A Symbolic Approach to Control via Approximate Bisimulations

Transcription

1 A Symolic Approch to Control vi Approximte Bisimultions Antoine Girrd Lortoire Jen Kuntzmnn, Université Joseph Fourier Grenole, Frnce Interntionl Symposium on Innovtive Mthemticl Modelling Tokyo, Jpn, Mrch 1st 2011 A. Girrd (LJK-UJF) A Symolic Approch to Control 1 / 31

2 Motivtion Algorithmic synthesis of controllers from high level specifictions: Physicl System Specifiction A. Girrd (LJK-UJF) A Symolic Approch to Control 2 / 31

3 Motivtion Algorithmic synthesis of controllers from high level specifictions: Physicl System = Specifiction Controller A. Girrd (LJK-UJF) A Symolic Approch to Control 2 / 31

4 Motivtion Specifictions cn e expressed using temporl logic (e.g. LTL): Sfety S (Alwys S) Rechility T (Eventully T ) Stility ( T ) Recurrence ( T ) Sequencing (T 1 T 2 ) Coverge T 1 T 2 Fult recovery (F = R) LTL formul dmits n equivlent (Büchi) utomton. A. Girrd (LJK-UJF) A Symolic Approch to Control 3 / 31

5 Motivtion Algorithmic synthesis of controllers from high level specifictions: Physicl System: ẋ(t) = f (x(t), u(t)) = Temporl Logic Specif.: Controller A. Girrd (LJK-UJF) A Symolic Approch to Control 4 / 31

6 Motivtion Algorithmic synthesis of controllers from high level specifictions: Physicl System: ẋ(t) = f (x(t), u(t)) = Temporl Logic Specif.: Controller:? The prolem is hrd ecuse the model nd the specifiction re heterogeneous. A. Girrd (LJK-UJF) A Symolic Approch to Control 4 / 31

7 Symolic Approch to Control Synthesis Approximte symolic (discrete) model tht is formlly relted to the (continuous) dynmics of the physicl system: Physicl System: ẋ(t) = f (x(t), u(t)) Symolic Model: A. Girrd (LJK-UJF) A Symolic Approch to Control 5 / 31

8 Symolic Approch to Control Synthesis Approximte symolic (discrete) model tht is formlly relted to the (continuous) dynmics of the physicl system: Physicl System: ẋ(t) = f (x(t), u(t)) Symolic Model: Discrete Controller: A. Girrd (LJK-UJF) A Symolic Approch to Control 5 / 31

9 Symolic Approch to Control Synthesis Approximte symolic (discrete) model tht is formlly relted to the (continuous) dynmics of the physicl system: Physicl System: ẋ(t) = f (x(t), u(t)) Symolic Model: Hyrid Controller: q(t + ) = g(q(t), x(t)) u(t) = k(q(t), x(t)) Refinement Discrete Controller: A. Girrd (LJK-UJF) A Symolic Approch to Control 5 / 31

10 Outline of the Tlk 1 Approximtion reltionships for discrete nd continuous systems 2 Symolic strctions of switched systems 3 Controller synthesis using pproximtely isimilr strctions A. Girrd (LJK-UJF) A Symolic Approch to Control 6 / 31

11 Trnsition Systems Unified modeling frmework of discrete nd (smpled) continuous systems. Definition A trnsition system is tuple T = (X, U, δ, Y, H) where X is (discrete or continuous) set of sttes; U is (discrete or continuous) set of inputs; δ : X U 2 X is trnsition reltion; Y is (discrete or continuous) set of outputs; H : X Y is n ouput mp. 0 1 X = {red, lue, green, yellow}, U = {, } Y = {0, 1, 2} 1 2 A. Girrd (LJK-UJF) A Symolic Approch to Control 7 / 31

