Stability analysis of LPV systems with piecewise differentiable parameters Corentin Briat and Mustafa Khammash - D-BSSE - ETH-Z¨ urich 2017 IFAC World Congress, Toulouse, France
Outline 1 Introduction 2 Stability analysis of LPV systems with piecewise differentiable parameters 3 Examples 4 Concluding statements C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 2 / 23
Outline 1 Introduction 2 Stability analysis of LPV systems with piecewise differentiable parameters 3 Examples 4 Concluding statements C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 2 / 23
LPV systems LPV systems LPV systems are generically represented as x ( t ) = A ( ρ ( t )) x ( t ) + B ( ρ ( t )) u ( t ) , x (0) = x 0 ˙ (1) where x and u are the state of the system and the control input ρ ( t ) ∈ P , P ⊂ R N compact, is the value of the parameter vector at time t The matrix-valued functions A ( · ) and B ( · ) are “nice enough”, i.e. continuous on P Rationale Can be used to approximate nonlinear systems [Sha88, BPB04] Can be used to model a wide variety of real-world processes [MS12, HW15, Bri15a] Convenient framework for the design gain-scheduled controllers [RS00] C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 3 / 23
Quadratic stability [SGC97] Definition The LPV system x ( t ) ˙ = A ( ρ ( t )) x ( t ) (2) x (0) = x 0 is said to be quadratically stable if V ( x ) = x T Px is a Lyapunov function for the system. Theorem The LPV system (2) is quadratically stable if and only if there exists a matrix P ∈ S n ≻ 0 such that the LMI A ( θ ) T P + PA ( θ ) ≺ 0 (3) holds for all θ ∈ P . Remarks All the possible trajectories ρ : R ≥ 0 �→ P are (implicitly) considered (together with the assumption of existence of solutions) Semi-infinite dimensional LMI problem (can be checked using various methods) C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 4 / 23
Robust stability [Wu95] Definition The LPV system x ( t ) ˙ = A ( ρ ( t )) x ( t ) (4) x (0) = x 0 ρ ( t ) ∈ D , for some given compact sets P , D ⊂ R N , is said to be with ρ ( t ) ∈ P and ˙ robustly stable if V ( x, ρ ) = x T P ( ρ ) x is a Lyapunov function for the system. Theorem The LPV system (4) is robustly stable if and only if there exists a differentiable matrix-valued function P : P → S n ≻ 0 such that the LMI N i ∂ θ i P ( θ ) + A ( θ ) T P ( θ ) + P ( θ ) A ( θ ) ≺ 0 � θ ′ (5) i =1 holds for all θ ∈ P and all θ ′ ∈ D . Remarks Trajectories of the parameters are continuously differentiable (can be relaxed) Infinite-dimensional LMI problem (can be approximately checked) C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 5 / 23
Summary Some remarks Two main classes of parameter trajectories associated with two main stability concepts But these classes are very far apart! Parameter trajectories are defined in a very loose/restrictive way The accuracy of the tools developed for periodic, switched and Markov jump systems stems from the fact that they are tailor-made C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 6 / 23
Summary Some remarks Two main classes of parameter trajectories associated with two main stability concepts But these classes are very far apart! Parameter trajectories are defined in a very loose/restrictive way The accuracy of the tools developed for periodic, switched and Markov jump systems stems from the fact that they are tailor-made Question What if we consider piecewise differentiable parameters? Robust stability not applicable and quadratic stability too conservative So, we need something else! C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 6 / 23
LPV systems with piecewise differentiable parameters Class of parameters Piecewise differentiable with aperiodic discontinuities Stability results Stability condition using hybrid systems method → minimum dwell-time condition Connections with quadratic and robust stability Examples C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 7 / 23
Outline 1 Introduction 2 Stability analysis of LPV systems with piecewise differentiable parameters 3 Examples 4 Concluding statements C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 7 / 23
