불안정 공정에 있어서 향상된 DMC의 강건성 분석

구도균, 손세훈, 이동권*, 정재학

영남대학교 화학공학및공업화학부

* 포항공과대학교 공정 산업의 지능 자동화 연구 센터

The Robustness of the Enhanced Strategy of DMC for Unstable Processes

Doe Gyoon Koo, Se Hoon Son, Dong Kwon Lee*, Jae Hak Jung

School of Chemical Engineering and Technology, Yeungnam University

* Automatic Research Center, POSTECH

Introduction

Model Predictive Control (MPC) has emerged as a powerful practical control technique during the last decade.  There are several representative MPCs such as Dynamic Matrix Control (DMC), Model Algorithmic Control (MAC), Internal Model Control (IMC), Quadratic Programming Solution of Dynamic Matrix Control (QDMC), etc.. The above-mentioned control algorithms have been applied successfully in many industrial chemical processes. These control techniques use the discrete convolution model to represent the process output, such as step response coefficients and impulse response coefficients of the reference models or real data models, and control the process by several optimization tools using the process outputs over some finite future time interval.  In the algorithms of MPCs an explicit dynamic model of the processes are usually used to predict the effect of future actions of the manipulated variables on the outputs.  We can determine the future moves of the manipulated variables using the optimization with the objective of minimizing the prediction error subject to operation constraints.

We already proposed the enhanced strategy of DMC for the open loop unstable processes in the last spring symposium. In this article, we studied the robustness of the enhanced strategy of DMC against the model mismatch and the noise effect of process outputs for the open loop unstable processes such as integrating processes and the process with Right-half Plan(RHP) poles. Gernerally the dynamic matrix control is hardly applicable to the noisy processes and poorly defined model.

Control Strategy and Modified Transfer Function

In this paper, we also use the enhanced strategy of DMC for the unstable processes including the integrating processes(Figure 1). As shown in Figure 1, we can calculate the following differential equations and the modified transfer function using the first order Taylor series expansion for the approximation of the time delay term, which is described in the equation (5) and equation (12). In the case 1, we develope the modified transfer function of integrating process and in the case 2, we also define the modified transfer function of the general unstable process.

Case 1. Integrating Process             Case 2. Process with RHP Poles

[eqn: tau {d^2 y} over dt^2 + dy over dt = K U_{n et} (t-theta)~]	(1)

[eqn: U_{n et}(t) = U(t) - K_i y(t)~]	(2)

[eqn: y(s) over u(s) = {{K e^{-theta s}} over {s(tau s +1)}}over{1+KK_i {e^{-theta s}} over {s(tau s +1)}}]	(3)

[eqn: = {Ke^{-theta s}}over {s(tau s+1)+KK_i e^{-theta s}}]	(4)

[eqn: = {Ke^{-theta s}}over {s(tau s+1)+KK_i (1-theta s)}]	(5)

[eqn: = {Ke^{-theta s}}over {tau s^2 +(1-KK_i theta)s+ KK_i }]	(6)

[eqn: = {1 over K_i e^{-theta s}}over {tau over KK_i s^2 +{1-KK_i theta} over KK_i s+ 1}]	(7)

[eqn: tau dy over dt -y(t) = K U_{n et} (t-theta)~]	(8)

[eqn: U_{n et}(t) = U(t) - K_i y(t)~]	(9)

[eqn: y(s) over u(s) ={{Ke^{-theta s}}over {tau s -1}}over{ 1+ KK_i e^{-theta s}over {tau s -1}}]	(10)

[eqn: = Ke^{- theta s } over {(tau s -1) + KK_i e^{-theta s}}]	(11)

[eqn: = Ke^{- theta s } over {(tau s -1) + KK_i (1-theta s)}]	(12)

[eqn: = { K over {KK_i -1} e^{-theta s}}over {{tau - KK_i theta}over {KK_i -1} s +1}]	(13)

Consequently, we can obtain the second(integrating process) and first(process with RHP poles) order plus time delay model as the overall transfer function. Considering the equation(7) and (13), we can obtain the dynamic matrix of the processes using this modified transfer function. And we suggest the  constraint of the value of inner feedback gain([eqn: K_i~]). From the equation (7) and (13), And we suggest the necessary and sufficient condition of the value of inner feedback gain([eqn: K_i ~]) to satisfy the stability of the modified transfer function.

