Research On Stabilization And H_∞Filtering Of Nonlinear Systems Based On Fuzzy Models

With the fast development of the science and technology, control systems inves-tigated by the designers becomes more and more complex, so it is hard to establish its mathematical model with satisfied precisions. In this case, the conventional con-trol theory is not applicable. With the purpose of resolving this problem, people have invented the fuzzy control theory, and it has also been successfully applied to many fields, such as industrial process control, signal processing, intelligent machine, pattern recognition, and medicine. Over the past two decades, fuzzy control theory has been an active research field. In particular, stabilization and H∞filtering of nonlinear systems based on fuzzy models have attracted wide attention from many investigators. They have been applied themselves to the improvement and gener-alization of existing results and many valuable productions have been achieved. It is worth noting that the existing conditions of fuzzy stabilization and H∞filtering will fail to work while the requirement of the control performance index gradually increases. Therefore, the research on exploiting more effective conditions for fuzzy stabilization and H∞, filtering is significant on academic and practical aspects. Based on the precious work of other researchers, this thesis further investigates the prob-lem of stabilization and H∞filtering of nonlinear systems via fuzzy models, and the main contributions of this dissertation are as follows:1. For the problem of quadratic stabilization of continuous-time T-S fuzzy sys-tems, two kinds of relaxed quadratic stabilization conditions are proposed. The first one is developed by considering the bounds of different fuzzy mem-bership functions’cross products. More free matrix variables are introduced and hence the conservatism is further reduced. The other one is based on the viewpoint of dividing the fuzzy membership functions space by applying the simplex edgewise subdivision approach. A new simplex edgewise subdivi-sion algorithm is proposed and thus the fuzzy membership functions space is divided into a lot of sub-spaces. Then, a novel kind of fuzzy switching con-troller is designed for relaxing the quadratic stabilization of continuous-time T-S fuzzy systems.2. For the problem of stability analysis and non-quadratic stabilization of continuous-time T-S fuzzy systems, relaxed stability conditions and non-quadratic stabilization conditions are proposed by applying a novel kind of augmented multi-indexed matrix approach. In the derivation process of sta-bility analysis of opened-loop T-S fuzzy systems, the famous Finsler lemma is used for enlarging the design space, and homogeneous matrix polynomial-type Lagrange multipliers, which is parameter-dependent on fuzzy membership function with a higher degree, are introduced. Then a new augmented multi-indexed matrix approach is developed for achieving less conservative stability conditions of opened-loop T-S fuzzy systems. In the derivation process of non-quadratic stabilization of closed-loop T-S fuzzy systems, a new kind of non-PDC control scheme, which is parameter-dependent on fuzzy member-ship function with a higher degree, is proposed for making good use of the information of fuzzy membership functions; thus relaxed non-quadratic stabi-lization conditions are obtained by applying both the homogenous polynomial technique and the augmented multi-indexed matrix approach while the con-servatism is further reduced.3. For the problem of non-quadratic stabilization of discrete-time T-S fuzzy sys-tems, less conservative non-quadratic stabilization conditions are developed by both improving the form of Lyapunov function and designing new fuzzy state feedback control scheme. A new kind of non-PDC control scheme(named as homogenous polynomial non-quadratic control scheme (HPNQCL) here) is proposed to stabilizing the closed-loop T-S fuzzy control systems; then a Lya-punuov function which is parameter-dependent on fuzzy membership functions with an higher degree is selected for implementing the control synthesis. The algebraic properties of fuzzy membership functions are exploited to convert the attained parameterized matrix inequality into a sequence of linear matrix inequalities. As the values of two design variables tend to sufficiently large, the exactness of ensuring the attained parameterized matrix inequality is cer-tificated by using the Polya’s theorem. Moreover, the advantages over those existing ones are also certificated in theory. It shows that the one in this thesis is both less conservative and more efficient in computations.4. For the problem of H∞filtering of a class of discrete-time T-S fuzzy systems, a novel fuzzy H∞filter is developed, i.e., homogenous polynomial parameter-dependent (HPPD) H∞filter. Unlike to the usual fuzzy H∞, filters which is motivated by the viewpoint of parallel distributed compensation, the one pro- vided in this thesis is parameter-dependent on fuzzy membership functions with a higher degree; more filtering matrix variables are introduced, and hence it has the potential of further reducing the conservatism. In the process of H^filtering, the positivity of the underlying homogenous polynomial parameter-dependent Lyapunov matrix is ensured by proposing a relaxed linear matrix inequality method; the information of fuzzy membership functions is effectively utilized and less conservative fuzzy H∞filtering condition is obtained.5. For the problem of H∞filtering of a class of continuous-time nonlinear stochas-tic systems, a kind of H∞filtering is proposed based on the fuzzy hyperbolic tangent model. Due to some characteristics of the fuzzy hyperbolic tangent model, the problem that the fuzzy membership functions’time derivatives are involved in the process of H∞, filtering via the T-S fuzzy Lyapunov functions are avoided. The obtained fuzzy H∞filter is with a simple structure and nicer performance. This is a good tradeoff between the performance and the required computational burden.6. For the problem of stabilization of discrete-time Roesser nonlinear two-dimensional(2-D) system, three kinds of control schemes are proposed based on the T-S fuzzy model, i.e., the PDC control scheme, the non-PDC con-trol scheme and the homogenous polynomial non-quadratic control scheme for2-D T-S fuzzy systems respectively. In the process of control synthesis, sev-eral kinds of new slack matrix variable approaches are proposed for relaxing the stabilization conditions of the Roesser2-D T-S fuzzy systems. Finally, a homogenous polynomial parameter-dependent Lyapunov function is proposed for conceiving relaxed stabilization of the the Roesser2-D T-S fuzzy systems. Unlike to the usual1-D T-S fuzzy system, there will be two different one-step-ahead fuzzy membership functions along the horizonal direction and the vertical direction respectively when the variation of the underlying Lyapunov function is calculated. This obstacle is overcome in control synthesis and some improved homogenous polynomial techniques are also developed for further re-ducing the conservatism.Finally, some unsolved problems for control synthesis and H∞filtering of non-linear systems based on fuzzy models are pointed out. Furthermore, the prospects of the future study are also given.