Smith predictor 에서의  모델 오차를 고려한 강건한 PID 조율법

이 동권, 성 수환*, 이 인범*, 이 문용

영남대학교 화학공학부, *포항공과대학교 화학공학과

A Robust PID Tuning Rule for Smith predictor

In the presence of Model Uncertainty

Dongkwon Lee, Suhwan Sung*, Inbeum Lee*, Moonyong Lee

School of Chem. Eng. & Tech., Yeungnam Univ., *Dept. of Chem. Eng., Pohang Univ.

Introduction

Time delay commonly occurs in usual process control problems in the presence of distance velocity lags, recycle loops and composition analysis loops.  The existence of the time delay in a process transfer function causes major difficulties in the design and implementation of control algorithms.  For this reason, extensive research effort has been done on developing efficient control strategies, Smith predictor (Smith 1957) or Dead Time Compensator (DTC) attracted many researchers and control engineers because of its simplicity and efficiency.  One of the main problems is its sensitivity to modeling errors. In the face of inevitable mismatches between the model and the actual process, the closed-loop performance can be very poor.  For this reason, recently, research effort has been focused on the robust tuning issues of the Smith predictor.   However, all these researches do not address simultaneous uncertainties in gain, time constant and  time delay.

In this article, we propose a robust PID tuning rule for the Smith predictor system coping with uncertainties in all parameters of the model. The proposed approximation technique using the equivalent gain and time delay provides a significant advantage in that it does not change the structure of the total transfer function while all plant/model mismatch effects are included.  As a consequence, the proposed method can utilize many existing PID tuning rules.

Theory

The Smith predictor control system is shown in Figure 1.

Figure 1. Block diagram of the Smith predictor

The blocks[eqn: {G sup *} sub p (s)] and [eqn: {G sup *} sub m (s)] are the delay-free parts of the process and the model, respectively,while[eqn: {G } sub p (s)] and[eqn: {G } sub m (s)] represent the process and the model, respectively.  The transfer function from the reference input [eqn: y sub sp (s)] to the process output [eqn: y (s)] is given by

[eqn: y(s) over {y sub sp (s)} = {G sub c (s)(G sub m sup * (s)+ G sub p (s) - G sub m (s))} over {1 + G sub c (s) G sub m sup * (s) + G sub c (s)(G sub p (s) - G sub m (s)) }]                     (1)

In order to address the plant/model mismatches, we would approximate [eqn: G sub m sup * (s) + G sub p (s) - G sub m (s)] by using the equivalent gain and time delay concept as follows :

[eqn: G sub m sup * (s) + G sub p (s) - G sub m (s) APPROX G sub w (s)]                                        (2)

where  the total transfer function [eqn: G sub w (s)] denotes the worst case process in all possible processes and its structure is the same as that of [eqn: G sub m (s)].

Then, the feedback system can be presented as

[eqn: y(s) over {y sub sp (s)} = {G sub c (s) G sub w ( s)} over {1 + G sub c (s) G sub w (s)}]                                            (3)

From eq. (3), we can recognize that [eqn: G sub c (s)] can be easily tuned based on [eqn: G sub w (s)] by using usual PID tuning rules such as ITAE and IMC-PID rules.

Here, the proposed equivalent gain and time delay [eqn: K sub eq e sup {-theta sub eq s}]  corresponding to [eqn: G (s)] is derived from the Taylor expansion as follows :

[eqn: G(s)=G(0) + dG(s) over ds vert sub s=0 s + 1 over 2 {d sup 2 G(s)} over ds sup 2 vert sub s=0 s sup 2 +cdots]                   (4)

[eqn: K sub eq e sup {-theta sub eq s} = K sub eq - K sub eq theta sub eq s + 1 over 2 K sub eq theta sub eq sup 2 s sup 2 + CDOTS]                             (5)

Comparing the first and second terms of eqs. (4) and (5).

[eqn: K sub eq = G(0)]                                                           (6)

[eqn: theta sub eq = -{1 over K sub eq} dG(s) over ds vert sub s=0]                                             (7)

As another approach, comparing the third terms instead of the second term in eqs. (4) and (5) gives

[eqn: theta sub eq = sqrt {1 over K sub eq {d sup 2 G(s) }over ds sup 2 vert sub s=0}]                                             (8)

FOPDT case

Assume the model and the process are the first order plus time delay systems as follows :

[eqn: G sub m (s) = {K sub m e sup {-theta sub m s}} over { tau sub m s + 1}]                                                  (9)

[eqn: G sub p (s) = {K sub p e sup {-theta sub p s}} over { tau sub p s + 1}]                                                   (10)

Using eqs. (6) and (7), we can obtain  following the equivalent gain and time delay.

