Abstraction-based synthesis: Challenges and victories

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1 Abstraction-based synthesis: Challenges and victories Majid Zamani Hybrid Control Systems Group Electrical Engineering Department Technische Universität München December 14, 2015 Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

2 Cyber-physical systems: Interaction of physical plants with embedded controllers Physical d = f(, )dt + or = f(, ) ( )dwt Cyber Specification: Automata on infinite strings, temporal logics such as LTL Main Problem: can we design (algorithmically & formally) appropriate control software providing υ such that the sampled output ξ satisfy the given specification? Challenge: Complex (even heterogeneous) dynamics and specifications Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

Cyber-physical systems: Interaction of physical plants with embedded controllers Physical d = f(, )dt + or = f(, ) ( )dwt Google Prototype Self-Driving Car Cyber Specification: Automata on infinite

3 Cyber-physical systems: Interaction of physical plants with embedded controllers Physical d = f(, )dt + or = f(, ) ( )dwt Google Prototype Self-Driving Car Cyber Specification: Automata on infinite strings, temporal logics such as LTL Main Problem: can we design (algorithmically & formally) appropriate control software providing υ such that the sampled output ξ satisfy the given specification? Challenge: Complex (even heterogeneous) dynamics and specifications Potential: Developing complex yet reliable systems at lower costs and times Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

4 General methodology: Symbolic methods Discrete abstraction d = f(, )dt + ( )dw t or = f(, ) Continuous dynamics Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

5 General methodology: Symbolic methods Discrete abstraction Hardware+Software Discrete controller d = f(, )dt + ( )dw t d = or f (, )dt + g ( )dw t = f(, ) Continuous dynamics Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

6 General methodology: Symbolic methods Discrete abstraction Hardware+Software Discrete controller d = f(, )dt + ( )dw t d = or f (, )dt + g ( )dw t = f(, ) Continuous dynamics q(k + 1) = g(q(k), (k )) (k ) =k( (k ),q(k)) Hybrid controller Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

7 General methodology: Symbolic methods Abstraction: key step in this approach; Discrete controller synthesis: use existing algorithms from algorithmic game theory or supervisory control of discrete-event systems; Sample specifications expressed using LTL: Safety φ (always φ) Reachability φ (eventually φ) Stability ( φ) (eventually always φ) Recurrence ( φ) (infinitely often φ) Sequencing (φ 1 φ 2 ) (visit φ 1 and then φ 2 ) Coverage φ 1 φ 2 (visit φ 1 and φ 2 ) Fault recovery (F R) (every time fault then eventually recover) motion planning ( Goal) ( i I Obs i ) (always avoid obstacles and reach and stay the goal set) Controller refinement: determined by the abstraction. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

8 General methodology: Symbolic methods Abstraction: key step in this approach; Discrete controller synthesis: use existing algorithms from algorithmic game theory or supervisory control of discrete-event systems; Sample specifications expressed using LTL: Safety φ (always φ) Reachability φ (eventually φ) Stability ( φ) (eventually always φ) Recurrence ( φ) (infinitely often φ) Sequencing (φ 1 φ 2 ) (visit φ 1 and then φ 2 ) Coverage φ 1 φ 2 (visit φ 1 and φ 2 ) Fault recovery (F R) (every time fault then eventually recover) motion planning ( Goal) ( i I Obs i ) (always avoid obstacles and reach and stay the goal set) Controller refinement: determined by the abstraction. How to construct an abstraction? Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

9 Two types of abstractions: Complete vs sound Finite system = " AS d = f(, )dt + or = f(, ) ( )dw t Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

10 Two types of abstractions: Complete vs sound Finite system d = f(, )dt + or = f(, ) ( )dw t Discrete controller Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

11 Two types of abstractions: Complete vs sound Finite system = " AS d = f(, )dt + or = f(, ) ( )dw t Discrete controller if and only if Refinement q(k + 1) = g(q(k), (k )) (k ) =k( (k ),q(k)) Hybrid controller Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

12 Two types of abstractions: Complete vs sound Finite system " AS d = f(, )dt + or = f(, ) ( )dw t Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

