Ind. Eng. Chem. Res. 1993.32, 2653-2661
2653
Experimental Comparison of Conventional and Nonlinear Model-Based Control of a Mixing Tank Kurt
E. Hnggblom'
Process Control Laboratory, Department of Chemical Engineering, Abo Akademi. 20500
Abo. Finland
In this case study concerning control of a laboratory-scale mixing tank, conventional multiloop single-input single-output (SISO) control is compared with "model-based" control where the nonlinearity and multivariable characteristics of the process are explicitly taken into account. It is shown, especially if the operating range of the process is large, that the two outputs (level and temperature) cannot be adequately controlled by multiloop SISO control even if gain scheduling is used. By nonlinear multiple-input multiple-output (MIMO) control, on the other hand, very good control performance is obtained. The basic approach to nonlinear control used in this study is first to transform the process into a globally linear and decoupled system, and then to design controllers for this system. Because of the properties of the resulting MIMO system, the controller design is very easy. Two nonlinear control system designs based on a steady-state and a dynamic model, respectively. are considered. In the dynamic case, both setpoint tracking and disturbance rejection can be addressed separately. 1. Introduction
Most industrial processes are multiple-input multipleoutput (MIMO) systems, but they are usually controlled with single-input single-output (SISO) controllers. Such a control system is relatively easy to design and operate, but it also has some obvious shortcomings. Linear quadratic control theory, for example, addresses some of the problems, but it has not been widely accepted in industry. One reason is that practically all processes are nonlinear. Furthermore, there are always a number of disturbances acting on the process. Even if it might be possible to measure or estimate some of the disturbances, it may not be obvious how information about the disturbances should be used in the control system design. Issues like these can be systematically addressed by means of "model-based" control theory, where the design of the control system in general is based on a nonlinear, multivariable process model. The basic approach in many design methods, both early ones (e.g., Hoo and Kantor, 1985; Ogunnaike, 1986) and more recently developed techniques such as "internal decoupling" (Balchen et al., 1988)."globally linearizing control" (Kravaris and Chung, 1987;Kravarisand Soroush, 1990),andextensions tothese (Daoutidisand Kravaris, 1989;Hensonand Seborg, 1990). is to perform a variable transformation that makes the system globally linear in terms of the new variables. A linear controller operating on the new variables is then designed. In "generic model control" (Lee and Sullivan, 1988) and the "reference system synthesis" approach (Bartusiak et al., 19891, the variable transformation and the controller design are not separated; instead the nonlinear process model is directly imbedded into the controller. Extensive reviews of these 'exact linearization" methods can be found in Kravaris and Kantor (1990a,b), McLellan et al. (1990), and Henson and Seborg (1991). Experimental applications have been reported by Cott et al. (1989). Nakamoto and Watanabe (1991). and Soroush and Kravaris (1992). The design method used in this study is based on a variable transformation such that the open-loop system in termsof the new variables behaveslikeagiven reference model. The reference model is selected to make it easy to control the process with the new variables. This means t
FAX +35&21-654479. E-mail:
[email protected].
MT
Figure 1. Mixing tank.
