Performance limitations in sensitivity reduction for nonlinear plants

This paper investigates performance limitations imposed by 'non-minimum phase' characteristics of a nonlinear time-varying plant. A performance criterion is defined which, in the linear case, is analogous to minimizing the sensitivity over a given frequency band. It is shown that if the nonlinear plant is 'non-minimum phase', then the frequency-weighted sensitivity cannot be made arbitrarily small while keeping the overall sensitivity bounded. The non-minimum phasedness of the plant is stated in terms of a deficiency in its range. These results extend the familiar 'push/pop' phenomenon in sensitivity optimization to a nonlinear time-varying setting. considered in [7,10,16]. For example it has been shown for plants with rational transfer functions that if the plant is minimum phase, then /~(P, K, ~2) can be made arbitrarily small while keeping the overall sensitivity bounded [16]. In case the plant has open right-half-plane zeros, then making /~(P, K, I2) arbitrarily small comes at the cost of making the overall sensitivity arbitrarily large [7]. This has been called the 'push/pop' or 'waterbed' phenomenon. These results were later extended to plants with irrational transfer functions in [10]. For further results on achievable performance, see [2,8,11]. In this paper, we consider analogous results for general nonlinear time-varying plants and com-pensators. Using a disturbance rejection interpretation of the above performance measure, an analogous performance measure is defined as follows. Let ~ be a given bounded class of finite-energy disturbances. Then define