Ind. Eng. Chem. Res. 1999, 38, 1575-1579
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Parallel Compensation with a Secondary Measurement Daewoong Chun, Jin Young Choi, and Jietae Lee* Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
A secondary measurement is usually used in the form of cascade control to effectively reject disturbances to the manipulated variable. In the cascade control system, the inner loop is tuned first and then the outer loop is tuned with the inner loop closed. When the dynamics of the inner process is slow, this tuning procedure does not provide the best control performance. Here, for slow inner processes, parallel compensation similar to the Smith predictor and the internal model control is shown to be more effective than the cascade control. A very simple tuning rule and a simple anti-reset windup are feasible in this control structure. It can be used in its own structure or to tune the cascade control system systematically. For automatic tuning of the proposed control system, an identification method which finds two process models with the relay feedback and P-control tests is also proposed. Introduction When a secondary measurement is available, cascade control is often used to improve the dynamic response of a feedback control system for disturbances to the manipulated variable and to reduce the static nonlinearity of the manipulated variable (Seborg et al., 1989). Krishnaswamy et al. (1990) showed that the performance of cascade control is always better than that of feedback control without the cascade loop for disturbances to both the inner process and the outer process. Hence, the cost of the additional measurement is cheap, so the cascade control is highly recommended. Yu and Luyben (1986) studied the stability of the cascade control system. To reduce the instability problem, they recommended not to use integral action in the inner loop (that is, P-only controller for the inner loop) and to use the derivative action in the outer loop. Krishnaswamy and Rangaiah (1992) recommended the integral action in the inner loop to enhance the control performance when the dynamics of the inner process is fast enough. It does not conflict with the stability result of Yu and Luyben (1986) because the stability problem is not serious for fast inner loop processes. Yu (1988) proposed a cascade control system for two parallel processes to reject disturbances. Shen and Yu (1990) considered a variable selection criterion for the secondary measurement. Wolff and Skogestad (1996) showed that the cascade control can reduce interaction in the multiloop control of the distillation column. Parrish and Brosilow (1985) used the secondary measurement to infer the controlled variable whose measurement is unavailable or very expensive. Hang et al. (1994) applied the relay autotuning method to tune the cascade control system automatically. Two relay feedback tests are used. The inner loop is put on relay feedback first and a P or PI controller is tuned. The outer loop is then placed on relay feedback with the inner loop closed, and a PI or PID controller is tuned for the outer loop. The ratio of two ultimate frequencies indicates the ratio of speeds of loops and * To whom all correspondence should be addressed. Tel.: +82-53-950-5620. Fax: +82-53-950-6615. E-mail: jtlee@ bh.kyungpook.ac.kr.
Figure 1. Classical cascade control system.
Figure 2. Proposed control system.
hence can be used to confirm the effectiveness of cascade control (Murrill, 1988) and this two-step autotuning method. When the inner process is slow, the cascade control is not so effective and the two-step tuning method as in the autotuning method of Hang et al. (1994) does not provide a well-tuned control system (Morari and Zafiriou, 1989). Here a new control structure and simple autotuning method are proposed for slow inner processes. The secondary measurement is used for the parallel compensation. With this new control structure, a simple tuning rule and better performance are obtained for slow inner processes. Proposed Control System When a secondary measurement is available, a cascade control system as in Figure 1 can be constructed to reject disturbance to the manipulated variable L2(s) effectively. If the dynamics of the inner process G2(s) is fast, the cascade control system of Figure 1 works well and tuning is also simple. The inner loop is tuned first with the inner process G2(s), and then the outer loop is tuned with the inner loop closed. However, when the speed of G2(s) is comparable to that of G1(s), the cascade
10.1021/ie980315b CCC: $18.00 © 1999 American Chemical Society Published on Web 03/09/1999
1576 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999
control is not so effective and the above two-step tuning does not work well. To avoid this problem, a new control structure such as that in Figure 2 is considered. For tuning of the proposed control system, the first order plus time delay models are assumed for both G1(s) and G2(s) as usual.
