Optimal control of a class of hybrid systems

This paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal linear programming (LP) problem. Its dual linear program of HamiltonJacobi-Bellman inequalities is also characterized. The LPs are then solved numerically with a converging hierarchy of primal-dual moment-sum-of-squares (SOS) linear matrix inequalities (LMI). Because of the special structure of switched systems, we obtain a much more efficient method than could be achieved by applying standard moment/SOS LMI hierarchies for general optimal control problems.

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“…Injecting this back in (11) shows that, in addition, m j=1ū j = 1,μ-almost everywhere. Define now µ ∈ M + (K × U ), with U as in (3), such that…”

Section: This Is Rigorously Justified By the Followingmentioning

confidence: 97%

Optimal switching control design for polynomial systems: an LMI approach

Henrion

Daafouz

Claeys

2013

52nd IEEE Conference on Decision and Control

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“…Many results can be found in the controlengineering literature. The optimal control of hybrid systems in manufacturing is addressed in [1,13,16], where the authors combine time-driven and event-driven methodologies to solve optimal control problems. An algorithm to optimize the switching sequences for a class of switched linear problems is presented in [20], where the algorithm searches for solutions that are arbitrarily close to the optimal ones.…”

Section: Introductionmentioning

confidence: 99%

Model predictive control of discrete-time hybrid systems with discrete inputs

2005

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This paper proposes and discusses a model predictive control approach to hybrid systems with discrete inputs only. The algorithm, which takes into account a model of a hybrid system, described as an MLD (mixed logical dynamical ) system, is based on a performance-driven reachability analysis. The algorithm abstracts the behavior of the hybrid system by building a "tree of evolution". The nodes of the tree represent the reachable states of a process, and the branches connect two nodes if a transition exists between the corresponding states. A cost-function value is associated with each node, and based on this value the exploration of the tree is driven. As soon as the exploration of the tree is finished, the corresponding input is applied to the system and the procedure is repeated.

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“…Thus, a main issue is the one of minimizing the "attention" that a control task requires, or which is the same, maximizing the dwell time (also called interexecution time), i.e., time between two consecutive executions [10]. We reframe the above challenge in the context of optimal impulse control problems (see, e.g., [5,7,8,18,22,23] and references therein) with uniformly distributed impulse instants.…”

Section: Introductionmentioning

confidence: 99%

“…We simulate the receding horizon procedure starting at x(0). In the experiment, we compare the solutions of (9) and (8). The first solution is obtained by solving the linear problem min u∈Pc u, P = {u ∈ R N : linear constraints (7), 0 ≤ u ≤ 1} , and returns a suboptimal cost.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

Impulsively-controlled systems and reverse dwell time: A linear programming approach

Bauso

2015

Nonlinear Analysis: Hybrid Systems

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We present a receding horizon algorithm that converges to the exact solution in polynomial time for a class of optimal impulse control problems with uniformly distributed impulse instants and governed by so-called reverse dwell time conditions. The cost has two separate terms, one depending on time and the second monotonically decreasing on the state norm. The obtained results have both theoretical and practical relevance. From a theoretical perspective we prove certain geometrical properties of the discrete set of feasible solutions. From a practical standpoint, such properties reduce the computational burden and speed up the search for the optimum thus making the algorithm suitable for on-line implementation in real-time problems. Our approach consists in approximating the optimal impulse control problem via a binary linear programming problem with a totally unimodular constraint matrix. Hence, solving the binary linear programming problem is equivalent to solving its linear relaxation. Then, given the feasible solution from the linear relaxation, we find the optimal solution via receding horizon and local search. Numerical illustrations of a queueing system are performed.

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Optimal control of a class of hybrid systems

Cited by 83 publications

References 20 publications

Optimal switching control design for polynomial systems: an LMI approach

Optimal switching control design for polynomial systems: an LMI approach

Model predictive control of discrete-time hybrid systems with discrete inputs

Impulsively-controlled systems and reverse dwell time: A linear programming approach

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