Ind. Eng. Chem. Res. 1999, 38, 2979-2983
2979
Control of Integrator Processes with Dominant Time Delay Yu-Chu Tian† and Furong Gao* Department of Chemical Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
A control scheme is proposed for integrator processes with dominant time delay. It contains a local proportional feedback to prestabilize the process, a proportional controller for set-point tracking, and a proportional-derivative controller for load disturbance rejection. The control allows a decoupled design of the load and set-point tracking responses. The tuning of the scheme is simple. Simulations show that the proposed scheme has fast set-point tracking and efficient load disturbance rejection. 1. Introduction Non-self-regulating integrator processes are common in process industries. An example of such a process is a liquid-level system with a pump attached to the outflow line. An integrator plus time delay can also approximate a process with a large time constant. The approximation has been shown by Chien and Fruehauf, through the control of an industrial distillation column, to give a good closed-loop system performance.1 The control of integrator processes with dominant time delay is difficult because of the severe limitations introduced by the time delay on system performance and stability. The Smith predictor,2 a popular time delay compensator, is ineffective for integrator processes because it cannot reject load disturbances.3,4 To improve the performance of the load disturbance rejection, Watanabe and Ito4 proposed a modification of the Smith predictor. The resulting set-point response was, however, very slow, and there was a steady-state error for a load change if the process delay time deviated from its nominal design value.5-7 To overcome these drawbacks, a new Smith predictor was proposed by Astro¨m et al.5 This new Smith predictor separated load response from set-point response and outperformed the scheme of Watanabe and Ito.4 The controller tuning was, however, rather complicated. A systematic tuning method was not given. This problem was investigated by Zhang and Sun (ZS),6 who improved the results of Astro¨m et al.5 Figure 1a shows the ZS structure, where R and L represent set-point and load, respectively. The process dynamics are characterized by P(s) ) (Kp/s)e-ds, where d is time delay. The process output and the model output are denoted by y and y*, respectively. The tuning of Kr was not discussed, while the direct synthesis method could be employed for the tuning. M(s) was recommended as
M(s) )
sM0(s)
, M0(s) ) 1 - K/pM0(s) e-d*s
(d* + 2λ)s + 1 K/p(λs + 1)
(1)
* To whom correspondence should be addressed. Tel: +8522358-7139. Fax: +852-2358 0054. E-mail:
[email protected]. † Current address: School of Chemical Engineering, Curtin University of Technology, G.P.O. Box U1987, Perth WA 6845, Australia. Fax: +61-8-9266 3554. E-mail:
[email protected]. edu.au.
Figure 1. (a) ZS scheme and (b) MM scheme.
where λ is a tuning parameter. Throughout this paper, the superscript asterisk indicates that the corresponding parameters are estimates. The ZS scheme retains the separation nature of the load response from the setpoint response, but eq 1 contains a positive feedback loop that is a potential instability source, resulting in limited robustness. Matausˇek and Micic (MM) modified the Smith predictor and also gave a simple controller tuning method.7 Figure 1b depicts the MM scheme, where K0 and Kr were recommended as
K0 )
1 1 , Kr ) / 2K/pd* KpTr
(2)
where Tr is a tuning time constant that decides the speed of the closed-loop set-point response. The MM scheme has a simple structure with three adjustable parameters. If the process velocity gain and delay are identified experimentally, only one parameter, Tr, has to be tuned manually. However, with the MM scheme,
10.1021/ie990033r CCC: $18.00 © 1999 American Chemical Society Published on Web 06/19/1999
2980 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999
The resulting phase margin, Φ0, is obtained as Φ0 ) π/2 - Kpd/2K/pd*. Setting K/p ) Kp and d* ) d gives the corresponding Φ0 ≈ 61.35°. The block G0(s) is designed to be the denominator of H0 in eq 3:
G0(s) ) 1 + Figure 2. Proposed control scheme.
