A controllability minimum principle

Grantham, Walter J.; Vincent, Thomas L.

doi:10.1007/bf00933917

Cited by 61 publications

(17 citation statements)

References 4 publications

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“…Furthermore the existence of such a control law is guaranteed for single input, controllable, two dimensional systems of the form given here [17]. The input u which will drive the system along the boundary of the reachable set for the retro system may be easily determined by applying the controllability maximum principle [4]. In particular, application of this principle to (53) yields…”

Section: Unstable Second Order Systemmentioning

confidence: 99%

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Some performance measures useful in the design of controllers for two dimensional dynamical systems subject to uncertain inputs

Vincent

Lin²

1995

Dynamics and Control

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Abstract. Classical control theory has always been concerned with uncertain inputs. However, this concern has generally been implicit rather than explicit. Modern control methods tend to be more explicit. In particular, methods based on game theory and methods based on a Lyapunov type of analysis have been proposed. One question which seems to be largely ignored is: how do these various methods compare when applied to a typical control problem? Indeed what performance measures can we use to compare various control designs dealing with uncertain inputs? This paper addresses both of these questions. We define performance measures called the v-reachable set and the v-attractive set. The v-reachable set measures the extent that the uncertain input can drive the system away from the target set and the v-attractive set measures the extent that the uncertainty diminishes the size of the controllable set to the target. An ideal control law would be one in which the v-reachable set is the target set and the v-attractive set is the controllable set. Five different controllers based on five different design philosophies are applied to two different control systems. We find that there can be marked differences in the above performance measures. However using these performance measures as guide to choosing feedback parameters, the differences between the various design methods can be minimized. It is found that for these problems a modified game theoretic approach and the Lyapunov type of approach tend to provide the best designs.

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Section: Unstable Second Order Systemmentioning

confidence: 99%

“…In particular, the boundary of the v-reachable (v-attractive) set may be found by satisfying the necessary conditions for a control which will drive the system along the boundary of the v-reachable (v-attractive) set [3,4], i.e. v must maximize (minimize) H = ~lgl(Xl,X2,v) @~292(Zl,X2,~ ).…”

Section: ~2 = 92(xlx2 V)mentioning

confidence: 99%

Some performance measures useful in the design of controllers for two dimensional dynamical systems subject to uncertain inputs

Vincent

Lin²

1995

Dynamics and Control

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“…More [12] Determination of a controllable set for a controlled dynamic system 175 precisely, we consider the following…”

Section: An Applicationmentioning

confidence: 99%

Determination of a controllable set for a controlled dynamic system

Péng

Yong

1991

J. Aust. Math. Soc. Series B, Appl. Math

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“…Recently, reachable sets have been used to increasing effect, for example, in problems in engineering design, in the study of the vulnerability of renewable resource systems, and in the stability of an ecosystem (for instance, a food chain) against external perturbations. See, for example, Grantham and Vincent (1975), Vincent and Anderson (1979), Goh (1976Goh ( , 1980, Fisher and Goh (1981), Vincent (1981) and Leitmann (1983). However, the theory of reachable sets for both linear and nonlinear .systems remains to be fully developed, and present techniques for calculating reachable set boundaries of systems of order three or more are generally difficult or impossible to apply.…”

Section: Introductionmentioning

confidence: 97%

“…However, the theory of reachable sets for both linear and nonlinear .systems remains to be fully developed, and present techniques for calculating reachable set boundaries of systems of order three or more are generally difficult or impossible to apply. The 'controllability method', in which a control is found to produce a trajectory in the boundary of the reachable set, is based on an application of a controllability minimum principle (Grantham 1973, Grantham andVincent 1975); a related 'minimum-time' approach consists of calculating the reachable-set boundary as an asymptotic limit of trajectories which approach the boundary in a minimum time (Vincent and Anderson 1979). An alternative approach is to use a Lyapunov method to estimate the boundary of a reachable set (Leifmann and Skowronski 1977), and it is this method we shall employ here.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov approximation of reachable sets for uncertain linear systems

Summers

1985

International Journal of Control

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A method is presented for calculating an approximation to the reachable set from the origin of a class of linear control systems. The method involves the construction of a Lyapunov-like function, and ultimately requires the numerical solution of a simple optimization problem. Advantages of the method are that it is applicable to n-dimensional systems and does not require calculation of orbits or a choice of boundary conditions. Further, the system need not be completely controllable for the method to be applicable. However, while the method ensures that the reachable set lies within a specified set, a closeness-of-fit criterion is not available.

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A controllability minimum principle

Cited by 61 publications

References 4 publications

Some performance measures useful in the design of controllers for two dimensional dynamical systems subject to uncertain inputs

Some performance measures useful in the design of controllers for two dimensional dynamical systems subject to uncertain inputs

Determination of a controllable set for a controlled dynamic system

Lyapunov approximation of reachable sets for uncertain linear systems

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