일반적인 Cascade 제어 구조에 대한 강화 제어

이용호*, 최재인**, 박선원**

(주)삼성 Engineering*, KAIST 화학공학과**

An Enhanced Control With A General Cascde Control Structure

Yongho Lee*, Jaein Choi** and Sunwon Park**

Samsung Engineering Co.*

Department of Chemical Engineering, KAIST**

1. Introduction

Cascade control is often used to improve single loop control performance when the disturbances are associated with the manipulated variable or when the final control element exhibits nonlinear behavior. The performance enhanced by cascade control is heavily depends on tuning of controllers in inner and outer loops. However, previous tuning methods for cascade control apears to be rather limited to the frequency response methods([3],[2],[1]) that are tedious to be applied due to the need of trial and error graphical calculations.

In this paper, a general structure for cascade control systems is proposed. With the general structure for cascade systems, the method by Lee et al.([5]) is extended to integrating and unstable processes. For more efficient performance, setpont filter for each loop in cascade systems is proposed. As a result, a general PID tuning rule for cascade control systems is developed.

2. Theory

As shown in figure 1, classical cascade control systems consist of two feedback loops. Lee et al([5]) proposed a PID tuning method for classical cascade control systems for general stable processes using IMC approach to design classical feedback controllers. The IMC structure in their IMC approach([7]) is shown in figure 2. With IMC parametrization of the classical feedback controllers and assumption that model is the same as the plant, the closed loop transfer functions for inner loop and outer loop are

[eqn: y_2 = q_2 p_2 r_2 + (1-q_2 p_2 )p_d2 d_2]

()

[eqn: y_1 = p_2 q_2 p_1 q_1 r_1 + (1-p_2 q_2 ) p_1 (1-p_2 q_2 p_1 q_1 )p_d2 d_2 + (1- p_2 q_2 p_1 q_1 ) p_d1 d_1]

()

With the structure shown in figure 2, Lee et al([6],[7]) designed PID controllers for general stable and unstable processes. In the structure with the setpoint filter, controllers can be designed with characteristics: efficient performance of disturbance rejection; elimination of overshoots for unstable processes.

Here, we generalize the cascade structure with a new structure as shown in figure 3 to eliminate overshoots, which often happen for unstable processes and to get better performance of disturbance rejection. A generalized feedback control structure corresponding to the generalized IMC structure is in fingure 4. With IMC parametrization, the closed loop transfer function for inner and outer loop become

[eqn: y_2 = q_2 p_2 q_f2 r_2 + (1- q_2 p_2 ) p_d2 d_2]

()

[eqn: y_1 = p_2 q_2 q_f2 p_1 q_1 q_f1 r_1 +(1-p_2 q_2 ) p_1 (1- p_2 q_2 q_f2 p_1 q_1 ) p_d2 d_2 +(1-p_2 q_2 q_f2 p_1 q_1 ) p_d1 d_1]

()

2.1 Design of secondary controller

The controller in the inner loop has to be desinged to reject the disturbances into the inner loop quickly. For general stable processes, the method by Lee et al ([6],[5])  can give a stable overdamping response.

The model for the inner loop can be often an integrator. By approximating the integrator as an unstable pole near zero, the integrator can be controlled without an offset to a setpoint change, which often happens for a level control. Further, to get enhanced performance of disturbance rejection and to handle unstable processes in the inner loop, the method by Lee et al([4]) is used.

Consider a general process model of the inner loop.

[eqn: p_2 (s) = p_2m (s) p_2a(s)]

()

where, the process model of the inner loop can be devided into two parts; [eqn: p_2m], the portion of the model inverted by the controller and [eqn: p_2a], the portion of the model not inverted by the controller. The noninvertible portion is the portion of the model with right half plane zeros and time delays.

unstable processes or stable processes with poles near 0

To get a good response, for unstable processes or stable processes with poles near zero, the IMC controller for the secondary loop should satisfy following conditions.

