On Topological Entropy of Switched Linear Systems with Pairwise - PowerPoint PPT Presentation

On Topological Entropy of Switched Linear Systems with Pairwise Commuting Matrices Guosong Yang and João P. Hespanha Center for Control, Dynamical Systems, and Computation University of California, Santa Barbara November 15, 2018 1 / 20

Motivation Topological entropy in systems theory Originated from [Kolmogorov, 1958], defined by [Adler, Konheim, and McAndrew, 1965], [Bowen, 1971], and [Dinaburg, 1970]. Essential idea: • The complexity growth of a system. • The information accumulation needed to approximate a trajectory. In control theory: • Topological feedback entropy [Nair, Evans, Mareels, and Moran, 2004] • Invariance entropy [Colonius and Kawan, 2009], exponential stabilization entropy [Colonius, 2012] • Estimation entropy [Savkin, 2006] and [Liberzon and Mitra, 2018] Minimal data rate for stabilizing linear time-invariant (LTI) system [Hespanha, Ortega, and Vasudevan, 2002], [Nair and Evans, 2003], and [Tatikonda and Mitter, 2004] 2 / 20

Motivation Switched linear system with pairwise commuting matrices Switching is ubiquitous in realistic system models. Stability under arbitrary switching: pairwise commuting matrices [Narendra and Balakrishnan, 1994] Neither minimal data rate for stabilization nor topological entropy is well-understood: • Sufficient data rate [Liberzon, 2014], [Yang and Liberzon, 2018], and [Sibai and Mitra, 2017] • Topological entropy [Yang, Schmidt, and Liberzon, 2018] 3 / 20

Switched System A finite family of continuous-time dynamical systems x = f p ( x ) , ˙ p ∈ P with the state x ∈ R n and an index set P . A switched system x = f σ ( x ) , ˙ x (0) ∈ K. Switching signal σ : R + → P is right-continuous and piecewise constant Initial set K is compact with a nonempty interior Modes { f p : p ∈ P} Denote by ξ σ ( x, t ) the solution at t with switching signal σ and initial state x 4 / 20

Entropy Definition A switched system x = f σ ( x ) , ˙ x (0) ∈ K. Given a time horizon T ≥ 0 and a radius ε > 0 , define the open ball: � x ′ ∈ K : max � t ∈ [0 ,T ] � ξ σ ( x ′ , t ) − ξ σ ( x, t ) � < ε B f σ ( x, ε, T ) := . A finite set E ⊂ K is ( T, ε ) -spanning if K = � x ∈ E B f σ (ˆ x, ε, T ) . ˆ Let S ( f σ , ε, T, K ) be the minimal cardinality of a ( T, ε ) -spanning set. The topological entropy with initial set K and switching signal σ is defined in terms of the exponential growth rate of S ( f σ , ε, T, K ) by 1 h ( f σ , K ) := lim ε ց 0 lim sup T log S ( f σ , ε, T, K ) . T →∞ 5 / 20

Alternative Entropy Definition A switched system x = f σ ( x ) , ˙ x (0) ∈ K. The topological entropy is defined by 1 h ( f σ , K ) := lim ε ց 0 lim sup T log S ( f σ , ε, T, K ) . T →∞ x ′ ∈ E , A finite set of points E ⊂ K is ( T, ε ) -separated if for all ˆ x, ˆ x ′ / � � ˆ ∈ B f σ (ˆ x, ε, T ) = x ∈ K : max t ∈ [0 ,T ] � ξ σ ( x, t ) − ξ σ (ˆ x, t ) � < ε . Let N ( f σ , ε, T, K ) be the maximal cardinality of a ( T, ε ) -separated set. Proposition 1. The topological entropy satisfies 1 h ( f σ , K ) = lim ε ց 0 lim sup T log N ( f σ , ε, T, K ) . T →∞ Proof. N ( f σ , 2 ε, T, K ) ≤ S ( f σ , ε, T, K ) ≤ N ( f σ , ε, T, K ) . 6 / 20