the set-point and load responses are no longer separated from each other and consequently cannot be independently optimized. The Smith predictor is a special case of the internal model control (IMC). The IMC could not been directly applied to an integrator process with dominant delay because the process is open-loop unstable. The model predictive control (MPC) is applicable to such a process with a discrete impulse response model. However, the existence of a dominant delay requires the model to be represented with a high dimension, resulting in extra complexity to system analysis and synthesis. A new delay compensator is proposed in this paper to control integrator processes with dominant time delay. It is a further development of the new Smith predictor of Astro¨m et al.5 The scheme retains the separation nature of the load response from the setpoint response and consequently allows independent optimization of the two responses. A simple controller tuning procedure is also presented. It will be shown that the proposed scheme has fast set-point tracking, efficient load disturbance rejection, and good robustness. 2. Control Scheme The block diagram of the proposed scheme is shown in Figure 2. To improve the load disturbance rejection performance of the closed-loop system, a local proportional feedback K0 is introduced to prestabilize the openloop unstable integrator process. The local closed-loop transfer function relating u1 to y (and also L to y) is
(Kp/s)e-ds Y(s) Y(s) H0(s) ) ) |u1)0 ) U1(s) L(s) 1 + (K0Kp/s)e-ds
The stability of the local feedback loop is determined by the following characteristic equation:
1 + W0(s) ) 0, W0(s) )
K0Kp -ds e s
(4)
The phase margin Φ0 of W0(s) for the local feedback loop is given by
|W0(s)| ) K0Kp/ω ) 1, Φ0 ) π + arg{W0(s)} ) π/2 - dω (5) Setting Φ0 ) 0, one can obtain the ultimate value K0u of K0: K0u ) π/2Kpd. For the prestabilization of the integrator process, the following K0 < K0u is recommended, as in Matausˇek and Micic:7
K0 )
1 2K/pd*
(6)
(7)
The introduction of G0 eliminates the effect of K0 on setpoint tracking. With predetermined K0 and G0, the other two controllers, Kr and Gc(s), are designed based on the following analysis. The transfer function from the set point R to the system output y is
Hr(s) )
Y(s) ) R(s)
KrKpe-ds
(s + K0K/pe-d* )(s + K/pGce-d*s) s
s + KrK/p s(s + K0Kpe-ds) + KpGce-ds(s + K0K/pe-d*s) (8) The transfer function from the load L to the system output y is
Hl(s) )
Y(s) ) L(s) sKpe-ds s(s + K0Kpe-ds) + (s + K0K/pe-d*s)KpGce-ds
(9)
Hl(s) is independent of Kr. If the process model is good, i.e., K/p ≈ Kp and d* ≈ d, Hr(s) and Hl(s) can be respectively simplified to
Hr(s) ≈ Hl(s) ≈
(3)
K0K/p -d*s e s
KrK/p s+
KrK/p
e-d*s
sKpe-ds (s + K0K/pe-d*s)(s + K/pGce-d*s)
(10)
(11)
Thus, Hr(s) is independent of K0 and G0. It is seen from eqs 10 and 11 that the set-point response and load response of the closed-loop system are decoupled. The former is determined by Kr, while the latter depends on Gc(s) and the previously determined K0 and G0. So, Gc can be designed for efficient load disturbance rejection and Kr can be tuned for fast set-point tracking. According to the simplified Hr(s) in eq 10, the tuning of Kr is easy and straightforward, which is for the local loop including Kr and K/p/s as shown in Figure 2. The direct synthesis method is employed. Let Hrd(s) denote the desired form of Hr(s). Hrd(s) is assumed to be
Hrd(s) ≈
1 e-d*s Trs + 1
(12)
Letting Hrd(s) ) Hr(s) yields the following tuning formula for Kr:
Kr )
1 K/pTr
(13)
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2981
Gp1(s) ) 1/s, d1 ) 5
To compensate the phase lag caused by integrator and time delay, a proportional-derivative Gc is suggested:
Gc(s) ) Kc(1 + Tds)
(14)
Two parameters, Kc and Td, are tuned based on the analysis of the characteristic equation of the transfer function Hl(s) in eq 11. A similar idea can be found in Matausˇek and Micic.7 Because K0 has been determined by eq 6, only the characteristic equation, given by
1 + Wc(s) ) 0, Wc(s) )
K/pKc(1 + Tds) -d*s e s
(15)
needs to be analyzed. The phase margin Φc of Wc(s) is determined by
KcK/px1 + T2dω2 ) 1, ω π Φc ) π + arg{W(s)} ) + tan-1(Tdω) - d*ω (16) 2
|Wc(s)| )
The parameters Kc and Td cannot be derived from the above relations because ω is unknown. To overcome this difficulty, Td is heuristically determined to be
Td ) Rd*
(17)
