Symbolic Control of Incrementally Stable Systems

Transcription

1 Symbolic Control of Incrementally Stable Systems Antoine Girard Laboratoire Jean Kuntzmann, Université Joseph Fourier Grenoble, France Workshop on Formal Verification of Embedded Control Systems LCCC, Lund, April A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 1 / 39

2 Motivation Algorithmic synthesis of controllers from high level specifications: Physical System Specification A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 2 / 39

3 Motivation Algorithmic synthesis of controllers from high level specifications: Physical System = Specification Controller A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 2 / 39

4 Motivation Specifications can be expressed using temporal logic (e.g. LTL): Safety S (Always S) Reachability T (Eventually T ) Stability ( T ) Recurrence ( T ) Sequencing (T 1 T 2 ) Coverage T 1 T 2 Fault recovery (F = R) LTL formula admits an equivalent (Büchi) automaton. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 3 / 39

5 Motivation Algorithmic synthesis of controllers from high level specifications: Physical System: ẋ(t) = f (x(t), u(t)) = Temporal Logic Specif.: Controller A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 4 / 39

6 Motivation Algorithmic synthesis of controllers from high level specifications: Physical System: ẋ(t) = f (x(t), u(t)) = Temporal Logic Specif.: Controller:? The problem is hard because the model and the specification are heterogeneous. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 4 / 39

7 Symbolic Approach to Control Synthesis Approximate symbolic (discrete) model that is formally related to the (continuous) dynamics of the physical system: Physical System: ẋ(t) = f (x(t), u(t)) Symbolic Model: A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 5 / 39

8 Symbolic Approach to Control Synthesis Approximate symbolic (discrete) model that is formally related to the (continuous) dynamics of the physical system: Physical System: ẋ(t) = f (x(t), u(t)) Symbolic Model: Discrete Controller: A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 5 / 39

9 Symbolic Approach to Control Synthesis Approximate symbolic (discrete) model that is formally related to the (continuous) dynamics of the physical system: Physical System: ẋ(t) = f (x(t), u(t)) Symbolic Model: Hybrid Controller: q(t + ) = g(q(t), x(t)) u(t) = k(q(t), x(t)) Refinement Discrete Controller: A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 5 / 39

10 Outline of the Talk 1 Behavioral metrics for discrete and continuous systems Language metric Approximate bisimulation and bisimulation metric 2 Symbolic abstractions of incrementally stable systems Incrementally stable switched systems State-space approaches: from uniform to multi-scale abstractions Input-space approach A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 6 / 39

11 Transition Systems Unified modeling framework of discrete and (sampled) continuous systems. Definition A transition system is a tuple T = (X, U, δ, Y, H, X 0 ) where X is a (discrete or continuous) set of states; U is a (discrete or continuous) set of inputs; δ : X U 2 X is a transition relation; Y is a (discrete or continuous) set of outputs; H : X Y is an ouput map; X 0 X is a set of initial states. The transition system is said to be discrete or symbolic if X and U are countable or finite. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 7 / 39

12 Transition Systems A trajectory of the transition system T is a finite or infinite sequence: s = (x 0, u 0 ), (x 1, u 1 ), (x 2, u 2 )... where x 0 X 0 and x k+1 δ(x k, u k ), k. The associated observed trajectory is o = (y 0, u 0 ), (y 1, u 1 ), (y 2, u 2 )... where y k = H(x k ), k. The set L(T ) of observed trajectories of T is the language of transition system T. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 8 / 39

13 Language Metric Traditional behavioral relationships for transition systems are based on language inclusion or equivalence. For systems observed over metric spaces, the distance between observed trajectories is more natural. Let T i = (X i, U, δ i, Y, H i, Xi 0 ), i {1, 2}, be transition systems with a common set of inputs U and outputs Y equipped with a metric d. For o 1 L(T 1 ), o 2 L(T 2 ), { sup d(y d(o 1, o 2 k 1 ) =, y k 2 ) if u1 k = u2 k, k k + otherwise A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 9 / 39

14 Language Metric Definition The language metric between T 1 and T 2 is given by { } d L (T 1, T 2 ) = max sup inf o 1 L(T 1 ) o 2 L(T 2 ) d(o 1, o 2 ), sup inf o 2 L(T 2 ) o 1 L(T 1 ) d(o 1, o 2 ) The language metric is generally hard to compute: The choice of trajectory o 2 approximating o 1 may require knowledge of the whole trajectory o 1. Easier if the approximating trajectory can be selected transition after transition: Bisimulation equivalence in the traditional setting. Natural extension given by the bisimulation metric. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 10 / 39