12 Trnsition Systems A trjectory of the trnsition system T is finite sequence: s = (x 0, u 0 ), (x 1, u 1 ),..., (x N 1, u N 1 ), x N where x k+1 δ(x k, u k ), k {0,..., N 1}. The ssocited oserved trjectory is o = y 0, y 1,..., y N 1, y N where y k = H(x k ), k {0,..., N}. The trnsition system is sid to e discrete or symolic if X nd U re countle or finite. Otherwise, it is sid to e uncountle. A. Girrd (LJK-UJF) A Symolic Approch to Control 8 / 31

13 Approximte Bisimultion Let T i = (X i, U, δ i, Y, H i ), i {1, 2}, e trnsition systems with common set of inputs U nd outputs O equipped with metric d. Definition Let ε R +, reltion R X 1 X 2 is n ε-pproximte isimultion reltion if for ll (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 tht (x 1, x 2 ) R; 3 u U, x 2 δ 2(x 2, u), x 1 δ 1(x 1, u), such tht (x 1, x 2 ) R. Definition T 1 nd T 2 re ε-pproximtely isimilr (T 1 ε T 2 ) if : 1 For ll x 1 X 1, there exists x 2 X 2, such tht (x 1, x 2 ) R; 2 For ll x 2 X 2, there exists x 1 X 1, such tht (x 1, x 2 ) R. A. Girrd (LJK-UJF) A Symolic Approch to Control 9 / 31

14 Approximte Bisimultion X 2 d(h 1 (x 1 ), H 2 (x 2 )) ε R x 1 X 1 A. Girrd (LJK-UJF) A Symolic Approch to Control 10 / 31

15 Approximte Bisimultion X 2 d(h 1 (x 1 ), H 2 (x 2 )) ε R x 2 x 1 X 1 A. Girrd (LJK-UJF) A Symolic Approch to Control 10 / 31

16 Approximte Bisimultion X 2 d(h 1 (x 1 ), H 2 (x 2 )) ε R x 2 x 1 x 1 δ 1 (x 1, u) X 1 A. Girrd (LJK-UJF) A Symolic Approch to Control 10 / 31

17 Approximte Bisimultion X 2 d(h 1 (x 1 ), H 2 (x 2 )) ε x 2 δ 2 (x 2, u) R x 2 x 1 x 1 δ 1 (x 1, u) X 1 A. Girrd (LJK-UJF) A Symolic Approch to Control 10 / 31

18 Approximte Bisimultion Proposition If T 1 ε T 2, then for ll trjectories of T 1, (x0 1, u 0),..., (xn 1 1, u N 1), xn 1, there exists trjectory of T 2, (x0 2, u 0),..., (xn 1 2, u N 1), xn 2 with the sme sequence of inputs, such tht k {0,..., N}, (x 1 k, x 2 k ) R. The ssocited oserved trjectories y 1 0,..., y 1 N nd y 2 0,..., y 2 N stisfy k {0,..., N}, d(y 1 k, y 2 k ) ε. For ε = 0, we recover the usul notion of isimultion reltion used in computer science for studying equivlence of discrete systems. A. Girrd (LJK-UJF) A Symolic Approch to Control 11 / 31

19 Outline of the Tlk 1 Approximtion reltionships for discrete nd continuous systems 2 Symolic strctions of switched systems 3 Controller synthesis using pproximtely isimilr strctions A. Girrd (LJK-UJF) A Symolic Approch to Control 12 / 31

20 Switched Systems Definition A switched system is tuple Σ = (R n, P, F) where: R n is the stte spce; 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 switching signl p : R + P, initil stte x R n, x(t, x, p) denotes the trjectory of Σ given y: ẋ(t) = f p(t) (x(t)), x(0) = x. A. Girrd (LJK-UJF) A Symolic Approch to Control 13 / 31

21 Switched Systems s Trnsition Systems Consider switched system Σ = (R n, P, F) nd time smpling prmeter τ > 0. Let T τ (Σ) e the trnsition system where: the set of sttes is X = R n ; the set of inputs is U = P; the trnsition reltion is given y x δ(x, p) x = x(τ, x, p); the set of outputs is Y = R n ; the output mp H is the identity mp over R n. The trnsition system T τ (Σ) is uncountle, cn we compute symolic strction? A. Girrd (LJK-UJF) A Symolic Approch to Control 14 / 31

