Safety Controller Synthesis for Switched Systems using Multiscale - PowerPoint PPT Presentation

Safety Controller Synthesis for Switched Systems using Multiscale Symbolic Models Antoine Girard Laboratoire des Signaux et Syst` emes Gif sur Yvette, France Workshop on switching dynamics & verification Paris, January 28-29, 2016 A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 1 / 33

Introduction Controller synthesis for a class of continuous-time switched systems Incrementally stable systems: the influence of initial condition asymptotically vanishes. Safety specification: controlled invariance. Approach based on the use of symbolic models Discrete (time and space) approximation of the switched system. Approach based on uniform discretization of time and space. [Girard, Pola and Tabuada, 2010] Distance between trajectories of incrementally stable switched system and of symbolic model is uniformly bounded, and can be made arbitrarily small. Safety controller synthesis using symbolic models via algorithmic discrete controller synthesis. A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 2 / 33

Motivation Limitations of the symbolic control approach Spatial and time resolution must be chosen carefully to achieve a given precision: fast switching requires fine spatial resolution; Uniform spatial discretization: excessive computation time and memory consumption. Overcome this problem with multiscale symbolic models Use of multiscale discretizations of time and space Incremental exploration of symbolic models during controller synthesis: The finer scales explored only if safety cannot be ensured at coarser level. A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 3 / 33

Outline 1 Incrementally stable switched systems 2 Multiscale symbolic models 3 Safety controller synthesis using multiscale symbolic models 4 Computational experiments A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 4 / 33

Switched systems Definition A switched system is a tuple Σ = ( R n , P , P , F ) , where R n is the state space; P = { 1 , . . . , m } is the finite set of modes; P is a subset of S ( R + 0 , P ), the set of functions from R + 0 to P with a finite number of discontinuities on every bounded interval of R + 0 ; F = { f 1 , . . . , f m } is a collection of smooth vector fields indexed by P . For a switching signal p ∈ P , initial state x ∈ R n , x ( ., x , p ) is the trajectory of Σ, solution of: x ( t ) = f p ( t ) ( x ( t )) , x (0) = x . ˙ S τ d ( R + 0 , P ) is the set of switching signals p with minimum dwell-time τ d ∈ R + : discontinuities of p are separated by at least τ d . A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 5 / 33

Incremental stability Definition Σ is incrementally globally uniformly asymptotically stable ( δ -GUAS) if there exists a KL function β such that for all x 1 , x 2 ∈ R n , p ∈ P , t ∈ R + 0 : � x ( t , x 1 , p ) − x ( t , x 2 , p ) � ≤ β ( � x 1 − x 2 � , t ) . x ( t , x 2 , p ) x ( t , x 1 , p ) t A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 6 / 33

Lyapunov characterization Definition V p : R n × R n → R + 0 , p ∈ P are multiple δ -GUAS Lyapunov functions for Σ if there exist κ, µ ∈ R + with µ ≥ 1, K ∞ functions α , α , such that for all x 1 , x 2 ∈ R n , p , p ′ ∈ P : α ( � x 1 − x 2 � ) ≤ V p ( x 1 , x 2 ) ≤ α ( � x 1 − x 2 � ); ∂ V p ∂ x 1 ( x 1 , x 2 ) f p ( x 1 ) + ∂ V p ∂ x 2 ( x 1 , x 2 ) f p ( x 2 ) ≤ − κ V p ( x 1 , x 2 ); V p ( x 1 , x 2 ) ≤ µ V p ′ ( x 1 , x 2 ) . Theorem Let τ d ∈ R + , Σ = ( R n , P , P , F ) with P ⊆ S τ d ( R + 0 , P ) admitting multiple δ -GUAS Lyapunov functions. If τ d > log µ κ , then Σ is δ -GUAS. A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 7 / 33

Additional assumption In the following, we will assume that there exists a K ∞ function γ such that for all x 1 , x 2 , x 3 ∈ R n | V p ( x 1 , x 2 ) − V p ( x 1 , x 3 ) | ≤ γ ( � x 2 − x 3 � ) , ∀ p ∈ P ; This is not restrictive if V p are smooth and we work on a bounded subset of R n . A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 8 / 33

Outline 1 Incrementally stable switched systems 2 Multiscale symbolic models 3 Safety controller synthesis using multiscale symbolic models 4 Computational experiments A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 9 / 33

Transition systems Definition A transition system is a tuple T = ( X , U , Y , ∆ , X 0 ) where X , U , Y , X 0 are the sets of states, inputs, outputs and initial states; ∆ ⊆ X × U × X × Y is a transition relation. T is metric if Y is equipped with a metric d , symbolic if X and U are finite or countable sets. ( x , u , x ′ , y ) ∈ ∆ is denoted ( x ′ , y ) ∈ ∆( x , u ); u ∈ U is enabled at x ∈ X , denoted u ∈ enab ( x ), if ∆( x , u ) � = ∅ ; If enab ( x ) = ∅ , then x is blocking, otherwise it is non-blocking; T is deterministic if for all x ∈ X and u ∈ enab ( x ), | ∆( x , u ) | = 1. A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 10 / 33