[eqn: K sub eq = K sub p over K sub m]                                                          (11)

[eqn: theta sub eq = ( tau sub p - tau sub m ) + theta sub p - K sub m over K sub p theta sub m]                                         (12)

If we assume that process parameters are bounded as follows :

[eqn: K sub m - delta sub K <= K sub p <=K sub m + delta sub K]                                              (13)

[eqn: tau sub m - delta sub tau <= tau sub p <=tau sub m + delta sub tau]                                                 (14)

[eqn: theta sub m - delta sub theta <= theta sub p <=theta sub m + delta sub theta]                                                (15)

Then the worst case should have the largest equivalent gain and time delay so that the following equivalent gain and time delay should be chosen as the worst case.

[eqn: K sub eq = { K sub m + delta sub K } over K sub m]                                                     (16)

[eqn: theta sub eq = K sub m over { K sub m + delta sub K } theta sub m + delta sub theta + delta sub tau]                                          (17)

SOPDT case

The process and the model are given as follows :

[eqn: G sub p (s) = {K sub p e sup {-theta sub p s}} over { a sub p s sup 2 + b sub p s + 1}]                                               (18)

[eqn: G sub m (s) = {K sub m e sup {-theta sub m s}} over { a sub m s sup 2 + b sub m s + 1}]                                             (19)

Using eq.(6), we can obtain the following relationship.

[eqn: K sub eq = { K sub m + delta sub K } over K sub m]                                                     (20)

If we choose the worst case process based on eq.(7), the equivalent time delay is as follows :

[eqn: theta sub eq = theta sub p + b sub p - b sub m - K sub m over K sub p theta sub m]                                           (21)

As another strategy, we can choose the worst case process based on eq. (8).

[eqn: theta sub eq = sqrt { theta sub p sup 2 + 2(b sub p -b sub m ) theta sub p + a sub m - a sub p + b sub p (b sub p - b sub m ) - K sub m over K sub p theta sub m sup 2}]              (22)

Here, we assume that the following bounds for the modeling errors are given

[eqn: K sub m - delta sub K <= K sub p <=K sub m + delta sub K]                                              (23)

[eqn: theta sub m - delta sub theta <= theta sub p <=theta sub m + delta sub theta]                                                (24)

[eqn: a sub m - delta sub a <= a sub p <=a sub m + delta sub a]                                                (25)

[eqn: b sub m - delta sub b <= b sub p <=b sub m + delta sub b {}]                                                (26)

Then, the maximum value of eq.(22) for the given modeling errors of eqs.(23)-(26) is as follows :

[eqn: theta sub eq = sqrt {delta sub K over {K sub m + delta sub K} theta sub m sup 2 + 2 theta sub m delta sub theta + 2 delta sub theta sup 2 + 2 delta sub b theta sub p + delta sub a + (b sub m + delta sub b ) delta sub b{} }]           (27)

Maximum value of (21) for the given modeling errors of eqs. (23)-(26) is as follows :

[eqn: theta sub eq =delta sub theta + delta sub b + delta sub K over {K sub m + delta sub K} theta sub m]                                          (28)

Simulation Result & Conclusions

Consider the process [eqn: G sub p = e sup -3s over {(s+1) sup 2 (2s +1)}].  The process model is obtained as [eqn: G sub m = e sup -4.8s over {(2.95s +1)}] based on the  on-line identification method by Chen (1989).  We assume that 10% modeling error both in the time delay and the time constant.  The approximated equation is obtained as [eqn: G sub m sup * K sub eq e sup {-theta sub eq s} = e sup -0.743s over {(2.95s +1)}].

Figure 2. Simulation result                Figure 3. Bode diagram

Figure 2. shows the control results of the PID controller with the IMC-PID tuning rule and the auto-tuned Smith predictor with IMC-PI.  Figure 3 shows the Bode plot of the [eqn: G sub m sup * (s) + G sub p (s) - G sub m (s)] for the process and the corresponding[eqn: G sub m sup * K sub eq e sup {-theta sub eq s}]. [eqn: G sub m sup * K sub eq e sup {-theta sub eq s}] shows a lower phase angle than[eqn: G sub m sup * (s) + G sub p (s) - G sub m (s)].  It shows the stable controller can be guaranteed.

Acknowledgement

Financial support from the Yeungnam university is gratefully acknowledged.

This study was also supported by Korea Science and Engineering Foundation through the Automatic Research Center at Postech..

Literature Cited

Z.J. Palmor and M. Bl며, An auto-tuner for Smith dead time compensator,     INT. J. CONTROL, Vol. 60, No. 1, 117-135, (1994)

Chen C., A Simple Method for On-line Identification and Controller Tuning,    AIChE J.,35, 2037