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.
Converse.pdf - Published Version
Restricted to Registered users only
Download (506Kb) | Request a copy
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 .The problem of instability of feedback systems with a single time-varying nonlinearity was initially considered by Brockett and Lee  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  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  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  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 , 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|
Actions (login required)