22 Computtion of the Symolic Astrction We strt y pproximting the set of sttes R n y: { [R n ] η = z R n z i = k i 2η n, k i Z, i = 1,..., n where η > 0 is stte smpling prmeter: x R n, z [R n ] η, x z η. }, A. Girrd (LJK-UJF) A Symolic Approch to Control 15 / 31

23 Computtion of the Symolic Astrction We strt y pproximting the set of sttes R n y: { [R n ] η = z R n z i = k i 2η n, k i Z, i = 1,..., n where η > 0 is stte smpling prmeter: x R n, z [R n ] η, x z η. Approximtion of the trnsition reltion = rounding : }, z x(τ, z, p) z A. Girrd (LJK-UJF) A Symolic Approch to Control 15 / 31

24 Computtion of the Symolic Astrction We define the trnsition system T τ,η (Σ) where : the set of sttes is X = [R n ] η ; the set of inputs is U = P; the trnsition reltion is given y z δ(z, p) z = rg min q [R n ] η ( x(τ, z, p) q ). the set of outputs is Y = R n ; the output mp is given y H(z) = z R n. The trnsition system T τ,η (Σ) is discrete nd deterministic. Are T τ (Σ) nd T τ,η (Σ) pproximtely isimilr? A. Girrd (LJK-UJF) A Symolic Approch to Control 16 / 31

25 Computtion of the Symolic Astrction We define the trnsition system T τ,η (Σ) where : the set of sttes is X = [R n ] η ; the set of inputs is U = P; the trnsition reltion is given y z δ(z, p) z = rg min q [R n ] η ( x(τ, z, p) q ). the set of outputs is Y = R n ; the output mp is given y H(z) = z R n. The trnsition system T τ,η (Σ) is discrete nd deterministic. Are T τ (Σ) nd T τ,η (Σ) pproximtely isimilr? Yes, if switched system Σ is incrementlly stle. A. Girrd (LJK-UJF) A Symolic Approch to Control 16 / 31

26 Incrementl Stility Definition The switched system Σ is incrementlly glolly uniformly symptoticlly stle (δ-guas) if there exists KL function β such tht for ll initil conditions x 1, x 2 R n, for ll switching signls p : R + P, for ll 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. Girrd (LJK-UJF) A Symolic Approch to Control 17 / 31

27 δ-gas Lypunov Functions Definition V : R n R n R + is common δ-guas Lypunov function for Σ if there exist K functions α, α nd κ R + such tht for ll 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 common δ-guas Lypunov function, then Σ is δ-guas. A. Girrd (LJK-UJF) A Symolic Approch to Control 18 / 31

28 δ-gas Lypunov Functions Definition V : R n R n R + is common δ-guas Lypunov function for Σ if there exist K functions α, α nd κ R + such tht for ll 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 common δ-guas Lypunov function, then Σ is δ-guas. Supplementry ssumption (true if working on compct suset of R n ): There exists K function γ such tht x 1, x 2, x 3 R n, V (x 1, x 2 ) V (x 1, x 3 ) γ( x 2 x 3 ). A. Girrd (LJK-UJF) A Symolic Approch to Control 18 / 31

29 Approximtion Theorem Theorem Let us ssume tht there exists V : R n R n R + which is common δ-guas Lypunov function for Σ. Consider smpling prmeters τ, η R + nd desired precision ε R +. If η min { γ 1 ( (1 e κτ )α(ε) ), α 1 (α(ε)) } then, the reltion R R n [R n ] η given y R = {(x, z) R n [R n ] η V (x, z) α(ε)} is n ε-pproximte isimultion reltion nd T τ (Σ) ε T τ,η (Σ). A. Girrd (LJK-UJF) A Symolic Approch to Control 19 / 31

