In this dissertation, uncertain linear systems with a class of time-varying real parameter uncertainties are considered based on the concept of quadratic stabilization, which relies on existence of a quadratic Lyapunov function. The real parameter uncertainties of interest are known as an effective way to describe the modeling error in state space. The dissertation has the focus on developing three control design methods - the robust LQR/LQG approaches, robust sliding mode control and constrained static output feedback stabilization.
In the presence of structured real parameter uncertainties, the notion of strongly quadratic stability is defined, which combines the scaling idea with quadratic stability in order to reduce conservatism in assessing the stability issue. The stability concept has advantages in treating time-varying structured uncertainties and, moreover, taking robust performance into account specially in time domain. The main framework of the dissertation - strongly quadratic stability and guaranteed cost control - does systematize some of control design methods under the optimal control as addressed below.
Regarding linear systems with the measurement and system noises, LQG control has been criticized due to the lack of robustness against parameter perturbations in system models. Based on strongly quadratic stability (and stabilizability), the LQR/LQG approaches can be extended to handle real parameter uncertainties. The proposed approach so-called robust LQG control seeks for an observer-based dynamic compensator with the same order of systems based on the guaranteed cost approach. While the full state feedback problem called robust LQR is completely solvable by using linear matrix inequalities (LMIs) methods, the robust LQG synthesis remains difficult to solve. To address this issue, a gradient search with the Block-Diagonal Riccati Approach is proposed. The Block-Diagonal Riccati Approach efficiently finds a quadratically stabilizing observer-based dynamic controller by separating the state feedback and observer designs. To show the effectiveness, robust LQG is applied to a flexible structure control problem with a number of states and uncertain parameters.
Unlike the systems discussed in the robust LQR/LQG approaches, many of uncertain systems have external disturbances as well, which may be assumed to be norm-bounded and matched. Faced with such circumstances, two nonlinear control methods - robust sliding mode control and constrained static output feedback stabilization - are newly formulated in terms of strongly quadratic stability and guaranteed cost control.
First, sliding mode control is revisited to investigate the effects of real parameter uncertainties that are mismatched. It is shown that some sliding hyperplanes do guarantee robust stability (strictly speaking, strongly quadratic stability) and robust performance in view of the guaranteed cost approach even for mismatched uncertainties. The necessary and sufficient condition for existence of robust sliding hyperplanes is shown by strongly quadratic stabilizability via full state feedback. According to the results, design of sliding modes can be accomplished by handling not the reduced order systems but the full order systems. In addition, the optimality issue is addressed for nominal systems in order to show that the proposed 'Lyapunov matrix partitioning' technique does not introduce any conservatism. The effectiveness of the proposed approach is demonstrated by an application to a motion stabilization problem.
Secondly, constrained static output feedback in which an equality constraint is satisfied with strongly quadratic stabilizability via static output feedback is considered when only the partial measurements are available. With the aid of the equality constraint, a nonlinear control can be designed to perfectly attenuate the effect of disturbances. The necessary and sufficient condition for solvability is newly proposed in terms of a dual LMIs problem which is solvable with existing numerical algorithms. Moreover, under invertibility of CB, where B and C are input and output matrices, respectively, the solvability condition forms an LMIs problem, which allows the convex approach. The proposed approach is combined with an integral control and applied to a vibration isolating system to reduce the induced vibration due to disturbances.
