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Ind. Eng. Chem. Res. 2003, 42, 484-489
PROCESS DESIGN AND CONTROL A Modified Smith Predictor for a Process with an Integrator and Long Dead Time C. C. Hang, Qing-Guo Wang,* and Xue-Ping Yang Department of Electrical & Computer Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore
In this paper, a new modified Smith predictor using a rapid load estimator scheme is proposed. The advantage of the design is that the load estimator does not involve the solution of a closedloop equation that contains dead time, and this can achieve a better load disturbance response. This scheme can be easily extended to the case of double integrators. Robust stability of the proposed method is analyzed, and a simple and effective robust controller design rule is derived. It is demonstrated by simulations that the new controller provides a similar or better setpoint response and a faster load disturbance rejection than the dead-time compensator in other papers. 1. Introduction predictor4
The original Smith is applicable only to open-loop stable systems. For an unstable system, an inner loop controller may first be used to stabilize the process before the Smith predictor is applied. The design of the main controller will then become more difficult because it has to be designed to control a modified process with a more complex transfer function. For a process with an integrator and long dead time, two modifications have been reported in the literature to overcome this problem. Wantanabe5 first proposed a modification using a mismatched process model. This requires the dead time to be accurately estimated in order to remove the offset caused by a step load disturbance. In refs 1 and 6 further modifications were made to improve load rejection without compromising the setpoint response. A disadvantage of this modified Smith predictor is that it is internally unstable and more care is needed in its implementation. An alternative approach in refs 2 and 7 that avoids the internally unstable signal is to use an estimate of the load disturbance. The disadvantage of this modification is that the load estimator involves the solution of a closedloop equation that contains dead time, and this would limit its performance when the dead time is relatively long. In this paper, a new modified Smith predictor using a rapid load estimator scheme is proposed. Compared with the original Smith predictor, this configuration is not only applicable to open-loop stable systems. In the case of the process with an integrator and significant time delay, it avoids the internally unstable signals and does not incur the solution of a closed-loop equation that contains dead time. A comparison of this performance with the dead-time compensator (DTC) in refs 1-3 will be made. * To whom all correspondence should be addressed. Email:
[email protected]. Tel: (+65) 874 2282. Fax: (+65) 779 1103.
The paper is organized as follows: An overview of the proposed modified Smith predictor is presented in section 2. In section 3, the robust stability of the proposed method is analyzed and the robust controller design rule is derived. Simulation results are given in section 4. Finally, some concluding remarks are presented in section 5. 2. Proposed Modified Smith Predictor The proposed modified Smith predictor makes use of the rapid load detector scheme8 and is shown in Figure 1 for the process of Ge-sL ) (kp/s)e-sL, where G(s) e-sL is the given process to be controlled, G ˆ e-sLˆ is a model of the process, and P1 and P2 are the two controllers. In this structure, P1 is optimized for setpoint response, while P2 is optimized for load disturbance. The output of P2 is effectively an estimate of the load disturbance. Because no dead time is involved in this generation and a much higher controller gain in P2 is allowed, this load estimator has a rapid response and can be used effectively in a feedforward control loop as shown in Figure 1. Both P1 and P2 can be simple proportional controllers, i.e., P1 ) kc1 and P2 ) kc2, without producing steady-state errors. The assumption is that the model is a perfect representation of the unknown plant, i.e., G ˆ (s) ) G(s) and L ˆ ) L. The setpoint and load disturbance responses are given by
Hyr(s) )
kpkc1 -sL e s + kpkc1
(1)
and
Hyd(s) ) kp[s + kpkc1(1 - e-sL)][s + kpkc2(1 - e-sL)] -sL (2) e s(s + kpkc1)(s + kpkc2) Because 1 - e-sL has s1 as its lowest power of s, the
10.1021/ie010881y CCC: $25.00 © 2003 American Chemical Society Published on Web 01/04/2003
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) Td2. Thus, the characteristic equation of setpoint response becomes
10kpkc1 10 2 s + 10kpkc1s + )0 Td1 Td1
s3 +
Figure 1. Proposed Smith predictor for a process with an integrator and a long dead time.
