Parameter estimation of fuzzy controller and its application to inverted pendulum

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Abstract

In this paper, a new approach to estimate scaling factors of the fuzzy PID controller is presented. The performance of the fuzzy PID controller is sensitive to the variety of scaling factors. The design procedure dwells on the use of evolutionary computing (more specifically, a genetic algorithm) and estimation algorithm. The tuning of the scaling factors of the fuzzy PID controller is essential to the entire optimization process. And then we estimate scaling factors of the fuzzy PID controller by means of three types of estimation algorithms such as HCM (Hard C-Means) clustering-based regression polynomial, neuro-fuzzy networks, and regression polynomials. Numerical studies are presented in detail along with a detailed comparative analysis.

Introduction

The ongoing challenge for advanced system control has resulted in a diversity of design methodologies and detailed algorithms. Fuzzy controllers have positioned themselves in the dominant role at the knowledge-rich spectrum of control algorithms. The advantages of the fuzzy controllers manifest by their suitability for nonlinear systems (as they are nonlinear mappings in the first place) and for high deviations from the set point. The intent of this study is to develop, optimize and experiment with the fuzzy controller (the fuzzy PD controller or the fuzzy PID controller). One of the difficulties in controlling complex systems is to derive the optimal control parameters such as linguistic control rules, scaling factors, and membership functions of the fuzzy controller (Wang and Kwok, 1992). With this regard, genetic algorithms (GAs) have already started playing an important role as a mechanism of global search of the optimal parameters of such controllers. However, in controlling a nonlinear plant such as the inverted pendulum of which initial states vary in each case, the performance of controllers may become poor, since the control parameters of the fuzzy controller cannot be easily adapted to the changing initial states such as angular position and angular velocity (Astrom and Wittenmark, 1995). To alleviate the above shortcoming, we use three types of estimation algorithms such as Hard C-Means (HCM) clustering-based regression polynomial, neuro-fuzzy networks, and regression polynomials, and then estimate the parameters of the controller in each case. The paper includes the experimental study dealing with the inverted pendulum. The performance of systems under control is evaluated from the viewpoint of Integral of the Time multiplied by the Absolute value of Error (ITAE) and overshoot (Oh, 2002).

Section snippets

The fuzzy PID controller

The block diagram of fuzzy PID controller is shown in Fig. 1. Referring to Fig. 1, we confine to the following notation. e denotes the error between reference and response (output of the system under control), Δe is the first-order difference of error signal while Δ2e is the second-order difference of the error (Hu et al., 1999). Note that the input variables to the fuzzy controller are transformed by the scaling factors (GE, GD, GH, and GC) whose role is to allow the fuzzy controller to “see”

Auto-tuning of the fuzzy controller by GAs

Genetic algorithms (GAs) are the search algorithms inspired by Nature in the sense that we exploit a fundamental concept of a survival of the fittest as being encountered in selection mechanisms among species. In GAs, the search variables are encoded in bit strings called chromosomes. They deal with a population of chromosomes with each representing a possible solution for a given problem. A chromosome has a fitness value that indicates how good a solution represented by it is. In control

Algorithm 1: HCM clustering based regression polynomial

In this algorithm, we use HCM clustering algorithm to classify the data and identify the divided data on each cluster by means of LMS method. We use the polynomial in the form given by (5) and estimate coefficients of the polynomial.ŷ(i)=C0+C1θ(i)+C2θ(i)2+⋯+Cnθ(i)n,where ŷ(i) is the ith output of polynomial model. θ(i) is the ith input of system while C0,C1,…,Cn are the coefficients of this polynomial. Given a set of data X={x1,x2,…,xn}, where xk=[xk1,…,xkm],n is the number of data and m is

Experimental studies

The proposed control scheme can be applied to a variety of control problems. In this section, we demonstrate the effectiveness of the fuzzy PID controller by applying it to the inverted pendulum system (Fig. 8).

The inverted pendulum system is composed of a rigid pole and a cart on which the pole is hinged (Jang, 1992; Wang, 1996). The cart moves on the rail tracks to its right or left, depending on the force exerted on the cart. The pole is hinged to the car through a frictionless free joint

Conclusions

In this paper, we propose the Fuzzy PID controller design based on the methodology of tuning of control parameters using GAs and estimating of control parameters using three types of estimation algorithms. First, to set the initial individual of GAs applied to controllers, we utilize the scaling factor estimation modes such as BM, CM and EM. Scaling factor estimation modes such as BM, CM and EM which are determined by means of relation between reference, process error and gain, respectively, is

Acknowledgements

This paper was supported by Wonkwang University in 2002.

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