A survey of techniques for approximating reachable and controllable sets

Gayek, J.E.

doi:10.1109/cdc.1991.261702

Cited by 33 publications

(12 citation statements)

References 29 publications

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“…See also [Gay86,Las87,SS90a,Las93] for further results on the topic. For further references, we refer to the survey [Gay91] or to [RKKM05a] for some more recent contributions concerning the computation and approximation of the minimal invariant set [RKKM05a].…”

Section: Historical Notes and Commentsmentioning

confidence: 99%

Set-Theoretic Methods in Control

Blanchini

Miani

2015

Systems &Amp; Control: Foundations &Amp; Applications

209

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Section: Historical Notes and Commentsmentioning

confidence: 99%

Set-Theoretic Methods in Control

Blanchini

Miani

2015

Systems &Amp; Control: Foundations &Amp; Applications

209

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“…Remark 1. Note that the minimization problem (16) can be written in the following LMI form: (19) Here, we apply the Kuhn-Tucker Theorem to solve the optimization problem of (16).…”

Section: Maximize the Level Set Under The Lyapunov Descent Critementioning

confidence: 99%

Stability of Linear Time‐invariant Open‐loop Unstable Systems With Input Saturation

Wang

Mukai

2004

Asian Journal of Control

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This paper analyzes the stability of a linear time-invariant open-loop unstable system subject to input saturation. First, we extend the idea of approximating the locally asymptotically stable region (controllable set) of the system for the case where the control is small enough to be unsaturated (inside the linear region), to the case when the control is allowed to saturate. It is shown that, when the Lyapunov descent criterion and the Kuhn-Tucker Theorem is applied, a superior locally asymptotically stable region is found. A technique for approximating the locally asymptotically stable region is presented.

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“…Such issues arise in many applied problems of control and computation, being an object of special interest [4,6,7,14,18,20].…”

Section: Introductionmentioning

confidence: 99%

On Ellipsoidal Techniques for Reachability Analysis. Part II: Internal Approximations Box-valued Constraints

Kurzhanski

Varaiya

2002

Optimization Methods and Software

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Following Part I, this article continues to describe the calculation of the reach sets and tubes for linear control systems with time-varying coefficients and ellipsoidal hard bounds on the controls and initial states. It deals with parametrized families of internal ellipsoidal approximations constructed such that they touch the reach sets at every point of their boundary at any instant of time. The reach tubes are thus touched internally by ellipsoidal tubes along some curves. The ellipsoidal tubes are chosen here in such a way that the touching curves do not intersect and that the boundary of the reach tube would be entirely covered by such curves. This allows exact parametric representation of reach tubes through unions of tight internal ellipsoidal tubes as compared with earlier methods based on constructing one or several isolated approximating tubes. The method of external and internal ellipsoidal approximations is then propagated to systems with box-valued hard bounds on the controls and initial states. It appears that the proposed technique may well work for nonellipsoidal, box-valued constraints. This broadens the range of applications of the approach and opens new routes to the arrangement of efficient numerical algorithms.

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A survey of techniques for approximating reachable and controllable sets

Cited by 33 publications

References 29 publications

Set-Theoretic Methods in Control

Set-Theoretic Methods in Control

Stability of Linear Time‐invariant Open‐loop Unstable Systems With Input Saturation

On Ellipsoidal Techniques for Reachability Analysis. Part II: Internal Approximations Box-valued Constraints

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