Robust compensation of a Cart–Inverted Pendulum system using a periodic controller: Experimental results

Abstract

Based on Das and Dey (2007), this paper designs and implements a periodic controller to achieve, via zero-placement, robustness of a physical Cart–Inverted Pendulum system with respect to differential gain variations in the output sensors. Experimental results that verify the superiority of this controller over linear time-invariant (LTI) ones are also presented.

Introduction

The well known Cart–Inverted Pendulum System (Fig. 1) has one input, the voltage ($u$) applied to the motor which provides the force ($F$) to the cart, and two sensor outputs, the cart position ($x$) and the pendulum angle with respect to the upward vertical ($\theta$). The standard control problem is to hold the pendulum in the upright position keeping the cart oscillations (about a desired mean position) within the rail limits. Typically this is achieved using an LTI pole placement controller that uses $x,\theta$ and their derivatives $\dot{x},\dot{\theta }$ as feedback. Such a design, the paper first observes, has a gain margin (GM) of about 100 with respect to variations in the gain, $k$, associated with input $u$, but if the gain, $K_x$, associated with feedback $x$ only is varied then the GM becomes only 8. (Note that a similar intolerance of differential perturbations in the output channels for a 1 input 2 output system was also observed in Yang and Kabamba (1994).) The explanation, it is next shown, lies in the fact that the loop involving $x$ feedback contains a right-half-plane (RHP) zero and a pair of open-loop poles at the origin, and is, therefore, a pathological problem so far as LTI controllers are concerned. (See Doyle, Francis, and Tannenbaum (1992) (ch.6) and Skogestad and Postelthwaite (2005) (ch.5) for discussions on the inability of LTI controllers regarding shifting of zeros leading to problems in compensating unstable plants with RHP zeros.) However, it has been shown first in Lee, Meerkov, and Runolfsson (1987), and later in Das and Dey (2007), that closed-loop, high frequency ($\omega$) periodic compensation may be used to effectively relocate the loop zeros to desired locations leading to improved gain and phase margins. (It may, however, be noted in this context that periodic compensation offers no advantage over LTI ones in respects of disturbance rejection and robustness to unstructured uncertainties, i.e, unmodeled dynamics, Shamma and Dahleh (1991). Also see Bittanti and Colaneri (2009) for a discussion on periodic systems.) Such a periodically compensated system, however, contains $\omega$ frequency oscillations superimposed on its averaged input and output. Now, the controller proposed in Lee et al. (1987) causes (i) the plant output to have $O\left(1\right)$ oscillations, and (ii) the plant input to have $O\left({\omega }^r\right)$ oscillations, where $r$ is the relative order of the plant. While the former signifies an unacceptable response, the latter is clearly non-implementable due to actuator limitations. Periodic controllers therefore remained only as theoretical constructs till Das and Dey (2007) proposed a new periodic controller that causes the plant input to have oscillations of only $O\left(1\right)$ and the plant output of $O(1/{\omega }^r)$.

This paper obtains a computer based multi-loop controller based on a cascade model of the Cart–Pendulum system. The inner loop uses the $\theta$ feedback and LTI compensation to achieve pole placement. The outer loop uses $x$ feedback and a periodic controller based on Das and Dey (2007) to achieve loop-zero placement. This design, moreover, takes into consideration the practical implementation constraints arising out of (i) the resolution of the sensors and the data acquisition system, (ii) the limitation on the upper limit of $\omega$ due to the sampling rate inherent in computer control, (iii) the actuator saturation limit, (iv) the rail limit, and has been implemented to control a physical Cart–Pendulum set-up. The experimental results show a marked improvement in the differential GM, thereby establishing the viability of using such periodic controllers in practical systems.