(5)
Because eq 5 is a third-order system, one can assume the desired characteristic equation with one pole at 1/T0 and two other poles at 2/T0. By pole placement, it is straightforward to compute the main controller P1
0.8 kpT02
(6)
Td1 , kc2 ) R2kc1 R
(7)
Td1 ) 2T0, kc1 ) Also P2 can be chosen as
Td2 )
Figure 2. Extension of the proposed scheme for a process with double integrators and a long dead time.
steady-state error can be eliminated when a constant static load disturbance is present. The design of both the setpoint response controller, P1, and the load estimator controller, P2, is very simple because the dead-time component is not involved in the closed loop. The design of the setpoint response (see eq 1) can be simply completed by choosing a pole at -1/T0, where T0 is the desired time constant of the closed-loop response. Also, a critically damped disturbance response can be achieved by choosing another pole in eq 2 at -R/ T0, where R > 0. We thus have
kc1 )
1 ; kc2 ) Rkc1 T0kp
(3)
In the case of a process having double integrators,9 i.e., Ge-sL ) (kp/s2)e-sL, the proposed scheme can still stabilize the output, with both P1 and P2 being of proportional-derivative controllers. However, it will yield a steady-state error when a step load disturbance is present. The proposed scheme needs to be further modified to avoid the steady-state error and internally unstable signal. By introduction of an additional controller C as shown in Figure 2, the setpoint response remains the same as that in Figure 1, while the load disturbance response is changed to be
Finally, it has been found from practice that a large overshoot in the setpoint response may occur and a simple solution is to filter the setpoint by a filter with a transfer function of (1 + γTd1s)/(1 + Td1s), and a default value of γ ) 0.4 is found to be adequate for most applications. 3. Robustness Analysis In practical situations, a model can never be a perfect representation of the actual process. It is, therefore, necessary to analyze the robustness of the proposed modified Smith predictor. Assume that the actual plant Ge-sL is described by a nominal model G ˆ e-sLˆ and -sL unstructured uncertainties ∆G, i.e., Ge )G ˆ e-sLˆ + ∆G. For the proposed scheme in Figure 1, the characteristic equation of the closed-loop system is
ˆ P2) + ∆G(P1 + P2 - G ˆ e-sLˆ LP1P2 + (1 + G ˆ P1)(1 + G 2G ˆ P1P2) ) 0 (8) and with the proposed tuning, the uncertainty normbound Π is defined1,10 by the following expression:
Π) )
Y(s) ) Hyd(s) ) D(s) [1 + GP1(1 - e-sL)][1 + GP2(1 - e-sLC)] -sL (4) Ge (1 + GP1)(1 + GP2) For Ge-sL ) (kp/s2)e-sL, P1 ) kc1(1+ Td1s)/(1 + Td1s/10), and P2 ) kc2(1 + Td2s)/(1 + Td2s/10), by the final value theorem, the steady-state output of y in response to a step load distance will diminish to zero provided 1 e-sLC has s2 as its lowest power of s. Obviously, C(s) ) 1, which has been used for a process having an integral mode, will no longer satisfy this condition. When C ) [1 + (T + L)s]/(1 + Ts) is chosen, this condition is met, so that the steady-state error caused by a step load disturbance will be zero. The influence of T on the characteristic equation can be canceled by selecting T
ˆ P2) (1 + G ˆ P1)(1 + G (P1 + P2 - G ˆ e-sLˆ P1P2 + 2G ˆ P1P2)
|
kˆ p(jωT0 + 1)(jωT0/R + 1)
jω[jω(T0 + T0/R) + 2 - e-jωLˆ ]
|
(9)
, ω>0 (10)
According to the general robust stability theorem,10 the closed-loop system is robustly stable for all of the plants if and only if |∆G| < Π. It is noted that the uncertainty norm-bound Π in eq 9 is a monotonic decreasing function of the frequency and has a constant value at high frequency. If a parameter β0 is defined as the minimum of the normalized norm-bound uncertainty, it follows that β0 ) |Πkˆ p-1|min ) |T0/(1 + R)|. As discussed in ref 1, the normalized error δG ) |∆G/ kˆ p| between Ges-1e-sL [Ge ) kp/(Ts + 1)] and kˆ ps-1e-sL˜ (L ˜ )L ˆ +T ˆ ) will be considered. Figure 3a shows that the value of δG remains constant at medium frequencies, which can be approximated by δG0 ) |(L - L ˆ ) + (T T ˆ )| and is nearly the same as the value of normalized
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Figure 3. δG (solid line) and normalized Π (dashed line) of the proposed scheme gain error ) 10%, and different dead-time errors: (a) G ˆ ) (1/s)e-20s, R ) 2; (b) G ˆ ) (1/s2)e-5s, R ) 1.
