Since the controller is part of the overall closed-loop system, it is necessary that the designed controller be able to tolerate some uncertainty in its coefficients. It is because the controller implementation may be subject to the imprecision such as AID and D/A conversion, finite word length, and round-off errors in numerical computations. Also, the designed controller is required to readjust because no scalar index can capture all the performance requirements of a control system. Therefore, an adequate stability and performance margin is required for the designed nominal controllers.
In this dissertation, we study the methods to design the non-fragile fixed-structured controller for real parametric uncertain systems. It is necessary to take the controller parameter perturbation into consideration when we design the robust controller. If not, the resulting controller may show the fragility property. When we impose the controller parameter perturbation, the structure of the controller must be given. Therefore, we assume that the controller has fixed-structure. The fixed-structure controller is practically necessary especially when the robust controller synthesis results in a high-order controller.
In SISO systems, we propose the robust controller design method using the Mapping theorem. In the method, the plant uncertainty and controller parameters are of the multilinear form in the stability and performance conditions. Then, the controller synthesis problem is easily recast to the Linear Programming Problem. There are many methods to solve the linear programming problem.
In MIMO systems, we consider the state-feedback problem. In the recent work, the effect of LQ/ $H_2$ robust synthesis of uncertain, static state feedback controllers, for linear systems with structured uncertainties in the dynamic matrix only. The result is represented in the form of non-convex problem when multiplicative structured uncertainties are allowed in the controller. A guaranteed-cost approach that turned out to be a worst-case gain margin optimal synthesis problem was formulated in a particular case using a linear matrix inequalities computational paradigm. In this paper, we address the design of State-feedback Robust Guaranteed-Cost Controllers with controller gain variations. Since the unstructured uncertainty is the most dominant uncertainty in the modeling of the plant, the plant is assumed to have the unstructured uncertainty. It is necessary to take the controller parameter perturbation into consideration when we design the robust controller. Otherwise, the resulting controller may show the fragility property. That is to say, the extremely small controller parameter variation may result in the instability of the overall closed-loop system. Therefore, the design purpose is that the maximum performance index is guaranteed in the presence of the unstructured plant uncertainty and controller parameter variations. The controller variations are of the unstructured and multiplicative types. In both case, we present the sufficient conditions for robust stability and performance, and the solving algorithms.
We also address the stabilization problem of the static output feedback with controller variations. The fixed-order dynamic output feedback problem with controller variations is treated as a special case of the static output feedback case. . We also present the sufficient conditions for stabilization of the static output feedback case and a stabilizing controller.
