About Polynomial Instability of Linear Switched Systems Paolo Mason - - PowerPoint PPT Presentation

About Polynomial Instability of Linear Switched Systems

Paolo Mason (CNRS / Laboratoire des Signaux et Systèmes)

(Joint work with Yacine Chitour and Mario Sigalotti)

Porquerolles, October 28th, 2010

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 1 / 19

Linear Switched Systems

Linear Switched System (continuous time) : (S) ˙ x(t) = A(t)x(t) x ∈ Rn, A(t) ∈ A ⊂ Rn×n. A(·) = any meas. function [0, +∞) → A; referred as a switching law. A compact.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 2 / 19

Linear Switched Systems

Linear Switched System (continuous time) : (S) ˙ x(t) = A(t)x(t) x ∈ Rn, A(t) ∈ A ⊂ Rn×n. A(·) = any meas. function [0, +∞) → A; referred as a switching law. A compact. Example : A = {A1, A2} or A = {λA1 + (1 − λ)A2 : λ ∈ [0, 1]} Remark : wlog A can be taken convex

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 2 / 19

Stability and Lyapunov exponent

Maximal Lyapunov Exponent of A defined as ρ(A) = sup

A(·),x(0)

• lim sup

t→∞

1 t log x(t)

• .
• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 3 / 19

Stability and Lyapunov exponent

Maximal Lyapunov Exponent of A defined as ρ(A) = sup

A(·),x(0)

• lim sup

t→∞

1 t log x(t)

• .

ρ(A) < 0 (S) Uniformly Globally Asymptotic Stable (UGAS) → Uniformly Exponentially Stable (UES), i.e. ∃M, λ > 0 s.t. ∀x(0) ∈ Rn , t ≤ 0, A(·) x(t) ≤ Me−λtx(0). ρ(A) = 0 (S) stable: all trajectories are bounded and there exists one traj. not converging to 0, (S) marginally unstable: ∃ unbounded traj. with non-expon. growth. ρ(A) > 0 (S) unstable: ∃ traj. going to ∞ exponentially.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 3 / 19

In general it is not easy to verify if a given switched system is stable at the origin, even for the simplest class of linear switched system i.e. single-input systems. ˙ x = u(t)A1x + (1 − u(t))A2x x ∈ R2, u(t) ∈ [0, 1] where A1, A2 are 2 × 2 matrices.

• 0.4 -0.2

0.2 0.4 0.6 0.8

• 0.5
• 0.25

0.25 0.5 0.75 1

• 0.5

0.5 1 1.5

• 0.4
• 0.2

0.2 0.4 0.6

• 1.5
• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

UNSTABLE ASYMPTOTICALLY STABLE ˙ x = A2x ˙ x = u(t)A1x + (1 − u(t))A2x ˙ x = A1x

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 4 / 19

Two-dimensional systems

The case of two-dimensional systems ˙ x = u(t)A1x + (1 − u(t))A2x x ∈ R2, u(t) ∈ [0, 1] has been completely solved by U. Boscain (2002), who gave easily verifiable necessary and sufficient conditions. → methods based on the notion of worst trajectory

worst trajectory : forms the smallest angle instability exponential stability with the exiting radial direction

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 5 / 19

General case

˙ x(t) = A(t)x(t) x ∈ Rn , A(t) ∈ A In general, (i.e. n ≥ 3) notion of worst trajectory not valid anymore (linked to Jordan Separation theorem). Classical to seek a Lyapunov function (e.g. polynomial, etc.). But,

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 6 / 19

General case

˙ x(t) = A(t)x(t) x ∈ Rn , A(t) ∈ A In general, (i.e. n ≥ 3) notion of worst trajectory not valid anymore (linked to Jordan Separation theorem). Classical to seek a Lyapunov function (e.g. polynomial, etc.). But, Theorem [U. Boscain, Y. Chitour, P. Mason] For each exp. stable linear switched system, ∃ common polynomial Lyapunov function. Degree of this common polynomial Lyapunov function is not uniformly bounded over all exp. stable linear switched system.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 6 / 19

Reduction to ρ(A) = 0

Computation of ρ(A) VERY HARD in general and even numerically. Up to a translation, A ❀ A − ρ(A)Id reduce to case ρ(A) = 0. ⇒ Study of the case ρ(A) = 0 crucial to understand stability properties.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 7 / 19

Invariant flags

Definition A irreducible if ∄V nontrivial subspace of Rn invariant wrt all matrices of A

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Polynomial instability of switched systems October 28th, 2010 8 / 19