Π, while in other frequency ranges, the normalized Π has a higher value of δG. Thus, to meet the robust stability condition, we can set β0 greater than δG0. We use β0 ) 1.5δG0 throughout this paper. Then T0 is given by T0 ) (1 + R)β0. Considering the extension of the proposed structure shown in Figure 2, the characteristic equation of the closed-loop system is changed to
(1 + G ˆ P1)(1 + G ˆ P2) + ∆G(P1 + P2 - G ˆ e-sLˆ P1P2C + ˆ P1P2C) ) 0 (11) G ˆ P1P2 + G The normalized norm-bound uncertainty Π is still a monotonic decreasing function of the frequency, as shown in Figure 3b. The minimum value of the normalized Π is β1 ) |Πkp-1|min ) |T02/8(1 + R2)|. The robustness can be analyzed by following the step used for the
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Figure 4. Nominal and robust responses for a process with an integrator and a long dead time: (a) nominal performance; (b) robust performance. s: proposed scheme. - -: DTC in ref 1. ‚‚‚: DTC in ref 2. - ‚ -: DTC in ref 3.
process of (kp/s)e-sL. However, in the case of the process having double integrators, δG at medium frequencies can be approximated as δG1 ) T0δG0 ) T0|(L - L ˆ ) +(T T ˆ )|. Taking into account the robustness performance, we still set β1 ) 1.5δG1 ) T0β0 throughout this paper. Then T0 is given by T0 ) 8(1 + R2)β0. 4. Simulation Results In this section, the proposed scheme will be compared with the DTC proposed in refs 1-3.
