## Impulse differential inclusions: A viability approach to hybrid systems (2002)

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Venue: | IEEE Transactions on Automatic Control |

Citations: | 39 - 5 self |

### BibTeX

@ARTICLE{Aubin02impulsedifferential,

author = {Jean-pierre Aubin and John Lygeros and Marc Quincampoix and Shankar Sastry and Nicolas Seube},

title = {Impulse differential inclusions: A viability approach to hybrid systems},

journal = {IEEE Transactions on Automatic Control},

year = {2002},

volume = {47},

pages = {2--20}

}

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### Abstract

Abstract. Impulse differential inclusions are introduced as a framework for modelling hybrid phenomena. Connections to standard problems in area of hybrid systems are discussed. Conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion. For sets that violate these conditions, methods are developed for approximating their viability and invariance kernels, that is the largest subset that is viable or invariant under the action of the impulse differential inclusion. The results are demonstrated on examples. 1.

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Citation Context ...w. Definitions 2 and 3 suggest that impulse differential inclusions are intimately related to other modeling languages found in the literature, such as different variants of hybrid automata (HA) [9], =-=[13]-=-, [36], [38] and hybrid input/output automata (HIOA) [7]. Many of the properties studied here for impulse differential inclusions can be easily extended to these different classes of models by assumin... |

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Citation Context ...veloped to exactly compute the reachable set of states whenever possible [20]–[23], compute conservative approximations for it [24]–[28], or at least help establish some of its properties deductively =-=[29]-=-. For continuous dynamical systems described by differential inclusions, questions of reachability have been addressed in the context of viability theory [30]. Viability theory deals with two fundamen... |

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Citation Context ...one can also define reachability controller synthesis problems, where the objective is to choose the values of the control inputs such that the run of the system never leaves the good set, [15], [10]–=-=[14]-=-, [16]. The solution to reachability controller synthesis problems comes down to computing controlled invariant subsets of [12], [30], that is subsets of for which there exist a choice for the control... |

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Citation Context ...em. The proof of Theorem 4 is based on the following abstract algorithm, which follows the standard iterative characterization of the greatest fixed point of a monotone operator on a complete lattice =-=[44]-=-. Algorithm 1 (Viability Kernel Approx.) initialization: , repeat until As shown in the proof of Theorem 4, the sets form a sequence of nested closed sets. Given a set it may, in general, be impossibl... |

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Citation Context ...id system. Theoretical methods have been developed for performing these computations [5], [6], [17], [7], [8], [18], [9], some of them supported by automated or semi-automated tools [20], [22], [23], =-=[26]-=-, [24], [25], [29], [21]. If control inputs are available, one can also define reachability controller synthesis problems, where the objective is to choose the values of the control inputs such that t... |

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Citation Context ...cal phenomena that involve both continuous evolution and discrete transitions. To distinguish the times at which discrete transitions take place we recall the notion of a hybrid time trajectory [12], =-=[37]-=-. Definition 1 (Hybrid Time Trajectory): A hybrid time trajectory is a finite or infinite sequence of intervals of the real line, such that • for , ; •if , then either ,or , possibly with ; • for all ... |

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Citation Context ...ethods have been developed for performing these computations [5], [6], [17], [7], [8], [18], [9], some of them supported by automated or semi-automated tools [20], [22], [23], [26], [24], [25], [29], =-=[21]-=-. If control inputs are available, one can also define reachability controller synthesis problems, where the objective is to choose the values of the control inputs such that the run of the system nev... |

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Citation Context ...ments of . This notation can be misleading since it may convey the impression that switching times are prescribed a priori. We mention it, however, to establish a connection with the notation used in =-=[40]-=-. F. Examples To illustrate how impulse differential inclusions can be used to characterize hybrid phenomena we consider two simple examples from the hybrid systems literature: a thermostat system and... |

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Citation Context ...ate how impulse differential inclusions can be used to characterize hybrid phenomena we consider two simple examples from the hybrid systems literature: a thermostat system and a bouncing ball system =-=[41]-=-. We will return to these examples in Section V, to illustrate the viability and invariance conditions for impulse differential inclusions. 1) Thermostat: The thermostat system (adapted from a simpler... |

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Citation Context ...lishing whether the set of reachable states is contained in a certain set [5]–[9], or, in the case of hybrid control systems, for synthesizing controllers that satisfy such safety specifications [10]–=-=[16]-=-. Since the reachability problem quickly becomes computationally infeasible, approxiManuscript received February 2, 2001; revised June 20, 2001. Recommended by Associate Editor A. Bemporad. This work ... |

8 | Approximation of Viability Kernels and Capture Basin for Hybrid Systems
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Citation Context ...also provide a procedure for establishing the viability and invariance kernels of a set of states. Numerical algorithms for implementing the procedure have been developed recently in a parallel study =-=[33]-=-. The material is arranged in five sections. In Section II, the impulse differential inclusion framework is introduced, and the basic concepts of viability theory are extended to it. Some examples are... |

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Citation Context ...clusions involve the notion of “viability with target.” This notion was introduced in [42] for continuous differential inclusions, motivated partly by target optimal control problems (see for example =-=[43]-=-). Viability with target provides conditions under which solutions of that remain viable in a set until they reach a target set below. exist. For completeness, conditions are summarized Lemma 1: Consi... |

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Citation Context ...chniques have been proposed to facilitate the analysis [17]–[19]. Based on the theoretical results, computational tools been developed to exactly compute the reachable set of states whenever possible =-=[20]-=-–[23], compute conservative approximations for it [24]–[28], or at least help establish some of its properties deductively [29]. For continuous dynamical systems described by differential inclusions, ... |

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Citation Context ...uaranteed for all initial conditions. A. Viability Conditions The viability conditions for impulse differential inclusions involve the notion of “viability with target.” This notion was introduced in =-=[42]-=- for continuous differential inclusions, motivated partly by target optimal control problems (see for example [43]). Viability with target provides conditions under which solutions of that remain viab... |

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Citation Context ... [17]–[19]. Based on the theoretical results, computational tools been developed to exactly compute the reachable set of states whenever possible [20]–[23], compute conservative approximations for it =-=[24]-=-–[28], or at least help establish some of its properties deductively [29]. For continuous dynamical systems described by differential inclusions, questions of reachability have been addressed in the c... |

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Citation Context ...e kernel) of . Numerical algorithms have been developed to compute these kernels (see [31] and the references therein), and have been used to compute, for example, basins of attraction for equilibria =-=[32]-=-. In this paper, we extend viability theory concepts to a wider class of systems, which we call impulse differential inclusions. Impulse differential inclusions capture a broad range of hybrid phenome... |

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Citation Context ...heoretical methods have been developed for performing these computations [5], [6], [17], [7], [8], [18], [9], some of them supported by automated or semi-automated tools [20], [22], [23], [26], [24], =-=[25]-=-, [29], [21]. If control inputs are available, one can also define reachability controller synthesis problems, where the objective is to choose the values of the control inputs such that the run of th... |

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Citation Context ...ns 2 and 3 suggest that impulse differential inclusions are intimately related to other modeling languages found in the literature, such as different variants of hybrid automata (HA) [9], [13], [36], =-=[38]-=- and hybrid input/output automata (HIOA) [7]. Many of the properties studied here for impulse differential inclusions can be easily extended to these different classes of models by assuming that the d... |