and every T > 0, RC(A, T, ρt ) < + for ρ < ρ. Idea of the proof: take K such that minᾱ [µ/t,1] min{ R(σ(A ᾱbk T ))} 1 and look for destabilizing α
32 Can the rate of convergence be made arbitrary large by suitably choosing K? Proposition There exists ρ (0, 1) such that for every controllable pair (A, b) M 2 (R) R 2 and every T > 0, RC(A, T, ρt ) < + for ρ < ρ. Idea of the proof: take K such that minᾱ [µ/t,1] min{ R(σ(A ᾱbk T ))} 1 and look for destabilizing α In particular, there exist controllable pairs (A, b) that are not (T, µ)-stabilizable for some T > µ > 0. A = J 2 + λid 2, λ large, T/µ < ρ
33 Can the rate of convergence be made arbitrary large by suitably choosing K? Proposition There exists ρ (0, 1) such that for every controllable pair (A, b) M 2 (R) R 2 and every T > 0, RC(A, T, ρt ) < + for ρ < ρ. Idea of the proof: take K such that minᾱ [µ/t,1] min{ R(σ(A ᾱbk T ))} 1 and look for destabilizing α In particular, there exist controllable pairs (A, b) that are not (T, µ)-stabilizable for some T > µ > 0. A = J 2 + λid 2, λ large, T/µ < ρ if we add the constraint that the (T, µ)-signal are M-Lipschitz for some fixed M > 0 we recover the arbitrary rate of stability [Y. C., G. Mazanti, M. Sigalotti, accepted SICON].
34 Can the rate of convergence be made arbitrary large by suitably choosing K? Proposition There exists ρ (0, 1) such that for every controllable pair (A, b) M 2 (R) R 2 and every T > 0, RC(A, T, ρt ) < + for ρ < ρ. Idea of the proof: take K such that minᾱ [µ/t,1] min{ R(σ(A ᾱbk T ))} 1 and look for destabilizing α In particular, there exist controllable pairs (A, b) that are not (T, µ)-stabilizable for some T > µ > 0. A = J 2 + λid 2, λ large, T/µ < ρ if we add the constraint that the (T, µ)-signal are M-Lipschitz for some fixed M > 0 we recover the arbitrary rate of stability [Y. C., G. Mazanti, M. Sigalotti, accepted SICON]. Open question: is this still true for n > 2?
35 Bifurcation phenomenon There is a bifurcation phenomenon at some ρ = ρ(a, T ) 0. (Arbitrary rate of convergence if µ/t > ρ(a, T ), bounded rate if µ/t < ρ(a, T ).) ρ(a, T ) > 0 if n = 2. The function T ρ(j n, T ) is constant.
36 Bifurcation phenomenon There is a bifurcation phenomenon at some ρ = ρ(a, T ) 0. (Arbitrary rate of convergence if µ/t > ρ(a, T ), bounded rate if µ/t < ρ(a, T ).) ρ(a, T ) > 0 if n = 2. The function T ρ(j n, T ) is constant. Lemma Let (A, b) M n (R) R n be a controllable pair. Then (i) T ρ(a, T ) is locally Lipschitz; (ii) there exist lim T + ρ(a, T ) = sup T >0 ρ(a, T ) and lim T 0 ρ(a, T ) = inf T >0 ρ(a, T ).
37 Bifurcation phenomenon There is a bifurcation phenomenon at some ρ = ρ(a, T ) 0. (Arbitrary rate of convergence if µ/t > ρ(a, T ), bounded rate if µ/t < ρ(a, T ).) ρ(a, T ) > 0 if n = 2. The function T ρ(j n, T ) is constant. Lemma Let (A, b) M n (R) R n be a controllable pair. Then (i) T ρ(a, T ) is locally Lipschitz; (ii) there exist lim T + ρ(a, T ) = sup T >0 ρ(a, T ) and lim T 0 ρ(a, T ) = inf T >0 ρ(a, T ). OPEN PROBLEMS is T ρ(a, T ) monotone? constant? are A lim T 0 ρ(a, T ) or A lim T ρ(a, T ) constant?
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