Positive systems analysis via integral linear constraints Sei Zhen Khong 1 , Corentin Briat 2 , and Anders Rantzer 3 1 Institute for Mathematics and its Applications University of Minnesota 2 Department of Biosystems Science and Engineering Swiss Federal Institute of Technology Zürich (ETH Zürich), Switzerland 3 Department of Automatic Control Lund University, Sweden IEEE Conference on Decision and Control 18 Dec 2015 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 1 / 16
Positive systems analysis Quadratic forms are widely used for systems analysis: Lyapunov inequality, Kalman-Yakubovich-Popov Lemma, integral quadratic constraints etc. Analysis can be simplified if systems are known to be positive Lyapunov inequality: ◮ ∃ P ≻ 0 such that A T P + PA ≺ 0 ◮ ∃ z > 0 (element-wise) such that Az < 0 Kalman-Yakubovich-Popov Lemma: � ∗ � ( j ω I − A ) − 1 B � ( j ω I − A ) − 1 B � ∀ ω ∈ [ 0 , ∞ ] Q ≺ 0 ◮ I I ◮ ∃ x , u , p ≥ 0 such that � � � A T � x Ax + Bu ≤ 0 + and Q p ≤ 0 B T u The theory of integral linear constraints (ILCs)? Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 2 / 16
Outline Positive closed-loop systems 1 Robust stability 2 Geometric intuition 3 Example 4 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 3 / 16
Positive closed-loop systems Outline Positive closed-loop systems 1 Robust stability 2 Geometric intuition 3 Example 4 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 3 / 16
Positive closed-loop systems Positive systems A system G is said to be positive if u ( t ) ≥ 0 ∀ t ≥ 0 = ⇒ y ( t ) = ( Gu )( t ) ≥ 0 ∀ t ≥ 0 d 2 y 1 u 1 G 1 ! d 1 y 2 u 2 G 2 ! Given a positive feedback interconnection of two positive systems G 1 and G 2 , is the closed-loop map ( d 1 , d 2 ) �→ ( u 1 , y 1 , u 2 , y 2 ) always positive? No ! Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 4 / 16
Positive closed-loop systems Positive systems A system G is said to be positive if u ( t ) ≥ 0 ∀ t ≥ 0 = ⇒ y ( t ) = ( Gu )( t ) ≥ 0 ∀ t ≥ 0 d 2 y 1 u 1 G 1 ! d 1 y 2 u 2 G 2 ! Given a positive feedback interconnection of two positive systems G 1 and G 2 , is the closed-loop map ( d 1 , d 2 ) �→ ( u 1 , y 1 , u 2 , y 2 ) always positive? No ! Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 4 / 16
Positive closed-loop systems Positive systems A simple counterexample: ! d 2 = 0 y 1 u 1 2 d 1 y 2 u 2 ! 1 1 d 1 �→ u 1 = 1 − 2 = − 1 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 5 / 16
Positive closed-loop systems Feedback interconnections d 2 y 1 u 1 G 1 ! d 1 y 2 u 2 G 2 ! G 1 ( s ) = C 1 ( sI − A 1 ) − 1 B 1 + D 1 ˆ G 2 ( s ) = C 2 ( sI − A 2 ) − 1 B 2 + D 2 ˆ A 1 and A 2 are Metzler and B 1 ≥ 0 , B 2 ≥ 0 , C 1 ≥ 0 , C 2 ≥ 0 , D 1 ≥ 0 , and D 2 ≥ 0 (element-wise) implies G 1 and G 2 are positive Positivity of closed-loop map [Ebihara et. al. 2011] If ρ ( D 1 D 2 ) < 1 , then ( d 1 , d 2 ) �→ ( u 1 , y 1 , u 2 , y 2 ) is positive Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 6 / 16
Positive closed-loop systems Feedback interconnections d 2 u 1 y 1 G 1 ! d 1 y 2 u 2 G 2 ! Suppose (nonlinear) G i : L 1 e → L 1 e are causal and positive, define � P T +∆ T ( G i x − G i y ) � 1 α ( G i ) := sup inf sup � P T +∆ T ( x − y ) � 1 ∆ T > 0 T > 0 x , y ∈ L 1 e ; PT x = PT y PT +∆ T ( x − y ) � = 0 Positivity of closed-loop map If α ( G 1 ) α ( G 2 ) < 1 , then ( d 1 , d 2 ) �→ ( u 1 , y 1 , u 2 , y 2 ) is positive Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 7 / 16
Robust stability Outline Positive closed-loop systems 1 Robust stability 2 Geometric intuition 3 Example 4 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 7 / 16
Robust stability Robust stability of feedback systems d 2 y 1 u 1 G 1 ! d 1 y 2 u 2 G 2 ! Integral quadratic constraints (IQCs) [Megretski & Rantzer 97] Given bounded, causal G 1 : L 2 e → L 2 e and G 2 : L 2 e → L 2 e , suppose there exists linear Π : L 2 → L 2 such that [ τ G 1 , G 2 ] is well-posed for all τ ∈ [ 0 , 1 ] ; �� u � � � ∞ v ( t ) T (Π v )( t ) dt ≥ 0 ∀ v ∈ G ( τ G 1 ) := ∈ L 2 : y = τ G 1 u , τ ∈ [ 0 , 1 ] ; 0 y � ∞ � ∞ | w ( t ) | 2 dt w ( t ) T (Π w )( t ) dt ≤ − ǫ ∀ w ∈ G ′ ( G 2 ) , 0 0 then [ G 1 , G 2 ] is stable Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 8 / 16