15 Approximate Bisimulation Definition Let ε R + 0, a relation R X 1 X 2 is an ε-approximate bisimulation relation if for all (x 1, x 2 ) R : 1 d(h 1 (x 1 ), H 2 (x 2 )) ε; 2 u U, x 1 δ 1(x 1, u), x 2 δ 2(x 2, u), such that (x 1, x 2 ) R; 3 u U, x 2 δ 2(x 2, u), x 1 δ 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 X1 0, there exists x 2 X2 0, such that (x 1, x 2 ) R; 2 For all x 2 X2 0, there exists x 1 X1 0, such that (x 1, x 2 ) R. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 11 / 39

16 Bisimulation Metric Definition The bisimulation metric between T 1 and T 2 is given by d B (T 1, T 2 ) = inf { ε R + 0 T 1 ε T 2 } Fixed-point computation of the bisimulation metric for symbolic systems. For other systems, computation of upper-bounds using the notion of bisimulation functions. Theorem The following inequality holds d L (T 1, T 2 ) d B (T 1, T 2 ). A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 12 / 39

17 A Simple Example T 1 T 2 T 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) Symbolic Control of δ-gas Systems 13 / 39

18 Outline of the Talk 1 Behavioral metrics for discrete and continuous systems Language metric Approximate bisimulation and bisimulation metric 2 Symbolic abstractions of incrementally stable systems Incrementally stable switched systems State-space approaches: from uniform to multi-scale abstractions Input-space approach A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 14 / 39

19 Switched Systems Continuous control systems with finite set of inputs: Definition A switched system is a tuple Σ = (R n, P, F) where: R n is the state space; P = {1,..., m} is the finite set of modes; F = {f p : R n R n p P} is the collection of vector fields. For a switching signal p : R + P, initial state x R n, x(t, x, p) denotes the trajectory of Σ given by: ẋ(t) = f p(t) (x(t)), x(0) = x. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 15 / 39

20 Incremental Stability Asymptotic forgetfulness of past history: Definition The switched system Σ is incrementally globally uniformly asymptotically stable (δ-guas) if there exists a KL function β such that for all initial conditions x 1, x 2 R n, for all switching signals p : R + P, for all t R + : x(t, x 1, p) x(t, x 2, p) β( x 1 x 2, t) t + 0. x(t, x 2, p) x(t, x 1, p) t A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 16 / 39

21 Examples of incrementally stable systems Power converters. Thermal dynamics in buildings. Road traffic. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 17 / 39

22 Lyapunov Characterization Definition V : R n R n R + is a common δ-guas Lyapunov function for Σ if there exist K functions α, α and κ R + such that for all x 1, x 2 R n : α( x 1 x 2 ) V (x 1, x 2 ) α( x 1 x 2 ), p P, V (x 1, x 2 )f p (x 1 ) + V (x 1, x 2 )f p (x 2 ) κv (x 1, x 2 ). x 1 x 2 Theorem If there exists a common δ-guas Lyapunov function, then Σ is δ-guas. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 18 / 39

23 Lyapunov Characterization Definition V : R n R n R + is a common δ-guas Lyapunov function for Σ if there exist K functions α, α and κ R + such that for all x 1, x 2 R n : α( x 1 x 2 ) V (x 1, x 2 ) α( x 1 x 2 ), p P, V (x 1, x 2 )f p (x 1 ) + V (x 1, x 2 )f p (x 2 ) κv (x 1, x 2 ). x 1 x 2 Theorem If there exists a common δ-guas Lyapunov function, then Σ is δ-guas. Supplementary assumption (true if working on a compact subset of R n ): There exists a K function γ such that x 1, x 2, x 3 R n, V (x 1, x 2 ) V (x 1, x 3 ) γ( x 2 x 3 ). A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 18 / 39

24 Switched Systems as Transition Systems Consider a switched system Σ = (R n, P, F) and a time sampling parameter τ > 0. Let T τ (Σ) be the transition system where: the set of states is X = R n ; the set of inputs is U = P; the transition relation is given by x δ(x, p) x = x(τ, x, p); the set of outputs is Y = R n ; the output map H is the identity map over R n ; the set of initial states is X 0 = R n. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 19 / 39

