Stability of networks of infinite-dimensional systems Andrii - PowerPoint PPT Presentation

Stability of networks of infinite-dimensional systems Andrii Mironchenko Faculty of Mathematics and Computer Science University of Passau IFAC World Congress 2020 Pre-Conference Workshop Input-to-state stability and control of infinite-dimensional systems 11 July 2020 www.mironchenko.com

Motivation: Large-scale systems Emerging technologies such as 5G, IoT, Clouds, make the networks larger and larger. Components may belong to different system classes The size is either fixed or unknown Safety and reliability need to be analytically verified Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 2 / 29

Under which conditions an interconnection of stable systems is stable? Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 3 / 29

Under which conditions an interconnection of stable systems is stable? Outline 1 part of the talk: couplings of 2 systems 2 part of the talk: infinite networks Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 3 / 29

Class of systems ˙ x ( t ) = Ax ( t ) + f ( x ( t ) , u ( t )) , x ( t ) ∈ D ( A ) ⊂ X . X = State space U = PC ( R + , U ) Ax = lim t → + 0 1 t ( T ( t ) x − x ) . x ∈ C ([ 0 , T ] , X ) is a mild solution iff � t x ( t ) = T ( t ) x 0 + T ( t − s ) f ( x ( s ) , u ( s )) ds . 0 Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 4 / 29

Comparison functions γ ( s ) ✻ γ ∈ K ∞ ✲ s 0 β ∈ KL � �� � β ( s , · ) β ( · , r ) ✻ ✻ K ✲ ✲ s r 0 0 Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 5 / 29

Input-to-state stability Definition (Sontag, 1989, for ODEs) ISS : ⇔ � x ( t ) � X ≤ β ( � x � X , t ) + γ ( � u � U ) , ∀ x , t , u . increasing in � x � X decreasing to 0 in t γ ( 0 ) = 0, increasing asymptotic gain β ( � x � X , t ) + γ ( � u � U ) β ( � x � X , t ) β ( � x � X , t ) � x ( t ) � X � x ( t ) � X γ ( � u � U ) t t (a) u ≡ 0 (b) u �≡ 0 Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 6 / 29

Integral input-to-state stability Definition (GAS uniform w.r.t. state (0-UGAS)) 0-UGAS : ⇔ ∃ β ∈ KL : ∀ x ∈ X , ∀ t ≥ 0 � φ ( t , x , 0 ) � X ≤ β ( � x � X , t ) . Definition (Integral input-to-state stability (iISS)) iISS : ⇔ ∃ β ∈ KL , θ, µ ∈ K : ∀ t ≥ 0, ∀ x ∈ X , ∀ u ∈ U � � t � � φ ( t , x , u ) � X ≤ β ( � x � X , t ) + θ µ ( � u ( s ) � U ) ds . 0 Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 7 / 29

Integral input-to-state stability Definition (GAS uniform w.r.t. state (0-UGAS)) 0-UGAS : ⇔ ∃ β ∈ KL : ∀ x ∈ X , ∀ t ≥ 0 � φ ( t , x , 0 ) � X ≤ β ( � x � X , t ) . Definition (Integral input-to-state stability (iISS)) iISS : ⇔ ∃ β ∈ KL , θ, µ ∈ K : ∀ t ≥ 0, ∀ x ∈ X , ∀ u ∈ U � � t � � φ ( t , x , u ) � X ≤ β ( � x � X , t ) + θ µ ( � u ( s ) � U ) ds . 0 Overview of the infinite-dimensional ISS theory Karafyllis, Krstic. Input-to-state stability for PDEs. Springer, 2019. M., Prieur. Input-to-state stability of infinite-dimensional systems: recent results and open questions. To appear in SIAM Review , 2020. Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 7 / 29

Lyapunov functions ˙ x ( t ) = Ax ( t ) + f ( x ( t ) , u ( t )) . Definition V : X → R + is an iISS-Lyapunov function iff ∃ ψ 1 , ψ 2 ∈ K ∞ and σ, α ∈ K : ψ 1 ( � x � X ) ≤ V ( x ) ≤ ψ 2 ( � x � X ) ˙ V u ( x ) ≤ − α ( V ( x )) + σ ( � u ( 0 ) � U ) , 1 ˙ V u ( x ) = lim t ( V ( φ ( t , x , u )) − V ( x )) . t → + 0 α ∈ K ∞ ⇒ V is an ISS-Lyapunov function. Theorem ∃ ISS/iISS Lyapunov function ⇒ ISS/iISS. Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 8 / 29