Active Time and Active Rates For a switching signal σ , define the following quantities. The active time of each mode over an interval [0 , t ] is � t τ p ( t ) := 1 p ( σ ( s )) d s, p ∈ P 0 with the indicator function 1 . Then � p ∈P τ p ( t ) = t . The active rate of each mode over [0 , t ] is ρ p ( t ) := τ p ( t ) /t, p ∈ P with ρ p (0) := 1 p ( σ (0)) . Then � p ∈P ρ p ( t ) = 1 . The asymptotic active rate of each mode is ρ p := lim sup ˆ ρ p ( t ) , p ∈ P . t →∞ It is possible that � p ∈P ˆ ρ p > 1 . 7 / 20

Entropy of Switched Linear Systems A switched linear system x = A σ x, ˙ x (0) ∈ K. Results from [Yang, Schmidt, and Liberzon, 2018]: Proposition 2. The topological entropy of the switched linear system is independent of the choice of the initial set K . Proposition 3. The topological entropy of the switched linear system satisfies � � lim sup tr( A p ) ρ p ( t ) ≤ h ( A σ ) ≤ lim sup n � A p � ρ p ( t ) t →∞ t →∞ p ∈P p ∈P with the active rates ρ p . Proof for the upper bound. � p � A p � τ p ( t ) � x ′ − x � . 1. The solutions satisfy � ξ σ ( x ′ , t ) − ξ σ ( x, t ) � ≤ e 2. Construct a ( T, ε ) -spanning set using a grid. Lack of “independence” between eigenspaces of different modes! 8 / 20

Switched Commuting Linear Systems A switched linear system x = A σ x, ˙ x (0) ∈ K with a commuting family { A p : p ∈ P} . If all A p are diagonalizable, then there is a change of basis under which all A p are diagonal. Every scalar component evolves independently (under the same switching signal). A formula for the entropy was established in [Yang, Schmidt, and Liberzon, 2018]. 9 / 20

Switched Commuting Linear Systems A switched linear system x = A σ x, ˙ x (0) ∈ K with a commuting family { A p : p ∈ P} . A well-known result: in general, there is a change of basis under which all A p are upper triangular. Each scalar component evolves in a “strict-feedback” fashion. An upper bound for the entropy was established in [Yang, Schmidt, and Liberzon, 2018]. Being simultaneously triangularizable is weaker than being pairwise commuting! 10 / 20

Switched Commuting Linear Systems A switched linear system x = A σ x, ˙ x (0) ∈ K with a commuting family { A p : p ∈ P} . First goal: a suitable change of basis for pairwise commuting matrices. Jordan–Chevalley Decomposition [Humphreys, 1972]. For each matrix A , there exist polynomials f and g , without constant term, such that f ( A ) is a diagonalizable matrix, g ( A ) is a nilpotent matrix, and A = f ( A ) + g ( A ) . A polynomial of a matrix A commutes with all matrices that commute with A . 11 / 20

A Change of Basis Proposition 5 For the commuting family { A p : p ∈ P} , there exists an invertible matrix Γ ∈ C n × n such that Γ A p Γ − 1 = D p + N p ∀ p ∈ P , where all D p ∈ C n × n are diagonal matrices, all N p ∈ C n × n are nilpotent matrices, and { D p , N p : p ∈ P} is a commuting family. Proof. 1. For each p , there are polynomials f p and g p such that f p ( A p ) is diagonalizable, g p ( A p ) is nilpotent, and A p = f p ( A p ) + g p ( A p ) . 2. The set { f p ( A p ) , g p ( A p ) : p ∈ P} is a commuting family. 3. There is an invertible Γ ∈ C n × n such that all D p := Γ f p ( A p )Γ − 1 are invertible. 4. All N p := Γ g p ( A p )Γ − 1 are nilpotent, and { D p , N p : p ∈ P} is a commuting family. 12 / 20

