Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Robust stability analysis of uncertain Linear Positive Systems via Integral Linear Constraints: L 1 and L ∞ -gain characterizations Corentin Briat KTH, Stockholm, Sweden December 14 th 2011 CDC 2011, Orlando, USA C. Briat [KTH / ] 1/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Outline ◮ Introduction ◮ Stability analysis and norm computation ◮ Robust stability analysis ◮ Conclusion and Future Works C. Briat [KTH / ] 2/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Introduction C. Briat [KTH / ] 3/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Linear positive systems Internally positive systems x ( t ) ˙ = Ax ( t ) (1) x (0) = x 0 ◮ ( P 1 ) Positive orthant R n + invariant: x 0 ∈ R n + ⇒ x ( t ) ∈ R n + , for all t ≥ 0 ◮ NSC: A is a Metzler matrix (nonnegative off-diagonal elements) Input/Output Positive systems x ( t ) ˙ = Ax ( t ) + Ew ( t ) z ( t ) = Cx ( t ) + Fw ( t ) (2) x (0) = x 0 ◮ ( P 1 ) holds ◮ ( P 2 ) For all w ( t ) ≥ 0 , we have z ( t ) ≥ 0 ◮ NSC: A is Metzler and E, C, F are nonnegative C. Briat [KTH / ] 4/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Stability analysis Quadratic Lyapunov Functions ◮ V ( x ) = x T Px , P = P T ≻ 0 ◮ Enough to pick a diagonal P A T P + PA ≺ 0 ◮ Semidefinite programming, LMIs ◮ Suitable for L 2 -gain analysis ( H ∞ -norm) Copositive linear Lyapunov Functions ◮ V ( x ) = λ T x , λ > 0 λ T A < 0 ◮ Linear programming ◮ Suitable for L 1 - and L ∞ -gain analysis C. Briat [KTH / ] 5/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Induced norms for positive systems ◮ h ( t ) ∈ R q × p : impulse response of the positive system Σ + ◮ Output z = h ∗ w nonnegative when w nonnegative L 1 -norm and L 1 -gain �� � � ∞ � ∞ 1 T w ( s ) ds || w || L 1 := || Σ || L 1 − L 1 := max h ij ( s ) ds j 0 0 i L ∞ -norm and L ∞ -gain � ∞ � || w || L ∞ := ess sup || w ( t ) || ∞ , || Σ || L ∞ − L ∞ := max h ij ( s ) ds i t 0 j || Σ ∗ || L 1 − L 1 = where � A T � C T Σ ∗ = . E T F T C. Briat [KTH / ] 6/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Stability analysis and norm computation C. Briat [KTH / ] 7/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion L 1 -gain analysis Theorem Let ( A, E, C, F ) be an input-output positive system. The following statements are equivalent: 1. The system is asymptotically stable and the L 1 -gain smaller than γ > 0 2. G ( s ) is asymptotically stable and 1 T q G (0) < γ 1 T q 3. G ( s ) is asymptotically stable and 1 T q ( F − CA − 1 E ) < γ 1 T q 4. There exists a vector λ > 0 such that the inequalities a. λ T A + 1 T q C < 0 b. λ T E − γ 1 T p + 1 T q F < 0 hold. Remarks ◮ Linear programming problem ◮ Actual L 1 -gain retrieved by minimizing γ > 0 C. Briat [KTH / ] 8/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion L ∞ -gain analysis Theorem Let ( A, E, C, F ) be an input-output positive system. The following statements are equivalent: 1. The system is asymptotically stable and the L ∞ -gain smaller than γ > 0 2. G ( s ) is asymptotically stable and G (0) 1 p < γ 1 p 3. G ( s ) is asymptotically stable and ( F − CA − 1 E ) 1 p < γ 1 p 4. There exists a vector λ > 0 such that the inequalities a. Aλ + E 1 p < 0 b. Cλ − γ 1 q + F 1 p < 0 hold. Remarks ◮ Linear program ◮ Convenient for control C. Briat [KTH / ] 9/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Robust stability analysis C. Briat [KTH / ] 10/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Uncertain systems and LFT Uncertain system x ( t ) ˙ = A u ( δ ) x ( t ) + E u ( δ ) w 1 ( t ) z 1 ( t ) = C u ( δ ) x ( t ) + F u ( δ ) w 1 ( t ) (3) δ := [0 , 1] N δ ∈ ◮ A u ( δ ) Metzler for all δ ∈ δ ◮ E u ( δ ) , C u ( δ ) and F u ( δ ) nonnegative for all δ ∈ δ Linear Fractional Representation x ( t ) ˙ = Ax ( t ) + E 0 w 0 ( t ) + E 1 w 1 ( t ) z 0 ( t ) = C 0 x ( t ) + F 00 w 0 ( t ) + F 01 w 1 ( t ) (4) z 1 ( t ) = C 1 x ( t ) + F 10 w 0 ( t ) + F 11 w 1 ( t ) w 0 ( t ) = ∆( δ ) z 0 ( t ) ◮ A Metzler ◮ C 0 , C 1 , E 1 , F 01 and F 11 nonnegative C. Briat [KTH / ] 11/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Integral linear constraints ◮ Σ ∈ Σ where Σ is a family of positive operators ◮ z = Σ w for some nonnegative input signal w ◮ Family can be characterized in terms of an Integral Linear Constraint (ILC) � ∞ ϕ T 1 z ( s ) + ϕ T 2 w ( s ) ds ≥ 0 (5) 0 for all z = Σ w , Σ ∈ Σ . ◮ Scaling factors ϕ 1 and ϕ 2 chosen accordingly ◮ Frequency domain interpretation ϕ T z (0) + ϕ T w (0) ≥ 0 1 � 2 � � � � ϕ T 1 � Σ(0) + ϕ T w (0) ≥ 0 � (6) 2 � ϕ T 1 � Σ(0) + ϕ T 2 ≥ 0 ◮ Only ω = 0 is important ◮ Last inequality contains parametric uncertainties only C. Briat [KTH / ] 12/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Examples Constant/Time-varying parameter uncertainty ◮ Parametric uncertainty δ ( t ) ∈ [0 , 1] , t ≥ 0 � ∞ ϕ T (1 − δ ( θ )) w ( θ ) dθ ≥ 0 ⇐ ⇒ ϕ ≥ 0 0 Constant delay operator z ( s ) = e − sh � ◮ z ( t ) = w ( t − h ) , � w ( s ) ϕ T Σ(0) + ϕ T ⇒ ϕ T 1 + ϕ T 1 � 2 ≥ 0 ⇐ 2 ≥ 0 Uncertain positive LTI system ◮ Uncertain asymptotically stable positive transfer function H ∈ H ◮ Static gain H (0) ∈ H 0 ϕ T 1 Z + ϕ T 2 ≥ 0 , Z ∈ H 0 (7) C. Briat [KTH / ] 13/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Robust stability conditions Theorem The uncertain linear positive system is asymptotically stable if there exist λ ∈ R n ++ , ϕ 1 ( δ ) , ϕ 2 ( δ ) ∈ R n 0 and γ > 0 such that the robust linear program λ T A + ϕ 1 ( δ ) T C 0 + 1 T 0 q C 1 < λ T E 0 + ϕ 2 ( δ ) T + ϕ 1 ( δ ) T F 00 + 1 T q F 10 < 0 (8) λ T E 1 − γ 1 T p + ϕ 1 ( δ ) T F 01 + 1 T q F 11 < 0 ϕ 1 ( δ ) T + ϕ 2 ( δ ) T ∆( δ ) ≥ 0 (9) is feasible for all δ ∈ δ . Moreover, in such a case, the L 1 -gain of the transfer from w 1 → z 1 is bounded from above by γ . ◮ Robust linear program ◮ Polynomial dependence → Handelman’s Theorem (preserve linear program structure) C. Briat [KTH / ] 14/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Examples C. Briat [KTH / ] 15/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Example 1 - Positive time-delay system Let us consider a positive time-delay system x ( t ) = Ax ( t ) + Bx ( t − h ) ˙ where A is Metzler and B is nonnnegative. Linear Fractional Transformation x ( t ) ˙ = Ax ( t ) + Bw 0 ( t ) z 0 ( t ) = x ( t ) ∇ h ( z 0 )( t ) w 0 ( t ) = where ∇ h is the constant delay operator with transfer function e − sh . Stability conditions λ T A + ϕ T < 0 λ T B − ϕ T < 0 for some ϕ ∈ R n . Condition equivalent to λ T ( A + B ) < 0 . C. Briat [KTH / ] 16/22
Introduction Stability analysis and norm computation Robust stability analysis Examples Conclusion Example 2 - Positive system with parametric uncertainty (1) Let us consider the positive uncertain system with constant parametric uncertainty δ ∈ [0 , 1] : ( A 0 + δA 1 + δ 2 A 2 ) x ( t ) + ( E 0 + δE 1 + δ 2 E 2 ) w 1 ( t ) x ( t ) ˙ = (10) ( C 0 + δC 1 + δ 2 C 2 ) x ( t ) + ( F 0 + δF 1 + δ 2 F 2 ) w 1 ( t ) z 1 ( t ) = ◮ 3 states, 2 inputs and 2 outputs ϕ 1 ( δ ) ϕ 2 ( δ ) constraints computed L 1 -gain time ϕ 0 ϕ 0 ϕ 0 1 ≥ 0 , ϕ 0 1 + ϕ 0 2 ≥ 0 133.95 2.7844s 1 2 ϕ 1 ϕ 0 ϕ 1 1 = − ϕ 0 1 δ 133.95 3.829s 2 2 ϕ 1 1 δ + ϕ 2 1 δ 2 ϕ 0 2 + ϕ 1 ϕ 1 1 = − ϕ 0 2 , ϕ 2 1 = − ϕ 1 2 δ 94.167 4.2758s 2 Table: L 1 -gain computation of the transfer w 1 → z 1 – Exact L 1 -gain: 92.8358 C. Briat [KTH / ] 17/22
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