**5. Control system example**

A fuzzy control system, as it is in the example, has the block diagram from Fig. 14. A fuzzy PI controller RF- is used in a speed control system of an electrical drive with the following elements: MCC - DC motor, CONV – power converter, RG-I – current controller, RF- speed controller, Ti – current sensor, T - speed sensor, CAN, CNA - analogue to digital and digital to analogue converters.

The fuzzy controller has the structure from Fig. 15. It is a quasi-fuzzy PI controller with summation at the output, with an internal fuzzy block BF with the structure presented at the beginning, and a correction circuit to insure stability. The controller has also an anti windup circuit.

Fig. 14. The block diagram of the fuzzy control system

Tuning Fuzzy PID Controllers 189

trr [s]

l-a 6,7 1 6,1 0,6 4,1 1,5 1,1 6,7 2,3 0,5 0,46 f-a 0 0,5 3,8 0,14 0 1,2 1,03

l-d 8,3 1,5 6,1 0,65 4,1 3 2,0 8,3 2,3 0,7 0,51 f-d 0 0,8 3,8 0,14 0 2,2 1,89

Based on a comparative analysis of the speed performance criteria, better results were there obtained with the fuzzy PI controller designed, using the above methods as it follows:


In this chapter, there were analyzed some digital controller, based on fuzzy blocks with

A pseudo-equivalence of them with linear PID controllers was made, based on the inputoutput transfer characteristics of the fuzzy block, obtained by digital computer calculation. The design of the fuzzy controller is based on the linearization of the fuzzy block around the origin, for the permanent regime. There is used the gain in the origin obtained as a limit in

For this type of controllers, the design relations were demonstrated. There was made an analysis of these design relations. There were also presented some observations related to

The results presented in this chapter are important in the practice design of the control systems based on PID fuzzy controllers. This method for equivalence is valid for all kind of fuzzyfication and defuzzification methods, all types of membership functions, all inference methods, because it is based on analytic transfer characteristic, which may be obtained using

origin of the gain function, obtained from the translated SISO transfer characteristic.

Table 2. The values of the quality criteria for the control system, for linear and fuzzy

 10-5 <sup>1</sup> [%]

1M [%]

tr [s]

trM [s]

Case <sup>1</sup>

[%]

tr [s] 1M [%] trM [s]

controllers, for tuned and detuned parameters of the electrical drive

Fig. 17. Transient characteristics for the current and speed


Mamdani structure and PID dynamics.

the influences of the scaling coefficients.

disturbance.

computer calculations.

**6. Conclusion** 


1r [%]

Fig. 15. The speed fuzzy PI controller, with anti-wind-up and correction circuit

A method to choose initial scaling coefficients based on the quality criteria of the control system is recommended, as it follows. The scaling coefficients were chosen after some iterative steps, using the quality criteria of the transient characteristics of the speed fuzzy control system at a step speed reference. The speed scaling coefficient *c*e had the same value *c*e=1/*e*M. The first value of the derivative scaling coefficient was *c*de=1/*de*M.


The adopted solution contains the values of the scaling coefficients from the sixth step. The transient characteristics obtained in the process of choosing the scaling coefficients are there presented in Fig. 16. The value of *c*de was decreased to the final value from the sixth step. Decreasing more this scaling coefficient, the fuzzy control system becomes unstable.

Simulations are made for the control system with fuzzy PI controller and also for linear PI controller, for tuned and detuned system parameters. The transient characteristics for the current and speed are to be presented in Fig. 17. With continuous line, there are represented the characteristics for fuzzy control, and with dash-dot line, there are represented the characteristics for conventional control. The regime consists in starting the process unloaded, with a constant speed reference. A constant load torque, in the range of the rated process torque, is also introduced. Then, the motor is reversed, maintaining the constant load torque.

Fig. 16. The transient characteristics for scaling coefficients determination

The quality criteria of the control system, with linear (l) and fuzzy controller (f), for tuned (a) and detuned (d) parameter are there presented in Tab. 2.

A method to choose initial scaling coefficients based on the quality criteria of the control system is recommended, as it follows. The scaling coefficients were chosen after some iterative steps, using the quality criteria of the transient characteristics of the speed fuzzy control system at a step speed reference. The speed scaling coefficient *c*e had the same value

2. An initial value for the output scaling coefficient is chosen *c*di1, based on controller

The adopted solution contains the values of the scaling coefficients from the sixth step. The transient characteristics obtained in the process of choosing the scaling coefficients are there presented in Fig. 16. The value of *c*de was decreased to the final value from the sixth step.

Simulations are made for the control system with fuzzy PI controller and also for linear PI controller, for tuned and detuned system parameters. The transient characteristics for the current and speed are to be presented in Fig. 17. With continuous line, there are represented the characteristics for fuzzy control, and with dash-dot line, there are represented the characteristics for conventional control. The regime consists in starting the process unloaded, with a constant speed reference. A constant load torque, in the range of the rated process torque, is also introduced. Then, the motor is reversed, maintaining the constant load torque.

Decreasing more this scaling coefficient, the fuzzy control system becomes unstable.

Fig. 15. The speed fuzzy PI controller, with anti-wind-up and correction circuit

*c*e=1/*e*M. The first value of the derivative scaling coefficient was *c*de=1/*de*M. 1. Initial values are chosen *c*e1 and *c*de1, based on operator knowledge.

3. With the above values for *c*e and *c*di it is calculated a value for *c*de2. 4. Maintaining the values of *c*e and *c*de and increasing the value of *c*di. 5. Maintaining the values of *c*e and *c*di and decreasing *c*de, and so on.

Fig. 16. The transient characteristics for scaling coefficients determination

(a) and detuned (d) parameter are there presented in Tab. 2.

The quality criteria of the control system, with linear (l) and fuzzy controller (f), for tuned

equivalence.


Table 2. The values of the quality criteria for the control system, for linear and fuzzy controllers, for tuned and detuned parameters of the electrical drive

Fig. 17. Transient characteristics for the current and speed

Based on a comparative analysis of the speed performance criteria, better results were there obtained with the fuzzy PI controller designed, using the above methods as it follows:

