A Gain-Margin and Phase-Margin Based Design of PID Controllers for Inverted Pendulum System

Abstract

The Cart-Inverted Pendulum System (CIPS)is a classical benchmark control problem. Its dynamics resembles with that of many real world systems of interest like missile launchers, pendubots, human walking and segways and many more. The control of this system is challenging as it is highly unstable, highly non-linear, non-minimum phase system and under actuated. Further, the physical constraints on the track position control voltage etc. also pose complexity in its control design. The thesis begins with the description of the CIPS together with hardware setup used for research, its dynamics in state space and transfer function models. In the past, a lot of research work has been directed to develop control strategies for CIPS. But, very little work has been done to validate the developed design through experiments. Also robustness margins of the developed methods have not been analysed. Thus, there lies an ample opportunity to develop controllers and study the cart-inverted pendulum controlled system in real-time. The objective of this present work is to stabilize the unstable CIPS within the different physical constraints such as in track length and control voltage. Also, simultaneously ensure good robustness. A systematic iterative method for the state feedback design by choosing weighting matrices key to the Linear Quadratic Regulator (LQR) design is presented. But, this yields oscillations in cart position. The Two-Loop-PID controller yields good robustness, and superior cart responses. A method for the calculation of all stabilizing PI controllers is presented. The method is based on plotting the stability boundary locus in the (𝑘𝑝,𝑘𝑖)-plane and then computing the stabilizing values of the parameters of a PI controller. The technique presented does not require sweeping over the parameters and also does not need linear programming to solve a set of inequalities. Thus, it offers several important advantages over existing results obtained in this direction. Beyond stabilization, the method is used to shift all poles to a shifted half plane that guarantees a specified settling time of response. Computation of stabilizing PI controllers, which achieve user specified gain and phase margins is studied. It is shown via an example that the stabilizing region in the (𝑘𝑝,𝑘𝑖)-plane is not a convex set. The method is also used to design PID controllers. The limiting values of a PID controller which stabilize a given system are obtained in the (𝑘𝑝,𝑘𝑖)-plane, (𝑘𝑝,𝑘𝑑)-plane and (𝑘𝑖,𝑘𝑑)-plane