Invariant flags

Definition A irreducible if ∄V nontrivial subspace of Rn invariant wrt all matrices of A Maximal invariant flag for A: {0} = E0 E1 · · · Ek−1 Ek = Rn where Ei invariant wrt each A ∈ A, ∄V invariant wrt A such that Ei−1 V Ei. Coordinate system adapted to the flag → A =        A11 A12 · · · A22 A23 · · · A33 A34 · · · . . . ... ... ... · · · · · · Akk        , ∀A ∈ A

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 8 / 19

Invariant flags

Definition A irreducible if ∄V nontrivial subspace of Rn invariant wrt all matrices of A Maximal invariant flag for A: {0} = E0 E1 · · · Ek−1 Ek = Rn where Ei invariant wrt each A ∈ A, ∄V invariant wrt A such that Ei−1 V Ei. Coordinate system adapted to the flag → A =        A11 A12 · · · A22 A23 · · · A33 A34 · · · . . . ... ... ... · · · · · · Akk        , ∀A ∈ A Call Ai = {Aii : A ∈ A}. Then Ai irreducible and ρ(A) = maxi ρ(Ai)

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 8 / 19

Marginal stability / instability

Assume from now on ρ(A) = 0 → marginal stability or instability Theorem (N. Barabanov) If A irreducible, then ∃ a norm v : Rn → [0, +∞) s. t.: v(x(t)) ≤ v(x(0)) for every switching law A(·) and initial cond. x(0); ∃ traj. x(·) s. t. v(x(t)) ≡ v(x(0)) ∀t ≥ 0 , ∀x(0). Thus if A irreducible the system is marginally stable!

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 9 / 19

Marginal stability / instability

Assume from now on ρ(A) = 0 → marginal stability or instability Theorem (N. Barabanov) If A irreducible, then ∃ a norm v : Rn → [0, +∞) s. t.: v(x(t)) ≤ v(x(0)) for every switching law A(·) and initial cond. x(0); ∃ traj. x(·) s. t. v(x(t)) ≡ v(x(0)) ∀t ≥ 0 , ∀x(0). Thus if A irreducible the system is marginally stable! Otherwise, if {0} = E0 E1 · · · Ek−1 Ek = Rn maximal invariant flag then from the block form and variation of constant we get x(t) ≤ C(1 + tk−1)x(0) In principle the system could be unstable with polynomial growth.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 9 / 19

Resonance

Definition Consider a reducible switched system A and denote by A1, . . . , Ak the subsystems corresponding to a maximal invariant flag. We say that Ai1, . . . , Air of A (i1 . . . ir distinct) are in resonance if they satisfy: (a) ρ(Ai1) = . . . = ρ(Air ) = 0 with vij(·) corresponding Barabanov norms; (b) ∃A(·) in A with associated switching laws Aijij(·) in Aij and corresp. trajectories γij(·) of Aij such that vij(γij(t)) = const for every t > 0 and for j = 1, . . . , r.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 10 / 19

Resonance

Definition Consider a reducible switched system A and denote by A1, . . . , Ak the subsystems corresponding to a maximal invariant flag. We say that Ai1, . . . , Air of A (i1 . . . ir distinct) are in resonance if they satisfy: (a) ρ(Ai1) = . . . = ρ(Air ) = 0 with vij(·) corresponding Barabanov norms; (b) ∃A(·) in A with associated switching laws Aijij(·) in Aij and corresp. trajectories γij(·) of Aij such that vij(γij(t)) = const for every t > 0 and for j = 1, . . . , r. Theorem (Y. Chitour, P.M., M. Sigalotti) Let A be convex compact. Assume that the linear switched system associated with A is marginally unstable. Then A is reducible and, for any maximal invariant flag, it admits two subsystems Aij, j = 1, 2, in resonance.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 10 / 19

Consequences

Simplest nontrivial case of reducible systems: A = conv{A0, A1} , A0 = A0

11

A0

12

A0

22

• ,

A1 = A1

11

A1

12

A1

22

• .

Assume A0, A1 Hurwitz and ρ(A) = 0.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 11 / 19

Consequences

Simplest nontrivial case of reducible systems: A = conv{A0, A1} , A0 = A0

11

A0

12

A0

22

• ,

A1 = A1

11

A1

12

A1

22

• .

Assume A0, A1 Hurwitz and ρ(A) = 0. n = 2, 3 A marginally unstable ⇒ 0 eigenvalue of A0 or A1 ⇒ contradiction

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 11 / 19

Consequences

Simplest nontrivial case of reducible systems: A = conv{A0, A1} , A0 = A0

11

A0

12

A0

22

• ,

A1 = A1

11

A1

12

A1

22

• .

Assume A0, A1 Hurwitz and ρ(A) = 0. n = 2, 3 A marginally unstable ⇒ 0 eigenvalue of A0 or A1 ⇒ contradiction n ≥ 4 assume A0

∗, A1 ∗ ∈ M2(R) Hurwitz with ρ({A0 ∗, A1 ∗}) = 0.