Preliminaries Let us consider the LPV system x ( t ) = A ( ρ ( t )) x ( t ) , x (0) = x 0 ˙ (6) with parameter trajectories ρ in P T where � ¯ � � � ρ ( t ) ∈ Q ( ρ ( t )) , t ∈ [ t k , t k +1 ) ˙ � P T := ρ : R ≥ 0 �→ P (7) � ¯ � T k ≥ ¯ T, ρ ( t k ) � = ρ ( t + k ) ∈ P , k ∈ Z ≥ 0 � where ρ ( t + k ) := lim s ↓ t k ρ ( s ) , t 0 = 0 (no jump at t 0 ), T k := t k +1 − t k , ¯ T > 0 , P =: P 1 × . . . × P N , P i := [ ρ i , ¯ ρ i ] , ρ i ≤ ¯ ρ i , i = 1 , . . . , N D =: D 1 × . . . × D N , D i := [ ν i , ¯ ν i ] , ν i ≤ ¯ ν i , i = 1 , . . . , N and Q ( ρ ) = Q 1 ( ρ ) × . . . × Q N ( ρ ) with D i if ρ i ∈ ( ρ i , ¯ ρ i ) , Q i ( ρ ) := D i ∩ R ≥ 0 if ρ i = ρ i , (8) D i ∩ R ≤ 0 if ρ i = ¯ ρ i . C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 8 / 23
Illustration Minimum dwell-time ¯ T = 3 . 3 Discontinuities separated by at least ¯ T = 3 . 3 seconds C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 9 / 23
System reformulation The key idea is to reformulate the system in a way that will allow us to capture the both the dynamics of the system and the dynamics of the parameters. C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 10 / 23
System reformulation The key idea is to reformulate the system in a way that will allow us to capture the both the dynamics of the system and the dynamics of the parameters. Hence, we propose the following hybrid system formulation [GST12] � x ( t ) ˙ = A ( ρ ( t )) x ( t ) � � ρ ( t ) ˙ ∈ Q ( ρ ( t )) if ( x ( t ) , ρ ( t ) , τ ( t ) , T ( t )) ∈ C � � τ ( t ) ˙ = 1 (eq. τ ( t ) < T ( t ) ) � ˙ � T ( t ) = 0 � (9) x ( t + ) � = x ( t ) � ρ ( t + ) � ∈ P if ( x ( t ) , ρ ( t ) , τ ( t ) , T ( t )) ∈ D � τ ( t + ) � = 0 (eq. τ ( t ) = T ( t ) ) � [ ¯ T ( t + ) � ∈ T, ∞ ) � where R n × P × E < , C = R n × P × E = D = (10) { ϕ ∈ R ≥ 0 × [ ¯ E � = T, ∞ ) : ϕ 1 � ϕ 2 } , � ∈ { <, = } and ( x (0) , ρ (0) , τ (0) , T (0)) ∈ R n × P × { 0 } × [ ¯ T, ∞ ) . (11) C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 10 / 23
Illustration Let the t k ’s be the time instants for which τ ( t k ) = T ( t k ) We consider a parameter trajectory given by ρ ( t ) = (1 + sin( t + φ ( t ))) / 2 where φ ( t ) = φ k , t ∈ [ t k , t k +1 ) and the φ k ’s are uniform random variables taking values in [0 , 2 π ] At each t k , a new value for φ k is drawn, which introduces a discontinuity in the parameter trajectory 3 = ( t ) T ( t ) 2 1 0 0 2 4 6 8 10 12 14 16 18 20 1 ; ( t ) 0.5 0 0 2 4 6 8 10 12 14 16 18 20 Time [s] C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 11 / 23
Main result Theorem (Minimum dwell-time) Let ¯ T ∈ R > 0 be given and assume that there exist a bounded continuously differentiable matrix-valued function S : [0 , ¯ T ] × P �→ S n ≻ 0 and a scalar ε > 0 such that the conditions N � ∂ τ S ( τ, θ ) + ∂ ρ i S ( τ, θ ) µ i + Sym[ S ( τ, θ ) A ( θ )] + εI � 0 (12) i =1 N � ∂ ρ i S ( ¯ T, θ ) µ i + Sym[ S ( ¯ T, θ ) A ( θ )] + εI � 0 (13) i =1 and S (0 , θ ) − S ( ¯ T, η ) � 0 (14) hold for all θ, η ∈ P , µ ∈ D and all τ ∈ [0 , ¯ T ] . Then, the LPV system (6) with parameter trajectories in P T is asymptotically stable. � ¯ For a square matrix M , we define Sym[ M ] = M + M T C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 12 / 23
Connection with quadratic and robust stability Theorem (Quadratic stability) When ¯ T → 0 in the minimum dwell-time theorem, then we recover the quadratic stability condition A ( θ ) T P + PA ( θ ) ≺ 0 , θ ∈ P . (15) C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 13 / 23
Connection with quadratic and robust stability Theorem (Quadratic stability) When ¯ T → 0 in the minimum dwell-time theorem, then we recover the quadratic stability condition A ( θ ) T P + PA ( θ ) ≺ 0 , θ ∈ P . (15) Theorem (Robust stability) When ¯ T → ∞ , then we recover the robust stability condition N ∂ ρ i P ( θ ) µ i + A ( θ ) T P ( θ ) + P ( θ ) A ( θ ) ≺ 0 , θ ∈ P , µ ∈ D . � (16) i =1 C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 13 / 23
Connection with quadratic and robust stability Theorem (Quadratic stability) When ¯ T → 0 in the minimum dwell-time theorem, then we recover the quadratic stability condition A ( θ ) T P + PA ( θ ) ≺ 0 , θ ∈ P . (15) Theorem (Robust stability) When ¯ T → ∞ , then we recover the robust stability condition N ∂ ρ i P ( θ ) µ i + A ( θ ) T P ( θ ) + P ( θ ) A ( θ ) ≺ 0 , θ ∈ P , µ ∈ D . � (16) i =1 C. Briat, D-BSSE@ETH-Z¨ urich 2017 IFAC World Congress, Toulouse 13 / 23
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