that it should ideally be linear, decoupled, and insensitive to disturbances. I t is then easy to synthesize controllers for this system such that a closed-loop system with desired properties is obtained. This 'direct synthesis" approach, which is similar to internal decoupling and globally linearizing control, is an extension of the control structure synthesisapproachoutlined in Hiiggblom (1988)for MIMO systems described by (nonlinear) steady-state models, especially distillation columns, and further developed (for systems described by linear models) in Haggblom and Waller (1990) and Higgblom (1992). A laboratory-scale mixing tank is used for illustration of design techniques and comparison of the following control strategies: conventional multiloop SISO control, multiloop SISO control with gain scheduling, nonlinear MIMO control based on a steady-state model, and nonlinear MIMO control based on a dynamic model. In the dynamic case, both setpoint tracking and disturbance rejection can be taken into account in the design. The process has some of the main characteristics of a "real" process: it is multivariable (2 X 2). nonlinear, and disturbances can be entered. Furthermore, hard input constraints have to be considered. 2. Process Description
The mixing tank is depicted in Figure 1. A hot water stream, with mass flow rate r n H and temperature 7".and
0888-5885/93/2632-2653$04.00/00 1993 American Chemical Society
2654 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
a cold water stream, with mass flow rate mc and temperature Tc,are mixed in the tank. A disturbance stream, with mass flow rate mD and temperature TD,can also be entered. The height of the liquid level in the tank, which has a constant cross-sectional area A, is denoted h. The outflow of liquid from the tank is governed by gravity. The outlet stream, with mass flow rate m and temperature T , flows through a pipe, whose outlet into atmospheric pressure is on a level h, below the bottom of the tank. The mass flow rates mH and rnc are manipulated through stepping-motor actuated valves with the inputs (voltage) U H and uc, respectively. 2.1. Nonlinear Process Model. Under reasonable assumptions, the followingmodel can be derived from total mass and energy balances for the mixing tank (Hiiggblom, 1991): dh pA - + B(h dt
+ h,)"'
= mH+ rnc
+ mD
Table I. Process Parameters A = 283.5 cm2 a H = 3.965 ac = 4.212 = 1.00 (kg/min)/cm1/2 y~ = 0.3446
yc 0.3244 h, = 109 cm mHo = 0.0484 kglrnin r n h = 0.5510 kg/min
Table 11. Nominal Temperatures of Inlet Streams TH 51 O C ?'c 17 "C T D = 19 O C
3 are nominal relationships). These factors limit the achievable control performance. 2.2. LinearizedProcess Model. It is useful to consider also the linearized process model because it gives information about the main characteristics of the process around a given operating point. Linearization of eqs 1 around a given steady-state yields
(la)
The assumptions employed are constant (and equal) specific heat capacities and densities of the entering streams and the liquid in the tank, perfect mixing in the tank (although the tank is not mechanically stirred), and turbulent flow in the outlet stream, that is, where is a constant. In the actual process, h, is large compared to h. Therefore, there is only a weak nonlinearity in eq la. Also, there is no direct coupling between h and Tin this equation. However, when control is applied, there will be a coupling through mH and mc. In eq lb, there is a stronger nonlinearity due to the factor (mH + mc + mD). In addition to the indirect coupling through mH and mc, there is also a direct (nonlinear) coupling between h and T. The relationships between the mass flow rates and the inputs to the stepping motors are also nonlinear. If the dynamics in these relationships are neglected, it is easy to eliminate mH and mc from the above equations to get the relationships between the outputs to be controlled (hand T )and the true control signals ( U H and uc). However, for a number of reasons it is more appealing to retain mH and mc in the model. A more fundamental process knowledge can then be developed and used in the control system design (e.g., h should ideally be controlled by mH + me). This will lead to a more general control system design, which is not, for example, dependent on how the flow rates are manipulated. (However, this may also be a weakness, of course.) In this study, the model-based control systems are therefore designed in terms of mH and mc. To apply the control, U H and uc are calculated from mH and rnc according to the experimentally determined static relationships
V = aH(" UH
kglmin - mHo)"
) kglmin
UC= a ( me - mc,
yc
V where CYH, YH, mHo,ac, yc, and mc,, are constant parameters. The fact that the dynamics between the flow rates and the control signals are neglected introduces modeling errors that become significant at high frequencies. Furthermore, the actuators and valves cause significant hysteresis (eqs
An overbar denotes a nominal steady-state value and 6 is a deviation operator (i.e., 6x = x - n). The relationships m = d H + rbc + d D and Ih = /3(h h,)lf2 have been used in the derivation of the expressions in eq 5. These expressions confirm the conclusions made on the basis of the nonlinear model. Because h, >> h (see Tables I and 111, rn and thus the time constant T h and the gain kh, are only weakly nonlinear functions of Fi. Furthermore, d(bh)/dt depends linearly on the mass flow rates. The gains and the time constant associated with T depend more strongly on the operating point (both h and 2'). Note also that the time constant T T is much smaller than 7h. 2.3. Process Data. The process parameters related to the design of the mixing tank are given in Table I. All parameters except the cross-sectional area A have been experimentally determined. The nominal temperatures of the inlet streams are given in Table 11. In practice, the temperatures T Hand TCvaried by about *0.5 O C and TD varied by about f 2 "C in different experiments. The process was operated at a number of different operating points in the range h = 10-30 cm, = 25-45 "C, m~ = 0 or 1.25 kg/min. The main (nominal) operating point is defined by h = 20 cm, T = 35 "C, d D = 0 kg/min. , and Ih can be The corresponding values for m ~ "c, calculated from eqs 1and 2 by setting the time derivatives equal to zero. The gains and the time constants at this operating point are given in Table 111. The density p = 1 kgldrns has been used in the calculations. Two other operating points of special interest in the experiments are defined by h = 20 cm, T = 45 "C,l f t ~ = 0 kglmin (the "upper" operating point with respect to T), and h = 20 cm, T = 25 "C,m~ = 0 kg/min (the "lower" operating point with respect to 7"). The gains and the time constants at these operating points are given in Tables IV and V.
