비선형 PI 제어기의 최적 조율법 개발 및 실험적 적용

구도균, 이동권*, 정재학

영남대학교 화학공학및공업화학부

* 포항공과대학교 공정산업의 지능자동화연구센터

The Optimal Tuning Method for Nonlinear PI Controller

and Its Eperimental Application

Doe Gyoon Koo, Dong Kwon Lee*, Jae Hak Jung

School of Chem. Eng. and Tech., Yeungnam Univ.

* Automation Reaserch Center, Dept. of Chem. Eng., POSTECH

Introduction

Generally, Most industrial chemical process control systems consist of the conventional PID controller even though many advanced control strategies have been proposed.. But some processes are highly nonlinear to be dealt with the PID controller.

In this work, we present the application guideline and tuning rules for the error square type nonlinear controllers which are currently most popular in process industries. We also propose the tuning rule for integrating processes. It can be applied to the general integrating processes without the change of strategy and the present transfer function. In addition to, we perform the real experimentation to the pilot scale processes. It shows that the new controller tuning methods can be applied to the industrial chemical processes. Also we can obtain a good control performance through the real application.

Error Square Controller Tuning

For suggesting the error square control tuning, the derivative term was removed, but for compensation of the loss of the responsiveness we used the controller to increase its proportional action according to the magnitude of the error [eqn: e(t){}].  Clark proposed replacing [eqn: K sub c] by [eqn: K sub c = K (1+beta{}|{}e{}| )].	So controller can be suggested following eq. 1.

Structure 1 :	[eqn: m(t) = K (1+beta{}|{}e{}| )[e + 1 over tau sub I int e{}dt{}]]	(1)

This type of controller would permit us to use a low value of gain so that the system is stable near the set point over a broad range of operating levels with changing process gains.  Another advantage of this nonlinear controller is that error square at a set point reduces the effects of noise. However it might not be reasonable error-square controller whose integral time constant changes with deviation. Therefore, the controller can be modified as following eq. 2.

Structure 2 :	[eqn: m(t) = K (1+beta{}|{}e{}| )e + 1 over tau sub I int e{}dt{}]	(2)

The above controller can be intended to maintain a constant damping factor. In order to develope tuning formulae for these two algorithms, we used a similar approach to  Lopez et al.(1967). For Considering the general feedback control loop, the controller [eqn: G sub c (s)~] is a two kind of error square controller and thus refers qualitatively to the transfer function relating the controller output [eqn: m(s)~] to the error [eqn: e(s)]. We can seek correlation among [eqn: (KK sub c )], [eqn: (tau sub I / tau )], and [eqn: (beta TIMES y sub sp )], for various values of [eqn: (theta/tau)] which gives optimal tuning.  Miler et al.'s (1967) recommendation of using ITAE optimal tuning criteria and this criteria demand as little oscillation as possible. The simulation was combined with a simplex search optimization algorithm, using random initial guesses.

ITAE Tuning for Nonlinear PI Controller

1. Load change

The optimal tuning formulae were obtained using linear regression of the dimensionless groups[eqn: (KK sub c )],[eqn: (tau sub I / tau )], and[eqn: (beta TIMES D sub max ){}].

Structure 1

[eqn: KK sub c{} ={}0.152{}(theta/tau) sup -1.11]	(3)

[eqn: tau sub I / tau{} ={}1.096{}(theta/tau) sup -0.4833]	(4)

[eqn: beta TIMES D sub max ={}5.335{}(theta/tau) sup -0.4013]	(5)

Structure 2

[eqn: KK sub c{} ={}0.1682{}(theta/tau) sup -1.164]	(6)

[eqn: tau sub I / tau{} ={}1.49{}(theta/tau) sup 1.308]	(7)

[eqn: beta TIMES D sub max ={}-6.164{}(theta/tau){} +{} 10.61]	(8)

2. Set point  changes

A similar approach was used for set point tuning formulae.