30 Approximtion Theorem Theorem Let us ssume tht there exists V : R n R n R + which is common δ-guas Lypunov function for Σ. Consider smpling prmeters τ, η R + nd desired precision ε R +. If η min { γ 1 ( (1 e κτ )α(ε) ), α 1 (α(ε)) } then, the reltion R R n [R n ] η given y R = {(x, z) R n [R n ] η V (x, z) α(ε)} is n ε-pproximte isimultion reltion nd T τ (Σ) ε T τ,η (Σ). Min ide of the proof: show tht ccumultion of successive rounding errors is contined y incrementl stility. A. Girrd (LJK-UJF) A Symolic Approch to Control 19 / 31

31 Outline of the Tlk 1 Approximtion reltionships for discrete nd continuous systems 2 Symolic strctions of switched systems 3 Controller synthesis using pproximtely isimilr strctions A. Girrd (LJK-UJF) A Symolic Approch to Control 20 / 31

32 Controllers for Sfety Specifictions Definition Let T = (X, U, δ, Y, H), stte-feedck controller for T is mp S : X 2 U. The dynmics of the controlled system is descried y the trnsition system T S = (X, U, δ S, Y, H) where the trnsition reltion δ S is given for ll x X, u U, x X y x δ S (x, u) ( u S(x) x δ(x, u) ). A. Girrd (LJK-UJF) A Symolic Approch to Control 21 / 31

33 Controllers for Sfety Specifictions Definition Let T = (X, U, δ, Y, H), stte-feedck controller for T is mp S : X 2 U. The dynmics of the controlled system is descried y the trnsition system T S = (X, U, δ S, Y, H) where the trnsition reltion δ S is given for ll x X, u U, x X y x δ S (x, u) ( u S(x) x δ(x, u) ). Definition Let Y s Y e set of outputs ssocited with sfe sttes. A controller S is sfe for specifiction Y s if, for ll x X with S(x), H(x) Y s (sfety); For ll u S(x), for ll x δ(x, u), S(x ) (dedend freedom). A. Girrd (LJK-UJF) A Symolic Approch to Control 21 / 31

34 Mximl Sfety Controller If for ll x X, S(x) =, then S is sfe... We need notion of est sfety controller. A. Girrd (LJK-UJF) A Symolic Approch to Control 22 / 31

35 Mximl Sfety Controller If for ll x X, S(x) =, then S is sfe... We need notion of est sfety controller. Definition Controller S 1 is more permissive thn controller S 2 (S 2 S 1 ) if, for ll x X, S 2 (x) S 1 (x). Definition S is the mximl sfety controller for specifiction Y s if, S is sfe nd for ll sfety controllers S, S S. The mximl sfety controller exists nd is unique. It cn e determined y fixed point computtion of the lrgest controlled-invrint of T, included in H 1 (Y s ). A. Girrd (LJK-UJF) A Symolic Approch to Control 22 / 31

36 Computtion of the Mximl Sfety Controller Informlly, on simple exmple: The lgorithm termintes in finite numer of steps for discrete trnsition systems if H 1 (Y s ) is finite. No gurntee of termintion for uncountle trnsition systems. A. Girrd (LJK-UJF) A Symolic Approch to Control 23 / 31

37 Computtion of the Mximl Sfety Controller Informlly, on simple exmple: The lgorithm termintes in finite numer of steps for discrete trnsition systems if H 1 (Y s ) is finite. No gurntee of termintion for uncountle trnsition systems. A. Girrd (LJK-UJF) A Symolic Approch to Control 23 / 31

38 Computtion of the Mximl Sfety Controller Informlly, on simple exmple: The lgorithm termintes in finite numer of steps for discrete trnsition systems if H 1 (Y s ) is finite. No gurntee of termintion for uncountle trnsition systems. A. Girrd (LJK-UJF) A Symolic Approch to Control 23 / 31

39 Computtion of the Mximl Sfety Controller Informlly, on simple exmple: The lgorithm termintes in finite numer of steps for discrete trnsition systems if H 1 (Y s ) is finite. No gurntee of termintion for uncountle trnsition systems. A. Girrd (LJK-UJF) A Symolic Approch to Control 23 / 31

