SUMMARYThere exists a well-known fundamental limitation upon the achievable setpoint tracking performance of a non-right-invertible plant. This limitation manifests itself, for example, in the cost associated with the cheap control tracking problem. In this paper, we provide a new interpretation of this limitation. We show that the cheap control cost may be decomposed into the sum of two terms. The first of these depends upon certain non-minimum phase zeroes that include the non-minimum phase plant zeroes. The second term depends upon the extent to which the plant direction varies with frequency. To state these results, we first develop a co-ordinate transformation that may be used to define the notion of frequency dependent plant direction and to display the relevant non-minimum phase zeroes. We also show that the cheap control cost is connected to an integral relation that constrains the performance of any stable closed-loop system (not necessarily under cheap control) for which the plant has a single control input and two performance outputs.
There has been a large literature on the feedback control of flexible and resonant systems. Such systems arise naturally when system weight and or response speed issues push designers toward lighter, faster structures for a range of mechanical systems. Feedback control of such systems is often proposed to ameliorate the effects of the resonance. In this paper, we investigate the extent to which the dynamic structure of a simple class of resonant systems limits the achievable feedback control performance for such systems. It turns out that in the class of systems considered, there is a trade off between three common control objectives, namely: (i) good initial transient response (that is the absence of large overshoot or undershoot in the initial rise time), (ii) fast response, (iii) good settling behaviour (that is, the absence of very slow modes in the step response).
A fundamental limitation exists in the achievable tracking performance of non-right-invertible systems. This limitation manifests itself in the cheap control tracking cost, which we show to be a function of the plant non-minimum phase zeros and of the variation with frequency of the plant direction. The cheap control tracking cost is further connected with an integral relation that constrains the performance of any stable closedloop system where the plant has a single input and two outputs.
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