Construction of Parameter-Dependent Lyapunov Functions for Robust Control

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1) In the first year of the project term, investigations were on ways to construct Lyapunov functions for linear systems represented by polynomials with uncertain coefficient parameters. This leads to an expected result that there actually exist a parameter-dependent Lyapunov function for the given polytopic uncertain polynomial. The Lyapunov function is also a polytope of Lyapunov functions that correspond to the extreme polynomials of the polytope of polynomials. We see a one-to-one correspondence between parameters of Lyapunov functions and those of polynomials.2) With this result, we then pass to uncertain nonlinear systems, anticipating some parallel results with the linear case. Deliberation together with numerical experiences, however, led us to conclude that for nonlinear systems seach for parameter-dependent Lyapunov functions was harder than expected. We were thus obliged to reconsider our idea and began to look for the "opposite-end" problem, so-called common Lyapunov function problem. This is because common Lyapunov functions for several systems make us easy to handle uncertain nonlinear systems.3) In the final year, we set out to consider this problem for linear systems in the first place. We could successfully identify some classes of discerte-time linear systems that have common quadratic Lyapunov functions. This yields a sufficient condition for the This could also give a scope to explore results for uncertain nonlinear systems Lyapunof function approach.