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. The performance of the proposed controllers has been compared with the conventional PID scheme in terms of various performance specifications such as rise time, maximum overshoot, settling time and steady-state error etc. Simulation results reveal that the proposed adaptive gain scheduling PID controllers provide better stabilization for all the three types of inverted pendulums while keeping the tracking at the same level as of conventional PID controllers. The mathematical model of a new three-dimensional x - y - z inverted pendulum in the form of state equations has been developed. The necessary and sufficient condition for vii stability of the proposed three-dimensional x - y - z inverted pendulum has been derived using the Lyapunov stability criterion. The stabilization and tracking control of the proposed x - y - z inverted pendulum has been obtained using conventional and proposed adaptive gain scheduling PID controllers. The simulation results show that the proposed adaptive gain scheduling PID controllers provide better performance than the conventional PID controllers in terms of different performance specifications. The effect of uncertainties (which are fast acting external disturbances, noise and frictional forces) on the performance of x , x - y , x - z and x - y - z inverted pendulums have been analysed using conventional and proposed adaptive gain scheduling PID controllers. The simulation results show an improvement in performance parameters with adaptive gain scheduling PID controllers in the presence of disturbance and noise in the controllers. Moreover, in case of friction, the conventional PID controllers perform poorly as compared to the proposed adaptive gain scheduling PID controllers, which perform quite satisfactorily in the presence of friction. CSTR has widespread applications in the process industry, for example, in wastewater treatment units (i.e. activated sludge reactors). A chemical reactor is a vessel where reactions are carried out to produce products from the reactants by means of one or more chemical reactions. The control objectives in a CSTR are the concentration and the temperature control of the product, which can be accomplished by controlling the inlet/outlet of the reactor/product or the coolant/heater flow rate. For the concentration control of a CSTR different hybrid control schemes based on fuzzy logic, artificial neural network, adaptive neuro fuzzy inference system and genetic algorithms have been used for PID controller tuning. Simulation results show that the best performance in terms of settling time and overshoot has been given by adaptive neuro fuzzy inference system tuned PID controller. Moreover, the magnitude of the inverse response behaviour in case of adaptive neuro fuzzy inference system tuned PID controller is less than the conventional Ziegler Nichols and fuzzy PID methods. Furthermore, to improve reference tracking and disturbance rejection  H based preview control scheme has been implemented for concentration control of a CSTR. The findings of this research work can be utilized to improve the existing control of nonlinear systems using gain scheduling or hybrid control methods.