where R is a coefficient. Thus, for a given R, the control gain Kc can be obtained from the relations of eq 16. For a dominant time delay process, we recommend taking R ) 0.5 and Φc ) 60°. It is noted that limsf0 Hr(s) ) 1 and limsf0 Hl(s) ) 0 hold true for the proposed scheme, implying that the closed-loop system will track the set point at steady state without offset for constant changes in set point and load. It is also noted that ur ) uc ) 0 at steady state. Thus, u1, the output of the unstable block G0(s), will not tend to infinity. It is easy to prove that u ) -L at steady state. As in a general proportional-integral-derivative control, integral windup needs to be considered as well. In summary, the proposed control scheme requires three parameters for controller tuning, K/p, Tr, and d*, for integrator processes with dominant time delay. As in Matausˇek and Micic,7 if the process velocity gain, K/p, and delay time, d*, are estimated experimentally, only one parameter, the time constant Tr determining the speed of the set-point tracking, needs to be tuned manually. The controller tuning can be divided into the following three steps: (1) Determine K0 from eq 6, to prestabilize the process. G0 is, thus, completely defined by eq 7. (2) Determine Kr from eq 13, with desired set-point tracking speed. (3) Tune Gc(s) ) Kc(1 + Tds) for load disturbance rejection, where Td is determined by eq 17 for a given R and Kc is determined by eq 16 for a given Φc. R ) 0.5 and Φc ) 60° are recommended for processes with dominant time delay. 3. Simulation Results Two integrator processes with dominant time delay, which have been investigated by Matausˇek and Micic,7 are considered with transfer functions Pi(s) ) Gpi(s) e-dis, i ) 1, 2, where
Gp2(s) )
(18)
1 , s(s + 1)(0.5s + 1)(0.2s + 1)(0.1s + 1) d2 ) 5 (19)
As in Matausˇek and Micic, the higher-order process P2(s) is modeled by a pure integrator plus time delay with d/2 ) 6.7 and K/p ) 1. Tr is set to be 1/0.6 and 1.7 for processes 1 and 2, respectively. Take R ) 0.5 and Φc ) 60° for both processes, as recommended. The controller settings are
Process 1: K0 ) 0.1, Kr ) 0.6, Td ) 2.5, and Kc ) 0.1758 Process 2: K0 ) 1/13.4, Kr ) 1/1.7, Td ) 3.35, and Kc ) 0.1312 The MM scheme is tuned based on eq 2. The ZS scheme is also well tuned under the same performance requirements, where λ is taken to be 4 for both processes and Kr is chosen to be 0.6 and 1/1.7 for processes 1 and 2, respectively. A unit step set-point change at time t ) 0 and a step load change L ) -0.1 at t ) 70 are introduced, as in refs 5-7. Throughout the paper, the solid, dashed, and dash-dotted lines respectively correspond to the proposed scheme, the ZS scheme, and the MM scheme. The simulation results for processes 1 and 2 are shown in Table 1 and Figures 3-7. Figure 3 shows that all three schemes have nearly the same set-point and load responses for process 1 in the nominal case. The proposed scheme has the best load response with the smallest ITAE index (Table 1). As shown in Figure 4 and Table 1, all three schemes have similar set-point responses with small differences in the ITAE index for process 2 in the nominal case. The load responses of the three schemes are acceptable, while the proposed scheme is slightly better than the MM scheme and slightly more sluggish than the ZS scheme. It should be pointed out that for these two processes the dominant time delay severely limits the achievable performance in set-point and load responses even in the nominal case. To show the applicability of the proposed scheme to the cases with model uncertainty, the process model and consequently the controller settings are kept unchanged for processes 1 and 2, while the process dynamics are changed in time delay and dynamic characteristics. Corresponding to d ) 4 for process 1, Figure 5 shows the effects of 20% time delay deviation, from the nominal value of d ) 5, on system performance. It is seen from the figure and Table 1 that the proposed scheme behaves slightly better than the MM scheme and much better than the ZS scheme. With a further increase of the time delay deviation, the ZS scheme tends to be unstable and the MM scheme becomes significantly more oscillatory, whereas the proposed scheme changes little. Figure 6 shows the responses of the three schemes for process 2 with a 20% time delay deviation (d ) 5 - d* × 0.2 ) 3.66). The ZS scheme strongly oscillates. Compared with the MM scheme, the proposed scheme has improved set-point tracking and load rejection with smaller ITAE indexes (Table 1). The changes of the dynamic characteristics of the delay-free part, Gp(s), of process 1 are also considered
2982 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 Table 1. ITAE Indexes [for Set point (Load)] process
MM
ZS
this work
1 (nominal case, Figure 3) 1′ (20% error in delay, Figure 5) 1′′ (variations in order, Figure 7) 2 (nominal case, Figure 4) 2′ (20% error in delay, Figure 6)
23.9 (131) 45.1 (156) 126.9 (319) 47.7 (295) 71.3 (318)
23.8 (127) 73.7 (184) unstable 48.1 (211) UAa
23.8 (111) 43.3 (136) 122.1 (278) 47.6 (250) 66.4 (279)
a
UA: unacceptable with oscillatory responses.
