It is well known that a large number of problems relevant to the control field can be formulated as optimization problems. For a long time, larger numbers of classical approaches have been used to find a closer form solution to the specific optimization problems. In recent years, the formidable increase of computational power has changed the feeling of theoreticians about what is meant by tractable problems. A main issue regard using homogeneous polynomial form to solve the convex optimization problems, and it has been proved to be a very effective method, especially in the framework of linear matrix inequality (LMI) problems. In this thesis, we study robust problems of uncertain systems by using homogeneous polynomial form, respectively. By constructing composite homogeneous polynomial Lyapunov functions and homogeneous parameterdependent quadratic Lyapunov function, we get less conservative results. The main research work of this thesis includes:1) Firstly, we present a new method to estimate the asymptotic stability regions for a class of nonlinear systems via composite homogeneous polynomial Lyapunov functions, where these nonlinear systems are approximated as a convex hull of some linear systems. Since the composite homogeneous polynomial Lyapunov functions have specific level sets, this new method can get less conservative asymptotic stability regions. Lastly, Numerical examples are used to illustrate the effectiveness of our method.2) The homogeneous parameter dependent quadratic Lyapunov function is used to estimate the attractive stability region for an impulsive switched linear system with saturated control input. Under the arbitrary switching rule and the dwell time switching rule, local stabilization conditions are obtained in terms of linear matrix inequalities. The corresponding optimization problems are formulated to obtain a larger attractive stability region. Using the simple ellipsoid method to estimate the attractive stability region produces very conservative results, because the attractive stability region is normally irregular. To solve this problem, a polyhedron constructed using a level set of the homogeneous parameterdependent quadratic Lyapunov function is used to estimate the attractive stability region. The polyhedron is closer to the attractive stability region than the simple ellipsoid. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.3) This chapter is concerned with the problem of robust Hâˆž filter design for switched linear discretetime systems with polytopic uncertainties. The condition of being robustly asymptotically stable for uncertain switched system and less conservative Hâˆž noise attenuation level bounds are obtained by homogeneous parameterdependent quadratic Lyapunov function. Moreover, a more feasible and effective method against the variations of uncertain parameter robust switched linear filter is designed under the given arbitrary switching signal. Lastly, simulation results are used to illustrate the effectiveness of our method.4) This chapter is concerned with the problem of the fault detection filter design for discretetime system with polytopic uncertainties. The proposed fault detection filter framework consists of the fault effect on the residual signal is maximised and the disturbance effects on the residual signals is minimized. Since model uncertainties may result in significant changes in the residual, homogeneous parameter dependent quadratic Lyapunov function method are used to reduce the effect of model uncertainties. Compared with common Lyapunov function method, homogeneous parameterdependent quadratic Lyapunov function method is more effective. Lastly, numerical examples are used to illustrate the effectiveness of our method.
