Control of Uncertain Systems: State-Space Characterizations

Citation

Lu, Wei-Min (1995) Control of Uncertain Systems: State-Space Characterizations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/dnxg-nz58. https://resolver.caltech.edu/CaltechETD:etd-03022006-131646

Abstract

A central issue in control system design has been to deal with uncertainty and nonlinearity in the systems. In this dissertation, an integrated treatment for both uncertainty and nonlinearity is proposed. This dissertation consists of two relatively independent parts. The first part deals with uncertain linear systems, while the second part treats uncertain nonlinear systems.

In the first part, the problem of control synthesis of uncertain linear systems is considered. A linear fractional transformation (LFT) framework is proposed for robust control design of uncertain linear control systems with structured uncertainty. Linear parameter-varying systems whose coefficients depend on some time-invariant unknown parameters are treated in a general algebraic framework; both the stabilization and the H_∞-control problems are considered. For uncertain linear systems under structured perturbations, robustness synthesis problems are characterized in terms of linear matrix inequalities (LMIs) in the LFT framework. A generalized PBH test is also used to characterize the robustness synthesis problems. Moreover, a separation principle for the control synthesis of uncertain linear systems is revealed. The machinery also streamlines a number of results concerning the analysis and synthesis of multidimensional systems.

In the second part, the problem of control synthesis for nonlinear systems is addressed; stabilization, L¹-control, H_∞-control, robustness analysis, and robustness synthesis problems for nonlinear systems are examined in detail. In particular, locally and globally stabilizing controller parameterizations for nonlinear systems are derived; the formulae generalize the celebrated Youla-parameterization for linear systems. Both nonlinear L¹-control and nonlinear H_∞-control are also considered for dealing with disturbance attenuation problems for nonlinear systems. The L¹-performance and L¹-control of nonlinear systems are characterized in terms of certain invariance sets of the state space; in addition, the relation between the L¹-control of a continuous-time system and the ℓ¹-control of the related Euler approximated discrete-time systems is established. A systematic treatment for H_∞-control synthesis of nonlinear systems is provided; the nonlinear H_∞-control problem is characterized in terms of Hamilton-Jacobi Inequalities (HJIs) and nonlinear matrix inequalities (NLMIs); a class of H_∞-controllers are parameterized as a fractional transformation of contractive stable parameters. Finally, the problems of stability and performance robustness analysis and synthesis for uncertain nonlinear systems subject to structured perturbations with bounded L₂-gains are introduced; they are characterized in terms of HJIs and NLMIs as well. Computational issues are also addressed; it is confirmed that the computation needed for robustness analysis and synthesis of nonlinear systems is of equivalent difficulty to that for checking Lyapunov stability.

Thesis Files