13 Two types of abstractions: Complete vs sound Finite system d = f(, )dt + or = f(, ) ( )dw t Discrete controller Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

14 Two types of abstractions: Complete vs sound Finite system " AS d = f(, )dt + or = f(, ) ( )dw t Discrete controller Sufficient Refinement q(k + 1) = g(q(k), (k )) (k ) =k( (k ),q(k)) Hybrid controller Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

15 Unified modelling of physical and cyber systems Definition (System) A system S is a sextuple (X, X 0, U, a set of states X ; a set of initial states X 0 X ; a set of inputs U; a transition relation X U X ; a set of outputs Y ; an output map H : X Y. A system is said to be:, Y, H) consisting of: metric, if the output set Y is equipped with a metric d : Y Y R + 0 ; countable, if X is a countable set; finite (or symbolic), if X is a finite set; Can write x u x instead of (x, u, x ). Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

16 Physical systems Definition: A control system Σ is a tuple Σ = (R n, U, U, f ) where: R n is the state space; U R m is the input set; U is a set of nice functions from R + 0 to U; f : R n U R n is a nice function; A curve ξ : R + 0 Rn is a trajectory of Σ if there exists υ U satisfying: ξ = f (ξ, υ). Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

17 Physical systems Definition: A control system Σ is a tuple Σ = (R n, U, U, f ) where: R n is the state space; U R m is the input set; U is a set of nice functions from R + 0 to U; f : R n U R n is a nice function; A curve ξ : R + 0 Rn is a trajectory of Σ if there exists υ U satisfying: ξ = f (ξ, υ). ξ xυ(t) denotes the value of the trajectory of Σ at time t under the input υ from initial condition x = ξ xυ(0). Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

18 Physical systems as systems Given Σ = (R n, U, U, f ) and a sampling time τ R +, the metric system S τ (Σ) = (X, X 0, U,, Y, H) associated with the sampled dynamics of Σ is given by: X = R n ; X 0 = R n ; U: all the curves in U of duration τ; x υ x iff ξ xυ(τ) = x ; Y = R n, equipped with the metric d(y, y ) = y y for any y, y R n ; H = 1 X. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

19 Physical systems as systems Given Σ = (R n, U, U, f ) and a sampling time τ R +, the metric system S τ (Σ) = (X, X 0, U,, Y, H) associated with the sampled dynamics of Σ is given by: X = R n ; X 0 = R n ; U: all the curves in U of duration τ; x υ x iff ξ xυ(τ) = x ; Y = R n, equipped with the metric d(y, y ) = y y for any y, y R n ; H = 1 X. S τ (Σ) is an infinite system! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

20 Physical systems as systems Given Σ = (R n, U, U, f ) and a sampling time τ R +, the metric system S τ (Σ) = (X, X 0, U,, Y, H) associated with the sampled dynamics of Σ is given by: X = R n ; X 0 = R n ; U: all the curves in U of duration τ; x υ x iff ξ xυ(τ) = x ; Y = R n, equipped with the metric d(y, y ) = y y for any y, y R n ; H = 1 X. S τ (Σ) is an infinite system! Can we replace S τ (Σ) with a finite complete or sound abstraction? Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

21 Two ways of constructing finite abstractions 1) Specification-free abstractions 2) Specification-guided abstractions Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

22 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

23 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

24 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

25 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

26 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

27 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

28 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

29 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

30 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

31 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

32 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

33 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

34 Specification-free abstractions: Stable dynamics Complete abstraction: η A. Girard, G. Pola, and P. Tabuada, Approximately bisimilar symbolic models for incrementally stable switched systems IEEE Transactions on Automatic Control, 55(1): , Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

35 Specification-free abstractions: Unstable dynamics Sound abstraction: Independent of the method for the computation of over-approximation of reachable sets! Tighter over-approximation results in less conservative abstraction (using monotonicity or mixed-monotonicity of dynamics) M. Zamani, G. Pola, M. Mazo Jr. and P. Tabuada, Symbolic models for nonlinear control systems without stability assumptions IEEE Transactions on Automatic Control, 57(7), pp , July η Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