.If the process model [eqn: p_2] has unstable poles, [eqn: up_1], [eqn: up_2],…, then [eqn: q_2] should have zeros at [eqn: up_1], [eqn: up_2],…

.If the process model [eqn: p_2] has unstable poles [eqn: dup_1],[eqn: dup_2],…, or has poles near zero, then [eqn: (1 -p_2 q_2 )] should have zeros at [eqn: dup_1],[eqn: dup_2],… or at the poles near zero.

When disturbance model has unstable poles, [eqn: p_2] = [eqn: p_d2] is assumed. IMC controller [eqn: q_2] is set as [eqn: q_2 = p_2m ^-1 f]. Then, automatically, the IMC controller satisfies the first condition because [eqn: p_2m ^-1] is the inverse of the model portion with the unstable poles. Through filter [eqn: f_2] design, the second condition should be satisfied. For this case, the filter is set as

[eqn: f_2 = {{SUM from { {i}=1} to m} alpha_i s^i +1} over {(lambda_2 s +1 )^m } }]

(6)

Here,[eqn: alpha_i] are determined to cancel the unstable poles or poles near zero and m is the number of poles canceled. Eq. (6) functions as a filter with adjustable time constant [eqn: lambda].

After obtaining the IMC controller, then, we get

[eqn: y_2 over r_2 = p_2 q_2 = {p_2a ( SUM from { { i}=0} to m alpha_i s^i +1) } over { (lambda_2 s +1 )^m } " " , " "y_2 over d_2 = (1- p_2 q_2 ) p_d2 ={ p_2a ( SUM from { { i}=1} to m alpha_i s^i + 1)} over { (lambda_2 s + 1)^m }]

(7)

The lead term in Eq. (7) causes an overshoot in the closed loop response. To eliminate the overshoot, we add a set point filter [eqn: q_f2] :

[eqn: q_f2 = 1 over { SUM from { { i}=1} to m alpha_i s^i + 1}]

(8)

stable processes

In the case that the process [eqn: p_2] doesn't have unstable poles or poles near zero, the IMC filter and the setpoint filter respectively become

[eqn: f_2 = 1 over { (lambda_2 s +1) ^m } ;" " q_f2 = 1]

(9)

where, m is the order of stable processes. With this setting , IMC controller is

[eqn: q_2 = {p_2m ^-1 } over {(lambda_2 s + 1 )^m }" "," " y = p_2 q_2 r_2 + (1- p_2 q_2 )p_2 d_2]

(10)

To get a stable response to a setpoint change and load change, the part of [eqn: p_2 q_2] and the part of [eqn: (1- p_2 q_2 ) p_2] should be stable. It is possible to make the closed loop response stable if IMC controller, [eqn: (q_2 )] and [eqn: (1- p_2 q_2 )] have zeros wherever [eqn: p_2] has RHP poles.

2.2 Design of primary controller

The PID controller tuned by the above procedure gives the closed loop response close to the desired one([6],[5],[4]). Thus, the closed loop transfer function for the inner loop can be approximated with equation 4. Through the design of the secondary controller, the closed loop transfer function of the outer loop becomes

[eqn: y_1 = p_2a over {(lambda_2 s + 1)^m } p_1 q_1 q_f1 r_1 + (1 - {p_2a ( SUM from { { i}=1} to m alpha_i s^i + 1 )} over {(lambda_2 s +1 )^m } p_d2 p_1 (1- p_2a over {(lambda_2 s +1) ^m} p_1 q_1 ) d_2 + (1 - p_2a over {(lambda_2 s +1)^m} p_1 q_1 ) p_d1 d_1]

(11)

here, note that for part of [eqn: d_2] in the closed loop transfer function, through the design of filter parameters [eqn: alpha_i], the poles near zero of [eqn: p_d2] is eliminated. Applying the similar procedure to the one desinging of the secondary controller, primary controller design is obtained for both unstable and stable processes.

