Using fuzzy logig to control the position of a two-coordinate system and its application to pneumatic drives.

Taratinsky, Ilya ; Mogilnikov, Pavel

1. INTRODUCTION

Since the mid 20th century, pneumatic control technology has been widely applied in the industry automation; in recent years, the technology is even applied in the different precision control machines, e.g., in the electronic industry. Owing to the compressibility characteristic of the air, the pneumatic control system has a very high level of nonlinearity and the system parameters are time variant with environmental change, therefore, the mathematical model isn't easy to derive and to obtain the accurate values of the coefficients of the model (Aldridge, 1992). The fuzzy theory and the application to the control technology have been successfully developed in recent years. The uncertainties affecting control problems can be solved through the fuzzy logic thinking and decision method.

For the pneumatic cylinder position control system, the dry friction forces, the compressibility of air, the pressure and the temperature are time variant uncertainties, affect the performance of position control. In different articles the authors (Kroll, 1996, Daugherity; Rathakrishnan & Yen, 1992) obtain the performance indices with rise-time, overshoot, settling time and steady state error; besides, with different searching methods to find the optimal control parameters and get the better performance. Adjusting the normalization factors for the optimal performance through trial and error is time-consuming and difficult.

Identification means computing the parameters of a given model structure assessing input/output data of the system considered. Various model structures are known. In general, linear models are easy to handle but will not yield satisfactory performance if the model validity is not restricted to small deviations from a fixed operating point. Nonlinear conventional models such as state-polynomial (e.g. bilinear or quadratic) ones require apriori knowledge for the choice of an appropriate model structure. In contrast, fuzzy models represent a rather general and flexible approach. They describe the input/output behaviour of a system using a set of If--THEN-production rules of the form: If condition Then action. Such models may be interpreted as a weighted combination of several local models resulting in a nonlinear global model.

They describe the input/output behavior of a system using a set of If--THEN-production rules of the form: If condition Then action. Such models may be interpreted as a weighted combination of several local models resulting in a nonlinear global model. Hence a mismatch between the nonlinearities of local models and process is less significant compared with a single nonlinear model. Therefore fuzzy modelling has been applied especially to modelling tasks with uncertain nonlinearities, uncertain parameters and/or high complexity.

This fuzzy controller was applied to the pneumatic two-coordinate system that was used for playing checkers.

2. SYSTEM DESCRIPTION

The experimental layout of the pneumatic cylinder position control system is shown in figure 1. The control signal is sent from the industrial controller to the servo valve. The air flow rate into the cylinder can be regulated through the valve. The pressure difference between the chambers of the cylinder is built up and then the motion of the cylinder can be controlled. The position of the cylinder is measured by a pulse scale and the load of the cylinder is controlled by a proportional pressure control valve.

1. Compressed air source 6. Pulse scale 2

2. F-R-L unit 7. Pneumatic cylinder 2

3. Servo valve 1 8. Servo valve 2

4. Pneumatic cylinder 1 9. Controller (Siemens S7-300)

5. Pulse scale 1 10. PC

[FIGURE 1 OMITTED]

3. DYNAMICAL FUZZY CONTROLLER

In this article functional fuzzy models (often referred to as models of the Sugeno type (Terano; Asai & Sugeno, 1992)) with multi-dimensional reference fuzzy sets as are described and used. Compared with the commonly used one-dimensional reference fuzzy sets the multi-dimensional approach is more flexible and promises higher model quality with less parameters.