A = conv{A0, A1} , A0 = A0

∗

Id A0

∗

• ,

A1 = A1

∗

Id A1

∗

• .

⇒ (x1, x2)(t) = (R∗(t, 0)x1(0) + tR∗(t, 0)x2(0) , R∗(t, 0)x2(0)) ⇒ polynomial instability

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 11 / 19

Numerical example

A0 = A0

∗

Id A0

∗

• ,

A1 = A1

∗

Id A1

∗

• Choose A0

∗ =

−1 −α α −1

• ,

A1

∗ =

−1 −α 1/α −1

• .

For a value α ∼ 4.5047 one has ρ(A∗) = 0 x = (x1, x2)

• 2 -1.5 -1 -0.5

0.5 1 1.5 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

• 4
• 2

2 4 6

• 4
• 2

2 4

1

x x

2

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 12 / 19

The 4D case

If n = 4 we have a partial converse result to the previous theorem A0 = A0

11

A0

12

A0

22

• ,

A1 = A1

11

A1

12

A1

22

• Theorem (Y. Chitour, P.M., M. Sigalotti)

Fix A1 = co{A0

11, A1 11} and A2 = co{A0 22, A1 22} and assume they admit

closed worst trajectories with pairwise equal switching times. (ie resonance) Then, there exists a set of pairs of matrices (A0

12, A1 12) which is open and

dense in M2(R) × M2(R), such that the system is polynomially unstable. (We actually have an explicit charact. of the set of matrices (A0

12, A1 12)...)

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 13 / 19

Worst polynomial behaviour

We know that a necessary condition for marginal instability is the existence

• f subsystems in resonance.

Q: What is the relation between resonances and max. polynomial growth of trajectories?

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 14 / 19

Worst polynomial behaviour

We know that a necessary condition for marginal instability is the existence

• f subsystems in resonance.

Q: What is the relation between resonances and max. polynomial growth of trajectories? A: The resonance degree L of A gives an estimate of the polynomial growth: x(t) ≤ C(1 + tL)x(0) L is the “maximum of the sum of the resonance degrees for disjoint resonances”

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 14 / 19

Resonance degree

A =              A11 A12 A22 A23 · · · · · · A33 A34 . . . A44 A45 . . . ... A55 A56 . . . A66 A67 · · · · · · · · · · · · A77             

B A B A C C C A

letters A, B, C denote subsystems in resonance each resonance has an associated resonance degree = nb of blocks in resonance

• − 1

In the example the resonance degree of the system is 4 = 2 + 2

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 15 / 19

Asymptotic behavior of trajectories

We have x(t) ≤ C(1 + tL)x(0) ∀t ≥ 0 Conversely, in several cases one can see that for any t > 0 there exists a switching law s.t. x(t) ≥ ˆ CtLx(0) and ˆ C > 0 does not depend on t → optimality of the estimate

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 16 / 19

Asymptotic behavior of trajectories

We have x(t) ≤ C(1 + tL)x(0) ∀t ≥ 0 Conversely, in several cases one can see that for any t > 0 there exists a switching law s.t. x(t) ≥ ˆ CtLx(0) and ˆ C > 0 does not depend on t → optimality of the estimate Surprisingly if we consider the behavior of traject. at infinity we find lim

t→∞

x(t) tL = 0 ∀x(·)

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 16 / 19

Discrete time systems

Discrete time switched systems: z(k + 1) = M(k) z(k) , where M(k) ∈ M , Stability is characterized by the Joint Spectral Radius: ρ := lim sup

k→∞

• max

M(1),...,M(k)∈M M(k) · · · M(1)1/k

All the results presented above can be easily adapted to discrete time switched systems.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 17 / 19

Nonnegative integer matrices

The discrete time case with M made by nonnegative integer matrices has already been studied in the literature. A complete characterization of the maximal polynomial growth of trajectories has been given by Jungers-Protasov-Blondel (2008). Note that the methods cannot be adapted to the general case considered here.

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 18 / 19

Conclusion and open problems

Main results: marginal instability ⇒ resonance of subsystems estimate of the maximal polynomial growth for marginal instability

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Polynomial instability of switched systems October 28th, 2010 19 / 19

Conclusion and open problems

Main results: marginal instability ⇒ resonance of subsystems estimate of the maximal polynomial growth for marginal instability Some open questions: Is the resonance degree L “generically” the best estimate for the polynomial growth? To understand resonances it is important to study the case A irreducible:

if A is irreducible is it true that there exists a periodic trajectory lying

• n the Barabanov sphere?

examples of “chaotic” behavior on the Barabanov sphere? Non-existence of periodic trajectories?

• P. Mason (CNRS / L2S)

Polynomial instability of switched systems October 28th, 2010 19 / 19