Ind. Eng. Chem. Res. 1990,29, 134-138
134
A New Control Algorithm for Single-Input-Single-Output and Multiple-Input-Multiple-Output Systems: Applications to Multiloop Control Systems A new control algorithm called Conservative Model-Based Control (CMBC) was presented in the literature recently, and its applications to single-input-single-output(SISO) systems were described by the authors. In this paper, we examine the potential of CMBC as a multiloop controller. Several simulated distillation columns, with dimensions ranging from 2 X 2 to 4 X 4,are selected for the study. T h e algorithms tested are CMBC and P I / P I D control tuned by the Biggest Log modulus Tuning (BLT) method. The results indicate the superior capability of CMBC as a multiloop controller in comparison with the other algorithms studied. The control system design procedure for multivariable systems consists of the following steps: (1) interaction analysis, to find the extent of interaction present and to select the proper pairings of controlled and manipulated variables from among the competing sets; (2) multiloop controller selection and design for modestly interacting systems; (3) decoupling procedures that provide explicit interaction compensation; and (4) use of a full multivariable control strategy that provides for inherent interaction compensation and constraint handling capabilities. The results of the interaction analysis may suggest the use of multiloop controllers. Indeed, industrial processes are often controlled by multiloop PID-type controllers. Luyben (1986) and Monica et al. (1988) described the tuning procedures for multiloop PI and PID controllers operating in a multivariable environment. These controllers contain up to three tuning constants per loop, and therefore, tuning them in the presence of modeling errors could present difficulties to the operator. Against this background, the recently developed Conservative Model-Based Control (CMBC) algorithm (Chawla et al., 1989) may offer significant advantages, as it contains a single tuning constant. Also, it has been shown to give superior performance in a comparison with PI and PID control for first-order with dead-time singleinput-single-output (SISO) processes.
Review of Conservative Model-Based Control The algorithm is based on the philosophy that, in the worst case, the closed-loop response should be at least as good as the normalized open-loop response. For a typical SISO sampled-data control system shown in Figure 1,the above condition may be stated mathematically as follows: DG, 1
+
.c, 6 " p =
Note that the z-transform operator has been omitted from all the equations in this section for brevity. Solving for D, we get 1
KP - G P but from Figure 1, we get D =M/E
(3)
From eq 2 and 3, we find that 1
M = -(E KP
+ G@)
0 is an adjustable tuning parameter which may be manipulated to satisfy a user-specified performance criterion such as ISE. The detailed derivation and discussion of the properties of the algorithm are given by Chawla et al. (1989). For first-order with dead-time SISO processes, Chawla et al. (1989) have shown that excellent servo-performance and regulatory performance can be obtained. A wide range of modeling errors was introduced by assuming that all the model parameters were off by as high as 60%, both on the positive and negative sides. In all the cases studied, effective control was obtained, giving a good indication of the robustness of the algorithm. Multiloop Control In multiloop control, the n X n multiple-input-multiple-output (MIMO) system is partitioned into n input/ output pairs, each of which is controlled by an SISO controller. We assume that, by some suitable interaction analysis method, the selection of controlled variables, manipulated variables, and variable pairings has already been made. Let the process transfer function matrix be formed of the elements gll,g12,...,gln,...,grin. Then, the process model, G, used in the CMBC algorithm is chosen as the diagonal matrix: = diag [~ll,g22,...,gnn1
1
D=-
Now, desirable properties such as robustness and dead-time compensation are built into the algorithm by multiplying a first-order lead term on the right side of eq 4, giving
(4)
where Gp may be represented with the aid of an impulse response model.
(6)
The omission of the off-diagonal elements may be looked upon as an imposition of modeling error. Satisfactory performance under these circumstances would further prove the capability of CMBC as a multiloop controller for multivariable systems.
Application to Distillation Columns The performance of CMBC has been tested on several simulated distillation columns. The open-loop transfer function matrices of these systems may be found in Luyben (1986). The column configurations range from 2 X 2 to 4 X 4 systems. Luyben (1986) presented the Biggest Log modulus Tuning (BLT-1) procedure for tuning multiloop PI controllers. Monica et al. (1988) extended this method to design multiloop PID controllers by the so-called BLT-4 procedure. All simulation work in this study was carried out on the VAX 8650 system. The use of impulse response repre-
0888-5885/90/2629-0134$02.50/0 0 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 135
Figure 1. Typical sampled-data control system.