Lyapunov functions ˙ x ( t ) = Ax ( t ) + f ( x ( t ) , u ( t )) . Definition V : X → R + is an iISS-Lyapunov function iff ∃ ψ 1 , ψ 2 ∈ K ∞ and σ, α ∈ K : ψ 1 ( � x � X ) ≤ V ( x ) ≤ ψ 2 ( � x � X ) ˙ V u ( x ) ≤ − α ( V ( x )) + σ ( � u ( 0 ) � U ) , 1 ˙ V u ( x ) = lim t ( V ( φ ( t , x , u )) − V ( x )) . t → + 0 α ∈ K ∞ ⇒ V is an ISS-Lyapunov function. Theorem ∃ ISS/iISS Lyapunov function ⇒ ISS/iISS. What about coupled systems? Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 8 / 29

Interconnections of two iISS systems � ˙ Σ 1 : x 1 = A 1 x 1 + f 1 ( x 1 , x 2 , u ) , x 1 ∈ X 1 Σ : ˙ Σ 2 : x 2 = A 2 x 2 + f 2 ( x 1 , x 2 , u ) , x 2 ∈ X 2 iISS-LF for Σ i V i : X i → R + is iISS-Lyapunov functions for Σ i , i = 1 , 2 iff ˙ V 1 ( x 1 ) ≤ − α 1 ( � x 1 � X 1 ) + σ 1 ( � x 2 � X 2 ) + κ 1 ( � u ( 0 ) � U ) , ˙ V 2 ( x 2 ) ≤ − α 2 ( � x 2 � X 2 ) + σ 2 ( � x 1 � X 1 ) + κ 2 ( � u ( 0 ) � U ) , Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 9 / 29

Interconnections of two iISS systems � ˙ Σ 1 : x 1 = A 1 x 1 + f 1 ( x 1 , x 2 , u ) , x 1 ∈ X 1 Σ : ˙ Σ 2 : x 2 = A 2 x 2 + f 2 ( x 1 , x 2 , u ) , x 2 ∈ X 2 iISS-LF for Σ i V i : X i → R + is iISS-Lyapunov functions for Σ i , i = 1 , 2 iff ˙ V 1 ( x 1 ) ≤ − α 1 ( � x 1 � X 1 ) + σ 1 ( � x 2 � X 2 ) + κ 1 ( � u ( 0 ) � U ) , ˙ V 2 ( x 2 ) ≤ − α 2 ( � x 2 � X 2 ) + σ 2 ( � x 1 � X 1 ) + κ 2 ( � u ( 0 ) � U ) , Lyapunov gains α ⊖ gain Σ 2 → Σ 1 := 1 ◦ σ 1 α ⊖ gain Σ 1 → Σ 2 := 2 ◦ σ 2 � ω − 1 ( s ) , if s ∈ Im ω ω ⊖ ( s ) := + ∞ , otherwise Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 9 / 29

Small-gain theorem for 2 interconnected iISS systems Theorem (A. Mironchenko, H. Ito, SICON, 2015) Let: V i ( x i ) = ψ i ( � x i � X i ) ∃ c > 1 : ∀ s ∈ R + : α ⊖ ◦ α ⊖ 1 ◦ c σ 1 2 ◦ c σ 2 ( s ) ≤ s . � �� � � �� � ≈ gain Σ 2 → Σ 1 ≈ gain Σ 1 → Σ 2 ⇒ Σ is iISS. If additionally α i ∈ K ∞ for i = 1 , 2 ⇒ Σ is ISS. � V 1 ( x 1 ) � V 2 ( x 2 ) V ( x ) = λ 1 ( s ) ds + λ 2 ( s ) ds . iISS-LF: 0 0 M., Ito. Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach. SICON, 2015. Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 10 / 29