Trajectories A trajectory of T is a finite or infinite sequence of transitions σ = ( x 0 , u 0 , y 0 )( x 1 , u 1 , y 1 )( x 2 , u 2 , y 2 ) . . . where ( x i +1 , y i ) ∈ ∆( x i , u i ), for all i ≥ 0. It is: initialized if x 0 ∈ X 0 ; maximal if it is infinite or it is finite and ends in a blocking state. x ∈ X is reachable if there exists an initialized trajectory reaching x . T is non-blocking if all initialized maximal trajectories are infinite or equivalently if all reachable states are non-blocking. A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 11 / 33

Approximate bisimulation Definition Let T i = ( X i , U , Y , ∆ i , X 0 i ), with i = 1 , 2 be metric transition systems with the same sets of inputs U and outputs Y equipped with the metric d . Let ε ∈ R + 0 , R ⊆ X 1 × X 2 is an ε -approximate bisimulation relation between T 1 and T 2 if for all ( x 1 , x 2 ) ∈ R , u ∈ U : ∀ ( x ′ 1 , y 1 ) ∈ ∆ 1 ( x 1 , u ) , ∃ ( x ′ 2 , y 2 ) ∈ ∆ 2 ( x 2 , u ) , d ( y 1 , y 2 ) ≤ ε and ( x ′ 1 , x ′ 2 ) ∈ R ; ∀ ( x ′ 2 , y 2 ) ∈ ∆ 2 ( x 2 , u ) , ∃ ( x ′ 1 , y 1 ) ∈ ∆ 1 ( x 1 , u ) , d ( y 1 , y 2 ) ≤ ε and ( x ′ 1 , x ′ 2 ) ∈ R . T 1 and T 2 are ε -approximately bisimilar, denoted T 1 ∼ ε T 2 , if X 0 1 ⊆ R − 1 ( X 0 2 ) and X 0 2 ⊆ R ( X 0 1 ). A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 12 / 33

Switched systems as transition systems Let Σ τ d = ( R n , P , P , F ) be a switched system with P = S τ d ( R + 0 , P ). We consider controllers that can select: a mode p ∈ P ; 1 a duration θ ∈ Θ N τ during which the mode remains active where 2 Θ N τ = { θ s = 2 − s τ | s = 0 , . . . , N } . where τ ∈ R + , N ∈ N are time sampling and scale parameters. We assume τ d = θ N d for some N d ∈ { 0 , . . . , N } , then Θ N d = { θ s ∈ Θ N τ | θ s ≥ τ d } . τ Let C ( I , R n ) denote the set of continuous functions from I to R n . A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 13 / 33

Switched systems as transition systems Let T N τ (Σ τ d ) = ( X , U , Y , ∆ , X 0 ) where: X = R n × P , z = ( x , p ) ∈ X consists of a continuous state x and an active mode p . U = P × Θ N τ , u = ( p , θ s ) ∈ U consists of a mode p and a duration θ s . Y = � s = N s =0 C ([0 , θ s ] , R n ) is a set of continuous functions, equipped with the metric: � � y − y ′ � ∞ if θ s = θ s ′ d ( y , y ′ ) = + ∞ if θ s � = θ s ′ X 0 = R n × P . A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 14 / 33

Switched systems as transition systems For z = ( x , p ) ∈ X , z ′ = ( x ′ , p ′ ) ∈ X , u = (¯ p , θ s ) ∈ U , y ∈ Y , p , θ s ) ∈ { p } × Θ N τ ∪ ( P \ { p } ) × Θ N d  (¯ τ  x ′ = x ( θ s , x , ¯ p ) and p ′ = ¯ ( z , u , z ′ , y ) ∈ ∆ ⇐ ⇒ p . y = x | θ s ( ., x , ¯ p )  x ′ = x ( θ s , x, ¯ p ) y = x | θ s ( ., x, ¯ p ) x T N τ (Σ τ d ) is deterministic and metric. A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 15 / 33

Computation of the symbolic model We approximate R n by a sequence of embedded multiscale lattices 2 − s +1 η � � � [ R n ] 2 − s η = q ∈ R n � √ n � q [ i ] = k i , k i ∈ Z , i = 1 , ..., n � where η ∈ R + is a state space sampling parameter. η : R n → [ R n ] 2 − s η such that We associate a multiscale quantizer Q s ⇒ q [ i ] − 2 − s η √ n ≤ x [ i ] < q [ i ] + 2 − s η Q s √ n , i = 1 , . . . , n . η ( x ) = q ⇐ Let X s η = [ R n ] 2 − s η × P , then X 0 η ⊆ X 1 η ⊆ · · · ⊆ X N η . η , U , Y , ∆ η , X 0 We define the symbolic model as T N τ,η (Σ τ d ) = ( X N η ). A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 16 / 33

Computation of the symbolic model For r = ( q , p ) ∈ X , r ′ = ( q ′ , p ′ ) ∈ X , u = (¯ p , θ s ) ∈ U , y ∈ Y ,  p , θ s ) ∈ { p } × Θ N τ ∪ ( P \ { p } ) × Θ N d (¯ τ  q ′ = Q s p )) and p ′ = ¯ ( r , u , r ′ , y ) ∈ ∆ ⇐ ⇒ η ( x ( θ s , q , ¯ p . y = x | θ s ( ., q , ¯ p )  q ′ = Q 1 η ( x ( θ 1 , q, ¯ p )) y = x | θ 1 ( ., q, ¯ p )) q A. Girard (L2S-CNRS) Synthesis using multiscale symbolic models 17 / 33