Normal forms of matrix words for stability analysis of discrete-time - PowerPoint PPT Presentation

Normal forms of matrix words for stability analysis of discrete-time switched linear systems Cyrille Chenavier 1 Rosane Ushirobira 2 Laurentiu Hetel 3 1 Johannes Kepler Universität, Linz, Austria 2 Inria, France 3 CNRS, U. Lille, France European Control Conference, ECC 2020 Saint Petersburg, Russia, May 12-15, 2020 C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 1 / 12

Motivation Discrete-time switched linear systems A discrete-time switched linear system is given by x 0 ∈ R n x k +1 = A σ ( k ) x k , k ∈ N , where • x : N → R n represents the state variable, x (0) = x 0 is the initial state • A 1 , · · · , A p ∈ R n × n are matrices representing stable subsystems • σ : N → { A 1 , · · · , A p } is the switching function (not known) Problem Analyse global uniform exponential stability (GUES) of such systems � do any trajectory converges to 0 with exponential decay? C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 2 / 12

Stability analysis methods Existing stability analysis methods • Joint spectral radius (Blondel) • Lie algebraic conditions (Liberzon, Gurvitz) • Set theoretic approach (Megretski, Kruszewski, Guerra) • Lyapunov functions (sufficient condition) Megretski’s method • Requires to solve LMIs problem • LMIs are indexed by matrix words C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 3 / 12

Word representation of trajectories Trajectories The trajectory associated to the switching σ (1) = i 1 ∈ { 1 , · · · , p } , σ (2) = i 2 ∈ { 1 , · · · , p } , · · · has the form x 0 → A i 1 x 0 → A i 2 A i 1 x 0 → A i 3 A i 2 A i 1 x 0 → · · · Matrix representation of finite trajectories If w = i k · · · i 1 is a k -length word over { 1 , · · · , p } � A w := A i k · · · A i 2 A i 1 • Example: � � � � 1 1 0 1 A 1 = , A 2 = 0 1 1 0 then � � � � � � 1 2 1 1 0 1 A 11 = A 1 A 1 = A 12 = A 1 A 2 = A 21 = A 2 A 1 = , , , 0 1 1 0 1 1 A 22 = A 2 A 2 = Id 2 C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 4 / 12

A LMIs criterion for GUES Theorem [Megretski, ’97] The discrete-time switched linear system is GUES if and only if ∃ N > 0 and P = P T ≻ 0 s.t. the following LMIs problem admits a solution P ≻ A T ∀ w = i N · · · i 1 w PA w , Remark: the size of LMIs grows exponentially ( p N words of length N ) Contribution of the work Use linear algebra methods to reduce the size of the LMIs problem • Motivation: assume that P = P T ≻ 0 solves LMIs for A w 1 , · · · A w r and r � A w 0 = λ i A w i i =1 Under which conditions P also solves the LMI for A w 0 ? C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 5 / 12

Normal forms matrices Definition Let N be an integer and R n × n d N := dim ( Vect ( A w : w = i N · · · i 1 )) ⊆ A free set of matrices A w 1 , · · · A w dN is called a set of normal form matrices Remark If A w is not a normal form matrix, it admits a unique decomposition d N � λ w A w = i A w i i =1 Question: how to use linear algebra to restrict LMIs to normal form matrices? C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 6 / 12

Candidates for a new LMIs problem 1st candidate for a new LMIs problem Let N > 0 and A w 1 , · · · A w dN be normal form matrices ∃ P = P T ≻ 0 s.t. P ≻ A T w i PA w i , 1 ≤ i ≤ d N Remark: the number of LMIs is bounded by the constant n 2 ( d N ≤ n 2 ) Problem If the decomposition of a non normal form matrix d N � λ w A w = i A w i i =1 involves "big" coefficients, P ≻ A T w PA w does not hold The LMIs problem has to take λ w i ’s into account C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 7 / 12

Candidates for a new LMIs problem Lemma If P is a solution to the LMIs problem ∃ P = P T ≻ 0 s.t. P ≻ A T i PA i , 1 ≤ i ≤ d N Then, P ≻ A T PA holds for every A is the convex hull of A i ’s 2nd candidate for a new LMIs problem Let N > 0 and A w 1 , · · · , A w dN be normal form matrices ∃ P = P T ≻ 0 s.t. P ≻ µ i A T w i PA w i , 1 ≤ i ≤ d N where µ i ’s are such that "linear combinations are transformed into convex decompositions" C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 8 / 12

Candidates for a new LMIs problem From linear to convex decompositions Start with a linear combination of a non normal form matrix d N � λ w A w = i A w i i =1 Letting n w := | λ w 1 | + · · · + | λ w n | , we get the following convex decomposition d N | λ w i | � ( ε ( λ w A w = i ) n w A w i ) n w i =1 Choices for µ i ’s First choice: all µ i ’s are equal to max( n w : A w is not a normal form matrix) A more optimal choice: µ i = max( n w : A w is not a normal form matrix and λ w i � = 0) C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 9 / 12

Main result Theorem Consider the discrete-time switched linear system x 0 ∈ R n x k +1 = A σ ( k ) x k , k ∈ N , (1) Let N be a strictly positive integer and let A 1 , · · · , A d N be normal form matrices. For every non normal form matrix A w , let us consider its unique decomposition d N � λ w A w = i A w i i =1 and for every 1 ≤ i ≤ d N , let µ i := max( n w : A w is not a normal form matrix and λ w i � = 0) If the following LMIs problem admits a solution ∃ P = P T ≻ 0 s.t. P ≻ µ i A T 1 ≤ i ≤ d N w i PA w i , then (1) is GUES C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 10 / 12

Numerical example Example Consider the discrete-time switched linear system defined with p = 2 and A i = exp( A c i T ), with T = 1, where � � � � − 1 − 1 − 1 − a A c A c 1 = , 2 = 1 1 − 1 − 1 a Changing the value of the parameter a , we get a=5 a=6 a=7 a=8 #LMI conditions N=1 � - - - 2 N=3 � � - - 9 N=8 257 � � � � where � means that a solution to the LMIs problem was obtained, and − not C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 11 / 12

Conclusion • We investigated stability of discrete-time switched linear systems using linear algebra techniques • Our approach may be used to reduce drastically the number of LMI’s conditions to check stability • The counter-part of the approach is that LMI’s are have higher numerical constraints THANK YOU! C.Chenavier, R.Ushirobira, L.Hetel ECC 2020, Saint Petersbourg May 12-15, 2020 12 / 12