k2 exp(-θ2s) τ2s + 1
(1)
As in the Smith predictor, the parallel compensator of the form
k1 Q1(s) ) (1 - exp(-θ1s) + τ2s) τ1s + 1
k1k2 exp(-θ2s) (3) T(s) ) G2(s) (G1(s) + Q1(s)) ) τ1s + 1 The feedback controller Q2(s) is designed for the compensated process T(s). The modified IMC-PID tuning rule is used as
(
K ˜ c2 )
)
1 τ˜ I2s
τ1(θ2 + 2λ) - λ2 k1k2(θ2 + λ)2
τ1(θ2 + 2λ) - λ2 τ˜ I2 ) θ 2 + τ1
C2(s) ) Q1(s) Q2(s)
C1(s) ) 1/Q1(s) )
(5)
≈
τ1s + 1 k1(1 - exp(-θ1s) + τ2s)
(
2τ1(θ1 + τ2) + θ12 2
2k1(θ1 + τ2)
(2)
is used. Here the time delay term, exp(-θ1s), is to cancel the large time delay of G1(s) in the feedback loop which enables tight tuning of the control system. The derivative term, τ2s, which is to cancel the fast pole in G2(s), will speed up the rejection rate of load disturbances in L2(s). The compensated process is
˜ c2 1 + Q2(s) ) K
C1(s) ) 1/Q1(s)
Hence, with the 1/1 Pade approximation of the time delay term and ignoring higher order terms in the series expansion, we can obtain controllers as
k1 exp(-θ1s) G1(s) ) τ1s + 1 G2(s) )
relationship between both control systems is
C2(s) ) ≈
where λ is the design parameter. Other tuning methods can also be used (Astrom and Hagglund, 1995). It is remarked that modeling errors do not cause offset errors because Q1(0) ) 0. Advantages of this control structure are that it has a simple tuning rule as above and it is very simple to implement the anti-reset windup function which is necessary for practical application of PID controllers. Morari and Zafiriou (1989) recommended the internal model control structure for the outer loop to avoid the reset windup problem. Cascade Control System Design The proposed control system of Figure 2 can be used in its own structure, and it has some advantages, as shown above. Sometimes, the control system is fixed to the classical cascade control and tuning of this cascade control system is required. For this, the tuning rule of eqs 2 and 4 for the proposed control system can also be used to tune the classical cascade control system. The
(
τ1 +
1 θ12
))
2(θ1 + τ2)
s
(6)
k1(1 - exp(-θ1s) + τ2s) Q2(s) τ1s + 1 k1(θ1 + τ2)K ˜ c2 τ˜ 12
(
τ˜ I2s + 1
τ1 +
θ12
)
2(θ1 + τ2)
(7) s+1
The controller C1(s) is a PID controller form and C2(s) is a lead and lag form. The classical cascade control system uses a P or PI controller for the inner loop controller C2(s). Because offset in the output of the inner loop is usually allowed and one of requirements for the inner loop is to achieve a large bandwidth (Eker and Johnson, 1996), the lead and lag module is better than the PI controller as the inner loop controller. Krishnaswamy and Rangaiah (1992) recommended the PI controller as the inner loop controller when the inner process is fast. Our tuning rule of eq 7 also becomes the PI-controller form as
C2(s) ≈ (4)
1+
k1(θ1 + τ2)K ˜ c2 τ1 +
θ12
(
1+
)
1 τ˜ I2s
2(θ1 + τ2)
if θ2 and λ are small. Classical design of the inner loop first results in C2(s), which depends only on the inner loop process G2(s). However, many researchers (Morari and Zafiriou, 1989; Eker and Johnson, 1994) have recommended C2(s), which accounts for the outer loop process G1(s) as in eq 7. All six process parameters of G1(s) and G2(s) are used to calculate C2(s) in eq 7. Process Identification with the Relay Feedback and P-control Method It is well-known that closed-loop identification is better than open-loop identification. The closed-loop identification method can be applied to find two process models for the cascade control. For this, Hang et al. (1994) used two relay feedback tests for finding the ultimate data of processes and tuning each controller sequentially. Here one relay feedback test is used to find process models as shown in Figure 3. After a stable oscillation has been reached, the relay is switched to
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1577
Figure 3. System for the relay feedback and P-control identification.
Figure 5. Bode plot for G1(s) and its approximation when R ) 0.8.
Figure 4. Typical responses of the relay feedback and P-control tests.
the P-control test to find the steady-state gains. Typical responses are shown in Figure 4. From the oscillation, exact frequency responses at the oscillation frequency can be obtained as (Sung and Lee, 1997)
∫tt y1(t) exp(-jωr) dt G1(jωr) ) t ∫t y2(t) exp(-jωr) dt 2
1
2
1
∫tt y2(t) exp(-jωr) dt G2(jωr) ) t ∫t u(t) exp(-jωr) dt 2
1
2
(8)
1
where ωr ) 2π/Pr. The steady-state process gains can be obtained from the responses of P-control:
Table 1. Identified Process Models G1(s)
k1 ) y1∞/y2∞ k2 ) y2∞/u∞
Figure 6. Load responses for R ) 0.2 (step changes in L2(s) and L1(s) at time 0 and 50, respectively).
(9)
From the steady-state gains of eq 9 and frequency responses of eq 8, first order plus time delay models of eq 1 can be obtained (Lee and Sung, 1993). Tests with the biased relay feedback (Shen et al., 1996) can also be used to find both the steady-state gains and frequency responses. The frequency of relay oscillation is near the critical frequency of G1(s) G2(s). Hence, it will be adequate to find the outer process model of G1(s) because the phase shift due to G2(s) is usually small. On the other hand, it can be too low to find the inner process model G2(s). However, it may be permissible because tuning for the inner loop is not tight and the operating frequency of the system is near that of this relay oscillation.
