Observer Design for a Variable Moment of Inertia System

Jerzy Baranowski

doi:10.3390/en14185850

Abstract

Variable moment of inertia systems are common, and a popular laboratory system of this type is the “ball-and-beam”. Such systems are, however, nonlinear and often unstable. Efficient control requires full state information (or at least partial velocities), which are generally difficult to measure. That is why the design of state observers is a relevant problem. In this paper, a new design of an observer is proposed. This new nonlinear observer uses partial output injection and the circle criterion to ensure semiglobal stability. Moreover, we present a complete modeling of the system and systematic testing of the observer in comparison to a baseline in the form of a linear observer. The results show that the designed observer outperforms its linear counterpart and does not impede control.

Highlights

What can be consistently observed among all cases is that the estimation of the linear velocity x2 was simple for both of the observers, and the transitional errors were small
The estimation of the linear velocity was still effective (Figure 7), but as we can see, the system stabilized after 5 s, twice as long as previously
The nonlinear observer introduced an effect, known as the “peaking phenomena”, where initially, the estimate had a large transitory error, which can be seen even in the estimates of the linear velocity (Figure 9). This transitional error vanished after approximately 0.25 s, which was consistent with the time constants of the linear part of the error dynamics −0.05 s

Summary

Introduction

Systems with a varying moment of inertia are quite common in nature and in engineering, so interest in their properties is natural. One of the more common laboratory examples of such systems is the “ball-and-beam” (or “ball-on-beam”). Such systems remain ever popular [1,2,3]. This system is a nontrivial example of nonlinearity and the fourth order of dynamics with a mechanical realization. The control of such systems involving complications requires full state information, which cannot be directly measured and has to be estimated

Methods

Results

Conclusion