Optimizing the Tracking Performance in Robust Control Systems

Abstract

A typical control engineering problem deals with the design of a control system subject to closed-loop stability and certain performance requirements. The requirements may include the figures of merit such as gain/phase margin, bandwidth, and tracking error to a reference command. The control system is required to achieve the design objectives against unknown or unmeasurable disturbances. The difficulty arises since the plant is often poorly modeled and the set of performance requirements is typically stringent. The robust control theory attempts to address the question of stability and performance of multivariable systems in the face of modeling errors and unknown disturbances (Zhou et al., 1996). In robust control theory, the question concerning the achievable performance limits is generally posed as an optimization problem in an appropriate mathematical setting. A major benefit of this approach is that it provides a means to optimize the system performance by trading off various stringent, and often conflicting, specifications against each other. In the last three decades, H∞ control theory has evolved as the primary multivariable optimization and synthesis tool that can effectively deal with the modeling errors and unknown disturbances (Skogesttad & Postlethwaite, 2007). In a tracking problem, the reference command is usually specified as a step or ramp signal. Accordingly, the tracking error is also specified in terms of such signals. This class of signals, however, does not model all command signals of interest. For example, a servo control system may be required to track a periodic signal of a fixed period. For this class of applications, the tracking performance must instead be specified in terms of a periodic command signal. Since every periodic signal can be represented by its Fourier series for all time, the steady state tracking performance of a linear feedback system with a periodic command signal can be studied in terms of the steady state tracking performance of each of its sinusoidal components. Design of the control systems that can track periodic reference signals falls in the category of repetitive control (Hara et al., 1998; Lee & Smith, 1998; Sugimoto & Washida, 1997). This has been an active area of research in the last three decades where many successful applications have been reported in the literature. However, applications of the results to certain high performance positioning systems have proved to be more challenging. For example, in (Broberg & Molyet, 1994) a robust repetitive control system is designed to improve the turn-around sinusoidal tracking performance of the imaging mirror system of 6