Switched linear discrete time systems EECI Graduate School on - PowerPoint PPT Presentation

Switched linear discrete time systems EECI Graduate School on Control 2009 Jamal Daafouz March 2009

Outline Switched Linear Discrete Time Systems ◮ Stability ◮ Structural Properties : Invertibility, Flatness Applications ◮ Digital Control : Hot Strip Mill, Networked Control ◮ Encryption

Spectral radius ◮ Spectral radius of A : largest modulus of its eigenvalues ρ ( A ) = max {| λ | : Av = λ v } ◮ Spectral radius of a matrix power ρ ( A k ) = ρ ( A ) k Convergence condition : k →∞ A k = 0 lim ⇐ ⇒ ρ ( A ) < 1 ◮ Spectral radius as a limit of norms k →∞ � A k � 1 / k ρ ( A ) = lim

Generalized spectral radius Consider a (non necessarily bounded) set A of n × n matrices A i � � � � A = A i : i ∈ I , I = 1 , ..., M The larget possible spectral radius of all products of the matrices k � � � ρ k ( A ) � sup ρ ( A i ) : A i ∈ A for 1 ≤ i ≤ k i = 1 ◮ GSR : The maximal asymptotic spectral radius of the product of matrices choosen freely in A � � 1 / k ρ ( A ) � lim sup ρ k ( A ) k →∞

Joint spectral radius The larget possible norm of all products of the matrices choosen in A k � � � ρ k ( A ) � sup ˆ � A i � : A i ∈ A for 1 ≤ i ≤ k i = 1 ◮ JSR : � � 1 / k ρ ( A ) � lim sup ˆ ρ k ( A ) ˆ k →∞ � � ◮ GSR = JSR = spectral radius when A = A ρ ( A ) = ˆ ρ ( A ) = ρ ( A ) ◮ GSR = JSR for any bounded set A ρ ( A ) = ˆ ρ ( A )

Stability of discrete switched systems Consider a set A of matrices A i , and the discrete linear inclusion � � x 0 ∈ R n arbitrary x k + 1 ∈ A σ k x k : A σ k ∈ A , (1) The sequence ( σ k ) is the switching signal depending on k and/or on x k . ( x 0 , x 1 , . . . , x k , . . . ) satisfying the inclusion (1) x k + 1 = A σ k x k , for some sequence σ k is a trajectory in the R n space. The set of all possible switchings signals defines a whole set of possible trajectories.

GSR/JSR and UFS (J. Theys et al. 2005) ◮ Uniform Asymptotic Stability : UFS Any trajectory ( x k ) of the discrete linear inclusion converges to the origin k →∞ x k = 0 , lim ∀ ( σ k ) As this is supposed to hold for any x 0 , it is equivalent to saying that all matrix products converge to 0 k →∞ A σ k A σ k − 1 . . . A σ 1 = 0 lim ◮ Theorem For a bounded set A of matrices ˆ ρ ( A ) < 1 ⇐ ⇒ discrete linear inclusion is UFS ρ ( A ) < 1 ⇐ ⇒ discrete linear inclusion is UFS

Stability of discrete switched systems V. D. Blondel and J. N. Tsitsiklis. The boundedness of all products of a pair of matrices is undecidable. Systems and Control Letters, 2000. Theorem Generalized/Joint spectral radius of a pair of matrices is not polynomial-time approximable. This is true even for a pair of matrices with { 0 , 1 } entries. It is NP-hard to decide whether all products of two given real matrices A 0 and A 1 are bounded. ◮ GSR/JSR complex to compute ◮ not suitable for design purpose

Stability of discrete switched systems ◮ Lyapunov functions : In general, a sufficient stability condition. A scalar function V such that V ( x k ) > 0 , ∀ x k � = 0 and the derivative of V along the system trajectories must be decreasing as time evolves. Linear systems : x 0 ∈ R n x k + 1 = Ax k , Quadratic Lyapunov function P = P T > 0 V ( x k ) = x T k Px k ,

Stability of discrete switched systems V ( x k + 1 ) < V ( x k ) , ∀ x k � = 0 x T k + 1 Px k + 1 < x T k Px k , ∀ x k � = 0 A T PA − P < 0 ⇐ ⇒ � λ ( A ) � < 1 Generalizing this to discrete time switched systems leads to the common quadratic Lyapunov function (CQLF) stability condition A T i PA i − P < 0 , ∀ i which should hold simultaneously for all matrices A i of the set A Existence of a CQLF is not a necessary condition.

Stability of discrete switched systems ◮ Example : (J. Theys et al. 2005) x k + 1 = A σ k x k , A σ k ∈ { A 1 , A 2 } � − 0 . 2 � − 0 . 2 � � − 0 . 4 − 2 . 4 A 1 = , A 2 = 0 . 4 − 0 . 2 1 / 15 − 0 . 2 No CQLF but 0 . 9275 ≤ ˆ ρ ≤ 0 . 9510

Classical results For more complex systems it is not clear : ◮ What form should the Lyapunov function have ? ◮ What form is necessary and sufficient for the stability ? ◮ How can we find it ? (analytical / numerical tool) Solution : ◮ Use simple forms for which we can get LMI conditions ⇒ only sufficient stability conditions ◮ Conservatism : V ( x ) does not exist but the system is stable

Stability Analysis of discrete time switched systems Goal : Analyse global stability of a switched system x k + 1 = A σ k x k where A σ k belongs to { A i : i = 1 , . . . , M } σ k is the switching rule meaning that at each instant of time k A σ k = A i , for some i ∈ { 1 , . . . , M } There is no complete solution for the stability problem even in the bidimensional case x k ∈ R 2 x k + 1 = A σ k x k A σ k ∈ { A 1 , A 2 } ,

