On the minimax reachability of target sets and target tubes

Abstract-This paper presents new results that allow one to compute the set of states that can be robustly steered in a finite number of steps, via state feedback control, to a given target set. The assumptions that are made in this paper are that the system is discrete-time, nonlinear and time-invariant and subject to mixed constraints on the state and input. A persistent disturbance, dependent on the current state and input, acts on the system. Existing results are not able to address state-and input-dependent disturbances and the results in this paper are, therefore, a generalization of previously published results. One of the key aims of this paper is to present results such that one can perform the relevant set computations using polyhedral algebra and computational geometry software, provided the system is piecewise affine and the constraints are polygonal. Existing methods are only applicable to piecewise affine systems that either have no control inputs or no disturbances, whereas the results in this paper remove this limitation. Some simple examples are also given that show that, even if all the relevant sets are convex and the system is linear, convexity of the set of controllable states cannot be guaranteed.

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“…In other words, is given by (17) with . Following a standard procedure [4], the sequence of sets may be calculated recursively as follows:…”

Section: B Reachability Analysis and Invariant Setsmentioning

confidence: 99%

Reachability Analysis of Discrete-Time Systems With Disturbances

Raković

Kerrigan

Mayne

et al. 2006

IEEE Trans. Automat. Contr.

154

108

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“…The topics of robust optimization and robust control have been studied, under different names, by a variety of academic groups, mostly in operations research (Ben-Tal and Nemirovski [2,3]; Ben-Tal et al [6]; Bertsimas and Sim [12,13]; Bertsimas et al [14]) and control theory (Bertsekas and Rhodes [10], Fan et al [19], El-Ghaoui et al [18], Grieder et al [22], Bemporad et al [1], Kerrigan and Maciejowski [25], Zhou and Doyle [33], Dullerud and Paganini [17]), with considerable effort put into justifying the assumptions and general modeling philosophy. As such, the goal of this paper is not to motivate the use of robust (and, more generally, distribution free) techniques.…”

mentioning

confidence: 99%

Optimality of Affine Policies in Multistage Robust Optimization

2010

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In this paper, we prove the optimality of disturbance-affine control policies in the context of one-dimensional, constrained, multistage robust optimization. Our results cover the finite-horizon case, with minimax (worst-case) objective, and convex state costs plus linear control costs. We develop a new proof methodology, which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function. Apart from providing an elegant and conceptually simple proof technique, the approach also entails very fast algorithms for the case of piecewise-affine state costs, which we explore in connection with a classical inventory management application.

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“…The invariant sets are computed for autonomous systems and can be used to "find, for a given feedback controller u = k(x), the set of states whose trajectory will never violate the system constraints". The following definitions are derived from [12], [9], [8], [22], [16].…”

Section: B Background On Invariant Setsmentioning

confidence: 99%

“…Control invariant sets are defined for systems subject to external inputs and can be used to "find the set of states for which there exists a controller such that the system constraints are never violated". The following definitions are derived from [12], [9], [8], [22].…”

Section: B Background On Invariant Setsmentioning

confidence: 99%

A reference governor approach for constrained piecewise affine systems

Falcone

Borrelli

Pekař

et al. 2009

2009 European Control Conference (ECC)

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Abstract-In this paper we present a novel approach to the design of reference tracking controllers for constrained, discrete-time piecewise affine systems and apply it to a vehicle yaw control problem. The paper is divided in two parts. In the first part, we present a methodology for designing reference tracking controllers for constrained, discrete-time piecewise affine systems. The approach follows the idea of reference governor techniques where the desired set-point is filtered by a system called the "reference governor". Based on the system current state, set-point, and prescribed constraints, the reference governor computes a new set-point for a low-level controller so that the state and input constraints are satisfied and convergence to the original set-point is guaranteed.In the second part of the paper, we apply the proposed approach to a vehicle dynamics control problem where the vehicle yaw rate have to be controlled through an Active Front Steering (AFS) system.

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On the minimax reachability of target sets and target tubes

Cited by 395 publications

References 6 publications

Reachability Analysis of Discrete-Time Systems With Disturbances

Reachability Analysis of Discrete-Time Systems With Disturbances

Optimality of Affine Policies in Multistage Robust Optimization

A reference governor approach for constrained piecewise affine systems

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