Robust Control Of Singularly Perturbed System

The inherited property of singularly perturbed systems (SPS) is two-time scale (TTS) property, which always leads to mathematical models that are "stiff". The "curse" of dimensionality coupled with stiffness poses formidable computational complexities for the analysis and control of the systems. To overcome these problems, the singular perturbation and time scale methods were put forward. These techniques had attained a certain level of maturity in theory of continuous control systems described by ordinary differential equations. In fact, the idea of time scale technique is to carry out the design procedure in each two different time scale. The two-frequency scale (TFS) property in frequency domain, which is in according to the property of TTS in time domain, poses high- and low-frequency categories. Here the slow subsystem of singular perturbed systems is sensitive to the low-frequency signals, while the fast subsystem is sensitive to the high-frequency signals. However, so far as we know, seldom research starts from the frequency decomposition points to deal with the related problem of SPS.The aim of this dissertation was to develop a frequency decomposition technique for the positive realness performance analysis and H_{∞}control design problem of SPS based on the generalized KYP lemma. Besides this, we made further study on singularly perturbed systems in unified time scale, which involves time delay singularly perturbed systems, T-S fuzzy singularly perturbed systems and discrete-time singularly perturbed systems, the frequency decomposition results need to be further studied.To sum up, the major works in this dissertation are as follows.Firstly, H_{∞}control of the singularly perturbed systems was studied from a new perspective. Unlike the traditional H_{∞}norm optimization in all frequencies, here we considered the optimization in finite frequencies. Based on generalized KYP lemma, a low frequency controller and a high frequency controller was designed, respectively, for the slow and fast subsystems. A composite controller was constructed by the solutions of the above reduced order systems. A tracking problem and a H_{∞}model matching problem were provided to show the priority to the traditional H_{∞}control of SPS.Secondly, the positive realness property of singularly perturbed systems was studied from a new perspective. Based on generalized KYP lemma, the positive realness property of the reduced subsystems in the corresponding frequency domain was studied. It was shown in this dissertation that the positive realness property in finite frequency ranges for the full-order system can be inferred by the reduced subsystems.Thirdly, in the unified time scale, the robust stability and robust stabilization of time delay singularly perturbed systems with uncertainty was studied. By using improved Lyapunov-Krasovskii functional, a sufficient condition in terms of LMI was proposed to ensure the singularly perturbed systems to be robustly stable. On the basis of which, a stabilizing controller was designed.Fourthly, in the unified time scale, the problem of robust fuzzy control for nonlinear singularly perturbed systems described by a T-S fuzzy model was studied. Two cases of the T-S fuzzy systems with parametric uncertainties, both continuous-time and discrete-time cases were considered. LMI-based sufficient conditions were proposed for stabilization of the controlled systems via fuzzy state feedback controllers. Especially for the discrete case, unlike the existing results, by introducing new matrix variables, LMI-based sufficient condition was proposed to ensure the fuzzy discrete-time systems robustly stable. Furthermore, generalized H₂ control for the fast sampling discrete-time nonlinear singularly perturbed systems described by T-S fuzzy model was studied.