36 Specification-free abstractions: Unstable dynamics Sound abstraction: Independent of the method for the computation of over-approximation of reachable sets! Tighter over-approximation results in less conservative abstraction (using monotonicity or mixed-monotonicity of dynamics) M. Zamani, G. Pola, M. Mazo Jr. and P. Tabuada, Symbolic models for nonlinear control systems without stability assumptions IEEE Transactions on Automatic Control, 57(7), pp , July η Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

37 Quotient system Definition Let S = (X, X 0, U,, Y, H) be a system and let Q be an equivalence relation on X such that (x, x ) Q implies ( that H(x) = H(x ). The quotient ) of S by Q, denoted by S /Q, is the system S /Q = X /Q, X /Q0, U /Q,, Y/Q, H /Q consisting of: X /Q = X /Q; X /Q0 = { x /Q X /Q x /Q X 0 } ; U /Q = U; x /Q u x /Q if there exists x /Q Y /Q = Y ; H /Q (x /Q ) = H(x) for some x x /Q. /Q u x in S with x x /Q and x x /Q; Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

38 Quotient system Definition Let S = (X, X 0, U,, Y, H) be a system and let Q be an equivalence relation on X such that (x, x ) Q implies ( that H(x) = H(x ). The quotient ) of S by Q, denoted by S /Q, is the system S /Q = X /Q, X /Q0, U /Q,, Y/Q, H /Q consisting of: X /Q = X /Q; X /Q0 = { x /Q X /Q x /Q X 0 } ; U /Q = U; x /Q u x /Q if there exists x /Q Y /Q = Y ; H /Q (x /Q ) = H(x) for some x x /Q. /Q u x in S with x x /Q and x x /Q; When the equivalence relation Q has finitely many equivalence classes, S /Q is guaranteed to be finite. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

39 Specification-guided abstractions: Complete abstraction Input: Partition P and system S Output: P P := P; while Fixed-point-condition do P := ; forall the P, P P do P a := P Pre(P); P b := P \Pre(P); P := P {P a, P b }; end P := P ; end W X, Pre(W ) = { x X x u } x for some u U and x W. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

40 Specification-guided abstractions: Complete abstraction Input: Partition P and system S Output: P P := P; while Fixed-point-condition do P := ; forall the P, P P do P a := P Pre(P); P b := P \Pre(P); P := P {P a, P b }; end P := P ; end W X, Pre(W ) = { x X x u } x for some u U and x W. When the algorithm terminates, one obtains: S /P = 0 AS S. R. Vidal, S. Schaffert, J. Lygeros, and S. Sastry, Controlled invariance of discrete time systems International Conference on Hybrid Systems: Computation and Control, pp , April Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

41 Specification-guided abstractions: Complete abstraction X P 0 P Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

42 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

43 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

44 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

45 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

46 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

47 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

48 Specification-guided abstractions: Complete abstraction X P 0 P The resulting bisimilar quotient system has 13 states! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

49 Specification-guided abstractions: Sound abstraction Input: Partition P, system S and k N Output: P P := P; i := 1; while i k do i := i + 1; P := ; forall the P, P P do P a := P Pre(P); P b := P \Pre(P); P := P {P a, P b }; end P := P ; end When the algorithm terminates, one obtains: S /P 0 AS S. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

50 SCOTS: A Tool for the Synthesis of Symbolic Controllers User Input: η, z R n + a, b R n User Input: µ R m + c, d R m τ > 0 ξ = f (ξ, u) ζ = L(u)ζ + w SymbolicSet SymbolicSet User Input: X 2 U 2 Init: η, z R n + a, b R n Init: (η, µ) R n+m + a, b c, d SymbolicModel SymbolicModelGrowthBound SymbolicSet K X 2 SymbolicSet FixedPoint H c X 2 U 2 F 2 X 2 U 2 X 2 writetofile controller.bdd Figure: The work flow in SCOTS to compute a symbolic model S 2 of a nonlinear control system S 1 and to synthesize a controller to enforce an invariance (reachability) specification where K X 2 is the safe (target) set. The tool and all conducted experiments are available at org/runmat/scots M. Rungger and M. Zamani, SCOTS: A Tool for the Synthesis of Symbolic Controllers International Conference on Hybrid Systems: Computation and Control, April 2016, under review. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