Stability of Switched systems / Polytopic systems Stability of a linear discrete time switched system x k + 1 = A σ k x k , A σ k ∈ { A i : i = 1 , . . . , M } is equivalent to the stability of the linear discrete-time polytopic system x k + 1 = A ξ k x k , A ξ k ∈ conv { A 1 , . . . , A M } that is A ξ k = � M � M i = 1 ξ i ( k ) A i and i = 1 ξ i = 1 , ξ i ≥ 0 One has to prove that ρ ( conv { A 1 , . . . , A M } ) = ρ ( { A 1 . . . A M } ) (P. Mason, M. Sigalotti and J. Daafouz, CDC 2007)

Stability of linear switched discrete time systems Different choices of parameter dependent quadratic Lyapunov functions for global stability analysis of a linear discrete-time polytopic system x k + 1 = A ξ k x k ( k ∈ N ) A ξ = � M i = 1 ξ i A i and � M i = 1 ξ i = 1 , ξ i ≥ 0 (convex combination of A i ). (1) Parameter dependent quadratic stability α 1 � x � 2 ≤ V ( x , ξ ) ≤ α 2 � x � 2 V ( x , ξ ) = x T P ( ξ ) x (2) Parameter and time dependent quadratic stability α 1 � x � 2 ≤ V ( k , x , ξ ) ≤ α 2 � x � 2 V ( k , x , ξ ) = x T P ( k , ξ ) x (3) Poly-quadratic stability M � V ( x , ξ ) = x T P ( ξ ) x with P ( ξ ) = ξ i P i i = 1

Using parameter dependent Lyapunov functions Proposition The previous three types of quadratic stability are equivalent. Proof : (P. Mason, M. Sigalotti and J. Daafouz, CDC 2007) Clearly: (3) Polyquadratic stability ( V ( x , ξ ) with P ( ξ ) = � M i = 1 ξ i P i ) ⇓ (1) Parameter dependent quadratic stability ( V ( x , ξ ) = x T P ( ξ ) x ) ⇓ (2) Parameter and time dependent quadratic stability ( V ( k , x , ξ ) = x T P ( k , ξ ) x ) It remains to show that (2) ⇒ (3).

Scheme of the proof (2) ⇒ (3) (i.e. ∃ a L.F. of the form x T P ( k , ξ ) x ⇒ ∃ a L.F. x T ( � M i = 1 ξ i P i ) x ) First step: Recall that x k + 1 = A ξ k x k where A ξ = � M i = 1 ξ i A i . Define P k , i = quadratic form associated with the vertex A i at time k Then one proves that M � Π k ,ξ ( x ) = x T ( ξ i P k , i ) x i = 1 is a (time and parameter dependent) quadratic LF. The key tool is the convexity of the function f ( A ) = A T PA for any positive definite matrix P .

Parameter dependent quadratic stability Second step: By a compactness argument one can choose suitable k and h with h < k such that k � Π ⋆ ξ ( x ) = Π k ,ξ ( x ) i = h is a LF and has the desired form. Remark more general than the classical "static” notion of quadratic stability P ( ξ ) ≡ P Asymptotic stability does not imply parameter dependent quadratic stability

LMI stability condition (J. Daafouz et al 2001 and 2002) ◮ There exist a polyquadratic Lyapunov function M � V ( x , ξ ) = x T P ( ξ ) x with P ( ξ ) = ξ i P i i = 1 for x k + 1 = A ξ k x k , A ξ k ∈ conv { A 1 , . . . , A M } ◮ There exist a switched Lyapunov function V ( x k , σ k ) = x T k P σ x k for x k + 1 = A σ k x k , A σ k ∈ { A i : i = 1 , . . . , M } ◮ There exist P i , i = 1 , . . . , M satisfying A T i P j A i − P i < 0 ∀ i = 1 , . . . , M , ∀ j = 1 , . . . , M (2)

Stability of Polytopic discrete time systems Necessary and sufficient LMI condition for the existence of M � V = x ⊤ ξ i P i x k k i = 1 or equivalently for the existence of V = x ⊤ k P σ x k � � A ⊤ P i i P j > 0 ∀ i = 1 , . . . , M , ∀ j = 1 , . . . , M (3) P j A i P j � G i + G i ⊤ − S i � G i ⊤ A ⊤ i > 0 , ∀ i = 1 , . . . , M , ∀ j = 1 , . . . , M A i G i S j (4) where S i = P − 1 , ∀ i = 1 , . . . , M i

Dwell Time (J.C Geromel et al 2006) Consider σ k = i ∈ { 1 , . . . , M } , k ∈ [ l q , l q + 1 ) where l q and l q + 1 are succesive switching times satisfying l q + 1 − l q ≥ ∆ ≥ 1 , ∀ q ∈ N Theorem Assume that, for some ∆ ≥ 1 , these exists a collection of positive definite matrices { P 1 , . . . , M } such that A T i P i A i − P i < 0 , ∀ i = 1 , . . . , M (5) ( A ∆ i ) T P j A ∆ i − P i < 0 , ∀ i � = j = 1 , . . . , M (6) The switched system is globally asymptotically stable with a dwell time ∆ .

Dwell Time An upper bound for the minimum dwell time ∆ ∗ can be computed by taking the minimum value of ∆ satisfying the Theorem conditions. One has to solve the optimization problem � min ∆ ≥ 1 , P 1 > 0 ,..., P M > 0 ∆ : ( 5 ) − ( 6 ) which, for ∆ ≥ 1 fixed, reduces to a convex programming feasibility problem with LMI constraints. ∆ = 1 : we recover the switched Lyapunov functions conditions A T i P j A i − P i < 0 ∀ i = 1 , . . . , M , ∀ j = 1 , . . . , M