Computational issues in the design of robust nonlinear controllers

Just like the algebraic Riccati equations (AREs) or inequalities (ARIs) in the linear {dollar}Hsb{lcub}infty{rcub}{dollar} control theory, the Hamilton-Jacobi equations (HJEs) or inequalities (HJIs) play an essential role in the nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} control theory. In this thesis, a successive algorithm for finding an approximate solution of HJE is proposed. Using the existing nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} controller formulas, a numerical difficulty could be encountered when a critical assumption is not satisfied. We propose modified nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} controller formulas to eliminate the numerical difficulty. The proposed nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} controller is identical to the existing nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} controller if the assumption is satisfied, and its linear version will be the same as that designed by conventional linear {dollar}Hsb{lcub}infty{rcub}{dollar} approaches.; An important application of the modified nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} controller formulas is the {dollar}Hsb{lcub}infty{rcub}{dollar} approximate input/output (I/O) linearization problem, which essentially is a nonlinear {dollar}Hsb{lcub}infty{rcub}{dollar} control problem without satisfying the critical assumption. Like the feedback I/O linearization, the {dollar}Hsb{lcub}infty{rcub}{dollar} approximate I/O linearization modifies a nonlinear system so that its I/O relationship resembles that of a linear system. However, the {dollar}Hsb{lcub}infty{rcub}{dollar} approximate I/O linearization can handle more general problem since it does not require minimum phase plant or full state feedback. The linearization technique together with {dollar}mu{dollar}-synthesis is used to design a robust nonlinear controller. An inverted pendulum control problem and a nonminimum phase longitudinal flight control problem are employed to illustrate the design of robust nonlinear controllers.