Numerical solution of large-scale problems in control system design

This thesis investigates the accurate and efficient solution of selected large-scale problems in control system design. Obviously, control system design is a huge field, with various models, controlling strategies and solution methods. However, the effectiveness of most designs will benefit, to varying extent, from advances in pole assignment (Footnote: For a smoother flow, results for pole assignment are presented in Appendix A.) and algebraic Riccati equations, especially for large systems. We first consider the robustness of the state-feedback pole assignment problem (SFPAP), which has been labelled ``intrinsically or generically'' ill-conditioned. We shall prove the exact opposite, that the SFPAP is not intrinsically ill-conditioned. We then move onto the optimal control of large systems, via algebraic Riccati equations (AREs). In the past decade, many authors have proposed algorithms based on the (inexact) Newton's method, requiring the solution of Lyapunov or Stein equations. These have not been fully successful, with structures in the problem not fully exploited. The core of the thesis is about applying the structure-preserving doubling algorithm (SDA) and its generalisations to large-scale AREs, as well as the closely related large-scale nonsymmetric algebraic Riccati equations (NAREs) and large-scale nonlinear matrix equations (NMEs). We shall illustrate that our adaptation of the SDA is efficient, with an efficient O(n) computational complexity and memory requirement per iteration and an almost quadratic convergence. In addition, the SDA is also generalized for solving rational Riccati equations (RREs) for the optimal control of (small) stochastic linear systems, possessing an O(n3) computational complexity per iteration and memory requirement. Lastly, the refinement of estimates of invariant and deflating subspaces (REIS/REDS) for large matrices and matrix-pencils, both related to NAREs, are considered. The associated algorithms, based on Newton's method, again possess an O(n) computational complexity under appropriate assumptions. All the algorithms in the thesis are analysed rigorously and tested thoroughly in numerical experiments.