This dissertation deals with the problem of designing a multiobjective controller for uncertain linear systems with sinusoidal disturbance of uncertain frequencies. The control objectives include the robust $H_{\infty}$ performance, the regional pole placement, and a performance to diminish the influence of the uncertain frequencies. A condition to meet each performance is also expressed by a linear matrix inequality and the controller design procedure is established by using the LMI convex optimization algorithm. The controller design method is applied to the track-following system of an optical disk drive and is evaluated through an experiment.
First, we design a robust $H_{\infty}$ controller with regional pole constraints and sinusoidal disturbance rejection under the assumption that the sinusoidal disturbance has constant known frequencies. The internal model principle is introduced to robustly reject the sinusoidal disturbance. A condition satisfying the regional pole constraints is expressed by an LMI using the Lyapunov theory and S-procedure. Consequently, the problem becomes a nonconvex optimization problem which is hard to solve, but we convert it into an LMI convex optimization problem by introducing the common Lyapunov matrix and new controller variables. In addition, we mention that the robust design can lead to a fragile controller, in the sense that very small perturbations of the controller coefficients can result in instability.
Secondly, we propose a controller design method for diminishing the effect of sinusoidal disturbance of uncertain frequencies. It is assumed that the uncertain frequencies vary within some bounds and the controller includes the dynamics corresponding to the nominal frequencies. A virtual system including the dynamics for the uncertain frequencies is introduced and a design method which minimizes the difference between system gains of the virtual system and the closed-loop system is presented. The advantages of the proposed design method are examined by comparing it with a design method that only minimizes the $H_{\infty}$norm of the transfer function between sinusoidal disturbance and measured output.
Lastly, we deal with a multiobjective control problem using an LMI approach and apply it to the track-following system of an optical disk drive. Owing to deficiencies in the track-geometry and eccentric rotation of the disk, disturbances acting on the track-following system contain a significant sinusoidal component which has the uncertain frequency varying directly as the angular velocity changes. A nonfragile tracking controller is obtained by applying the proposed design method and is implemented digitally on the DSP96002. Several simulation and experimental results present that the designed track-following system satisfies the desired transient performance and diminishes the effect of sinusoidal disturbance of uncertain frequency.
