Analysis and Design of Control Systems Using Piecewise Linear Lyapunov Functions

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The main results obtained through the research are summarized as follows.1.Generalization of the class of piecewise linear Lyapunov functions.We proposed a new class of piecewise linear Lyapunov functions (PWLLFs) and derived stability results. A candidate of PWLLF has parameters corresponding to piecewise linear function defined in the region divided by hyperplanes. The set of stability conditions are formulated as Linear Programming Problem (LP) in terms of parameters inserted by piecewise linear functions. If the computed optimal value is negative, we construct a PWLLF using the solution. When the optimal value of the LP is nonnegative, we modify the PWLLF candidate by adding appropriate hyperplanes to introduce more freedom in the LP formulation and arrive at the desired result. We derived a new condition for generating hyperplanes such that the optimal value of the new LPs is less than that of the old LP. This condition is an improvement of the previous result. By adopting this me … More thod, we can generate a PLLF for some systems, for which we could not generate a PLLF.2.Enlargement of estimates of stability regions.In this research project, we are interested in semi-global stability rather than the global stability. In this respect, it is very important issues to compute larger estimates of stability regions or to design controller so that the closed system has large stability region. We proposed a method to achieve this.3.Design of nonlinear servo systems.We proposed a design method of nonlinear servo systems by using PLLFs. This method reduces conservativeness included in previous results. To improve the transient response characteristic, we proposed a scheme, which adopts idea based on the reference governor and the linear quadratic regulator theory.4.Fast solving methods for bilinear optimization problems.When we design controller using PLLFs, we need to solve bilinear optimization problems, which are not convex problems. To solve bilinear optimization problems, we applied the Zoutendijk's method, a genetic algorithm based on Imanishi's evolution theory and the probabilistic approach for some examples. Each method has both the advantage and the disadvantage. Further research on this issue is needed. Less