+
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2655 T40CV
Table 111. Gains and Time Constants at Nominal Operating Point (h = 20 cm, 2' = 36 "C, &)= 0 kg/min) 22.7 cm kh, = TT = 0.50 rnin T h = 6.44 min kg/min 1.41 "C -1.58 "C -1.41 "C k-=kg/min k h o = k gmin / k%=kg/min k r r , = 0.529 kmc 0.471 k m D= 0 Table IV. Gains and Time Constants at Upper Operating Point ( h = 20 cm, 2' = 45 "C, e = 0 kg/min) 22.7 cm kh, = T h = 6.44 min TT = 0.50 rnin kg/min -2.46 "C 0.528 "C ---2.28 "C kfic=kTmH = kg/min kg/min ' w - kg/min k m H= 0.824 kmc = 0.176 krr, = 0 Table V. Gains and Time Constants at Lower Operating Point (A = 20 cm, 2' = 25 "C, e = 0 kg/min) 22.7 cm k,, = Th = 6.44 min TT = 0.50 rnin kg/min 2.29 "C -0.704 "C k%=- -0.528 "C k," = kg/min kTmc = kg/min kg/min k p H 0.235 kmc = 0.765 kmD 0
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3. Multiloop SISO Control The conventional way of controlling a 2 X 2 MIMO process is to use two SISO controllers. In this control structure, one of the process outputs is "paired" with one of the inputs and the other output is "paired" with the remaining input. The question then is which input to pair with which output. 3.1. Variable Pairing. Both eqs l a and 4a show that mH and mc have identical effects on h. Apparently, one should then control Twith the stream that has the stronger effect on T and h with the other stream. However, which stream has the stronger effect on T is dependent on the temperatures of the two streams (THand Tc), as shown by eq lb, OJ, equivalently, on the nominal temperature (setpoint) T, as shown by the expressions for kTmH and kTmc in eq 5. This is also indicated by the values of these gains in Tables 111-V. A t the upper operating point one should obviously control T with mc, but equally obvious is that one should control T with mH at the lower operating point. At the nominal operating point there is no clear-cut choice. The same conclusions can be drawn from relative gain analysis (Bristol, 1966). It can be shown that the relative gain for pairing T with mc is
-
T - Tc hTmc= TH- Tc
(6)
which holds over the whole frequency range when a frequency dependent relative gain is considgred. When > 0.5, or equivalently, when T >_0.5(T~_+ Tc), T should be controlled with mc, and when T < 0 . 5 ( T ~+ Tc), T should be controlled with mH. This means that T should be controlled wiih the stfeam whose temperature is farther from T. When T = 0 . 5 ( T ~+ Tc), there is no good variable pairing for multiloop SISO control with this set of input and output variables. In the following study of multiloop SISO control, the controllers are directly manipulating U H and uc. Therefore, the above analysis should actually be done with U H and uc as process inputs. However, the result of the analysis would be the same, because the replacement of 6mH and 6mc by ~ U and H 6uc, respectively, only has the effect of rescaling the process inputs. 3.2. Control with Correct Variable Pairing.- Consider the operating point defined by the setpoints h = 20
0
10
30
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40
50
60
Time (rnin)
Figure2. ConventionalSISO PI control with correct variable pairing at = 40 "C. (Dotted line, setpoint; solid line, temperature, cold water flow rate;dashed line, level, hot water flow ratq dash-dot line, disturbance flow rate.)