Example 1. The real process is given by
Ge-sL )
kp
e-sL
s(0.9s + 1)(0.5s + 1)(0.3s + 1)(0.1s + 1)
Thus, the equivalent time constant is computed as Te ) 1.8. For the process model, the estimated static gain is kˆ p ) 1 and the estimated dead time is L ˆ ) 20. Then the equivalent dead time can be computed as L ˜ )L ˆ + Te ) 21.8. Considering there is a 10% error in estimating L and kp (L ) 18 and kp ) 0.9), for robust
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Figure 5. Nominal response (solid line) and robust response (dashed line) for a process with double integrators and a long dead time. Table 1. Summary of Simulation Results proposed scheme Mp (%) tr ts IAE IAEload
DTC in ref 1
DTC in ref 2
DTC in ref 3
nominal
robust
nominal
robust
nominal
robust
nominal
robust
0.38 13.54 50.07 27.95 38.67
10.01 34.71 144.06 33.69 41.91
3.61 13.52 82.86 29.10 43.98
12.01 33.30 160.23 33.89 46.49
0.42 13.54 49.80 27.95 42.35
5.78 35.80 135.20 33.40 44.88
0.13 13.54 50.30 27.91 63.71
4.94 36.60 158.10 33.51 66.91
performance β0 is chosen as β0 ) 1.5|L - L ˆ | ) 3. With R ) 1, T0 ) (1 + R)β0 ) 6 in the proposed scheme, while T0 ) β0[1 + (1 + L/β0)1/2] ) 11.62 for the DTC in ref 1. Because the guideline for selecting the desired time constant of the closed-loop response, T0, is not available in refs 2 and 3, we set T0 ) 6 for both cases. The nominal and robust performances are compared in Figure 4. To have a fair and comprehensive assessment of the controller performance, most performance indices popularly used in process control, such as overshoot in percentage (Mp), rising time (from 10% to 90%) in seconds (tr), setting time (to 1%) in seconds (ts), integral absolute error of the setpoint response (IAE), and integral absolute error of the load disturbance response (IAEload), are considered. The results are tabulated in Table 1. It is clearly demonstrated that the proposed modified Smith predictor has a similar or better performance than the DTC proposed in refs 1-3. Example 2. Consider a process with double integrators
Ge-sL )
kp -sL e s2
For the process model, the estimated static gain is kˆ p ) 1 and the estimated dead time is L ˆ ) 5. Considering there is a 5% error in estimating L and kp (L ) 4.75 and kp ) 0.95), for robust performance β0 is chosen as β0 ) 1.5|L - L ˆ | ) 0.375. With R ) 0.4, T0 ) 8(1 + R2)β0 ) 3.48 in the proposed scheme. The nominal and robust performances are shown in Figure 5. It is clearly demonstrated that the proposed modified Smith predictor is excellent in both the setpoint and load disturbance responses.
5. Conclusion A new modified Smith predictor using a rapid load estimator scheme for a process with an integrator and long dead time is proposed. Based on robust stability analysis, a robust controller tuning rule is derived. The proposed method does not incur the solution of a closedloop equation that contains dead time. An extension of the proposed scheme to the double integrator process with long dead time is also studied, and a satisfactory performance can be obtained according to the proposed robust controller tuning guideline. Simulation results have shown that, compared with the DTC in refs 1-3, the proposed scheme could achieve a faster load disturbance rejection and a similar or better setpoint response under the same robustness index. Literature Cited (1) Normey-Rico, J. E.; Camacho, E. F. Robust tuning of deadtime compensators for processes with an integrator and long deadtime. IEEE Trans. Autom. Control 1999, 44 (No. 8), 15971603. (2) Matausˇek, M. R.; Micic´, A. D. On the modified Smith predictor for controlling a process with an integrator and long dead-time. IEEE Trans. Autom. Control 1999, 44 (No. 8), 16031606. (3) Majhi, S.; Atherton, D. P. Online tuning of controllers for an unstable FOPDT process. IEE Proc. Control Theory Appl. 2000, 147 (No. 4), 421-427. (4) Smith, O. J. A controller to overcome dead time. ISA J. 1959, 6 (No. 2), 28-33. (5) Watanabe, K.; Ito, M. A process-model control for linear systems with delay. IEEE Trans. Autom. Control 1981, 26 (No. 6), 1261-1266. (6) A° stro¨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 (No. 2), 343-345.
Ind. Eng. Chem. Res., Vol. 42, No. 3, 2003 489 (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 (No. 8), 1199-1203. (8) Hang, C. C.; Wong, F. S. Modified Smith predictors for the control of processes with dead time. Proc. Instrum. Soc. Am. Annu. Conf. 1979, Oct, 33-34. (9) Mazenc, F.; Mondie, S.; Niculescu, S. I. Global asymptotic stabilization for chains of integrators with a delay in the input. IEEECDC 2001.
(10) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: London, 1989.
Received for review October 26, 2001 Revised manuscript received July 23, 2002 Accepted November 7, 2002 IE010881Y
