A PI2D Feedback Control Type for Second Order Systems

Abstract

Second order autonomous systems are key systems in the study of non linear systems because their solution trajectories can be represented by curves in the plane (Khalil, 2002), which helps in the development of control strategies through the understanding of their dynamical behaviour. Such autonomous systems are often obtained when considering feedback control strategies, because the closed loop system might be rewritten in terms of the state system and perturbation terms, which are function of the state as well. Thus, analyses of stability properties of second order autonomous systems and their convergence are areas of interest on the control community. Moreover, several applications consider nonlinear second order systems; there are various examples of this: 1. In mechanical systems the pendulum, the inverted pendulum, the translational oscillator with rotational actuator (TORA) and the mass-spring systems; 2. In electrical systems there are examples such as the tunnel diode circuit, some electronic oscillators as the negative-resistance twin-tunnel-diode circuit; and finally 3. Other type of these systems are mechanical-electrical-electronic combinations, for example a two degree of freedom (DOF) robot arm or a mobile planar robot and among every degree of freedom on a robotic structure can be represented by a second order nonlinear system. Therefore, due to the wide applications in second order nonlinear systems, several control laws have been proposed, which comprises from simple ones, like linear controllers, to the more complex, like sliding mode, backstepping approach, output-input feedback linearization, among others (Khalil, 2002). Despite the development of several control strategies for nonlinear second order systems, it is not surprising that for several years and even nowadays the classical PID controllers have been widely used in technical and industrial applications and even on research fields. This is due to the good understanding that engineers have of them. Moreover, the PID controllers have several important functions: provide feedback, has the ability to eliminate steady state offset through integral action, and it can anticipate the future through derivative action. PID controllers are sufficient for many control problems, particularly when system dynamics are favourable and the performance requirements are moderate. These types of