Approximate methods for nonlinear output regulation problem

One of the important control problems is to design a control law for a plant such that the output of the plant can asymptotically track a class of reference trajectories and/or reject a class of external disturbances. When the reference trajectories and the disturbance are generated by an autonomous differential equation, the problem is called output regulation, or servomechanism problem. Output regulation is becoming more and more challenging and interesting because of its ability of coping with uncertainties and dealing with a large class of complex systems. For the class of linear systems, the output regulation problem was thoroughly studied in the 1970s and 1980s. For the class of nonlinear systems, the output regulation problem has attracted extensive attention since the 1990s. So far, many fruitful theoretical results have been obtained. However, the key issue of the practical computation of the control law has not been well addressed because the control law based on output regulation theory relies on the solution of a set of mixed nonlinear partial differential and algebraic equations known as the regulator equations. Since it is almost impossible to obtain the closed form solution for the regulator equations due to the nonlinearity and complexity, it is necessary to develop effective approximation methods to solve the output regulation problem in order to make the output regulation theory a practical design tool. This thesis will propose two approximation methods for solving the output regulation problem as described below: (1) We will present an approximation method for solving the regulator equations based on a class of feedforward neural networks. We will show that a three layer neural network can solve the regulator equations up to a prescribed arbitrarily small error, and this small error can be translated into a guaranteed steady state tracking error for the closed-loop system. The method will lead to an effective control strategy to solving the nonlinear output regulation problem approximately and practically. (2) We will also present an approximation method that does not rely on the solution of the regulator equations. This approach will approximately solve the output regulation problem by directly approximating a feedforward function using a class of artificial neural networks. Further, a control configuration is developed that allows the reduction of the tracking error by the on-line adjustment of the parameters of the neural networks. A strength of the output regulation theory is its capability of handling a large class of nonlinear systems that cannot be controlled by other methods. We will apply the above two approaches to some benchmark nonlinear control problems such as the asymptotic tracking of the inverted pendulum on a cart system, the ball and beam system and the disturbance rejection problem for rotational/translational actuator (RTAC). These designs will be thoroughly evaluated and compared.