Stabilization Of Nonlinear Time-Delay Systems In Triangular Form

In this dissertation, the controller design of triangular (lower triangular and upper triangular) structural nonlinear time-delay systems is researched. Asymptotic stabilization by state and output feedback of triangular structural nonlinear systems in the absence of delay has been studied by many researchers during the last decade. While the progress of asymptotic stabilization by state or output feedback of triangular structural nonlinear time-delay systems seems less significant. It is well known that the backingstepping method and the forwarding method are powerful tools to design the stability controllers for triangular system without delay. But, it is not easy to apply them designing stability controllers for triangular system with delay. It is also well known that output feedback control is a hard problem for nonlinear systems because of the lack of systematic approach for designing nonlinear observers and the failure of separation principle which may make observers not applicable for output feedback design. For the first time, this dissertation provides systematically alternative constructive control techniques instead of the backingstepping method or the forwarding method to design stability controllers for triangular system with delay, and proposes nonseparation principle paradigms in designing output feedback control for triangular systems with delay.There are three chapters in this dissertation.In Chapter 1, we review some fundamental theories of time-delay systems and the background of time-delay systems, point out some open problems in this field, and introduce two constructive methods (backstepping and forwarding) which have been widely applied in designing controller for the triangular (lower-triangular and upper-triangular) structural nonlinear systems without delay.In Chapter 2, we study the problem of global stabilization for large-scale lower-triangular systems with delays in the state. It is provided constructively state feedback controller and output feedback controller for such systems. All controllers given in Chapter 2 are delay-independent. Large-scale systems are comprised of several interconnected subsystems. The fundamental approaches for the control of large-scale complex systems are decentralized control and dynamical hierarchial control. They decompose the control of large-scale systemsas a set of independent control of subsystems, which can only access the local information or the information of a high-level auxiliary system. Based on the decentralized control and the backstepping method, it was constructed the state feedback controller for large-scale lower-triangular systems in [1][2]. While the method used here is quite deferent from the backstepping method, which is widely used in dealing with lower triangular system. For lower triangular systems, the gains of stabilizing controller given by many people (see[1][2][3][4][5]) are very high, and the high gains may make the system cause a undesirable transient behavior. Comparing our design schemes proposed here with the backstepping method, the gains of our controllers are lower and the design procedures are much simpler and more efficient because no recursive computation is involved here.In section 2.1 and section 2.2. constructive control techniques have been proposed for controlling large-scale nonlinear lower triangular systems with delayed state interconnections using state feedback and output feedback, respectively. The uncertain nonlinearities are assumed to be bounded not only by polynomial functions of the outputs, but also by polynomial functions of the states or delayed states. The nonlinear systems considered in section 2.1 are more general than conventional lower triangular systems (see[3][4][5][6][7]), and they could be viewed as generalized lower triangular systems. Based on the use of a memoryless hierarchial high gain controller (or, the use of a memoryless hierarchial high gain observer in combination with a memoryless hierarchial high gain controller) and choosing appropriate Lyapunov-Krasovskii functionals (LKF), the delay-independent hierarchial state (or output) feedback controller achieving global asymptotic stabilization of the large-scale nonlinear time delay systems is explicitly constructed. The hierarchial state (or output) feedback controller includes a high-level subsystem with all states (or outputs) of the large-scale systems as its input and a low-level subsystem whose gains are from high-level subsystem. For the first time, the approaches are proposed to design the stabilizing controller for large-scale nonlinear lower-triangular systems with delays in the state. Simulation examples are given in every section of Chapter 2 to demonstrate the effectiveness of the proposed design procedure.In Chapter 3, we study the problem of global stabilization for a class of nonlinear systems with delays in the input or output. All controllers given in Chapter 3 are delay-dependent. The nonlinear systems considered here are more general than the upper-triangular systems widely considered in many papers (see [8] [9] [10] [11]). Hence our nonlinear systems could be viewed as generalized upper triangular systems. Asymptotic stabilization by state feedback of upper triangular system in the absence of delay has been studied by many researchers (see [12][13][14][15]). However, the stabilization of upper triangular systems with delays in the input or output has not been fully investigated, and few paper has considered the problem of the output feedback stabilization for such systems. In this paper, Based on the constructingappropriate LKF and applying the model transformation of time-delay systems, we shall propose constructive control techniques for controlling upper triangular nonlinear systems with delays in the input or output. The designed controllers have a very simple structure and do not involve any saturation or recursive computation, which are widely applied in designing control of upper triangular systems. By using the transformation of coordinates and the property of Hurwitz polynomial, the problem of designing controller can be converted into the problem of finding a parameter, which can be solved by solving optimization problem with linear matrix inequalities (LMIs) constraints. This is the main idea of the proposed design methods. All LMIs in this chapter always have solutions. Hence, the proposed design approach is quite different from the existed LMIs design method (see [14][16][17][18][19]). in which the sufficient condition for the existence of controller is always given in terms of LMIs. Therefore, the methods proposed here are all constructive methods.In section 3.1. section 3.2 and section 3.3, it is considered the state feedback stabilization of upper triangular nonlinear systems with delays in the input, output feedback stabilization of upper triangular nonlinear systems with delays in the input, and output feedback stabilization of upper triangular nonlinear systems with delays in the output, respectively. Examples are given in every section of Chapter 3 to illustrate the effectiveness of the proposed method.