Figure 5. Control of process 1 with a 20% estimating error in delay time (K/p ) Kp ) 1, d* ) 5, and d ) 4). Dash-dotted line: MM. Dashed line: ZS. Solid line: this work.
Figure 3. Control of process 1 with a good process model (K/p ) Kp ) 1 and d* ) d ) 5). Dash-dotted line: MM. Dashed line: ZS. Solid line: this work.
Figure 6. Control of process 2 with a 20% estimating error in delay time (K/p ) Kp ) 1, d* ) 6.7, and d ) 5 - 6.7 × 0.2 ) 3.66). Dash-dotted line: MM. Dashed line: ZS. Solid line: this work.
Figure 4. Control of the higher-order process 2 with a good lowerorder process model (K/p ) Kp ) 1, d* ) 6.7, and d ) 5). Dashdotted line: MM. Dashed line: ZS. Solid line: this work.
in the simulations. Figure 7 gives the responses of the three control schemes for process 1 when the denominator of Gp1(s) changes from s to s(s2 + 0.9s + 1), which corresponds to an increase in the dynamic order with complex poles in Gp(s). The ZS scheme becomes unstable. The proposed scheme behaves best with fewer oscillations and the smallest ITAE indexes (Table 1). In addition to the above simulations, other simulations are also performed with deviations in process gain Kp and a time delay larger than the nominal value. Results indicate that the MM and the proposed scheme are much better than the ZS scheme, and the proposed scheme behaves slightly better than the MM scheme. 4. Discussions and Conclusions Among the ZS, the MM, and the proposed schemes for integrator processes with dominant time delay, the MM scheme has the simplest structure, and its performance is acceptable in most cases. The ZS scheme gives
Figure 7. Control of process 1 with a variation in dynamic order (d* ) d ) 5). Dash-dotted line: MM. Dashed line: ZS. Solid line: this work.
a good performance with a perfect model. The proposed scheme behaves with a good performance in the nominal case and in the presence of process dynamics variations. The tuning of the proposed scheme is simple and requires three parameters, K/p, Tr, and d. If the process gain, K/p, and delay time, d*, are estimated experimentally, only one parameter, the time constant Tr, needs to be tuned. It should be pointed out that attention should be paid to processes with strong measurement noises because the proposed scheme contains a block
Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2983
Gc(s) with derivative action. In this case, a simple firstorder filter may be added before the proportionalderivative controller to smooth the measurement. In conclusion, a new control scheme with a simple tuning procedure has been proposed for integrator processes with dominant delay. Simulations have shown that the proposed control has fast set-point tracking and efficient load disturbance rejection, in the presence of process dynamics variations. This suggests that the proposed control is a promising tool for the control of integrator processes with dominant delay. Acknowledgment The authors acknowledge the support of the Hong Kong Government under Grant RI95/96.EG03. Literature Cited (1) Chien, I. L.; Fruehauf, P. S. Consider IMC tuning to improve controller performance. Chem. Eng. Prog. 1990, 86, 33.
(2) Smith, O. J. M. Closer control of loops with dead time. Chem. Eng. Prog. 1957, 53, 217. (3) Hang, C. C.; Wong, F. S. Modified Smith predictors for the control of processes with dead time. Proc. ISA Annu. Conf. 1979, 33. (4) Watanabe, K.; Ito, M. A process-model control for linear systems with delay. IEEE Trans. Autom. Control 1981, 26, 1261. (5) Astro¨m, K. J.; Hang, C. C.; Lim, B. C. A new Smith predictor for controlling a process with an integrator and long dead-time. IEEE Trans. Autom. Control 1994, 39, 343. (6) Zhang, W. D.; Sun, Y. X. Modified Smith predictor for controlling integrator/time delay processes. Ind. Eng. Chem. Res. 1996, 35, 2769. (7) Matausˇek, M. R.; Micic, A. D. A modified Smith predictor for controlling a process with an integrator and long dead-time. IEEE Trans. Autom. Control 1996, 41, 1199.
Received for review January 12, 1999 Revised manuscript received May 6, 1999 Accepted May 10, 1999 IE990033R
