Theory Of Nonlinear Singularly Perturbed Control Systems: Study And Applications

A class of systems is ofen encountered in the practical engineering field of aviation, electric power, and mechanism ect, which many small time constant, conductance or capacitance will appear during modeling process. The existence of these parameters produces high order of the model and morbid numerical characteristics of the differential equation. The neglect of sunch uncertain small parameters of system model will bring about the reduction of system dimension (order). The systems that possesses above characteristics are called singularly perturbed systems, and singular perturbation method is the major means to deal with such systems.Along with the rapid development of singular perturbation theory in mathematics field, it also achieves breakthrough progress in control area and obtains constant perfection with the development of control theory. More and more scholars are interested in the singular perturbation theory and many results are obtained. Accordingly, based on singular perturbation theory, the purpose of this dissertation is to study the stability, robustness of the nonlinear singularly perturbed systems and some applications of singulry perturbation in nonaffine systems. The main works is generalized as follows:The state feedback controller is designed for a kind of singularly perturbed system, with the fast system is linear and the slow system is partially input-output linearizable. The Lyapunov functions are established for the linear part and zero dynamics of slow subsystem and for the boundary layer system. The sufficient condition of asymptotically stability for the system is obtained though calculating the derivative of the composite Lyapunov function along original trajectory, and then the upper bound expression of perturbation parameter is given.The problem of L2 gain disturbance attenuation is considered for a class of nonlinear singularly perturbed system. Through dividing the system into the fast-slow subsystems and according to the assumption of system structure, a state feedback controller with asymptotic convergence is obtained for the fast subsystem. Then, based on Lyapunov function and the backstepping design technique, a state feedback controller for the slow subsystem is constructed, which enables the closed-loop system to be internally stable for all bounded interference and satisfies the arbitrarily small bounded L2 gain from exogeneous interference input to its output. The strictly dissipative inequality for the whole system can be deduced through recursive method after getting that for the first subsystem. So, the design process of controller can be done without the solution to Hamilton-Jacobi equation.The problem of semiglobally practical stabilization is considered for nonlinear singularly perturbed systems with unknown parameters. The composite Lyapunov function for the full systems is established by both of that for the slow subsystem and the boundary layer system, a state feedback control law for the linear part of the slow subsystem and boundary layer system is proposed which renders the whole closed-loop system semiglobally stable. It also verifies that the controller is robust as long as the variation of unknown parameters maintains the same relative degree. The upper bound expression of perturbation parameter is given to obtain the condition of asymptotically stability for the system.The problem of robust control for uncertain nonlinear systems with high-gain observer is considered. Using singularly perturbed theory, we have proven that the closed loop system based on high-gain output feedback controller is asymptotically stable as long as the state feedback system is asymptotically stable under the meaning of stability in the first approximation. The states will enter the positively invariant subset within finite time under some definite initial conditions, and the controller is robust to all uncertain terms that belong to a known compact set. The simulation results show the output feedback controller recur the performance of the state feedback controller.A novel control method for asymptotical stabilization based on singular perturbation theory combined with inverting design is considered for a class of nonaffine nonlinear systems. The resulting control signal is defined as a solution of a "fast" dynamical equation, and the state of the original nonaffine-in-control system is shown to track the reference model with zero tracking error. Moreover, from our proof, the state trajectory enters the positively invariant subset of the region of attraction in finite time and achieves asymptotical stabilization ultimately. The control law can be easily solved and the simulations illustrate the theoretical results.A novel control synthesis method for output regulation based on singular perturbation theory combined with inverting design is considered for a class of nonaffine nonlinear systems. The resulting control signal which is defined as a solution of fast subsystem inverts a series error model between output and reference signal, and whose states are exponentially stable. It is shown that, under the satisfaction of the assumptions in Tikhonov’s theorem from singular perturbation theory, this problem is solvable with o(ε) tracking error if and only if a set of first-order nonlinear partial differential equations are solvable. However, from the above discussion results, there are still accuracy deficiencies in Tikhonov’s theory. So, Isidori’s theorem is further applied, and then zero track error is accomplished between the system output and reference signal. It is more ideal than o(ε) outcome with Tikhonov’s theory. Moreover, the above results are all extended to MIMO systems.The conclusion is drawn for the whole dissertation, and the next step research aspect is put forward.