Conventional and Intelligent Control of Nonlinear Systems

Abstract

In the real world, most of the systems are nonlinear by nature. Nonlinearities can be inherent or intentionally introduced into the system. The control of a nonlinear system can be achieved using linear and nonlinear models. In this work, two nonlinear systems are considered for control, one an inverted pendulum system, while the other a continuous stirred tank reactor (CSTR). The inverted pendulum system uses the nonlinear state equation model while the CSTR uses the linear transfer function model. An inverted pendulum is a renowned benchmark problem in control literature because the control of many real time systems such as segways, rocket launchers, crane lifting containers and self-balancing robots, resembles the inverted pendulum system. It is a highly nonlinear, under-actuated and non minimum phase system. In this, the control objective is to keep the inverted pendulum in the upright position while following a desired reference trajectory by the base thus resulting into one ( x ), two ( x - y and x - z ) and three ( x - y - z ) dimensional inverted pendulum problem. For this system (one, two and three-dimensional inverted pendulum) conventional fixed gain proportional integral derivative (PID) controller may not produce satisfactory performance under all operating regions. Therefore, adaptive controller is preferred over a conventional controller. For the tuning of PID controller, an adaptation mechanism using gain scheduling as a function of time and error has been proposed in this work. The gain scheduling depends upon the transient and the steady state part of the response. The proposed time as well as error adaptive gain scheduling PID controllers have been implemented in the MATLAB environment for the stabilization and tracking control of x , x - y and x - z inverted pendulums. The stability analysis of these different types of inverted pendulums with the proposed controllers has been performed using the Lyapunov stability criterion.