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Ind. Eng. Chem. Res. 1999, 38, 1580-1588
One-Parameter Method for a Multiloop Control System Design Jinhee Jung, Jin Young Choi, and Jietae Lee* Department of Chemical Engineering, Kyungpook National University, Taegu 702-701, Korea
A multiloop control system can be designed with the independent design method by assuming trial controllers with design parameters and adjusting the design parameters so as to satisfy stability and performance bounds. Various stability and performance bounds have been available and were studied for less restrictive ones. On the other hand, little attention has been paid to trial controllers even though they greatly affect the final control performances. Controllers with one design parameter are the usual choice. Here, a one-parameter method for 2 × 2 multiloop control systems is extended to higher dimensional processes in which the design parameter is the time constant of the set point response of the paired output. Control performances are compared with the well-known biggest log modulus tuning method and the sequential autotuning method. Introduction Practical chemical processes have at least two controlled variables of concentration and flow rate (product quality and productivity) (Seborg et al., 1989). Many methods for designing and tuning multivariable processes have been available. Among them, multiloop control systems having multiple single-input singleoutput controllers are often used in industry (Campo and Morari, 1994). They are easily understood by control engineers and require fewer parameters to tune than multivariable controllers. Another advantage of the multiloop control system is that it is relatively easy to obtain the loop failure tolerance. Because some loops can be in manual mode or the manipulated variables of some loops can be saturated to their limits, the loop failure tolerance is important for practical applications. Mayne (1973) proposed the sequential loop-closing method to design multiloop control systems. This method sequentially designs controllers. The first controller is designed for the selected input/output (SISO) pair and this loop is closed. The second controller is designed for the second pair while the first loop is closed. In this manner, controllers up to the final controller are designed. Because each controller is designed by using SISO methods, it is systematic and simple. A potential disadvantage is that loop failure tolerance is not guaranteed automatically when loops designed earlier fail. Also, the performance of the method depends strongly on which loop is designed first and how the first controller is designed. Usually the faster loops are designed first. Sometimes, for better performance control, redesigns of earlier loops are required (Hovd and Skogestad, 1994). Chiu and Arkun (1992) discussed how to handle some properties such as robustness, but these modifications make the method complex. Single variable autotuning methods can be used for tuning of each controller (Shen and Yu, 1994; Loh et al., 1993). Similarly, Lee et al. (1997) proposed a multiloop control system design method in which the proportional gains and integral times are designed sequentially. Their method is limited to multiloop PI controllers. * To whom all correspondence should be addressed. Telephone: +82-53-950-5620.Fax: +82-53-950-6615.E-mail: jtlee@bh. kyungpook.ac.kr.
Sourlas and Manousiouthakis (1995) proposed an optimization-based design method with parametrization of all decentalized stabilizing controllers, inspired by the parametrization of all stabilizing controllers for the multivariable control system design. It can design the best multiloop control system having nice properties such as robust performance. However, a multiloop control system with finite numbers of states cannot achieve the optimal solution and computations are very complex. The independent design method finds each controller based on the paired transfer function while satisfying some constraints due to the process interactions (Grosdidier and Morari, 1986). The stability constraints imposed on the individual design are given by criteria such as the µ-interaction measure of Grosdidier and Morari (1986). Usually, loop failure tolerance is automatically obtained in this method. One disadvantage of the method is its conservatism because of an inherent assumption of independent design that does not exploit the information about controllers in the other loops (Skogestad and Morari, 1989). To reduce this conservatism, Hovd and Skogestad (1993) have proposed an independent design method for IMC type multiloop controllers where the design parameters are the time constants of IMC filter. Because the stability and performance bounds of the IMC filter constants can be calculated in terms of the real structured singular values, the conservatism of the independent design method can be considerably reduced. Lee and Edgar (1998) have proposed a phase stability condition which can be used together with the µ-interaction measure and reduces the conservatism of the independent design method. Skogestad and Morari (1989) also have proposed some useful bounds for robust performances. The independent design method begins by specifying the form of each controller or each individual closedloop transfer function. The usual choices of the trial controllers are the IMC-PID (Morari and Zafiriou, 1989) type controllers with one design parameter (Grosdidier and Morari, 1986, 1987; Skogestad and Morari, 1989). As the IMC-PID design parameter varies, only the gain of each controller varies while the integral times and the derivative times remain constant. Lee and Choi (1993) proposed a multiloop control system design method which uses this characteristic of the IMC-PID
10.1021/ie980314j CCC: $18.00 © 1999 American Chemical Society Published on Web 03/03/1999
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1581
Figure 1. A multiloop control system. Table 1. Multiloop Controllers by the DLT Method and the Proposed Methods PI process WBa
DLT λ Kc τI τD
6 0.22 -0.14 17.2 15.9
τF WWb
λ Kc τI τD
15 33.33 -21.67 63. 39.
