Fundamental limitations on the time-domain shaping of response to a fixed input

Abstract

In this paper, we examine time-domain limitation-of-performance problems in feedback control system design. The main result is a theorem which gives a dual formulation to the problem of determining absolute limits on the time-domain shaping of the response to a fixed input. The duals that arise are particularly illuminating. They have a simple interpretation as optimizing the impulse response of a transfer function whose poles are readily constructed from the data. Much insight is obtainable from the dual, for example, in helping to classify classes of plants with the same quantitative behavior. New results on classical performance measures, such as overshoot and undershoot, are presented. We also examine minimization of the difference between the maximum and minimum values of the error response, a quantity we term fluctuation.

Continuous-time performance limitations for overshoot and resulted tracking measures

“…The constraints for the dual formulation presented here can be interpreted as arising from an open-loop dynamic system. These structural insights are exploited to derive new results on the influence of plant pole/zero locations, or rise-time constraints on achievable performance, giving results of identical form to those obtained for discrete time in [5]. The results presented here are applicable for general reference inputs ( [9,10] assumes step-input) and to systems with more general exogenous inputs, for example a fixed disturbance entering at the plant output.…”

“…whose proof follows by trivial modification of the proof of its discrete counterpart in [5]. Later, we will consider some simple time-domain constraints, so require treatment of functionals of form f 0 +ι T for appropriate choices of sets T representing these additional conditions.…”

“…We then have µ = µ + + µ − and the variation of µ (denoted |µ|) is defined by |µ| := µ + − µ − . Note that our sign convention has been chosen to conform with that used in the discrete-time analysis of [5], but differs from the usual choice in measure theory, where µ + and µ − are both non-negative, whereas here we have µ − ≤ 0. Recall that we assume M(R + ) to be normed by µ := |µ|(R + ).…”

“…The simultaneous imposition of hard bounds on the output, in conjunction with overshoot minimisation, can also be handled, allowing consideration of the inevitable trade-off between rise-time and overshoot performance. These issues have been examined in a discrete-time setting in [5] etc. The investigation of time-domain performance limitations is mathematically more challenging in continuous time than in discrete time.…”

“…Further, we consider a general class of performance functionals and set up a general duality framework for the analysis of such problems (in the style of [5] for the case of discrete time). These functionals include as special cases the wellknown classical criteria of overshoot, undershoot and others.…”