Computational Approaches to Analysis and Control of Hybrid Systems - PowerPoint PPT Presentation

Computational Approaches to Analysis and Control of Hybrid Systems Antoine Girard Laboratoire Jean Kuntzmann Universit´ e Joseph Fourier, Grenoble Habilitation ` a Diriger des Recherches Universit´ e de Grenoble 19 Novembre 2013 A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 1 / 43

Research activity Positions: 01-04: PhD in applied mathematics, Laboratoire de Mod´ elisation et Calcul, Grenoble. 04-06: Postdoctoral researcher University of Pennsylvania, Dept. of ESE, Philadelphia (10/04-12/05); Verimag, Grenoble (01/06-08/06). 06-...: Maˆ ıtre de conf´ erences , Universit´ e Joseph Fourier, Research appointment at Laboratoire Jean Kuntzmann. Publications, projects and research supervision: 20 journal and 49 conference papers; 6 funded local or national research projects; 6 PhD students and 5 postdoctoral researchers. A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 2 / 43

Hybrid systems Dynamical systems with continuous and discrete behaviors. In the physical sciences: Models are traditionally continuous (ODE, PDE...); Numerous sources of hybrid behaviors: mechanical impacts, electrical diodes, biological switches... In computer science: Models of reactive systems are discrete (Automata, DEDS...); Models become hybrid when reactive systems are subject to timing constraints (Timed Automata) or are interacting with the physical world (Hybrid Automata). Theory of hybrid systems has rapidly grown since the 90s at the interface of computer science and control theory. A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 3 / 43

Hybrid systems Computation & Control Automata Differential equations Logics Stability Abstraction Robustness Model checking Lyapunov functions A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 4 / 43

Outline of the talk 1 Approximation metrics for discrete and continuous systems Approximate bisimulation Symbolic approach to controller synthesis 2 Reachability analysis of hybrid systems Approximation of reachable sets of linear systems Efficient set representation and algorithm A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 5 / 43

Systems approximation In control theory, approximation is often captured through metrics e.g. between transfer functions ( H 2 , H ∞ , Hankel norms): Mostly for linear systems; Extensions to hybrid behaviors not straightforward. In computer science, approximation is characterized by behavioral relationships (language inclusion or equivalence, bi-simulation): Extended to continuous and hybrid systems... For these systems with real-valued variables, metrics seem to be more suitable. Our contribution: a new approximation framework, based on metrics and capturing these behavioral relationships as special cases. [TAC 2007, Axelby Award 2009]. A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 6 / 43

Transition systems Abstract description of discrete, continuous or hybrid dynamical systems. Definition A transition system T = ( X , U , S , X 0 , Y , O ) is given by a set of states X ; a set of inputs U ; a transition relation S : X × U → 2 X ; a set of initial states X 0 ⊆ X ; a set of outputs Y ; an ouput map O : X → Y . A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 7 / 43

Trajectories A state trajectory of the transition system T is a sequence: s = x 0 , u 0 , x 1 , u 1 . . . where x 0 ∈ X 0 , x k +1 ∈ S ( x k , u k ) , ∀ k . The associated output trajectory is o = y 0 , u 0 , y 1 , u 1 , y 2 , u 2 . . . where y k = O ( x k ) , ∀ k . The set L ( T ) of observed trajectories is the language of T . If Y is equipped with a metric d , we can define the distance between two output trajectories o 1 and o 2 as: d ( y 1 k , y 2 if u 1 k = u 2 � sup k ) k , ∀ k d ( o 1 , o 2 ) = k + ∞ otherwise A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 8 / 43

Language metric Let T i = ( X i , U , S i , X 0 i , Y , O i ), i ∈ { 1 , 2 } , be transition systems with a common set of inputs U and outputs Y equipped with a metric d . Definition The language metric between T 1 and T 2 is the Hausdorff distance between L ( T 1 ) and L ( T 2 ): � � o 2 ∈ L ( T 2 ) d ( o 1 , o 2 ) , o 1 ∈ L ( T 1 ) d ( o 1 , o 2 ) d L ( T 1 , T 2 ) = max sup inf sup inf . o 1 ∈ L ( T 1 ) o 2 ∈ L ( T 2 ) Language metric = matching game on trajectories; Non anticipative matching strategies lead to bisimulation metric. A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 9 / 43