τF ORc
λ
20
Kc
0.606 -0.138 0.394 8.0 6.5 6.848
τI τD τF
a
proposed 6 (ω ) 0.001-0.333) 0.19 -0.099 8.51 8.58
PID DLT 3 0.34 -0.14 17.2 15.9 0.49 1.36 0.38 0.75 10
15 (ω ) 0.001-0.132) 19.10 31.25 -10.81 -18.06 23.48 63. 16.09 39. 2.86 3.59 1.875 2.22 20 10 (ω ) 0.0001-0.1) 0.0062 0.962 -0.0036 -0.212 0.24 2.371 0.0457 8.0 0.1084 6.5 2.98 11.0 1.089 1.154 3.223 1.032 1.154 11.610
proposed 3 (ω ) 0.001-0.667) 0.27 -0.103 6.91 5.90 3.935 1.88 1.81 0.175 10 (ω ) 0.001-0.2) 37.83 -38.86 28.09 30.62 33.02 31.36 39.01 68.11 10 (ω ) 0.0001-0.2) 1.59 -0.44 1.24 6.58 7.403 8.08 6.66 5.50 3.835 10.22 11.27 5.57
Wood and Berry. b Wordle and Wood. c Ogunnaike and Ray.
design rule. For some processes, fixed integral times and derivative times are not desirable. Luyben (1986) proposed a simple multiloop control system design method called biggest log modulus tuning (BLT). In this method, one design parameter which divides the gains of controllers and multiplies the integral times is introduced to detune the multiloop PI control system which is designed for the paired transfer functions and ignores the process interactions. The detuning parameter is increased until the biggest log modulus should be a given value for robust stability. This detuning technique can be used as the trial controller for the independent design. The success of independent design is highly dependent on the choice of trial controllers, as well as the stability bounds. Nevertheless, not many investigations of trial controllers have been made yet. Here, a oneparameter method for the 2 × 2 multiloop control systems (Jung et al., 1998) is extended to higher dimensional processes. The method requires more computations than other similar methods such as the BLT
Figure 2. Amplitude ratios and phase angles of the ideal controllers and approximating PI controllers for the Wood and Berry column process and λ ) 6.
method. However, computations are not a problem because they can be done without the designer’s intervention. Control performances are compared with the well-known BLT method and the sequential autotuning method (SAT). Independent Design Method Consider the multiloop control system in Figure 1, in which an n × n process, G(s) ) {gij(s)}, is controlled by a diagonal controller, C(s) ) diag{cii(s)}. In the independent design method, each controller, cii(s), is designed for the paired transfer function, gii(s). Instead of designing cii(s) directly, as in the IMC-PID method for tuning of single variable control systems (Morari and Zafiriou, 1989), we may design the individual closedloop transfer function of
{
H ˜ (s) ) diag{h ˜ ii(s)} ) diag
gii(s)cii(s)
}
(1)
1 + gii(s)cii(s)
From h ˜ ii(s), the multiloop controller cii(s) can be calculated as
{
˜ -1(s) - I ) diag C(s) ) G ˜ -1(s) H
h ˜ ii(s)
}
gii(s)(1 - h ˜ ii(s))
(2)
where G ˜ (s) ) diag{gii(s)}. It may be somewhat convenient to design h ˜ ii(s) because it is bounded at the frequency zero even though the integral action is used for the offset free operation. ˜ ii(s) Because of the process interactions, cii(s) or h should satisfy some constraints such as the µ-interaction measure for stability. For this, Grosdidier and Morari (1986) proposed the µ-interaction measure which bounds the amplitude of h ˜ ii(jω) for the stability of multiloop control system. The µ-interaction measure is somewhat conservative because the phase of h ˜ ii(jω) is not considered. To reduce the conservatism, Lee and Edgar (1998) proposed a phase stability condition which can be used together with the µ-interaction measure. Most of the examples for independent design methods (Grosdidier and Morari, 1986, 1987; Skogestad and
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Figure 3. Stability bounds and corresponding quantities of PI and PID controllers for the Wood and Berry column process (λ ) 6 for PI controllers and λ ) 3 for PID controllers).