cm and T = 40 "C. The linearized model with U H and uc as inputs becomes d(6h') 6.44 - bh' = 63.0 8uH' dt
+
0.50
dt'
+ 35.9 6 ~ c+' 22.7 bmD'
(7a)
+ 6T' = 2.69 ~ u H ' - 3.20 6vc'- 1.85 bmD' + 0.662 6TH' + 0.338 6Tc' (7b)
where t' = t/min, h' = hfcm, T = T/"C, TH' = TH/"C, Tc' = Tc/"C, UH' = UH/V,uc' = uc/V, and mD' = mD/(kg/min) are dimensionless variables. This model is obtained when 6mH and bmc are eliminated from eqs 4 by means of the linearized forms of eqs 3. According to the above analysis, the correct variabl: pairing is to-control h with U H and T with uc because T > 0 . 5 ( T ~+ Tc) = 34 "C. The relative gain AT, = 0.68. This variable pairing is confirmed by the gains in eqs 7. Figure 2 shows an experiment (TIOCV), where the indicated variable pairing is used at the operating point defined above. Step changes around the operating point are made in the setpoints for the temperature T and the level h and in the flow rate of a disturbance stream mD. The control system can bring or keep the outputs close to the desired setpoints, but the control performance is not very good. The controllers used are sampled proportional-integral (PI) controllers implemented as incremental controllers with a sampling period of 0.1 min. Table VI shows the controller parameters used in this and all other SISO experiments. In all cases, the "fine-tuning" was made by
2656 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 Table VI. Controller Parameters for Multiloop SISO Control
sn
9 40-" Y
0.20 0.30 0.25 0.25 0.25
T40CV T3OCV T35CV T35DCV T35GCV
2.50 2.50 2.50 1.00 1.00
0.00 0.00 0.00 0.25 0.25
-0.35 -0.35 -0.30 -0.30 -0.30
1.00 1.00 1.00 1.00 1.00
0.00 0.00 0.00 0.25 0.25
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trial and error. The controller outputs VH and vc were constrained between 1 and 9.9 V. Because of the incremental form, this could be done without causing integral windup. 3.3. Control with Incorrect Variable Pairing. Consider now another eperating point defined by the setpoints h = 20 cm and T = 30 "C. The linearized model becomes d(6h') 6.44 -+ ah' = 43.2 SvH' dt
+ 58.7
+ 22.7 6mD'
1
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0 12
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Bo
1
(8a)
d(6T') 0.50 - 6T' = 3.52 BvH' - 2.96 6vC' - 0.968 amD' + dt 0.368 6TH' + 0.632 6Tc' (8b)
+
The correct variable pairing is now to control h with uc and T with VH ( X T = ~ 0.38). ~ Suppose, however, that the process normally is operated at 7' > 34 "C. If it is impractical to change the variable pairing when the setpoint is changed (and it usually is), the better variable pairing for 7' > 34 "C would then be used over the whole operating range. Figure 3 shows an experjment (T30CV), whe_re the correct variable pairing for T > 34 "C is used for T < 34 OC. The parameters for the temperature controller are the same as in the previous experiment (the gain between T and uc has not changed much), but the gain for the level controller has been increased by 50% to compensate for the change in the gain between h and VH. As seen, the control performance is poor. When the experiment was carried out with the same controller parameters as in TIOCV, a setpoint change of the temperature resulted in a very large disturbance in the level, which the level controller was unable to eliminate in a reasonable time. 3.4. Control When Neither Variable Pairing Is Superior. At the operating point defined by h = 20 cm and T = 35 "C, neither variable pairing is clearly better ~ 0.53). ~ This is also seen from the than the other ( X T = linearized model, which in this case is 6.44
do + 6h' = 53.6 8vH' + 47.9 bvCl + 22.7 amD' (9a) dt'
0.50
do + ST' = 3.32 SuH' - 3.34 6vc' - 1.41 6mD'+ dt' 0.515 6TH'
+ 0.485 6Tc'
(9b)
Figure 4 shows an experiment (T35CV),where the same variable pairing is used as in the previous experiments (SH controls h, uc controls 7').As shown in Table VI, the controllers are slightly detuned compared to those used in the previous experiment (T30CV). Note also that the setpoint changes of the temperature are twice as large as previously. The control performance can be improved by including derivative action in the controllers. Figure 5 shows such an experiment (T35DCV). Because the experiments shown in Figures 4 and 5 serve as a reference for comparisons with the model-based
OJ 0
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Time (min) Figure 3. Conventional SISO PI control with incorrect variable pairing at T = 30 OC. (Dotted line, setpoint; solid line, temperature, cold water flow rate; dashed line, level, hot water flow rate; dash-dot line, disturbance flow rate.)