25 Computation of the Symbolic Abstraction We start by approximating the set of states R n by: { [R n ] η = z R n z i = k i 2η n, k i Z, i = 1,..., n where η > 0 is a state sampling parameter: x R n, z [R n ] η, x z η. Approximation of the transition relation = rounding : }, z x(τ, z, p) z A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 20 / 39

26 Approximation Theorem Theorem Let us assume that there exists V : R n R n R + which is a common δ-guas Lyapunov function for Σ. Consider sampling parameters τ, η R + and a desired precision ε R +. If η min { γ 1 ( (1 e κτ )α(ε) ), α 1 (α(ε)) } then, the relation R R n [R n ] η given by R = {(x, z) R n [R n ] η V (x, z) α(ε)} is an ε-approximate bisimulation relation and T τ (Σ) ε T τ,η (Σ). Main idea of the proof: show that accumulation of successive rounding errors is contained by incremental stability. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 21 / 39

27 Comments on the Approximation Theorem Based on sampling (gridding) of time and space: simple to compute. For a given time sampling parameter τ, any precision ε can be achieved by choosing appropriately the state sampling parameter η (the smaller τ or ε, the smaller η). Uniform time and space discretization: excessive computation time and memory consumption. Overcome this problem with multi-scale symbolic abstractions: on-the-fly refinement where fast switching needed, guided by controller synthesis. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 22 / 39

28 Switched Systems in a Multi-Scale Setting Consider a switched system Σ = (R n, P, F), time and scale sampling parameters τ > 0 and N N. We change the control paradigm: the (aperiodic) controller chooses a mode and a duration during which it will be applied. Let T N τ (Σ) be the transition system where: the set of states is X = R n ; the set of inputs is U = P Θ N τ where Θ N τ = {2 s τ s = 0,..., N}; the transition relation is given by x δ(x, (p, 2 s τ)) x = x(2 s τ, x, p); the set of outputs is Y = R n ; the output map H is the identity map over R n ; the set of initial states is X 0 = R n. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 23 / 39

29 Multi-Scale Symbolic Abstraction The set of states R n is approximated by a sequence of embedded lattices Q 0 Q 1... Q N R n with: { Q s = [R n ] 2 s η = z R n 2 s+1 } η z i = k i, k i Z, i = 1,..., n n where η > 0 is a state sampling parameter: Approximation of the transition relation: z z z x(z, p, τ) x(z, p, τ/2) Fine scales reached only by transitions of shorter duration. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 24 / 39

30 Approximation Theorem Theorem Let us assume that there exists V : R n R n R + which is a common δ-guas Lyapunov function for Σ. Consider sampling and scale parameters τ, η R +, N N and a desired precision ε R +. If { η min min s=0...n [ ( )] 2 s γ 1 (1 e κ2 sτ )α(ε), α 1 (α(ε)) then, the relation R R n Q N given by { } R = (x, z) R n Q N V (x, z) α(ε) is an ε-approximate bisimulation relation and T τ (Σ) ε T τ,η (Σ). } A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 25 / 39

31 Controller Synthesis using Multi-Scale Abstractions Multi-scale abstractions are computed on the fly during controller synthesis using depth first search algorithm: Start from initial states: elements of the coarsest lattice. Explore transitions of longer duration first and transitions of shorter duration only if specification cannot be met by transitions of longer durations: fine lattices are explored only when necessary. For safety specifications: notion of maximal lazy safety controller. Tool CoSyMA: Controller Synthesis using Multi-Scale abstractions. multiscale-dcs.gforge.inria.fr A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 26 / 39

32 Example: DC-DC Converter Power converter with switching control: r x l l i l s 2 s 1 Incrementally stable. v s r c r 0 v 0 Safety specification: [1.15, 1.55] [5.45, 5.85]. v c x c Uniform abstraction T τ,η(σ) τ = 0.5, η = , ε = 0.05 Time 9.2s Size (10 3 ) 936 Cont. ratio 93% Multi-scale abstraction Tτ,η N (Σ) N = 6, τ = 32, η = 0.018, ε = 0.05 Time 0.6s Size (10 3 ) 6 Durations 4 (33%), 2 (9%), 1 (50%), 0.5 (8%) Cont. Ratio 92% A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 27 / 39

33 Example: Boost DC-DC Converter Uniform abstraction T τ,η (Σ): A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 28 / 39