Motion planning v0 v α δ a b The vehicle model, which is not incrementally stable, is as follows: cos(α+θ) ẋ = v 0, ( ) cos(α) sin(α+θ) a tan(δ) Σ : ẏ = v 0 cos(α), where α = arctan b

51 Motion planning v0 v α δ a b The vehicle model, which is not incrementally stable, is as follows: cos(α+θ) ẋ = v 0, ( ) cos(α) sin(α+θ) a tan(δ) Σ : ẏ = v 0 cos(α), where α = arctan b θ = v 0 b tan(δ), X = [0, 10] [0, 10] [ π, π], (v 0, δ) U = [ 1, 1] [ 1, 1], and ε = 0.2. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

52 Motion planning Specification: Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

53 Motion planning Some numbers: CPU [GHz] #F 2 t abs [sec] t syn[sec] Pessoa Core2Duo SCOTS #1 (BDD) i SCOTS #2 (array) i UniBW (not public) i Table: Comparison of SCOTS with other tools. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

54 The Achilles heel of the proposed techniques Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

The Achilles heel of the proposed techniques Curse of Dimensionality: In view of all that we have said in the foregoing sections, the many obstacles we appear to have surmounted, what casts the pall

55 The Achilles heel of the proposed techniques Curse of Dimensionality: In view of all that we have said in the foregoing sections, the many obstacles we appear to have surmounted, what casts the pall over our victory celebration? It is the curse of dimensionality, a malediction that has plagued the scientist from the earliest days. Richard E. Bellman. Adaptive Control Processes: A Guided Tour. Princeton University Press, Figure: Image courtesy Time Inc. Photographer Alfred Eisenstaedt. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

56 First approach: A state-space discretization free scheme Curse of dimensionality due to the discretization of state and input sets! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

57 First approach: A state-space discretization free scheme Curse of dimensionality due to the discretization of state and input sets! The sizes of the symbolic models grow exponentially with the dimension of the continuous spaces! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

58 First approach: A state-space discretization free scheme Curse of dimensionality due to the discretization of state and input sets! The sizes of the symbolic models grow exponentially with the dimension of the continuous spaces! In practical applications, the state-space dimension is usually much larger than the input set dimension! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

59 First approach: A state-space discretization free scheme Curse of dimensionality due to the discretization of state and input sets! The sizes of the symbolic models grow exponentially with the dimension of the continuous spaces! In practical applications, the state-space dimension is usually much larger than the input set dimension! Can we get rid of the state-space discretization? Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

60 Specification-free abstractions: Stable dynamics (without state-space discretization) x s M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

61 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 1 x s M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

62 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 2 x 1 x s M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

63 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 2 x 1 x 3 x s M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

64 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 2 x 1 x s x 3 x 4 M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

65 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 5 x 2 x 1 x s x 3 x 4 M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

66 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 2 x 1 x 5 x 6 x s x 3 x 4 M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

67 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 2 x 1 x 5 x 6 x s x 3 x 4 x 7 M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

68 Specification-free abstractions: Stable dynamics (without state-space discretization) " x 2 x 1 x bn x 5 x 6 x s x 3 x 4 x 7 M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

69 Specification-free abstractions: Stable dynamics (without state-space discretization) u 1 " x bn u 4 x 2 x s x 3 x 4 x 1 u 2 x 5 x 6 u 3 x 7 M. Zamani, I. Tkachev, and A. Abate, Bisimilar symbolic models for stochastic control systems without state-space discretization International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, A. Abate, and A. Girard, Symbolic models for stochastic switched systems: A discretization and a discretization-free approach Automatica, 55, pp , May Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

