| The class of uncertainties that may be present in dynamical systems includes, but is not limited to, unknown disturbance inputs, unpredictable abrupt structural changes, unknown parameters, and unmodeled fast dynamics. A number of different stochastic and deterministic models that incorporate these uncertainties have been proposed in the literature, and these have led to different robust control and filtering problem formulations, methodologies, and resulting robust designs. This thesis presents novel formulations of three different classes of problems featuring the types of uncertainties described above, studies them from the points of view of model reduction and robust controller design, and develops new theory and methodology for their solutions. The common thread that runs through these three classes of problems (which involve both deterministic and stochastic systems) is the use of "singular perturbations," either as a modeling tool or as a computational tool. The reason for adopting the singular perturbations framework is threefold. First, we must study robustness with respect to fast unmodeled dynamics of different control or filtering designs. The second reason is to incorporate appropriate knowledge on fast dynamics into the design, without using the full-order system, so as to improve performance and to alleviate the numerical ill-conditioning associated with the computation of the full-order solution. Third, we must obtain "reliable" reduced-order models for large-scale systems that exhibit two or multi time-scale behavior, such that the solution to the reduced-order model can achieve a desired performance for the full-order system.;The thesis can naturally be divided into three parts. In the first part, the problem of optimal control and model reduction for singularly perturbed linear Gaussian systems under an exponential-of-quadratic performance index is addressed, for which a complete solution is obtained that achieves all three objectives stated above. In the second part, which comprises two chapters, solutions to the problem of model reduction for large-scale jump linear systems are presented. The original system is again assumed to exhibit a two-time-scale behavior, and two types of time-scale separations are studied in detail in two separate chapters. Finally, in the last part of the thesis, which also comprises two chapters, solutions to problems of robust parameter identification and robust adaptive control are presented. For these classes of problems, the original system may not exhibit a two-time-scale behavior at the outset, but time-scale decomposition arises naturally due to singularity in the measurement scheme. A singular perturbation analysis is then employed to construct reduced-order identifiers and controllers and to prove their optimality. Again, the three objectives are achieved for the closed-loop system. The thesis ends with some concluding remarks, delineation of its main contributions, and some discussions on future extensions of these results. |