Structure 1

[eqn: KK sub c{} ={} 0.02586{}+ {}0.0982{}(theta/tau) sup -1.049] (9)

[eqn: tau sub I / tau{} ={}0.997{}+{}0.08{}(theta/tau) sup 2.351]      (10)

[eqn: beta TIMES Delta y sub sp ={}4.259{}+{}0.5362{}(theta/tau) sup 0.667]  (11)

Structure 2

[eqn: KK sub c{} ={}0.1245{}(theta/tau) sup -0.965]	(12)

[eqn: tau sub I / tau{} ={}1.57{}(theta/tau){} +{}0.09]	(13)

[eqn: beta TIMES Delta y sub sp {}={}1.05{}(theta/tau) sup 0.763{} +{} 4.12]  (14)

3. Integrating Processes

In this session, we propose the optimal tuning method that is obtained by the same strategy in the above open loop stable processes.

Load Change Based on ITAE Criteria

[eqn: KK_c ^' =0.665(theta / tau )^-1.004 ~]	(15)

[eqn: tau_I / tau = 3.682(theta /tau )^1.004 ~]	(16)

[eqn: beta times D_max =0.1588(theta/tau)^-0.9806 ~]	(17)

Load Chang Based on ISE Criteria

[eqn: KK_c ^' =2.818(theta / tau )^-0.396 -2.104~]   (18)

[eqn: tau_I / tau = 3.859(theta /tau )^1.017 +0.044~]	(19)

[eqn: beta times D_max =0.165(theta/tau)^-0.864 ~]	(20)

Simulation Results

1. Control Response of Set point Change and Load change for FOPTD Model

Several different sets of responses were simulated for both disturbance and set point changes. Let's consider the following process transfer function :

[eqn: G sub p (s){} = {}e sup -3s over {(s+1) sup 2 (2s+1)}]	(21)

The process model is obtained by the following equation based on the on-line identification method by Chen (1989).

[eqn: G sub m (s){} = {}e sup -4.48s over {2.95s+1}]	(22)

We compares the responses of the PID controller tuned by the Internal Model Control (IMC) tuning rule and the nonlinear PI controller tuned by the proposed tuning rule. The response performance of nonlinear PI controller shows similar control performance to that of the PID controller as shown in Fig. 1.

Figure 1. Autotuning results for set-point change and load change[eqn: (theta/tau =0.5)]

Experimental Study

1. Introduction and Control Structure

Experimental Study of the proposed controller tuning method is presented. The schematic diagram of a level control for a continous tanks level system is shown in the Figure 2. The system consists of two noninteracting series tanks. Each tank has a constant diameter, and the level of the store tank also has a constant value. The difference of the height between the control valve and the last tank is about 1.5 meters, so this control suructure contains a large time delay. The control objective is to keep the level of the last tank at the desired value by adjusting the opening time of the on-off valve attached below the store tank.  Therefore, the         Figure 2. Schematic Diagram manipulated and controlled variables are the opening        of Level Control System

time of the actuator and the average flow rate of the valve.

2. Process Identification

In this article, we use the open loop step change of the controller output or manipulated variable as the test signal. So we can obtain the following transfer function of eq.(23). we can get the process informations, [eqn: K_m~]=3.816, [eqn: tau_m ~]=181.1 sec and [eqn: theta_m ~]=97.47 sec for FOPTD model of eq. (23).

[eqn: G_m = {K_m e^{-theta_m s}} over {tau_m s + 1}]	(23)

We use this transfer function model to calculate the parameters of the conventional PID controllers as well as the nonlinear PI controllers and the result of process identification is as shown in Fig. 3.

3. Real Application and Experiment

First, we experimented the set point tracking. From the above process information, we can use the Equation (3),(4) and (5) for nonlinear controller Structure 1. At the steady state, DP Cell indicate the output value as 2.46 Volt. Also we determined the set point as 2.895 Volts that indicate 20cm of real tank. As a result, we can get the controller parameter, [eqn: K_c^' ~]=0.0561, [eqn: tau_I ~]=183.9252 sec, and [eqn: beta~]=6.0085. The controlled process output is shown in the Figure 4.

Figure 3. Step Change of Manipulated      Figure 4. Process Output Using

Variable for Process Identification           the Nonlinear PI Controller

Conclusions

The nonlinear PI controller with the tuning rule provides a superior control performance for set point change in the several processes. From an experimental study, it can be noted that the proposed method gives the reliability and efficiency of the application to the industrial processes. Our experimental process and control system has a lot of nonlinearity, a long time delay and a large time constant. It also means that the proposed controller tuning method is more suitable for the highly nonlinear and industrial systems.

Acknowledgment

This work was supported by the Korea Science and Engineering Foundation through the Automation Research Center at POSTECH.

References

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