A functional fuzzy model with M inputs [X.sub.j] and one output y is described by a set of c rules such as (Zadeh, 1965):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

R1: if [e.sub.k] is [P.sub.1] then [DELTA][u.sub.k] is [P.sub.u1]

R2: if [e.sub.k] is [N.sub.1] then [DELTA][u.sub.k] is [N.sub.u1]

R3: if [DELTA][e.sub.k] is [P.sub.2] then [DELTA][u.sub.k] is [P.sub.u2]

R4: if [DELTA][e.sub.k] is [N.sub.2] then [DELTA][u.sub.k] is [N.sub.u2]

R5: if [[DELTA].sup.2][e.sub.k] is [P.sub.3] then [DELTA][u.sub.k] is [P.sub.u3]

R6: if [[DELTA].sup.2][e.sub.k] is [N.sub.3] then [DELTA][u.sub.k] is [N.sub.u3]

where: [e.sub.k] = r - [y.sub.k] is the error; [DELTA][e.sub.k] = [e.sub.k] - [e.sub.k-i] is the error, multiple of 1st; [[DELTA].sup.2][e.sub.k] is the error, multiple of 2nd; [[DELTA][u.sub.k] = [u.sub.k] - [u.sub.k-1]; r is the reference input, [y.sub.k] is the output.

Comparing to PID, rules R1 and R2 correspond to integral action; R3 and R4 correspond to proportional action; R5 and R6 correspond to differential action based on fuzzy rules.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Membership functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of fuzzy numbers [P.sub.i], [N.sub.i], [P.sub.ui], [N.sub.ui] are shown on Fig. 2.

[DELTA][u.sub.1] = 1/[pi] x (arctg([d.sub.1] x [e.sub.k]) + arctg([d.sub.2] x [DELTA][e.sub.k]))/([g.sub.1] + [g.sub.2]), (6)

[DELTA][u.sub.2] = 1/[pi] x (arctg([d.sub.2] x [DELTA][e.sub.k]) + arctg([d.sub.3] x [[DELTA].sup.2][e.sub.k]))/([g.sub.2] + [g.sub.3]), (7)

[DELTA][u.sub.3] = 1/[pi] x (arctg([d.sub.3] x [[DELTA].sup.2][e.sub.k]) + arctg([d.sub.1] x [e.sub.k]))/([g.sub.3] + [g.sub.1]), (8)

[d.sub.i] = tg(0.45 x [pi])/[a.sub.i], [g.sub.i] = 1/2 x [b.sub.i]. (9)

The fuzzy controller can be expressed as following equations: [DELTA][u.sub.k.sup.*] = medium {[DELTA][u.sub.1], [DELTA][u.sub.2], [DELTA][u.sub.3]}. The membership function of [DELTA][u.sub.k.sup.*]. can be defined as follows (Fig. 3):

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

4. CONCLUSIONS

Modelling of systems with uncertain nonlinearities and little physical insight is a domain of those black box models showing universal approximating capabilities such as fuzzy models. Pneumatic drives belong to this kind of systems. Through the above explained controller design method and the experimental results, one can make the following conclusions:

1. The effect of uncertain factors, e.g., the dry friction force, the time response and the steady state error of the pneumatic cylinder position control with the designed controller are minimal. The experimental results have shown the advantages of such self tuning controllers.

2. The adaptivity of the external disturbance effect with the designed self tuning neural fuzzy controller is much better than that of controller without tuning capabilities. The fuzzy controller, which normalizes uncertain nonlinearities, was developed and implemented in industrial controller Siemens S7-300.

5. REFERENCE

Aldridge J. (1992). Automated tuning and generation of fuzzy control system, ISATrans.31, pp. 15-17.

Daugherity W.; Rathakrishnan B. & Yen J. (1992), Performance evaluation of a self -tuning fuzzy controller, IEEE Internet Conf., pp. 389--397.

Kroll A. (1996). Fuzzy-modelling of systems with uncertain nonlinearities and its application on pneumatic drives, in IEEE Int. Conf. on System, Man and Cybernetics, Peking, 14.-17.10.1996.

Terano T.; Asai K. & Sugeno M. (1992). Fuzzy systems theory and its applications, Academic Press, INC., San Diego.

Zadeh L.A. (1965). Fuzzy sets, Information and Control, Vol. 8, pp. 338-353.