Delay-feedback using derivatives for minimal time linear control problems

Asner, Bernard A.; Halanay, A.

doi:10.1016/0022-247x(74)90230-3

Cited by 18 publications

(3 citation statements)

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“…Derivative feedback has been considered previously for the problem of stabilization of linear neutral systems in [2], [39]. Since the characteristic equation for the homogeneous system may be written …”

Section: Extension To Unstable D-operatorsmentioning

confidence: 99%

A several complex variables approach to feedback stabilization of linear neutral delay-differential systems

Byrnes

Spong

Tarn

1984

Math. Systems Theory

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Abstract. In this paper we consider the problem of uniform asymptotic stabilization of neutral delay differential systems by a suitable choice of linear feedback law, where the controller is static and at time t o has full knowledge of x(t 0) ~ R" as well as x(t o -d) for finitely many delays present in the system. Using complex analysis in several variables in a Banach algebra context we present a solution to this problem in the form of a sufficient condition for the existence of such a stabilizing feedback gain. This condition is a weak form of (algebraic) reachability together with a condition, which we call resolvability, on the solution of an associated algebraic Riccati equation. In the case of commensurable delays our results apply to neutral systems having a D-operator which is stable in the sense of Cruz and Hale. In the non-commensurate case, our results hold for systems with a D-operator satisfying a stronger condition on the spectrum of the evolution equation, which we call formal stability. A graphical criterion, in the spirit of the multi-dimensional Nyquist theory, is presented for formal stability of D. We find these results especially interesting in light of recent work [39] which shows that, for a special class of systems, formal stability is necessary for stabilization by a feedback gain of the type considered here. This, in fact, gives a counterexample [39] to the well known condition of Pandolfi [40].Included among our results is an extension to the neutral case of the result in [11] which states that if the pointwise Kronecker indices are constant then the system is feedback equivalent to a delay-free system. As corollaries to this result we obtain some existing results [33] on canonical forms for time-delay systems, and, in addition, exhibit a large class of systems which are resolvable.

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Section: Extension To Unstable D-operatorsmentioning

confidence: 99%

A several complex variables approach to feedback stabilization of linear neutral delay-differential systems

Byrnes

Spong

Tarn

1984

Math. Systems Theory

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“…A large number of investigations, devoted to problems of optimal control for objects described by differential equations with deviating argument of neutral type, are devoted to a certain extent to the transfer of the maximum principle to this case [19,102], [231,232,243,278,280,288,297], etc. In a series of papers (see, for example, [232,243,279,280]) one investigates the controlled system (6) where A1, A 2, A 3, B are constant or variable matrices and h is a constant lag.…”

Section: [Xl --~ F (T X (T) X' (T) X (T--) X' (T --~)) Dtmentioning

confidence: 99%

“…Linear problems of optimal control have been considered also in [231,288,295]. One has considered problems of finite control for systems described by higher-order differential equations of neutral type [61], the problem of the control of a system of neutral type with several inputs of the group of ordinary dynamic regulators, the problem of the optimal control for a hyperbolic system of partial differential equations with a deviating argument of neutral type [19] and certain other problems.…”

Section: Various Problems Of Optimalmentioning

confidence: 99%