A Formula for Entropy The switched commuting linear system becomes x = ( D σ + N σ ) x, ˙ x (0) ∈ K, where D p = diag( a 1 p , . . . , a n p ) are diagonal, N p are nilpotent, and { D p , N p : p ∈ P} is a commuting family. Theorem 6 The topological entropy of the switched commuting system satisfies n 1 � � Re( a i h ( D σ + N σ ) = lim sup T max p ) τ p ( t ) t ∈ [0 ,T ] T →∞ i =1 p ∈P with the active times τ p . The entropy only depends on the diagonal part, i.e., h ( D σ + N σ ) = h ( D σ ) . 13 / 20

A Formula for Entropy Proof. 1. The solutions satisfy � � � p ∈P D p τ p ( t ) ( x ′ − x ) � p ∈P N p τ p ( t ) e � ξ σ ( x ′ , t ) − ξ σ ( x, t ) � = � e � . � � 2. Lemma 2. Consider the commuting family of nilpotent matrices { N p : p ∈ P} . For each δ > 0 , there is a constant c δ > 0 such that for all v ∈ C n , � � � p ∈P N p τ p ( t ) v c − 1 δ e − δt � v � ≤ � ≤ c δ e δt � v � � � e � for all t ≥ 0 with the active times τ p . 3. Given a radius ε > 0 , there is a constant c ε > 0 such that � p ∈P Re( a i p ) τ p ( t ) | x ′ · · · ≤ � ξ σ ( x ′ , t ) − ξ σ ( x, t ) � ≤ c ε e εt i =1 ,...,n e max i − x i | . 4. For the upper/lower bound, construct a ( T, ε ) -spanning/separated set using a grid. 14 / 20

The Non-Switched Case The formula for entropy yields the following well-known result [Bowen, 1971]: Corollary 7 The topological entropy of the linear time-invariant (LTI) system x = Ax, ˙ x (0) ∈ K equals the sum of the positive real parts of the eigenvalues of A , that is, � h ( A ) = max { Re( λ ) , 0 } . λ ∈ spec( A ) Proof. 1. The spectrum spec( A ) = { a 1 , . . . , a n } . 2. The entropy n n 1 � � t ∈ [0 ,T ] Re( a i ) t = max { Re( a i ) , 0 } . h ( A ) = lim sup T max T →∞ i =1 i =1 15 / 20

More General Upper and Lower Bounds for Entropy # Formula/upper bounds Sw CoB n 1 � � Re( a i (1) = lim sup T max p ) τ p ( t ) τ p Yes t ∈ [0 ,T ] T →∞ i =1 p ∈P n � � � � Re( a i (2) ≤ max lim sup p ) ρ p ( t ) , 0 ρ p Yes t →∞ i =1 p ∈P � (3) ≤ lim sup h ( D p ) ρ p ( t ) ρ p No t →∞ p ∈P ≤ � (4) h ( D p )ˆ ρ p ρ p ˆ No p ∈P (5) ≤ max p ∈P h ( D p ) N/A No (6) ≤ lim sup � n � A p � ρ p ( t ) No ρ p t →∞ p ∈P � (7) ≥ lim sup tr( A p ) ρ p ( t ) No ρ p t →∞ p ∈P 16 / 20

Numerical Example Let P = { 1 , 2 } and � − 1 � � 3 � 0 0 D 1 = , D 2 = . 0 2 0 0 (ˆ ρ 1 , ˆ ρ 2 ) (1) (2) (3) (4) (5) (6) (7) No switch (1 , 0) 2 2 2 2 3 4 1 Periodic switches (0 . 5 , 0 . 5) 2 2 2 . 5 2 . 5 3 5 2 Switches w/ set-points (0 . 9 , 0 . 9) 2 . 8 4 . 4 2 . 9 4 . 5 3 5 . 8 2 . 8 17 / 20