Efficient Computation of Reachable Sets of Linear Time-Invariant Systems with Inputs Colas Le Guernic ∗ ´ Ecole Normale Sup´ erieure joint work with Antoine Girard ∗ Oded Maler University of Pennsylvania VERIMAG March 29, 2006 ∗ currently at VERIMAG Colas Le Guernic HSCC 2006 – 1 / 20
Motivations Discrete Linear Time Invariant System: Introduction ■ Motivations DLTI x k +1 = Φ x k + u k x 0 ∈ Ω 0 , ∀ i u i ∈ U The wrapping effect A new algorithm Experimental Results Conclusion Colas Le Guernic HSCC 2006 – 2 / 20
Motivations Discrete Linear Time Invariant System: Introduction ■ Motivations DLTI x k +1 = Φ x k + u k x 0 ∈ Ω 0 , ∀ i u i ∈ U The wrapping effect A new algorithm Experimental ◆ Obtained by discretisation of a continuous system Results ◆ Input can take into account errors due to linearisation Conclusion and discretisation Colas Le Guernic HSCC 2006 – 2 / 20
Motivations Discrete Linear Time Invariant System: Introduction ■ Motivations DLTI x k +1 = Φ x k + u k x 0 ∈ Ω 0 , ∀ i u i ∈ U The wrapping effect A new algorithm Experimental ◆ Obtained by discretisation of a continuous system Results ◆ Input can take into account errors due to linearisation Conclusion and discretisation Reachable sets: ■ ◆ Set of points reachable from a specified initial set with the considered dynamic under any possible input ◆ Computation required for safety verification, controller synthesis,. . . Colas Le Guernic HSCC 2006 – 2 / 20
Motivations Discrete Linear Time Invariant System: Introduction ■ Motivations DLTI x k +1 = Φ x k + u k x 0 ∈ Ω 0 , ∀ i u i ∈ U The wrapping effect A new algorithm Experimental ◆ Obtained by discretisation of a continuous system Results ◆ Input can take into account errors due to linearisation Conclusion and discretisation Reachable sets: ■ ◆ Set of points reachable from a specified initial set with the considered dynamic under any possible input ◆ Computation required for safety verification, controller synthesis,. . . We will not detail here how Ω 0 , Φ and U can be obtained from a continuous time system. Colas Le Guernic HSCC 2006 – 2 / 20
DLTI We want to compute the N first sets of the sequence defined by: Introduction Motivations DLTI Ω n +1 = ΦΩ n ⊕ U The wrapping effect A new algorithm Experimental Ω 0 is the set of initial points ■ Results U is the set of inputs ■ Conclusion Φ is a d × d matrix ■ ⊕ is the Minkowski sum ■ A ⊕ B = { a + b | a ∈ A and b ∈ B } = ⊕ Colas Le Guernic HSCC 2006 – 3 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Φ Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion ⊕ Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Φ ⊕ Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Φ ⊕ Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Φ ⊕ Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Φ ⊕ Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Colas Le Guernic HSCC 2006 – 4 / 20
A naive algorithm Direct use of the recurence relation: Introduction The wrapping effect A naive algorithm Ω n +1 = ΦΩ n ⊕ U Usual Solution: Approximation Tight approximation For that, we need a class of sets closed under linear Example transformation and Minkowski sum, for example: convex A new algorithm polytopes represented by their vertices. Experimental Results But: Conclusion Φ ⊕ Colas Le Guernic HSCC 2006 – 4 / 20
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