Figure 4. Set point responses of the multiloop PI control systems for Wood and Berry column.
Morari, 1989; Lee et al., 1998) are based on
h ˜ ii(s) )
g+ ii (s) (λs + 1)r
(3)
where h ˜ ii(s) is the nonminimum phase part of gii(s) whose gain is 1 and r is the relative order of gii(s). As in the IMC tuning for single-input single-output processes, λ is the design parameter to be determined. A stable
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1583
Figure 5. Set point responses of the multiloop PID control systems for Wood and Berry column.
control system which satisfies the stability conditions of the µ-interaction measure and the phase stability bound can be designed by adjusting λ. The F detuning in the BLT method (Luyben, 1986) can also be used as trial controllers for the independent design method. That is, controllers
cii(s) )
Ki (1 + 1/τiFs) F
(4)
where Ki and τi are the gain and integral time of the controller designed for the paired transfer function gii(s) ignoring the process interaction, can be trial controllers. The detuning parameter, F, is the design parameter to be determined for the multiloop control system to satisfy the stability conditions. The trial transfer functions of eq 3 and the trial controllers of eq 4 have only one design parameter and so are very convenient to obtain stable multiloop control systems. Because the trial controllers or the trial transfer functions can greatly affect the final control performances, they are as important as less conservative stability bounds. Here, new trial transfer functions which have one design parameter are studied. One-Parameter Design Method The transfer function matrix between the set points and the outputs of the multiloop control system in Figure 1 is
{
H ˆ (x)) diag
g+ ii (s)
}
(λs + 1)r
) diag((I + G(s)C(s))-1G(s)C(s))
(6)
When there are no interactions (G(s) ) G ˜ (s)), the ˆ -1(s) - I) can satisfy multiloop controller C(s) ) G ˜ -1(s)(H eq 6. In general, the controller does not guarantee eq 6. A slightly different C(s) which has a dummy diagonal matrix Q(s) is introduced as
C(s) ) G ˜ -1(s)(H ˆ -1(s)Q(s) - I)-1
(7)
If the dummy diagonal matrix Q(s) satisfies
Y(s)) (I + G(s)C(s))-1G(s)C(s)R(s) ≡ H(s)R(s)
be much different from H ˜ (s) and performances are different from what are expected at the design stage. For a given controllers C(s), the closed-loop transfer function H(s) can be calculated as in eq 5, and it represents responses for the set point changes. Conversely, if H(s) is given, controller C(s) can be calculated. However, in this case, the controller C(s) may not be a diagonal form for the multiloop control system. Because the multiloop controller has n components, basically n elements in H(s) can be determined arbitrarily. That is, there will exist multiloop controllers which result in n desired diagonal elements of H(s). It guarantees the performance of output for the paired set point change. Here, a computational method for the controllers which provide a desired diagonal elements of H(s) is derived. Specifically, we find multiloop controllers satisfying
(5)
When process interactions are severe, the controller eq 2 with trial functions eq 3 does not guarantee the band width of the closed loop system. Even diag{H(s)} can
diag((E(s) + I)(Q(s) + H ˆ (s)E(s))-1) ) I
(8)
the multiloop controller of eq 7 satisfies eq 6. Here,
E(s) ) (G(s) - G ˜ (s))G ˜ -1(s)
(9)
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Figure 6. Set point responses of the multiloop PI control systems for Ward and Wood column.