Approximate bisimulation Definition Let ε ≥ 0, a relation R ⊆ X 1 × X 2 is an ε -approximate bisimulation relation if for all ( x 1 , x 2 ) ∈ R : 1 d ( O 1 ( x 1 ) , O 2 ( x 2 )) ≤ ε ; 2 ∀ u ∈ U , ∀ x ′ 1 ∈ S 1 ( x 1 , u ), ∃ x ′ 2 ∈ S 2 ( x 2 , u ), such that ( x ′ 1 , x ′ 2 ) ∈ R ; 3 ∀ u ∈ U , ∀ x ′ 2 ∈ S 2 ( x 2 , u ), ∃ x ′ 1 ∈ S 1 ( x 1 , u ), such that ( x ′ 1 , x ′ 2 ) ∈ R . Definition T 1 and T 2 are ε -approximately bisimilar ( T 1 ∼ ε T 2 ) if : 1 For all x 1 ∈ X 0 1 , there exists x 2 ∈ X 0 2 , such that ( x 1 , x 2 ) ∈ R ; 2 For all x 2 ∈ X 0 2 , there exists x 1 ∈ X 0 1 , such that ( x 1 , x 2 ) ∈ R . A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 10 / 43

Bisimulation metric Definition The bisimulation metric between T 1 and T 2 is given by d B ( T 1 , T 2 ) = inf { ε ≥ 0 | T 1 ∼ ε T 2 } . Characterization of bisimulation metric using Lyapunov-like functions called bisimulation functions . Theorem The following inequality holds (equality for deterministic systems): d L ( T 1 , T 2 ) ≤ d B ( T 1 , T 2 ) . A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 11 / 43

A simple example T 1 T 2 T 3 0 0 0 1 1 1 1 2 4 2 4 3 d L ( T 1 , T 2 ) = 0 , d B ( T 1 , T 2 ) = 2 . d L ( T 1 , T 3 ) = 1 , d B ( T 1 , T 3 ) = 1 . A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 12 / 43

Applications of approximate bisimulation Model reduction of continuous and hybrid systems; [Automatica 2007, DEDS 2007]. Reported applications in biology, mechanics, multi-agent robotics... Hierarchical hybrid control of linear and differentially flat systems; [HSCC 2007, Automatica 2009, ECC 2013] . Trajectory based verification; [HSCC 2006, FORMATS 2006, TECS 2012] . Symbolic approaches to controller synthesis. A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 13 / 43

Symbolic approaches to controller synthesis Controller synthesis using a discrete system ( symbolic model ) which is approximately bisimilar to the dynamics of the physical system: Physical System: Symbolic Model: ≈ x ( t ) = f ( x ( t ) , u ( t )) ˙ A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 14 / 43

Symbolic approaches to controller synthesis Controller synthesis using a discrete system ( symbolic model ) which is approximately bisimilar to the dynamics of the physical system: Physical System: Symbolic Model: x ( t ) = f ( x ( t ) , u ( t )) ˙ Discrete Controller: A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 14 / 43

Symbolic approaches to controller synthesis Controller synthesis using a discrete system ( symbolic model ) which is approximately bisimilar to the dynamics of the physical system: Physical System: Symbolic Model: ≈ x ( t ) = f ( x ( t ) , u ( t )) ˙ Hybrid Controller: Discrete Controller: q ( t + ) = g ( q ( t ) , x ( t )) u ( t ) = k ( q ( t ) , x ( t )) Refinement A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 14 / 43

Switched systems We consider a switched system Σ: x ( t ) = f p ( t ) ( x ( t )) , x ( t ) ∈ R n , p ( t ) ∈ P = { 1 , . . . , m } . ˙ Let τ > 0, the sampled dynamics of Σ is described by transition system T τ (Σ) = ( R n , P , S , R n , R n , id) where the transition relation is given by x ′ = S ( x , p ) ⇐ ⇒ x ′ = x ( τ ) , where ˙ x ( t ) = f p ( x ( t )) , x (0) = x . A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 15 / 43

Computation of the symbolic model We start by approximating the set of states R n by: � � 2 η � [ R n ] η = q ∈ R n � √ n , k i ∈ Z , i = 1 , ..., n � q i = k i , � where η > 0 is a state sampling parameter. Approximation of the transition relation by quantization: S a ( q, p ) S ( q, p ) A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 16 / 43

Computation of the symbolic model In general, the computed symbolic model T τ,η (Σ) and T τ (Σ) are not approximately bisimilar. It holds when Σ satisfies an incremental stability property: intuitively, asymptotic forgetfulness of past history. x ( t , x 2 , p ) x ( t , x 1 , p ) t Incremental global uniform asymptotical stability ( δ -GUAS) of switched systems can be proved using a common δ -GAS Lyapunov function . A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 17 / 43

Approximation theorem Theorem Let V : R n × R n → R + 0 be a common δ -GUAS Lyapunov function for Σ . Consider time sampling parameter τ > 0 and a desired precision ε > 0 , then there exists a state sampling parameter η > 0 such that T τ (Σ) ∼ ε T τ,η (Σ) . The ε -approximate bisimulation relation is given by R = { ( x , q ) ∈ R n × [ R n ] η | V ( x , q ) ≤ α ( ε ) } . Explicit relation between τ , ε , and η . Main idea of the proof: show that accumulation of errors due to successive quantizations is contained by incremental stability. A. Girard (LJK-UJF) Analysis and Control of Hybrid Systems 18 / 43