40 Computtion of the Mximl Sfety Controller Informlly, on simple exmple: The lgorithm termintes in finite numer of steps for discrete trnsition systems if H 1 (Y s ) is finite. No gurntee of termintion for uncountle trnsition systems. A. Girrd (LJK-UJF) A Symolic Approch to Control 23 / 31

41 Sfety Controller Synthesis vi Symolic Astrctions Mximl sfety controllers re esy to compute for symolic strctions... We need controller refinement procedure! Definition Let Y Y nd ϕ 0. The ϕ-contrction of Y is the suset of Y is C ϕ (Y ) = {y Y y Y, d(y, y ) ϕ = y Y }. C ϕ (Y ) ϕ Y A. Girrd (LJK-UJF) A Symolic Approch to Control 24 / 31

42 Sfety Controller Synthesis vi Symolic Astrctions Theorem ( Correct y design ) Let T 1 ε T 2, let R X 1 X 2 denote the ε-pproximte isimultion reltion etween T 1 nd T 2. Let S2,ε e the mximl sfe controller for T 2 for the specifiction C ε (Y s ). Let S 1 e the controller for T 1 given y x 1 X 1, S 1 (x 1 ) = S2,ε(x 2 ) x 2 R(x 1 ) where x 2 R(x 1 ) mens (x 1, x 2 ) R. Then, S 1 is sfe for specifiction Y s. Theorem ( Optiml in the limit ) Let S 1 nd S 1,2ε e the mximl sfe controllers for T 1 for specifictions Y s nd C 2ε (Y s ), respectively. Then, S 1,2ε S 1 S 1. A. Girrd (LJK-UJF) A Symolic Approch to Control 25 / 31

43 Exmple: DC-DC Converter Power converter with switching control: r x l l i l s 2 v s s 1 v c r c x c r 0 v 0 Stte vrile: x(t) = [i l (t), v c (t)] T. System dynmics: ẋ(t) = A p x(t) +, p {1, 2}. Common δ-guas Lypunov function of the form: V (x, y) = (x y) T M(x y). A. Girrd (LJK-UJF) A Symolic Approch to Control 26 / 31

44 Sfety Controller for the DC-DC Converter Astrction prmeters: τ = 1, η = 10 3 = ε = Y s = [1.1, 1.6] [5.4, 5.9] = C ε (Y s ) = [1.15, 1.55] [5.45, 5.85]. The symolic strction hs sttes, the synthesis lgorithm termintes in 2 itertions S 2,ε A. Girrd (LJK-UJF) A Symolic Approch to Control 27 / 31

45 Sfety Controller Refinement Using the controller refinement eqution: S 1 (x 1 ) = S 2,ε(x 2 ). x 2 [R n ] η V (x 1,x 2 ) α(ε) A. Girrd (LJK-UJF) A Symolic Approch to Control 28 / 31

46 Switching Controller for the DC-DC Converter The synthesized controller is non-deterministic. Severl implementtions of the controller re possile. Possiility to ensure posteriori secondry control ojective S 1 A. Girrd (LJK-UJF) A Symolic Approch to Control 29 / 31

47 Switching Controller for the DC-DC Converter Using two possile implementtions: Lzy control Stochstic control Both implementtions stisfy the sfety specifiction. A. Girrd (LJK-UJF) A Symolic Approch to Control 30 / 31

48 Conclusions Approximtely isimilr symolic strctions: A rigorous tool for controller synthesis: = Controllers re correct y design, optiml in the limit. Allow us to leverge efficient lgorithmic techniques from discrete systems to continuous nd hyrid systems. Computle for interesting clsses of systems: switched systems, continuous control systems... Ongoing nd future work: Multiscle nd dptive symolic models. Controller synthesis for other type of specifictions. Compositionl symolic models. Complexity reduction of synthesized controllers. A. Girrd (LJK-UJF) A Symolic Approch to Control 31 / 31