Example  ∂ t ( l , t ) = ∂ 2 x 1 ∂ x 1 ∂ l 2 ( l , t ) + x 1 ( l , t ) x 4 2 ( l , t ) ,       x 1 ( 0 , t ) = x 1 ( π, t ) = 0 ;  � 1 � � � 2  x 2 ∂ t = ∂ 2 x 2 2  ∂ x 2 ∂ x 2 ∂ l 2 + ax 2 − bx 2 +  1 ,  ∂ l 1 + x 2   1  x 2 ( 0 , t ) = x 2 ( π, t ) = 0 . Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 11 / 29

Example  ∂ t ( l , t ) = ∂ 2 x 1 ∂ x 1 ∂ l 2 ( l , t ) + x 1 ( l , t ) x 4 2 ( l , t ) ,       x 1 ( 0 , t ) = x 1 ( π, t ) = 0 ;  � 1 � � � 2  x 2 ∂ t = ∂ 2 x 2 2  ∂ x 2 ∂ x 2 ∂ l 2 + ax 2 − bx 2 +  1 ,  ∂ l 1 + x 2   1  x 2 ( 0 , t ) = x 2 ( π, t ) = 0 . For what a , b is this system UGAS? Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 11 / 29

Example  ∂ t ( l , t ) = ∂ 2 x 1 ∂ x 1 ∂ l 2 ( l , t ) + x 1 ( l , t ) x 4 2 ( l , t ) ,       x 1 ( 0 , t ) = x 1 ( π, t ) = 0 ;  � 1 � � � 2  x 2 ∂ t = ∂ 2 x 2 2  ∂ x 2 ∂ x 2 ∂ l 2 + ax 2 − bx 2 +  1 ,  ∂ l 1 + x 2   1  x 2 ( 0 , t ) = x 2 ( π, t ) = 0 . For what a , b is this system UGAS? X 2 := H 1 X 1 := L 2 ( 0 , π ) 0 ( 0 , π ) Strategy x 1 -subsystem is iISS 1 x 2 -subsystem is ISS 2 Interconnection is UGAS 3 Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 11 / 29

x 1 -subsystem is iISS � � 1 + � x 1 � 2 iISS-LF for Σ 1 : V 1 ( x 1 ) := ln L 2 ( 0 ,π ) 2 � x 1 � 2 ˙ L 2 ( 0 ,π ) + 8 � x 2 � 4 Lie derivative of V 1 : V 1 ( x 1 ) ≤ − H 1 1 + � x 1 � 2 0 ( 0 ,π ) L 2 ( 0 ,π ) � �� � � �� � σ 1 ( � x 2 � H 1 0 ( 0 ,π ) ) α 1 ( � x 1 � L 2 ( 0 ,π ) ) Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 12 / 29

x 1 -subsystem is iISS � � 1 + � x 1 � 2 iISS-LF for Σ 1 : V 1 ( x 1 ) := ln L 2 ( 0 ,π ) 2 � x 1 � 2 ˙ L 2 ( 0 ,π ) + 8 � x 2 � 4 Lie derivative of V 1 : V 1 ( x 1 ) ≤ − H 1 1 + � x 1 � 2 0 ( 0 ,π ) L 2 ( 0 ,π ) � �� � � �� � σ 1 ( � x 2 � H 1 0 ( 0 ,π ) ) α 1 ( � x 1 � L 2 ( 0 ,π ) ) x 2 -subsystem is ISS � � 2 � π ∂ x 2 dl = � x 2 � 2 ISS-LF for Σ 2 : V 2 ( x 2 ) = H 1 0 ∂ l 0 ( 0 ,π ) Lie derivative of V 2 : � � � x 1 � 2 0 ( 0 ,π ) − 2 b + π � � L 2 ( 0 ,π ) ˙ 1 − a − ω � x 2 � 2 3 π � x 2 � 4 V 2 ≤ − 2 H 1 H 1 2 0 ( 0 ,π ) 1 + � x 1 � 2 ω L 2 ( 0 ,π ) � �� � � �� � α 2 ( � x 2 � H 1 0 ( 0 ,π ) ) σ 2 ( � x 1 � L 2 ( 0 ,π ) ) Andrii Mironchenko Stability of networks of infinite-dimensional systems IFAC WC, 2020 12 / 29