Computing the viability kernel using maximal reachable sets

Kaynama, Shahab; Maidens, John; Oishi, Meeko; Mitchell, Ian M.; Dumont, Guy A.

doi:10.1145/2185632.2185644

Cited by 52 publications

(53 citation statements)

References 35 publications

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“…This section shows that both problems proposed in Section 1 can be expressed as the computation of the largest positive invariant set [10] which is included inside a given set X.…”

Section: Resultsmentioning

confidence: 99%

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

Mézo

Jaulin

Zerr

2018

Applied Mathematics and Computation

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In this paper, we present a new method for bracketing (i.e., characterizing from inside and from outside) all solutions of an ordinary differential equation in the case where the initial time is inside an interval and the initial state is inside a box. The principle of the approach is to cast the problem into bracketing the largest positive invariant set which is included inside a given set X. Although there exists an efficient algorithm to solve this problem when X is bounded, we need to adapt it to deal with cases where X is unbounded.

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“…This section shows that both problems proposed in Section 1 can be expressed as the computation of the largest positive invariant set [10] which is included inside a given set X.…”

Section: Resultsmentioning

confidence: 99%

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

Mézo

Jaulin

Zerr

2018

Applied Mathematics and Computation

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“…For example [15] utilizes a recursive method to under-approximate the viability kernel for continuous-time systems, and proposes a scalable piecewise ellipsoidal algorithm (based on ellipsoidal techniques for maximal reachability [17]) for linear time-invariant (LTI) dynamics.…”

Section: Related Workmentioning

confidence: 99%

“…The following lemma describes how to construct such a subset, called A ↓ . LEMMA 1 ( [15]). Suppose that f is uniformly bounded on A in some norm · p 1 by M .…”

Section: From Continuous To Pointwise Feasibilitymentioning

confidence: 99%

Sampling-based approximation of the viability kernel for high-dimensional linear sampled-data systems

Gillula

Kaynama

Tomlin

2014

Proceedings of the 17th International Conference on Hybrid Systems: Computation and Control

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Proving that systems satisfy hard input and state constraints is frequently desirable when designing cyber-physical systems. One method for doing so is to compute the viability kernel, the subset of the state space for which a control signal exists that is guaranteed to keep the system within the constraints over some time horizon. In this paper we present a novel method for approximating the viability kernel for linear sampled-data systems using a sampling-based algorithm, which by its construction offers a direct trade-off between scalability and accuracy. We also prove that the algorithm is correct, that its convergence properties are optimal, and demonstrate it on a simple example. We conclude by briefly describing additional results which are omitted due to space constraints.

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“…In [19], a dynamic programming technique is proposed and reachable sets are approximated by using ellipsoidal techniques. In [20], a Lagrangian method is proposed for computing the viability kernel via ellipsoidal representation. In [14], this method is extended by using polytopic and support vector representations.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the work in [20], and by adopting the control purpose of our previous work [21], this paper proposes a Lagrangian method based on a zonotopic set representation. Different from the existing literature, this paper uses the backward reachable sets to estimate and enlarge the RA by designing an optimal controller.…”

Section: Introductionmentioning

confidence: 99%

On enlarging backward reachable sets via Zonotopic set membership

Han

Rizaldi

El-Guindy

et al. 2016

2016 IEEE International Symposium on Intelligent Control (ISIC)

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Abstract-While a number of efficient methods have been proposed for approximating backward reachable sets, no synthesis method via backward reachable sets has been developed for estimating and enlarging the region of attraction (RA). This paper shows how to use backward reachable sets to enlarge the estimate of the RA of linear discrete-time systems, by using an optimal static feedback controller. Two controller design methods are provided: the first method enlarges the estimate of the RA via invariant sets, whose existence is ensured by zonotope containment; the second method provides the optimal control input by using Lyapunov stability and quadratic stabilization. The backward reachable set is represented by zonotopes which give a good compromise between accuracy and efficiency. The effectiveness of both methods is illustrated by a numerical example.

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Computing the viability kernel using maximal reachable sets

Cited by 52 publications

References 35 publications

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

Sampling-based approximation of the viability kernel for high-dimensional linear sampled-data systems

On enlarging backward reachable sets via Zonotopic set membership

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