70 Example: Thermal model of a six-room building Heater Room 5 Room 1 Room 2 Room 3 Room 4 Room 6 Σ : Heater Figure: A schematic of the six-room building. dξ 1 = (α 21 (ξ 2 ξ 1 ) + α 31 (ξ 3 ξ 1 ) + α 51 (ξ 5 ξ 1 ) + α e1 (T e ξ 1 ) + α f 1 (T f 1 ξ 1 )δ p2 )dt + (σ 1,1 δ p1 + (1 δ p1 )σ 1 ) ξ 1 dwt 1, dξ 2 = (α 12 (ξ 1 ξ 2 ) + α 42 (ξ 4 ξ 2 ) + α e2 (T e ξ 2 ))dt + (σ 2,1 δ p1 + (1 δ p1 )σ 2 )ξ 2 dwt 2, dξ 3 = (α 13 (ξ 1 ξ 3 ) + α 43 (ξ 4 ξ 3 ) + α e3 (T e ξ 3 ))dt + (σ 3,1 δ p1 + (1 δ p1 )σ 3 )ξ 3 dwt 3, dξ 4 = (α 24 (ξ 2 ξ 4 ) + α 34 (ξ 3 ξ 4 ) + α 64 (ξ 6 ξ 4 ) + α e4 (T e ξ 4 ) + α f 4 (T f 4 ξ 4 )δ p3 )dt +(σ 4,1 δ p1 + (1 δ p1 )σ 4 )ξ 4 dwt 4, dξ 5 = (α 15 (ξ 1 ξ 5 ) + α e5 (T e ξ 5 ))dt + (σ 5,1 δ p1 + (1 δ p1 )σ 5 )ξ 5 dwt 5, dξ 6 = (α 46 (ξ 4 ξ 6 ) + α e6 (T e ξ 6 ))dt + (σ 6,1 δ p1 + (1 δ p1 )σ 6 )ξ 6 dwt 6, where T e = 10, T f 1 = T f 4 = 100, α 21 = α 12 = α 13 = α 31 = α 42 = α 24 = α 34 = α 43 = α 15 = α 51 = α 46 = α 64 = , α e1 = α e4 = , α e2 = α e3 = α e5 = α e6 = , and α f 1 = α f 4 = Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

71 Example: Thermal model of a six-room building We assume that at most one heater is on at each instant of time: both heaters switched off (p = 1), 1st heater (T f 1 ) on and the 2nd one (T f 4 ) off (p = 2), or vice versa (p = 3). Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

72 Example: Thermal model of a six-room building We assume that at most one heater is on at each instant of time: both heaters switched off (p = 1), 1st heater (T f 1 ) on and the 2nd one (T f 4 ) off (p = 2), or vice versa (p = 3). By choosing ε = 1, the size of the abstraction based on the state-space discretization free is 3 14 = Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

73 Example: Thermal model of a six-room building We assume that at most one heater is on at each instant of time: both heaters switched off (p = 1), 1st heater (T f 1 ) on and the 2nd one (T f 4 ) off (p = 2), or vice versa (p = 3). By choosing ε = 1, the size of the abstraction based on the state-space discretization free is 3 14 = By working in a compact set D = [ ] 6, the size of the abstraction based on the state-space discretization is ! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

74 Example: Thermal model of a six-room building Specification: D, where D = [19 22] 6. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

75 Example: Thermal model of a six-room building Specification: D, where D = [19 22] 6. Figure: A few realizations of the solution process ξ x0 υ (top panel) and the corresponding evolution of the obtained switching signal υ (bottom panel), where x 0 = [11.7, 11.7, 11.7, 11.7, 11.7, 11.7] T. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

76 Second approach: A compositional scheme Interconnected System y 31 u 1 Σ 1 y 14 Σ 3 y 33 u 2 Σ 2 y 23 Σ 4 y 44 y 42 Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

77 Second approach: A compositional scheme Interconnected System Subsystem Σ i with abstraction ˆΣ i y 31 u 1 Σ 1 y 14 Σ 3 y 33 ˆΣ i AS Σ i u 2 Σ 2 y 23 Σ 4 y 44 y 42 Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

