The LMI (linear matrix inequality)-based H-infinity controller synthesis theory guarantees that if the controller is allowed to have the same order as the plant, and every system matrix of the controller is freely tunable, then the H-infinity optimization problem can be solved by convex optimization and the global optimum can be always found. This dissertation focuses on an extension of H-infinity optimization theories to the problems that cannot be parameterized as a convex optimization problem.

This dissertation first presents an extension of the LMI-based H-infinity controller synthesis algorithm for full-order controllers to fixed structure controllers. A critical limitation of the LMI-based full-order H-infinity controller synthesis algorithm is that it allows no additional constraint to be imposed on the problem; the closed-loop H-infinity norm constraint must be the only constraint imposed on the problem in order for it to be globally solvable by convex optimization. In the case where the controller has a fixed structure and only its parameters are tunable, the problem cannot be reparameterized as a convex optimization problem. The proposed algorithm starts from the transformation of the fixed-structure H-infinity controller optimization problem into an H-infinity optimization problem of a static output feedback controller. Then, the cone complementarity linearization algorithm is used to locally solve the H-infinity optimization problem of a static output feedback controller.

Based on the proposed H-infinity optimization algorithm, this dissertation demonstrates a tuning method of controller parameters to explicitly design frequency responses of the closed-loop system. The proposed approach offers an intuitive and efficient way to re-tune controller parameters, which were finely tuned by an expert engineer, and improve the control performance. The following three practical application examples are presented: 1) the tuning of a single-input single-output (SISO) PID (Proportional plus Integral plus Derivative) controller for head positioning of a magnetic hard disk drive (HDD), 2) the tuning of a discrete-time observer feedback controller for head positioning of an HDD, and 3) the tuning of a multi-input single-output (MISO) PI (Proportional plus Integral) controller for the lateral control of an automated heavy-duty vehicle (HDV). The effectiveness of the proposed re-tuning method is demonstrated by simulation and experimentation.

Secondly, this dissertation considers the BMI (bilinear matrix inequalities) formulation of H-infinity optimization problems. The BMI framework offers an unified approach to formulate a further general class of H-infinity optimization problems with arbitrary constraints or additional optimization objectives. BMI problems are generally nonconvex optimization problems and are proven to be NP-hard. This dissertation proposes a novel local search approach for solving general BMI problems. The proposed algorithm is based on the semidefinite programming (SDP) relaxation approach to BMI problems and the linearization-based local search algorithm, which is analogous to the algorithm employed to solve reduced-order H-infinity controller synthesis problems. Four numerical experiments are conducted to show the search performance of the proposed approach.

Finally, the proposed fixed-structure H-infinity optimization algorithm is applied to the H-infinity optimization problem of state observers. First, the H-infinity optimization algorithm of Luenberger-type state observers is presented. The proposed approach is applied to the design of fault detection filters for lateral control of automated passenger vehicles. The H-infinity optimization of Luenberger state observers is then extended to the design of more general mismatched state observers (i.e. system matrices of the observer do not necessarily coincide with those of the plant model), and a novel application of H-infinity-optimal mismatched state observers to the observer-based feedback control is presented. The mismatched state observer is tuned by using H-infinity optimization such that it not only provides good estimation of state variables of the plant, but also stabilizes the overall closed-loop system under the feedback linearization control scheme. As an application example, the proposed approach is applied to lateral control of HDVs.