untstable processes or stable processes with poles near zero

IMC controller : [eqn: q_1 = p_pm ^-1 {( SUM from { {i }=1} to m beta_i s^i +1)} over {(lambda_1 s +1 )^n }]                                             (12)

set point filter [eqn: q_f1] as :[eqn: q_f1 = 1 over { SUM from { { i}=1} to m beta_i s^i + 1}]                                            (13)

stable processes

IMC controller : [eqn: q_1 = p_pm ^-1 over {(lambda_1 s +1 )^n}]                                                  (14)

2.3 PID controller settings for FOPDT model

With the conversion procesdure([6]) from IMC controllers to PID controllers, we can find tuning for FOPDT models.

3. Simulation Study

Example : unstable cascade process

To test the capability of the proposed method for unstable processes, the following process models were considered. The process of the outer loop is unstable while the inner loop is  stable. These models are typical examples to represent a CSTR system which includes highly exothermic reaction([8]).

[eqn: G_P1 = G_D1 = e^-4s over { (-20s+1)}" "," " G_P2 = G_D2 = 2 e^-2s over {(20s +1)}]

(15)

The closed loop time constants are chosen as [eqn: lambda_1 = 1] and [eqn: lambda_2 = 8]. The tuning parameters are in Table 1.

Table 1. Resulting Tuning Values for Example

	Kc	□I	□D	setpointfilter
inner loop	6.9262	4.6034	0.7940	qf2=1/(3.6677s+1)
outer loop	-3.3188	36.2231	3.0845	qf1=1/(32.9145s +1)

Because any other methods can't handle these kinds of unstable cascade processes, the simulation results are not compared with those of other methods. Figures 5 and 6 show the closed loop response of the unstable processes given by Eq.(15) to load changes in the inner and the outer loop.

4. Conclusion

We generalize the IMC approach for cascade control systems with a general cascade control structure and show how to obtain PID tuning parameter values for general process models. General processes are divided into two parts. One includes integrating, unstable processes and stable processes with poles near zero. The other includes normal stable processes. For these processes, IMC controllers to give desired closed loop responses are obtained and converted to PID controllers. The PID controllers tuned by the proposed method give superior performances to the performances of the PID controllers tuned by other design methods.

References

[1] T.F. Edgar, R.C. Heeb, and J.O. Hougen. Computer-aided process control system design using interactive graphics. Comp. and Chem. Engr., 5(4):225, 1982.

[2] J.O. Hougen, Measurement and control applications. In Ph.D. Thesis, Pittsburgh, PA, 1979. Monograph No. 28, 1973.

[3] F.D. Jury. Fundamentals of three-mode controllers. Fisher Controls Company: Technical Monograph No. 29, 1973.

[4] Y. Lee, J. Lee, and S. Park. PID controller tuning for integrating and unstable processes with time delay. Chem. Eng. Sci., 2000. in press.

[5] Y. Lee, M. Lee, and S. Park. PID controller tunning to obtain desired closed-loop responses for cascade control systems. Ind. Eng. Chem. Res., 37(5):1859-1865, 1998.

[6] Y. Lee, M. Lee, S. Park, and Brosilow C. PID controller tuning for desired closed-loop responses for si/so systems. AIChE J., 44(1):106-115, 1998.

[7] M. Morari and E. Zafiriou. Robust Process Control. Prentice Hall, 1989.

[8] LP. Russo and BW. Bequette. Effect of process design on the open-loop behavior of a jacketed exothermic CSTR. Comp. and Chem. Engr., 20(4):417-426, 1996.

Figure 1: Classical cascade control system

Figure 2: IMC structue for cascade control system

Figure 4: The proposed generalized feedback control structure for cascade control

Figure 3: A generalized IMC structure for cascade control

Figure 6 : Closed loop response due to a load change for example (L1)

Figure 5 : Closed loop response due to a load change for example.(L2)