Discrete-Time H2 Guaranteed Cost Control
In this dissertation, we first use the techniques of guaranteed cost control to derive an upper bound on the worst-case H2 performance of a discrete-time LTI system with causal unstructured norm-bounded dynamic uncertainty. This upper bound, which we call the H2 guaranteed cost of the system, can be computed either by solving a semi-definite program (SDP) or by using an iteration of discrete algebraic Riccati equation (DARE) solutions. We give empirical evidence that suggests that the DARE approach is superior to the SDP approach in terms of the speed and accuracy with which the H2 guaranteed cost of a system can be determined.
We then examine the optimal full information H2 guaranteed cost control problem, which is a generalization of the state feedback control problem in which the H2 guaranteed cost is optimized. First, we show that this problem can either be solved using an SDP or, under three regularity conditions, by using an iteration of DARE solutions. We then give empirical evidence that suggests that the DARE approach is superior to the SDP approach in terms of the speed and accuracy with which we can solve the optimal full information H2 guaranteed cost control problem.
The final control problem we consider in this dissertation is the output feedback H2 guaranteed cost control problem. This control problem corresponds to a nonconvex optimization problem and is thus "difficult" to solve. We give two heuristics for solving this optimization problem. The first heuristic is based entirely on the solution of SDPs whereas the second heuristic exploits DARE structure to reduce the number of complexity of the SDPs that must be solved. The remaining SDPs that must be solved for the second heuristic correspond to the design of filter gains for a estimator.
To show the effectiveness of the output feedback control design heuristics, we apply them to the track-following control of hard disk drives. For this example, we show that the heuristic that exploits DARE structure achieves slightly better accuracy and is more than 90 times faster than the heuristic that is based entirely on SDP solutions.
Finally, we mention how the results of this dissertation extend to a number of system types, including linear periodically time-varying systems, systems with structured uncertainty, and finite horizon linear systems.