This can be proven easily by substituting eqs 7 and 8 into eq 6. Because G(s)G ˜ -1(s) ) E(s) + I, we have
which is similar to the notation vec(.) (Horn and Johnson, 1991). Applying this notation to eq 12, we have
G(s)C(s)(I + G(s)C(s))-1 ) G(s)G ˜ -1(s)(H ˆ -1(s)Q(s) - I)-1(I + G(s)G ˜ -1(s) (H ˆ -1(s)Q(s) - I)-1)-1
vecd(∆Qk) ) P-1 vecd(-I + (E + I)(Qk + H ˆ E)-1) (13)
) (E(s) + I)(H ˆ -1(s)Q(s) - I)-1(I + (E(s) + I) (H ˆ -1(s)Q(s) - I)-1)-1 ) (E(s) + I)(H ˆ -1(s)Q(s) + E(s))-1
and eq 6 follows. For n ) 2, analytical solution is available for eq 8 (Jung et al., 1998). In general, a numerical method should be used. Here, the Newton-Raphson method is applied. We solve the equation
diag((E(jω) + I)(Q(jω) + H ˆ (jω)E(jω))-1) ) I (10) at each frequency ω and obtain the frequency response of Q(s). Let (the argument jω is omitted for brevity)
(11)
Then, ignoring higher orders of ∆Qk, we have
diag((E + I)(Qk + H ˆ E)-1∆Qk(Qk + H ˆ E)-1) ) ˆ E)-1) (12) diag(-I + (E + I)(Qk + H Equation 12 is linear for ∆Qk. To obtain an explicit equation for ∆Qk, we introduce the notation
vecd(A) ≡ (a11,a22,...,ann)T
P ) {pij} ) {(BT X A)n(i-1)+1,n(j-1)+1} ˆ E)-1 A ) (E + I)(Qk + H B ) (Qk + H ˆ E)-1
) (E(s) + I)(Q(s) + H ˆ (s)E(s))-1H ˆ (s)
Qk+1 ) Qk + ∆Qk
where
and X means the Kronecker product (Horn and Johnson, 1991). Iterations are continued until |∆Qk| is small enough. Initial value is set to Q0 ) I. When there is no interaction, Q0 ) I is the solution. The iteration will converge when the interaction is not too severe. At each frequency, we apply this iteration and obtain the frequency response of Q(jω) and consequently C(jω). Here, λ is the design parameter and determines the speed of response of output for the paired set point change. By adjusting λ, the multiloop control system can be made to meet the stability bounds for loop failure tolerance and robustness. For this, some iterations to find an appropriate λ may be required. For a λ which is not too small, the responses of the other outputs which are not specified may also be desirable because fast and smooth response of the paired output will not disturb the other outputs much. Approximation with PID Controllers The above controllers are given by their frequency responses. Usually PID controllers are used for multiloop control systems. PID controllers approximating the
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1585
Figure 7. Set point responses of the multiloop PID control systems for Ward and Wood column.
frequency responses are calculated. A PID controller with the filter
(
cPID(s) ) Kc 1 +
)
1 1 + τDs τIs τFs + 1
(14)
is used. An objective function for fitting is chosen as
J)
|c(jω) - cPID(jω)| + |∠c(jω) - ∠cPID(jω)| ∑ ω
(15)
A PID controller which minimizes the above cost is obtained. The frequency range for fitting is up to the frequency about the closed loop bandwidth (ω ) 2/λ). The PI controller has two fitting parameters, and the PID controller has four fitting parameters. Better fitting with the PID controller is possible, and hence, smaller λ can be used for the multiloop PID controllers. Angles of c(jω) over the fitting frequencies may be beyond the range of (-π/2, π/2) for overly small λ, which cannot be fitted with PID controllers. Poor fitting does not provide an overdamped response of eq 6. In that case, λ should be increased, and the design should be repeated. The proposed method has several weak points. The stability and loop failure tolerance of the resulting control systems are not guaranteed automatically. They should be checked with stability conditions such as the µ-interaction measure (Grosdidier and Morari, 1986) and the phase stability condition (Lee and Edgar, 1998). The off-diagonal elements of H(s) can severely degrade the control performances. However, this disadvantage is expected to be overcome by adjusting λ. Case Studies Four methods for designing multiloop control systems are compared. The first is the BLT method of Luyben (1986). The second is the SAT method of Loh et al.