78 Second approach: A compositional scheme Interconnected System Subsystem Σ i with abstraction ˆΣ i y 31 u 1 Σ 1 y 14 Σ 3 y 33 ˆΣ i AS Σ i u 2 Σ 2 y 23 Σ 4 y 44 y 42 Provide conditions so that ŷ 31 y 31 û 1 ˆΣ 1 ŷ 14 ˆΣ 3 ŷ 33 u 1 Σ 1 y 14 Σ 3 y 33 AS û 2 ˆΣ 2 ŷ 23 ˆΣ 4 ŷ 42 ŷ 44 u 2 Σ 2 y 23 Σ 4 y 42 y 44 M. Rungger and M. Zamani, Compositional construction of approximate abstractions International Conference on Hybrid Systems: Computation and Control, pp , April M. Zamani, M. Rungger, and P. Mohajerin Esfahani, Approximations of Stochastic Hybrid Systems: A Compositional Approach arxiv: Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

79 Example y 31 u 1 Σ 1 y 14 Σ 3 y 33 u 2 Σ 2 y 23 Σ 4 y 44 y 42 Σ 1, Σ 2 are 2D systems Σ 3, Σ 4 are 3D systems Interconnected System I(Σ 1,..., Σ 4) 10D with 2 external inputs u 1, u 2 2 outputs y 33 and y 44 Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

80 Example u 1 Σ 1 y 14 Σ 3 y 31 y 33 û 1 ˆΣ 1 ŷ 14 ˆΣ 3 ŷ 33 ŷ 44 u 2 Σ 2 y 23 Σ 4 y 44 û 2 ˆΣ 2 ŷ 23 ˆΣ 4 y 42 Σ 1, Σ 2 are 2D systems Σ 3, Σ 4 are 3D systems Interconnected System I(Σ 1,..., Σ 4) 10D with ˆΣ 1, ˆΣ 2 are 1D systems ˆΣ 3, ˆΣ 4 are 2D systems Interconnected System I(ˆΣ 1,..., ˆΣ 4) 6D (2 disjoint 3D) with 2 external inputs u 1, u 2 2 external inputs û 1, û 2 2 outputs y 33 and y 44 2 outputs ŷ 33 and ŷ 44 Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

81 Example u 1 Σ 1 y 14 Σ 3 y 31 y 33 û 1 ˆΣ 1 ŷ 14 ˆΣ 3 ŷ 33 ŷ 44 u 2 Σ 2 y 23 Σ 4 y 44 û 2 ˆΣ 2 ŷ 23 ˆΣ 4 y 42 Σ 1, Σ 2 are 2D systems Σ 3, Σ 4 are 3D systems Interconnected System I(Σ 1,..., Σ 4) 10D with ˆΣ 1, ˆΣ 2 are 1D systems ˆΣ 3, ˆΣ 4 are 2D systems Interconnected System I(ˆΣ 1,..., ˆΣ 4) 6D (2 disjoint 3D) with 2 external inputs u 1, u 2 2 external inputs û 1, û 2 2 outputs y 33 and y 44 2 outputs ŷ 33 and ŷ 44 I(ˆΣ 1,..., ˆΣ 4) AS I(Σ 1,..., Σ 4) Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

82 Example Specification: D, where D = [0 5] 2. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

83 Example Specification: D, where D = [0 5] 2. 1 ζ 1(t ) ˆζ 1(t ) ζ 2(t ) ˆζ 2(t ) ν 1(t ) 1 ν 2(t ) Figure: Top two plots: One realization of ζ 1 (resp. ζ 2 ) ( ) and ˆζ 1 (resp. ˆζ 2 ) ( ). The middle plot: the corresponding realization of external inputs ν 1 ( ) and ν 2 ( ) of Σ. The 2nd plot from bottom: Square root of the average values (over 1000 experiments) of the squared distance of the output trajectory of Σ to the one of ˆΣ. The solid black line indicates the computed theoretical error bound. Bottom plot: Square root of the average values (over 1000 experiments) of the squared distance of the output trajectory of Σ to the safe set S. Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

84 Question Thanks for your attention! Majid Zamani (TU München) 54th IEEE Conference on Decision and Control December 14, / 34

Symbolic Control of Incrementally Stable Systems

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