On the Value Functions of the Discrete-Time Switched LQR Problem

Zhang, Wei; Hu, Jianghai; Abate, Alessandro

doi:10.1109/tac.2009.2031574

Cited by 90 publications

(60 citation statements)

References 13 publications

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“…Thus, in order to make some comparisons, Figure 1 shows the state space trajectories for the switching law given by (30) and the optimal one. The later is obtained by NL programming in a suitable formulation taking into account singular arcs [24].…”

Section: Example 1 : a Regular Casementioning

confidence: 99%

“…Several approaches have been used to tackle this problem, one can cite for example [18,26,29,30] for dynamic programming approaches, [1,2,27] for variational approaches, or [6,8,12] for Lyapunov based approaches. This problem is not easy, even numerically [24,27] and the design of a stabilizing feedback law based on the optimization of a criterion is a challenging task.…”

Section: Introductionmentioning

confidence: 99%

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An LQ sub-optimal stabilizing feedback law for switched linear systems

Riedinger

Vivalda

2014

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

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The aim of this paper is the design of a stabilizing feedback law for continuous time linear switched system based on the optimization of a quadratic criterion. The main result provides a control Lyapunov function and a feedback switching law leading to sub optimal solutions. As the Lyapunov function defines a tight upper bound on the value function of the optimization problem, the sub optimality is guaranteed. Practically, the switching law is easy to apply and the design procedure is effective if there exists at least a controllable convex combination of the subsystems.

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Section: Example 1 : a Regular Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An LQ sub-optimal stabilizing feedback law for switched linear systems

Riedinger

Vivalda

2014

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

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“…Kozyakin [24,25] provides a counter example to the famous finiteness conjecture that includes an interesting and deep analysis of the "most unstable" switching law. Another related direction is the study of optimal control problems for DT linear switched systems using dynamic programming [52].…”

Section: 2010 Draftmentioning

confidence: 99%

Analysis of Discrete-Time Linear Switched Systems: A Variational Approach

Monovich¹,

Margaliot²

2011

SIAM J. Control Optim.

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A powerful approach for analyzing the stability of continuous-time switched systems is based on using tools from optimal control theory to characterize the "most unstable" switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific "most unstable" switching law. More generally, this so-called variational approach was successfully applied to derive nice-reachability-type results for both linear and nonlinear continuous-time switched systems.Motivated by this, we develop in this paper an analogous approach for discrete-time linear switched systems. We derive and prove a necessary condition for optimality of the "most unstable" switching law. This yields a type of discrete-time maximum principle (MP). We demonstrate using an example that this MP is in fact weaker than its continuous-time counterpart. To overcome this, we introduce the auxiliary system of a discrete-time linear switched system, and show that regularity properties of time-optimal controls for the auxiliary system imply nice-reachability results for the original discretetime linear switched system. Using this approach, we derive several new Lie-algebraic conditions guaranteeing nice-reachability results. These results, and their proofs, turn out to be quite different from their continuous-time counterparts.

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“…In the last decade, many contributions have addressed the design of stabilizing switching laws for switched systems both in continuous and discrete time (see for examples [1], [2], [3], [4] for dynamic programming approaches, [5], [6], [7] for variational approaches, and [8], [9], [10] for Lyapunov based approaches). This problem cannot be considered trivial even numerically [11], [7] and the goal to design closed loop control, based on the optimization of a criteria, is a challenging task.…”

Section: Introductionmentioning

confidence: 99%

“…Up to now, the exact solution of a switched LQ problem is not available. Only approximations of the state feedback switching law have been proposed in the literature using the dynamic programming [1], [2], [3], [4]. The practical obstacle to the application of these methods relies on the difficulties to compute numerically a good approximation of the solution not only for small dimensional problems.…”

Section: Introductionmentioning

confidence: 99%

A Switched LQ Regulator Design in Continuous Time

Riedinger

2014

IEEE Trans. Automat. Contr.

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In this paper, the design of a LQ regulator for linear switched systems in continuous time is investigated. From a relaxation of the optimal control problem, a Lyapunov based switching law is provided. Even if the subsystems are all unstable, the state feedback switching law can be applied subject to a positiveness condition. In any cases, the real cost is always upper bounded by the Lyapunov function value. The optimality of the switching law is also discussed and we prove that the switching conditions are optimal in some generic cases. This point explains why the obtained results over examples approach finely the optimal solutions. Finally, a design strategy is also given that extends the results to the cases where the subsystems are controlled linear systems.

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On the Value Functions of the Discrete-Time Switched LQR Problem

Cited by 90 publications

References 13 publications

An LQ sub-optimal stabilizing feedback law for switched linear systems

An LQ sub-optimal stabilizing feedback law for switched linear systems

Analysis of Discrete-Time Linear Switched Systems: A Variational Approach

A Switched LQ Regulator Design in Continuous Time

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