[eqn: 0<KK_i < 1 over theta] integrating processes	(14-1)

[eqn: 1<KK_i < tau over theta] unstable processes	(14-2)

So we have to determine the value of [eqn: K_i~] in the constraint of equation (14). To calculate the value of inner gain, we can use the  following equation.

[eqn: K_i = 0.2 over K 1 over theta](Sung and Lee 1996) for integrating process	(15-1)

[eqn: K_i = 1 over K root{tau over theta}] for unstable process	(15-2)

Simulation Study

Example 1.(Modeling Error) Let us consider the modeling error of the reference model for integrating processes. If the reference models(for calculating the modified transfer function) have ±10% and ±20% model mismatch in the gain of the transfer function, the gains and time constants will be changed in the modified transfer function. For example, we consider the following two process transfer function and they have the [eqn: theta /tau~]values of 0.5 and 0.1 each other.

[eqn: G_p = {2 e^-2s } over s(4s+1) ~]    [eqn: G_p = 2*e^-s over 10s-1]	(16)

The modified transfer functions([eqn: G_{p,n ew}]) which have each ±10% and ±20% model mismatch will be changed as shown in the Table 1.

Table 1. Each Modified Transfer Functions with Model Mismatch

	Integrating Process	Unstable Process
-10% Model mismatch	[eqn: 36e^-2s over{44.444s^2 +9.111s+1}]	[eqn: 3.0981e^-s over 3.8759s+1]
-20% Model mismatch	[eqn: 32e^-2s over{50s^2 +10.5s+1}]	[eqn: 2.7539^-s over 4.8839s+1]
+10% Model mismatch	[eqn: 44e^-2s over{32.364s^2 +7.091s+1}]	[eqn: 3.7866e^-s over 2.6317s+1]
+20% Model mismatch	[eqn: 48e^-2s over{33.333s^2 +6.333s+1}]	[eqn: 4.1308e^-s over 2.2207s+1]

However, the control performance against the effect of the model mismatch in the proposed method for set point change is shown in the Figure 3.

Example 2.(Noise Effect) Consider the following 3 process transfer functions with 1 RHP pole. They have the [eqn: tau/theta~]values of 0.1, 0.5, 0.9 respectively.

[eqn: {2 e^-s } over 10s-1], [eqn: {2 e^-s } over 2s-1], [eqn: {4 e^-s } over 2.222s-1 ~]

From equation (8), we can obtain the values of inner gain of each processes, 1.581, 0.707, 0.264 respectively. From (6), the corresponding modified transfer functions are obtained as following equations.

[eqn: {3.442 e^-s } over 3.163s+1], [eqn: {4.831 e^-s } over 1.415s+1], [eqn: {71.428 e^-s } over 1.964s-1]

Using above equations, we can obtain the step response coefficients and impulse response coefficients. The control performance of the proposed method for set point change is shown in the Figure 4. We added up 10 % random deviation noise of each process outputs to research the robustness of proposed method. The control performance against the noise effect in proposed method for set point change is shown in the Figure 4.

Conclusions

We proposed a enhanced control strategy of DMC using inner feedback gain for integrating and unstable processes.. We also defined the reasonable constraints of inner feedback gain to satisfy the stability of the closed loop system.  The unstable processes and integrating processes were stabilized by inner feedback gain, and we could easily apply to industrial unstable and integrating processes using the proposed enhanced control strategy method. The enhanced control strategy was easily applied to MPCs without change of the control shame in the industrial processes and we can directly obtain the step response coefficient and impulse response coefficients with the modified transfer function.

Finally, the proposed control strategy gave superior control performance, as shown in Figure 1 and the enhanced control strategy is robust against the noise effect, as shown in Figure 2. On the other hand, we can also obtain the robust control performance against the model mismatch at not only the integrating process but also the unstable process, as shown in Figure 4.

Acknowledgment

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

Figure 1. Schemetic Diagram for      Figure 2. Control Performance of Closed-Loop

Enhanced Strategy of DMC          Response for Set Point Change with Noise

Figure 3. Control Performance of Model     Figure 4. Control Performance of Model

Mismatch in the Process Model for Set      Mismatch in the Process Model for

Point Change (Integrating Process)          set point change (Unstable Process)