(1993). The third is the independent design method with the trial controllers of eq 4, the decentralized λ tuning (DLT) method. The fourth is that with the proposed trial controllers. Example 1. Consider the Wood and Berry column (Luyben, 1986):
[
12.8(exp(-s) -18.9exp(-3s) 16.7s + 1 21s + 1 G(s) ) 6.6exp(-7s) -19.4exp(-3s) 10.9s + 1 14.4s + 1
]
Figure 2 shows amplitude ratios and phase angles of the ideal controllers for λ ) 6. The design parameter λ is such that responses are comparable with the other competitive methods such as the BLT method. The frequency responses of ideal controllers are fitted with PID controllers. Table 1 shows the fitting results. Responses are compared with those of three existing methods of BLT, SAT, and DLT. Figure 3 shows the two stability bounds of 1 and 3 and the corresponding quantities for PI and PID controllers, respectively. We can see that both control systems satisfy the magnitude stability bound of the µ-interaction measure. Figures 4 and 5 show responses of the PI and PID control systems, respectively. For the BLT method, although equal detuning is applied, unbalanced responses between loops are seen. For the SAT method, loops designed earlier are usually faster than later loops. Iterations as in Hovd and Skogestad (1994) and Shen and Yu (1994) can resolve this problem but require long experiments. If loops are unbalanced in speed, the load responses of faster loops are good, but those of slower loops are very sluggish. Unless some loops are less important, such unbalanced responses are not recommended. The proposed method provides control systems with well-balanced and fast responses.
1586 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999
Figure 8. Set point responses of the multiloop PI control systems for Ogunnaike and Ray column.
Responses with the PI controllers are somewhat oscillatory. This is mainly due to poor fitting at high frequencies. With the PID controllers, as expected, smoother responses are obtained. Example 2. Consider the Wardle and Wood column (Luyben, 1986):
[
0.126exp(-6s) -0.101exp(-12s) (48s + 1)(45s + 1) 60s + 1 G(s) ) 0.094(exp(-8s) -0.12exp(-8s) 38s + 1 35s + 1
]
Tuning results are in Table 1. Figures 6 and 7 show the responses of PI and PID control systems, respectively. The same conclusion can be drawn as that from the above Wood and Berry column example. Other 2 × 2 examples in Luyben (1986) have been also simulated, but they are not shown here because results are very similar to examples 1 and 2 (Jung et al., 1998). Example 3. Consider the Ogunnaike and Ray column (Luyben, 1986)
[
G(s) ) -0.61exp(-3.5s) -0.0049exp(-s) 0.66exp(-2.6s) 6.7s + 1 8.64s + 1 9.06s + 1 -2.36exp(-3s) -0.01exp(-12s) 1.11exp(-6.5s) 3.25s + 1 5s + 1 7.09s + 1 0.89(11.61s + 1)exp(-s) -34.68exp(-9.2s) 46.2exp(-9.4s) (3.89s + 1)(18.8s + 1) 8.15s + 1 10.9s + 1
]
Tuning results are in Table 1. Figures 8 and 9 show the responses of PI and PID control systems, respectively. The same conclusion can be drawn as that from the above 2 × 2 examples. Conclusion In the independent design method, multiloop control systems are designed by assuming trial controllers with design parameters first, and then the design parameters are determined so that some stability bounds due to the process interactions should be satisfied. Stability bounds such as the µ-interaction measure have been studied extensively. However, although the form of trial controllers greatly affect the final control performances of
Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1587
Figure 9. Set point responses of the multiloop PID control systems for Ogunnaike and Ray column.
multiloop control systems, few studies of trial controllers have been made. Here, a one-parameter method for multiloop control systems which guarantees responses between the set points and the corresponding outputs is proposed. The design parameter is the closed-loop time constant between the set points and the corresponding outputs.