G2(s)
R
k1
τ1
θ1
k2
τ2
θ2
0.2 0.8
1.005 1.004
1.782 1.531
1.510 1.510
1.001 1.004
0.200 0.800
0.218 0.805
Simulation Study The following process is considered (Hang et al., 1994):
G1(s) ) G2(s) )
exp(-s) (s + 1)2
exp(-Rs) Rs + 1
Two values of R (0.2 and 0.8) representing the fast and slow inner processes are simulated. First, process
1578 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 Table 2. Tuning Results method
R
PI-P cascade (refined Z-N tuning)
0.2 0.8
PI-PI cascade (refined Z-N tuning)
0.2 0.8
controllers and compensators
1 ( 2.325s ) 1 C (s) ) 0.659(1 + 2.452s) 1 C (s) ) 0.566(1 + 1.812s) 1 C (s) ) 0.465(1 + 2.348s) 1 C (s) ) 0.432(1 + 2.448s) 1 C (s) ) 0.874(1 + 2.023s) C1(s) ) 0.789 1 + 1
C2(s) ) 1.131
1
C2(s) ) 0.576 1 +
1
PI-LL cascade (proposed tuning)
0.2
1
0.8
1
proposed control system
0.2
0.8
C2(s) ) 1.131
1 ( 0.199s ) 1 C (s) ) 0.576(1 + 0.794s) 1.382s + 1 C (s) ) 2.829( 2.448s + 1) 1.171s + 1 C (s) ) 2.668( 2.023s + 1) 2
2
2
1.005 (1 - exp(-1.510s) + 0.200s) 1.782s + 1 1 Q2(s) ) 2.286 1 + 1.382s 1.004 (1 - exp(-1.510s) + 0.800s) Q1(s) ) 1.531s + 1 1 Q2(s) ) 1.350 1 + 1.171s Q1(s) )
(
)
(
)
Figure 7. Load responses for R ) 0.8 (step changes in L2(s) and L1(s) at time 0 and 50, respectively).
Figure 8. Load responses when R ) 0.2 for the model and R ) 0.24 for the process (step changes in L2(s) and L1(s) at time 0 and 50, respectively).
models are obtained by applying the relay feedback and P-control tests. Results are shown in Table 1. Because the inner process G2(s) is of the form of FOPTD, identified models for G2(s) are nearly exact as shown in Table 1. Bode plots for G1(s) and its identified model are shown in Figure 5. The proposed control system and the cascade control system with the proposed tuning rule of eqs 6 and 7 are compared with the cascade control systems with the PI-P controller and the PI-PI controller tuned via the refined Ziegler-Nichols rule of Hang et al. (1994). The cascade control system with the proposed tuning rule is called the PI-LL cascade. Table 2 shows tuning results. The design parameter λ is set to 0.4τ1 for the proposed control system and 0.5τ1 for the PI-LL cascade system. Simulations are for the step load changes of L2(s) and L1(s) at t ) 0 and 50, respectively. Figure 6 shows responses for R ) 0.2. For the load change in L2(s), similar responses are obtained for the
proposed control system, PI-LL cascade and PI-PI cascade systems. For the load change in L1(s), the proposed control system and PI-LL cascade systems have better results. As shown by Krishnaswamy and Rangaiah (1990), responses with the PI-P cascade system are poor. Figure 7 shows responses for R ) 0.8. Far better responses are obtained with the proposed control system and the PI-LL cascade system. There is no substantial error in G2(s) used in the simulation. To illustrate the robustness of the proposed control system and the proposed PI-LL cascade system for the inner process mismatches, load responses for the process with R 20% higher than that of the model are simulated. Figures 8 and 9 show results for step load changes. We can see that performance degradation is not much.
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1579
Acknowledgment Financial support from the Korea Science and Engineering Foundation through the Automation Research Center at POSTECH is gratefully acknowledged. Literature Cited
Figure 9. Load responses when R ) 0.8 for the model and R ) 0.96 for the process (step changes in L2(s) and L1(s) at time 0 and 50, respectively).
Conclusion Cascade control is often used in industry to effectively reject disturbances to the manipulated variable and to enhance performances of control systems. Although control performances of the cascade control system are always better than those of the feedback-only control, the cascade control system is not so effective and its tuning is difficult when the inner process is slow. Here, to enhance control performances of the cascade control systems for such slow inner processes and to make tuning easy, a new control structure which uses the secondary measurement for the parallel compensator like the Smith predictor is proposed. The tuning rule for the proposed control system can also be used to tune the classical control system. Simulations show that, compared with the classical cascade control system, the proposed method and the cascade control system tuned with the proposed tuning rule have similar performances for fast inner processes and better performances for slow inner processes.
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Received for review May 21, 1998 Revised manuscript received December 11, 1998 Accepted December 14, 1998 IE980315B
