Model Reduction For Discrete-Time T-S Fuzzy Time-Delay Systems

High-order complex mathematical modeling of physical systems and processesin many areas of engineering is frequently encountered, which brings severe hardshipto the analysis and synthesis of the concerning systems. Thereby, considerable atten-tion has been paid to solve the problem of simplifying these models with respect tocertain criterion. Given a high-order system, it is desirable to find a lower-order sys-tem to approximate the original one without significant error being introduced. In thepast decades, an army of efficient approaches have been developed to settle the diffi-culties of model approximation, such as the H_∞approach, the L2-L∞approach, the H2approach and the Hankel-norm approach. Very recently, the linear matrix inequalitytechnique has been explored to solve the model approximation problem for differentclasses of systems in effect, involving linear systems , time-delay systems, Markovianjump systems and linear parameter-varying systems.Complex nonlinear systems are ubiquitous in chemical processes, robotics sys-tems, automotive systems and many manufacturing processes, which introduce seri-ous difficulties in system analysis and design. It is an effective method to analysisand synthesis nonlinear systems with the advent of the Takagi-Sugeno (T-S) fuzzymodel. It has been proved that the T-S fuzzy model can be employed to appropri-ate a nonlinear system. On the other hand, it has been well known that time-delaysappear commonly in various practical systems such as communication, networks andchemical processes. Their existence may introduce instability, oscillation and poorperformance.Owing to its significance in practical applications, the research in this area shouldbe of both theoretical and practical importance. In this thesis, based on the T-S fuzzymodel and linear matrix inequality method, the problems of stability analysis, con-troller design and model approximation of nonlinear time-delay systems have beeninvestigated:1. Delay-dependent stability analysis for discrete-time T-S fuzzy systems withtime-varying state delay is studied. Based on a novel fuzzy Lyapunov-Krasovskiifunction, a delay partitioning method has been developed for the delay-dependent sta- bility analysis of T-S fuzzy time-varying state delay systems. A delay-dependent sta-bilization approach based on a nonparallel distributed compensation scheme is givenfor the closed-loop fuzzy systems. Based on the delay-dependent stability analysisfor discrete-time T-S fuzzy systems with time-varying state delay, we can obtainthe delay-dependent stability results for discrete-time T-S fuzzy systems with time-constant state delay.2. Model approximation of discrete-time T-S fuzzy systems with time-constantstate delay has been investigated. For a given stable T-S fuzzy system, our attention isfocused on the construction of a reduced-order model, which not only approximatesthe original system well in an H_∞performance but also is translated into a linearlower-order system. By applying the delay partitioning approach, a delay-dependentsufficient condition is proposed for the asymptotic stability with an H_∞error perfor-mance for the error system. Then, the H_∞model approximation problem is solved byusing the projection approach, which casts the model approximation into a sequentialminimization problem subject to linear matrix inequality constraints by employing thecone complementary linearization algorithm. Moreover, by further extending the re-sults, H_∞model approximation with special structures is obtained, that is, delay-freemodel and zero-order model. Finally, the H_∞model approximation problem is solvedby using the convex linearization approach, which casts the model approximation intothe convex optimization problems subject to linear matrix inequality constraints.The proposed conditions are formulated in terms of linear matrix inequality,which can be solved efficiently by using existing optimization techniques.