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Converse hölder inequality and the Lp-instability of nonlinear time-varying feedback systems

Venkatesh, YV (1988) Converse hölder inequality and the Lp-instability of nonlinear time-varying feedback systems. In: Nonlinear Analysis:Theory, Methods and Applications, 12 (3). pp. 247-258.

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Official URL: http://dx.doi.org/10.1016/0362-546X(88)90111-3

Abstract

WE DEAL with the Lp-instability of a nonlinear time-varying feedback system governed by the pair of equations: v(t) = x(t) - k(t)cp (y(t)),y(t) = (qJv)(t) (1) := ~ g i v ( t - ri) + g( O v ( t - r) dr i=1 for all t I> 0, where x, v, and y are respectively the input to the system, the error signal and output of the system; N is a time-invariant linear operator; qv is a first and third quadrant continuous, memoryless (monotone) nonlinearity; and k is a time-varying gain. For assumptions on the components of (1), see Section 2 below. We derive Lp-instability conditions in terms of the frequency response of ~3 and a general causal + anticausal multiplier function. The derivation is based on the converse H61der inequality and the "energy balance" argument as used in [1].The problem of instability of feedback systems with a single time-varying nonlinearity was initially considered by Brockett and Lee [2] who, in a Lyapunov-Chetaev setting, derived an instability version of the circle criterion [3, 4] under certain assumptions on the related linear time-invariant systemwith a constant gain in the feedback loop. It is found that this instability theorem is conservative [5, Section 5]. See [5] for references to other instability results. As far as the analysis of instability of feedback systems by functional methods is concerned,different types of results are available. Willems [6] extends the domain of operators and imbedsthe system causal operator in a noncausal operator in an attempt to prove the noncausality of the inverse of the original operator by contradiction. This technique has been explicitly used [7] but there do exist certain unresolvable difficulties [8; 9, Chapter 7]. Similar difficulties are encountered in the results of Takeda and Bergen [10, 11] and Steding and Bergen [12], who consider a subclass of inputs over which the linear time-invariant part of the system is well- behaved (and satisfies some additional conditions), and prove instability by contradiction. See [9, Chapter 7] for a complete analysis of these contributions in which, more importantly, the assumptions made on the linear time-invariant part of the system are too severe.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Elsevier Science Ltd.
Keywords: Instability;Lp-instability;Hölder inequality;feedback systems;nonlinear;time-varying;converse Hölder inequality; Riesz-Thorin theorem
Department/Centre: Division of Electrical Sciences > Electrical Engineering
Date Deposited: 01 Jul 2004
Last Modified: 09 Jan 2012 07:50
URI: http://eprints.iisc.ernet.in/id/eprint/492

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