Extended-Space Control Design with Parameter-Dependent Lyapunov Functions

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This research attempts to establish a new framework of control design that allows for parameter-dependent Lyapunov functions. In this research, we particularly consider multiobjective control, robust control for polytopic uncertainties, and gain-scheduled control for LPV (Linear Parameter-Varying) systems. Through inverse use of the elimination lemma, we translate a given BMI(Bilinear Matrix Inequality) problem into an extended LMI(Linear Matrix Inequality) problem while introducing a virtual parameter. In this translation, the original BMI variables are split into two different LMI terms at the expense of generating some new BMI terms with the virtual parameter, and we can solve the problem with distinct Lyapunov solutions. In other words, we can solve mixed problems or matrix polytope problems with parameter-dependent Lyapunov functions. Based on this idea, we develop a theory described in detail and verify the applicability of the proposed method through some numerical design exampl … More es. In 2002, we have formulated extended LMI representations of Lyapunov stability, H2 control, and H∞ control. Combining them, we have successfully solved multiobjective control design problems with distinct Lyapunov solutions. In 2003, we have formulated extended LMI formulations of both robust control for polytopic uncertainties and gain-scheduled control for LPV systems. In every case, the past conservatism of common Lyapunov solutions has been considerably improved. In robust H2 control in matrix polytope problems, we cannot know which vertex achieves the worst H2 cost in advance. In 2004, we have derived an excellent formulation in which the worst H2 cost is minimized even though we cannot which one achieves it in advance. Moreover, a condition regarding dP/dt should be included in the formulation for gain-scheduled control, where the Lyapunov function is given by Ψ=x'Px. In 2004, we have successfully included a condition of dP/dt and reduced it to a set of conditions at vertices. Less