Dynamical system design from a control perspective: finite frequency positive-realness approach

Nonminimum phase zeros are well known to limit the best achievable control performance when the control gain is allowed to be arbitrarily high. On the other hand, the phase crossover appears to be a limiting factor for performance when high-gain controllers are not allowed. In particular, the positive-realness in a finite frequency range seems crucial for achieving good performance in the presence of control constraints. This paper will first give multiple reasons to support this conjecture, and then develop a systematic method for designing mechanical systems to achieve the finite frequency positive-real (FFPR) property. Specifically, we present a state-space characterization of the FFPR property by generalizing the well known KalmanYakubovich-Popov lemma to deal with a class of frequency domain inequalities that are required to hold within a finite frequency interval. The result is further extended for uncertain systems to give a sufficient condition for satisfaction of a robust FFPR property. The (nominal) FFPR result is interpreted in the time-domain in terms of input/output signals. Finally, we show that certain sensor/actuator placement problems to achieve the FFPR property can be reduced to finite dimensional convex problems involving linear matrix inequalities. The method is applied to the shape design of a swing-arm for magnetic storage devices with the objective of maximizing the control bandwidth achievable with a limited actuator power.