펄스응답을 이용한 공정해석과 복합제어기의 튜닝

함 태원, 김 영한

동아대학교 화학공학과

Process Identification Using Pulse Response

and PID Controller Tuning

Tae Won Ham and Young Han Kim

Dept. of Chem. Eng., Dong-A University

INTRODUCTION

The process reaction curve method is simple and fast procedure, and a first order plus dead time model (FOPDT) is obtained to be employed in tuning. But it leaves an off-set from the initial steady state during the test. On the other hand, closed-loop techniques, such as the Ziegler-Nichols tuning and the relay feedback method, guarantee stability without the off-set problem. However, they take long test time and give only gain margin information unless test input is manipulated for other than phase angle of -180。.

While gain and phase margins are based on open-loop characteristic of the control system, maximum closed-loop modulus is closed-loop system guideline. Though the closed-loop modulus is preferred for the controller tuning, the open-loop guideline is often employed in many tuning procedures, e.g., the Ziegler-Nichols tuning and the relay feedback method, since its implementation is easier than the closed-loop guideline and no process model is necessary.

In this study, a process identification method using square pulse response is proposed and the identified model is employed to find the parameters of PID controller by applying combined open-loop and closed-loop tuning guidelines. Also, a new tuning guideline for the maximum closed-loop modulus is suggested.

TUNING

Three tuning guidelines, two from open-loop characteristic and one from closed-loop characteristic, are utilized in combination as follows:

[eqn: IP= LEFT | AR_1 -1 RIGHT | + LEFT | AR_2 -0.5 RIGHT | + LEFT | M_p -M_r RIGHT |]

where AR1 is the amplitude ratio, modulus, of open-loop frequency response at phase angle of -135。 and AR2 is the amplitude ratio at phase angle of -180。. The Mp is maximum modulus of closed-loop frequency response and its reference value is Mr. Since the phase margin of 45。 and the gain margin of 2 are suggested for chemical process control, the first and second terms are introduced.

As the reference value of maximum closed-loop modulus, the value of 1.26 corresponding to 2 of log modulus is recommended in several literatures. For the improved performance, however, different values of the reference amplitude for systems having different damping coefficients are introduced. An adjusted reference value from the following equation is utilized in the tuning of PID controllers from the result of the analysis between maximum modulus and damping coefficient.

[eqn: M_r = 1.433 - 0.433 zeta]

EXAMPLES

Four example systems are used for the performance test of the proposed tuning method of this study. Also, the performances are compared with the results of Yuwana and Seborg [1982] and Lee et al. [1990]. The simulation of the control system is carried out with the MATLAB.

Process I

[eqn: { e^-3s} over {(s+1)^2 (2s+1)}]

Process II

[eqn: {1} over {(s+1)^5 }]

Process III

[eqn: { e^-s} over {9s^2 +2.4s+1}]

Process IV

[eqn: { e^-s} over {(2s^2 +2s+1) (s+1) }]

The first three processes are adopted from Yuwana and Seborg [1982] and the last process is newly introduced to see the effect of damping coefficient on controller tuning. While the first two are overdamped systems, remaining two are underdamped systems. Especially the last system has intermediate damping coefficient between the first (and second) and the third processes.

RESULTS AND DISCUSSION

A square pulse having amplitude of one and width of one is introduced and the response is analyzed to find the parameters of a SOPDT model.

Using the parameters of the identified model, the optimization of the performance index is conducted to find the control parameters. For the comparison of control performances, the responses of step set-point change and regulatory performances are compared. In order to examine regulatory performance, an off-set value of 1 is added to the input of process as disturbance and output deviation is investigated.

This study and the tuning of Lee et al. show similar performance, while the tuning of Yuwana and Seborg leads to highly oscillatory response. The outcome of this study gives faster response than that of Lee et al. The regulatory performances show similar performance comparison to the set-point tracking. In process II, comparable responses of all three tunings are found in both set-point tracking and regulation, though the highest deviation is observed in the result of Yuwana and Seborg. Notice that this process has no process delay. As the highest deviation is attained from the tuning of this study of process III, the fastest response and settlement are found. Alike outcome is yielded for process IV. In regulatory performances of the two processes, the tuning of this study apparently leads to the smallest deviation among three tunings.

Since the tuning of process IV is not investigated in other studies, it is conducted by the instructions given in the studies. This process has the intermediate damping coefficient between processes I (and II) and III. Therefore, in Lee et al.'s tuning, Mr = 1.02 is used because it gives higher frequency.

Though fast response produces high deviation, the fast tracking to new set point is favorable unless the deviation is too large. In other words, the tracking has to be compromised for proper deviation leading the best performance. One way to measure the performance of different tunings is to check sum of deviations. Numerical comparison of the performance is carried out by computing the integral of absolute errors (IAE) between set point and output. The outcome of this study has the smallest errors except the regulatory performance in process II.

CONCLUSION

The proposed identification method takes short test time and leaves no disturbance to process. Moreover, since process model is available, any controller tuning guideline can be applied.

The analysis between maximum closed-loop modulus and the damping coefficient of process for a typical system results in the linear relationship between them which is implemented in the combined guideline including gain and phase margins. The performance of the proposed tuning procedure is examined in four different processes and the numerical comparison with other tuning techniques indicates that the tuning of this study gives the best performance.

ACKNOWLEDGMENT

Financial supports from the Dong-A University and the Korea Science and Engineering Foundation through the Automation Research Center at POSTECH are gratefully acknowledged.

LITERATURE CITED

Lee, J., Cho, W. and Edgar, T. F., "An Improved Technique for PID

Controller Tuning from Closed-Loop Tests",  AIChE J., 36, 1891 (1990).

Yuwana, M. and Seborg, D. E., " A New Method for On-Line Controller

Tuning", AIChEJ., 28, 434 (1982).