34 Example: Boost DC-DC Converter N (Σ): Multiscale abstraction Tτ,η Modes A. Girard (LJK-UJF) Durations Symbolic Control of δ-gas Systems 29 / 39

35 Example: 4 Room Building 4 dimensional thermal model: Incrementally stable. At most one heater on at every instant. Safety specification: [20, 22] 4. Heater 1 Heater 2 Room 1 Room 2 Room 4 Room 3 Heater 4 Heater 3 Multi-scale abstractions Tτ,η(Σ) N N = 4, τ = 80, η = 0.14, ε = 0.2 Time 39s Size (10 3 ) 232 Durations 20 (2%), 10 (91%), 5 (7%) Cont. Ratio 99% A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 30 / 39

36 Example: 4 Room Building Control maps (mode 1): T 3 = 20.5 T 3 = T 4 = T 4 = A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 31 / 39

37 Example: 4 Room Building Control maps (durations): T 3 = 20.5 T 3 = T 4 = T 4 = A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 32 / 39

38 Mode Sequences as Symbolic States State-space approaches suffer from the curse of dimensionality. Alternative: input-space approach Incremental stability = asymptotic forgetfulness of past history, Use mode sequences of given length N, representing the latest applied modes, as symbolic states of symbolic model T τ,n (Σ), The transition relation is given for w = p 1 p 2... p n and p P by w δ(w, p) w = p 2... p n p. The output map is defined for w = p 1 p 2... p n as H(w) = x(nτ, x s, p w ) where p w (t) = p i, t [(i 1)τ, iτ). where x s R n is a source state. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 33 / 39

39 Approximation Theorem Theorem Let us assume that there exists V : R n R n R + which is a common δ-guas Lyapunov function for Σ. Consider time sampling parameter τ R +, sequence length N N and a desired precision ε R +. Let ( ( γ e ε α 1 Nκτ θ(x s ) ) ) 1 e κτ where θ(x s ) = max p P V (x(τ, x s, p), x s ). Then, the relation R R n P N given by { } R = (x, w) R n P N V (x, H(w)) α(ε) is an ε-approximate bisimulation relation between T τ (Σ) and T τ,η (Σ). A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 34 / 39

40 Comments on the Approximation Theorem The source state can be chosen so as to minimize θ(x s ). For a given time sampling parameter τ, any precision ε can be achieved by choosing appropriately the sequence length N. Number of symbolic states grows exponentially with the sequence length N. Asymptotic estimates show that for a given precision ε, the input-space approach leads to a smaller number of symbolic states than the (uniform) state-space approach as soon as ln( P ) κτn. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 35 / 39

41 Example: Road Traffic 5 dimensional model: Incrementally stable. At least one green light. Safety specification: [0, 15] 5. Fairness constraint: red light no longer than 3 time units Sequence length N Size (10 3 ) Precision ε A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 36 / 39

42 Example: Road Traffic Periodic schedule for light coordination: traffic density 16 Maximal density = cell 5 cell 1 cell 2 cell cell applied modes A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 37 / 39

43 Conclusions Approximately bisimilar symbolic abstractions: A rigorous tool for controller synthesis: Synthesized controllers are correct by design. Allow us to leverage efficient algorithmic techniques from discrete systems to continuous and hybrid systems. Computable for interesting classes of systems: switched systems, continuous control systems... Several approaches can help to reduce the computation burden. Ongoing and future work: Tool CoSyMA: Controller Synthesis using Multi-scale Abstractions. Multi-scale input-space approaches. Symbolic models for infinite dimensional systems. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 38 / 39

44 References Behavioral metrics for discrete and continuous systems: G. and Pappas, Approximation metrics for discrete and continuous systems. IEEE TAC, Symbolic abstractions of incrementally stable systems: Pola, G. and Tabuada, Approximately bisimilar symbolic models for nonlinear control systems. Automatica, G., Pola and Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems. IEEE TAC, Camara, G. and Goessler, Safety controller synthesis for switched systems using multi-scale symbolic models. CDC, Mouelhi, G. and Goessler, CoSyMA: a tool for controller synthesis using multi-scale abstractions. HSCC, Le Corronc, G. and Goessler, Mode sequences as symbolic states in abstractions of incrementally stable switched systems. Submitted. A. Girard (LJK-UJF) Symbolic Control of δ-gas Systems 39 / 39