ON PID CONTROLLER DESIGN USING KNOWLEDGE BASED FUZZY SYSTEM

The designing of PID controllers is a frequently discussed problem. Many of the design methods have been developed, classic (analytical tuning methods, optimization methods etc.) or not so common fuzzy knowledge based methods. The aim of design methods is in designing of controllers to achieve good setpoint following, corresponding time response etc. In this case, the new way of designing PID controller parameters is created where the above mentioned knowledge system based on the relations of Ziegler-Nichols design methods is used, more precisely the combination of the both Ziegler-Nichols methods. The proof of efficiency of the proposed method and a numerical experiment is presented including a comparison with the conventional Ziegler-Nichols method.


Introduction
As it is written in [1] two expert system approaches exist.The first one, the fuzzy rule base way for controlling processes for which suitable models do not exist or are inadequate.The rules substitute for conventional control algorithms.The second way originally suggested in [2] is to use an expert system to widen the amounts of classical control algorithms.
The system of designing a PID controller (its parameters) with a knowledge base is created which is built on know-how obtained from the Ziegler-Nichols designing methods, the combination of the frequency response and step response Ziegler-Nichols methods and its mutual conversion.The system created in this way is determined to design parameters of a classical PID controller, which is considered in closed feedback control.

Ziegler-Nichols Design Methods
These classical methods for the identification of the parameters of PID controllers were presented by Ziegler and Nichols in 1942.Both are based on determination of process dynamics.The parameters of the controller are expressed as a function by simple formulas [3].
The transfer function of the PID controller using these methods is expressed as:

The Frequency Response Method
The first Ziegler-Nichols method is based on the frequency response of the system.The essence is to find the point -the ultimate point P u on the Nyquist curve of the transfer function of the system, where the Nyquist curve intersects the real negative axis (Fig. 1).Frequency corresponding to the ultimate point is then f Pu .This point is characterized by the parameters K u and T u -called the ultimate gain and the ultimate period.The ultimate gain K u may be determined as: the ultimate period as: In the other way the parameters K u and T u can be obtained so that controller is connected to the system, parameters are set as T i = ∞ and T d = 0 (it means that the control action is proportional).The gain is then increased until the system starts to oscillate.The period of this oscillation is T u and the gain when the oscillations appear is K u .
Parameters of a PID controller from the Ziegler-Nichols frequency response method are expressed as (Tab.1).
Fig. 1: The ultimate gain Ku, and the ultimate period Tu defined in the Nyquist diagram [4].
There are also some limitations using this method -the Nyquist curve may restrict the real axis only in one point [3].

The Step Response Method
The time domain method is based on a registration of the open-loop unit step response of the system.The parameters of a PID controller are directly given from the function of the parameter a, and dead time L which are obtained so that the tangent is drawn at the maximum of the slope of the unit step response (Fig. 2).The intersection of the tangent with axis y and the distance of this intersection and the beginning of the coordinate axis determine the parameter a.The dead time L i s t h e distance of the intersection of the tangent with time axis (axis x) and also the beginning of the coordinate axis [3], [4].The parameters of a PID controller from the Ziegler-Nichols step response method are expressed as (Tab.2).

Used Mutual Conversion of Both Ziegler-Nichols Methods
To determine parameters of PID controller the relations of the frequency response method are used expressed as (Tab.3).
Tab.3: PID controller parameters obtained from the Ziegler-Nichols frequency response method [3].But the ultimate gain K u and the ultimate period T u (whose importance is described above) are obtained from unit step response (Fig. 3) of the controlled system and one of the conversion is used by the dead time L and the delay time D. For understanding it is important to explain these two times L and D and how to obtain these constants from unit step response of the controlled system.It is necessary to draw a tangent at the maximum of the slope of the step response.The dead time L has the same importance as in the step response design method.The delay time D is the distance between the intersection of the tangent and the time axis and the point p on the time axis.The point p is created so that in the intersection of the tangent and the maximum of the unit step response of the controlled system is the perpendicular to the time axis raised and the intersection of the perpendicular and the time axis is the point p [3].
The ultimate gain K u and the ultimate period T u are then defined as:

The Definition of the Controlled System
The first thing is to define the controlled system.It could be defined in many ways; in this case the controlled system is defined by a transfer function.The determining parameters are constants A, B and C from the denominator of the transfer function of the controlled system in a form: () The constants A, B and C are the inputs variables of the knowledge base system which as it was explained above, is used for identifying the parameters of a classical PID controller.The constant A can be from the range from 0 to 22, the B from 0 to 20 and the C from 0 to 28.