Acknowledgment
The first loop in the sequential loop-closing method is usually faster than the other loops. Load responses of the faster loop are good, but those of the other loops are not. Similarly, although loops in the BLT method and the usual independent design method are equally detuned, responses are unbalanced and very different from what are expected at the design stage because of the process interactions. On the other hand, wellbalanced responses are obtained with the independent design method and the proposed trial controllers. Together with stability bounds, the proposed trial controllers can be used to design robust multiloop control systems.
Campo, P. J.; Morari, M. Achievable Closed-Loop Properties of Systems under Decentralized Control: Conditions Involving the Steady-State Gain. IEEE Trans. Autom. Control 1994, AC-39, 932-943. Chiu, M. S.; Arkun, Y. A Methodology for Sequential Design of Robust Decentralized Control Systems. Automatica 1992, 28, 997-1001. Doyle, J. Analysis of Feedback System with Structured Uncertainties. IEE Proc. Part D. 1982, 129, 242-250. Garcia, C. E.; Morari, M. Internal Model Control. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 472-474. Grosdidier, P.; Morari, M. Interaction Measures under Decentralized Control. Automatica 1986, 22, 309-319. Grosdidier, P.; Morari, M. A Computer Aided Methodology for the Design of Decentralized Controllers. Comput. Chem. Eng. 1987, 11, 423-433.
The financial support of the Automation Research Center at POSTECH designated by KOSEF is gratefully acknowledged. Literature Cited
1588 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 Horn, R. A.; Johnson, C. R. Topics in Matrix Analysis; Cambridge University Press: 1991. Hovd, M.; Skogestad, S. Improved independent design of robust decentralized controllers. J. Process Control 1993, 3, 43-51. Hovd, M.; Skogestad, S. Sequential Design of Decentralized Controllers. Automatica 1994, 30, 1601-1607. Jung, J.; Choi, J. Y.; Lee, J. A Decentralized Controller Tuning Method with One Design Parameter of the Closed-Loop Time Constant. KIChE J. 1998, in revision (Korean). Lee, J.; Choi, J. Y. Design of Multiloop PI Controllers. KIChE J. 1993, 31, 272-278 (Korean). Lee, J.; Cho, W.; Edgar, T. F. Multiloop PI Controller Tuning for Interacting Multivariable Processes. Comput. Chem. Eng. 1997, in press. Lee, J.; Edgar, T. F. Phase Conditions for Stability of Multiloop Control Systems. 1998, submitted for publication. Loh, A. P.; Hang, C. C.; Quek, C. K.; Vasnani, V. U. Autotuning of Multivariable Proportional-Integral Controllers Using Relay Feedback. Ind. Eng. Chem. Res. 1993, 32, 10021007.
Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 654-660. Mayne, D. Q. The Design of Linear Multivariable Systems. Automatica 1973, 9, 201-207. Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: New Jersey, 1989. Seborg, D. E.; Edgar, T. F.; Mellichamp, D. A. Process Dynamics and Control; John Wiley: New York, 1989. Shen, S. H.; Yu, C. C. Use of Relay-Feedback Test for Autotuning of Multivariable Systems. AIChE J. 1994, 40, 627-646. Skogestad, S.; Morari, M. Robust Performance of Decentralized Control Systems by Independent Design. Automatica 1989, 25, 119-125. Sourlas, D. D.; Manousiouthakis, V. Best Achievable Decentralized Performance. IEEE Trans. Automatic Control 1995, AC-40, 1858-1871.
Received for review May 21, 1998 Accepted December 11, 1998 IE980314J
