4. Motion Control Using an \(\mathcal {H}_\infty \)-Control-Based Approach

This chapter presents the key contents of the monograph, that is, the \(\mathcal {H}_\infty \)-control-based approach for the OBM robust control problems. This control methodology consists of several schemes, specifically nonlinear state-feedback control to reduce the nonlinearity, parametric model uncertainty representation, \(\mathcal {H}_\infty \) control with weighting functions, and additional linear state-feedback control to compensate for the \(\mathcal {H}_\infty \) control scheme. In particular, in order to less conservatively and effectively represent parametric model uncertainties due to the payload variations, we have developed a machinery the “extended matrix polytope,” which is an extension of the conventional matrix polytope, in the aim of representation of a non-convex parameter space. In this chapter, we begin with brief review of the basic notion of \(\mathcal {H}_\infty \) control, and introduce the extended matrix polytope. Then, we present the control design method with some design examples considering four types of \(\mathcal {H}_\infty \) controllers, and perform analyses on the designed control systems in terms of properties such as system poles and zeros, frequency response, and robustness by utilizing two different approaches, one of which is based on \(\mu \)-analysis with the extended matrix polytope, and the other one is a Lyapunov-theory-based one with a state-dependent coefficient (SDC) form. From the results of analyses, the designed control system reveals favorable properties in terms of disturbance rejection, tracking control, and robustness. Hence, their practical control performances can be also expected, which will be shown in the next chapter.