Knowledge Based System of Identification of Parameters of a PID Controller
The constants A, B and C as the inputs of the knowledge based system are represented in (Fig. 4) so as the outputs KKNOW, TIKNOW and TDKNOW w h i c h a r e a l s o constants and are used as parameters of the classical PID controller.

Description of Linguistic Variables
The inputs of knowledge based system A, B and C are the linguistic variables expressed by fuzzy sets, for each linguistic variable by three linguistic values -small (S), medium (M) and large (L) (Fig. 5, Fig. 6 and Fig. 7).
The membership functions of all linguistic values have a triangular shape.The shape could be described in three numbers.The first one is the point of the triangle where the membership degree is equal to zero, the same as the third number, and second number is the point, where the membership degree is equal to one.Description of the shape of the linguistic values of the linguistic variable A: • small (S) [0 6 11], • medium (M) [6 11 16], • large (L) [11 16 22].
So according to the description of the importance of three numerical descriptions, the membership function of linguistic value M of input linguistic variable A has the shape of an isosceles triangle, and shape of the membership functions for linguistic value S and L is a general triangle.Description of the shape of the linguistic values of the linguistic variable B: • small (S) [0 5 10], • medium (M) [5 10 15], • large (L) [10 15 20].
Membership functions of all linguistic values of linguistic variable B have the shape of an isosceles triangle.Description of the shape of the linguistic values of the linguistic variable C: • small (S) [0 9 14], • medium (M) [9 14 19], • large (L) [14 19 28].
The membership function of linguistic value M of input linguistic variable C has the shape of an isosceles triangle, and the others are general triangles.
The first thing for determining the values of the outputs KKNOW, TIKNOW and TDKNOW is to define the crisp values of the constants K, T i and T d , which are identified for every combination of input linguistic values by combination of Ziegler-Nichols methods described above.In total, 27 crisp values (a single element fuzzy set) for each output value K, T i and T d are created.The constants K, T i and T d are the concrete values (Tab.4), whose outputs KKNOW, TIKNOW and TDKNOW can take but the resulting values of those outputs are given according to the active rules and then according to the weighted mean.

Knowledge Base of the Fuzzy System
The core of the knowledge base is the definition of the linguistic IF-THEN rules of the Takagi-Sugeno type [2]: where r = 1, 2,…,R is a number of the rule.
The fuzzy conjunction in antecedent of the rule is interpreted as a minimum.The crisp output is determined using the weighted mean value [2], where µ Ai (x) is the membership degree of the given input x.
Each output is defined as: where

concrete values of inputs variables. S o t h e o u t p u t o f t h e s y s t e m i s t h e c r i s p v a l u e a n d defuzzification is not required.
In total, r = 27 rules of the Takagi-Sugeno fuzzy model are used:

Experimental Verification of the Created System
For the experiment the controlled system is selected with a transfer function in the form: () For this system two PID controllers are determined, one using only the combination of the Ziegler-Nichols design methods and the second one using the created knowledge based system.Both PID controllers are inserted into a closed feedback loop with an appropriate system and the time response is assessed for the unit step of both closed feedback loops.The timing is displayed in the time response of closed feedback loop with the PID controllers with parameters determined using created knowledge based system and a combination of classical Ziegler-Nichols methods for a unit step (Fig. 8).For evaluation of a time response the 3 % standard deviation from the steady-state value is chosen.For our selected controlled system the difference in time of stabilization as a time response for the unit step is 1 second (Tab.5).So for this concrete example the difference is approximately 25 %.
The second very noticeable difference is in the overshoot.Using the PID controller tuned by combination of the Ziegler-Nichols method the overshoot is approximately 20 %.Using PID controller identified by the knowledge based system the time step response is in this case without overshoot.

Conclusion
The fuzzy knowledge based system which was created is the new non-conventional tool for designing the parameters of PID controllers.It was experimentally verified and it was found that it is very usable for systems with the transfer function in shape (6), where the parameter A is rather small while the parameters B and C rather medium or large.In future research some new rules will be added to extend the class of usable controlled systems types.

Fig. 3 :
Fig. 3: Characterization of a unit step response of the controlled system with the representation of the dead time L and the delay time D [3], [5].

Fig. 4 :
Fig. 4: The representation of inputs and outputs of the created knowledge based system.

Fig. 5 :
Fig. 5: The shape of the membership function of linguistic values for input linguistic variable A.

Fig. 6 :
Fig. 6: The shape of the membership function of linguistic values for input linguistic variable B.

Fig. 7 :
Fig. 7: The shape of the membership function of linguistic values for input linguistic variable C.

Fig. 8 :
Fig. 8: Time response of the closed feedback loop with a PID controller with parameters determined using created the knowledge based system and a combination of classical Ziegler-Nichols methods for a unit step.
Crisp values of the constants K, Ti and Td.