sentation considerably simplified the computations. Minimization of ISE was used as the performance criterion for tuning the CMBC controllers. The tuning parameters giving the least ISE were obtained by using an optimization procedure described by Luus and Jaakola (1973). Several distillation columns were simulated. These are referred to as VL (Vinante and Luyben, 1972),WB (Wood and Berry, 1973), OR (Ogunnaike et al., 1983), and A1 and A2 (Alitiqi, 1985). The PI controller parameters tuned by the BLT-1 procedure were obtained from Luyben (1986). The responses of the above processes to a unit step change in distillate composition under PI and CMBC control were simulated. The graphs for both PI and CMl3C control are presented in Figures 2-6. Also presented are the movements of the manipulated variables. Deviation variables have been used in these figures. The ISE's indicated on these figures are the sum of the ISE's of all the variables. Table I summarizes the results of the above comparative study. It may be noted that the performance of CMBC is definitely superior to the PI algorithm in terms of ISE. The controlled variables can be seen to reach the desired
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Table 11. BLT-4 Tuning Parameters and ISE's BLT-4 parameters process KO 71 ~n BLT-4 ISE 16.315 0.074 6.14 Wood-Berry 0.191 0.161 10.856 0.890 Ogunnaike-Ray 1.213 20.35 0.319 371 -0.477 0.706 11.04 0.297 4.879 3.57 Alatiqi case 1 5.130 32.1 2.537 580 0.964 0.0383 22.877 5.132 0.167 1.705 3.889 14.391 0.831
set points much faster than in the case of PI control. Also, the movements of the manipulated variables do not indicate any undesirable trends. The improvement in per-
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136 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 a
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formance of the CMBC algorithm over PI is particularly notable as the order of the system increases. The PID settings for the WB, OR, and A1 columns obtained by the BLT-4 method are given by Monica et al. (1988). The responses of these columns on PID control to a unit step change in the distillate composition are included in Figures 3-5. Table I1 gives the parameters and the ISE's obtained for these simulations. The ISE's obtained by PID control, in the WB and OR cases, are comparable to those obtained by CMBC. In the A1 case, the response by PID control is highly oscillatory and is not satisfactory.
The use of PI/PID controllers involves the use of up to three tuning parameters per loop. In the case of CMBC, a single parameter is used per control loop. The advantage of having a single tuning parameters would become particularly evident as the order of the system increases and the interactions between the various loops become significant. Conclusions
The applicability of CMBC as a multiloop controller operating in a multivariable environment has been dem-
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 137
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onstrated. A comparison of the results with several distillation columns shows that CMBC is superior to PI control. The performance of PID control is comparable with CMBC in some cases but is inferior in one case studied. Since CMBC contains a single tuning constant and has been shown to give excellent performance, it may
be the preferred choice as a multiloop controller in process applications. Nomenclature D = digital control algorithm
E = error
138 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
g,, = transfer function of the ith input-ith output G , = process transfer function hi = impulse response coefficient at the ith instant
ISE = integral of the square of errors K , = proportional constant of a PID controller K = process gain L4 = side-stream draw-off (in figures) M = manipulated variables QB = boiler heat supply (in figures) RR = reflux ratio (in figures) Tb = temperature of the bottoms plate (in figures) AT = temperature difference across the side-stream draw-off tray (in Figures) XB = bottoms composition (in figures) XD = distillate composition (in figures) XS = side-stream composition (in figures) Greek Symbols @ = CMBC tuning parameter T,, = derivative constant of a PID controller i I= integral constant of a PID controller
Literature Cited Alatiqi, I. Composition Control of Distillation Systems Separating Ternary Mixtures with Small Intermediate Feed Concentrations. Ph.D. Dissertation, Lehigh University, Bethlehem, PA, 1985. Chawla, V. K. Development of an Expert System Framework for Multivariable Control System Design. Ph.D. Dissertation, University of Louisville, Louisville, KY, 1988. Chawla, V. K.; Prasad, P. R.; Deshpande, P. B. A New Digital Control Algorithm for SISO and MIMO systems. Hydrocarbon ProC P S b . 1989. Oct.
Luus, R.; Jaakola, T. H. I. Optimization by Direct Search and Systematic Reduction of the Size of Search Region. AIChE J. 1973, 19 (4), 760-66. Luyben, W. L. Simple Method for Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Process Des. Deu. 1986, 25,654-660. Monica, T. J.; Yu, C.; Luyben, W. L. Improved Multiloop Singleinput Single-output (SISO) Controllers for Multivariable Processes. Ind. Eng. Chem. Res. 1988, 27, 969-973. Ogunnaike, B. A.; Lemaire, J. P.; Morari, M.; Ray, W. H. Advanced Multivariable Control of a Pilot-plant Distillation Column. AIChE J . 1983,29, 632. Prasad, P. R. Control Strategies for Multivariable Processes: Distillation Columns and Polymerization Reactors. M.S. Thesis, University of Louisville, Louisville, KY, 1989. Vinante, C. D.; Luyben, W. L. Kern. Teollisuus 1972, 29, 499. Wood, R. K.; Berry, M. W. Terminal Composition Control of a Binary Distillation Column. Chem. Eng. Sci. 1973, 28, 1707.
* To whom correspondence should be addressed. +Presentaddress: Chemical Engineering Department, The Ohio State University, Columbus, OH 43201. Present address: Technology Applications Inc., Jacksonville, FL 32216. f
P. Ramanathan Prasad,' Vipin K. Chawla' Pradeep B. Deshpande* Department of Chemical Engineering University of Louisville Louisville, Kentucky 40292 Received for reuiew February 21, 1989 Revised manuscript received October 6 , 1989 Accepted October 27, 1989