Robust stability Integral quadratic constraint (IQC) examples Structure of G 1 Π Condition � 0 � I G 1 is passive I 0 � x ( j ω ) I � 0 � G 1 � ≤ 1 x ( j ω ) ≥ 0 0 − x ( j ω ) I � X ( j ω ) Y ( j ω ) � X = X ∗ ≥ 0 , Y = − Y ∗ G 1 ∈ [ − 1 , 1 ] Y ( j ω ) ∗ − X ( j ω ) � X � Y X = X ∗ ≥ 0 , Y = − Y ∗ G 1 ( t ) ∈ [ − 1 , 1 ] Y T − X G 1 ( s ) = e − θ s − 1 , � x ( j ω ) ρ ( ω ) 2 � 0 ρ ( ω ) = 2 max sin ( θω/ 2 ) for θ ∈ [ 0 , θ 0 ] − x ( j ω ) 0 | θ |≤ θ 0 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 9 / 16
Robust stability Robust stability of positive feedback systems y 1 d 2 u 1 G 1 ! d 1 y 2 u 2 G 2 ! Integral linear constraints 1 e → L p 1 e and G 2 : L p Given bounded, causal, linear G 1 : L m 1 e → L m 1 e , suppose there exists Π ∈ R 1 × m + p such that [ τ G 1 , G 2 ] is well-posed and positive for all τ ∈ [ 0 , 1 ] ; �� u � � � ∞ Π v ( t ) dt ≥ 0 ∀ v ∈ G + ( τ G 1 ) := ∈ L 1 + : y = τ G 1 u , τ ∈ [ 0 , 1 ] ; 0 y � ∞ � ∞ ∀ w ∈ G ′ Π w ( t ) dt ≤ − ǫ | w ( t ) | dt + ( G 2 ) , 0 0 then [ G 1 , G 2 ] is stable When G 1 and G 2 are LTI, conditions can be stated as � ˆ � � � I G 2 ( 0 ) Π ≥ 0 and Π < 0 τ ˆ G 1 ( 0 ) I Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 10 / 16
Geometric intuition Outline Positive closed-loop systems 1 Robust stability 2 Geometric intuition 3 Example 4 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 10 / 16
Geometric intuition Geometric interpretation of integral quadratic constrains G ( G 1 ) G ′ ( G 2 ) Feedback stability G ( G 1 ) + G ′ ( G 2 ) = L 2 ; G ( G 1 ) ∩ G ′ ( G 2 ) = { 0 } Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 11 / 16
Geometric intuition Geometric interpretation of integral quadratic constraints G ( G 1 ) G ′ ( G 2 ) Integral quadratic constraints (IQCs) � ∞ v ( t ) T (Π v )( t ) dt ≥ 0 ∀ v ∈ G ( G 1 ) ; 0 � ∞ � ∞ | w ( t ) | 2 dt ∀ w ∈ G ′ ( G 2 ) w ( t ) T (Π w )( t ) dt ≤ − ǫ 0 0 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 12 / 16
Geometric intuition Geometric interpretation of integral linear constraints G + ( G 1 ) G ′ + ( G 2 ) Feedback stability G + ( G 1 ) + G ′ + ( G 2 ) = L 1 + ; G + ( G 1 ) ∩ G ′ + ( G 2 ) = { 0 } Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 13 / 16
Geometric intuition Geometric interpretation of integral linear constraints G + ( G 1 ) G ′ + ( G 2 ) Integral linear constraints � ∞ Π v ( t ) dt ≥ 0 ∀ v ∈ G + ( G 1 ) ; 0 � ∞ � ∞ ∀ w ∈ G ′ Π w ( t ) dt ≤ − ǫ | w ( t ) | dt + ( G 2 ) 0 0 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 14 / 16
Example Outline Positive closed-loop systems 1 Robust stability 2 Geometric intuition 3 Example 4 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 14 / 16
Example LTI systems d 2 u 1 y 1 G 1 ! d 1 y 2 u 2 G 2 ! G 1 ( s ) = C 1 ( sI − A 1 ) − 1 B 1 + D 1 ˆ G 2 ( s ) = C 2 ( sI − A 2 ) − 1 B 2 + D 2 ˆ A 1 and A 2 are Metzler, Hurwitz and B 1 ≥ 0 , B 2 ≥ 0 , C 1 ≥ 0 , C 2 ≥ 0 , D 1 ≥ 0 , and D 2 ≥ 0 Robust stability [Ebihara et. al. 2011] [Tanaka et. al. 2013] If ρ (ˆ G 1 ( 0 )ˆ G 2 ( 0 )) < 1 , then [ G 1 , G 2 ] is stable Can be recovered with integral linear constraint theorem with Π := z T � ˆ � G 1 ( 0 ) − I , where z T (ˆ G 1 ( 0 )ˆ G 2 ( 0 ) − I ) < 0 Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 15 / 16
Example Conclusions: Sufficient condition for positivity to be preserved under feedback Developed integral linear constraints theory for analysis of feedback interconnections with positive closed-loop mappings Many extensions possible: ◮ Positive coprime factorisations ◮ Integral linear constraints with time-varying multipliers ◮ LMI conditions for verifying integral linear constraints ◮ Stabilisation of open-loop unstable dynamics? Khong, Briat, Rantzer (UMN, ETH, Lund) Integral linear constraints CDC 12/18/2015 16 / 16
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