designs presented in section 4, considerable effort was made to obtain the best possible result with the conventional multiloop SISO control structure. 3.5. Control with Gain Scheduling. One possible way of improving the control performance is to apply gain scheduling so as to counteract the nonlinearity of the process. A straightforward strategy is to keep the controller gains inversely proportional to the process gains. As shown in H" gblom (1991), this is achieved if the ~ adjusted according to controller gainsYuHhand k u c are
where kUHhand kucT are the controller gains at the nominal operating point. Figure 6 shows an experiment (T35GCV) where the controller gains are adjusted according to eqs 10. The other controller parameters, that is, the integration and derivative times, are kept constant. The contr_ollergains kVHhand kucT at the nominal operating point (h = 20 cm, T = 35 "C) are the same as in the previous experiment. As can be seen from Figure 6, the control performance is worse than in experiment T35DCV. It would, of course, be possible to improve the performance to at least the same as in T35DCV by modification of the gain scheduling mechanism, but it is not clear how this should be done. If the controller gains were assumed to be inversely proportional to the process gains when "the other" output
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2657 T35CV 1
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Figure 4. Conventional SISO PI control when neither variable pairing is superior at ?' = 35 OC. (Dotted line, setpoint; solid line, temperature, cold water flow rate; dashed line, level, hot water flow rate; dash-dot line, disturbance flow rate.)
Figure 5. Conventional SISO PID control when neither variable pairing is superior at P = 35 O C . (Dotted line, setpoint; solid line, temperature, cold water flow rate; dashed line, level, hot water flow rate; dash-dot line, disturbance flow rate.)
is perfectly controlled, the ri ht-hand sides of eqs 10should be multiplied by (T - T'c)/( - Tc),but as can be judged from Figure 6, this modification would degrade the performance even more. It seems that the effects of U H and uc in eqs 10are too strong. However, the main problem is that the interactions between the control loops cannot be adequately handled by multiloop SISO control whether (simple) gain scheduling is used or not.
u the vector of available control variables, and w a vector of disturbance variables. Ideally, one would like the process to behave like (Haggblom, 1988)
%
4. Model-Based MIMO Control In general, good control of a MIMO process requires that it also is treated as such a process in the control system design. This was not done above. The multivariable process model was only used to select the best variable pairing from two preselected alternatives and to aid in the tuning of the controllers. A MIMO design, where also the nonlinearities of the process are taken into account, requires a more rigorous "model-based" approach. It might seem that such an approach would result in a complex control system which is difficult to design, tune, and maintain. However, the contrary is true, which is quite natural since the "design part" should become easier when a better process description is used. Through suitable variable transformations it is possible to derive a system which is decoupled and linear in terms of the new variables. It is also possible to make the process outputs insensitive to disturbances that can be measured or estimated. It is then very easy to design and tune a linear control system which operates on these variables. 4.1. Steady-State Design. Consider a MIMO process, where y is the vector of output variables to be controlled,
y = Iu + ow (11) where I is an identity matrix and 0 a zero matrix. The process would then be linear, completely decoupled, and insensitive to the disturbances w. According to eq 11,the relationship between y and u would also be static. Dynamically, this is unrealistic, of course. However, by suitable variable transformations it might be possible to obtain a set of control variables u that would make the steady-state behavior of the process follow eq 11. Consider the model for the mixing tank, eqs 1. The steady-state values of h and T are given by
T=
mHTH+ mcTc + mDTD mH+ mc + m,
(12b)
These relationships are nonlinear and coupled when mH and mc are used as control variables. However, if Uh and UT defined as
UT
=
mnTH + mcTc + mDTD mH+ mc + m,
(13b)
2658 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 T35GCV
dh + B(h + dt
pA -
= B(u,
+ h,)lI2
(16a)
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As can be seen, the steady-state part of the system is linear and decoupled, and the outputs are unaffected by disturbances in mD. The gains and the time constants at h = h are
60 I
where m has been introduced according to eq 2. The time constants, which are independent of T', are the same as for the untransformed system (see eq 5). In practice, 7 h is constant (h,>> h),while T T is approximately proportional to h. It is easy to synthesize controllers for the process since the transformed system is essentially linear and decoupled. For a first-order system with the gain k and the time constant 7 , a PI controller with the gain k , = 7 / ( 7 , k ) and the integration time ~i = 7 results in a first-order closedloop system with the time constant 7,. For the transformed system, the expressions become 0
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Time (min)
Figure 6. SISO PID control with gain scheduling when neither variable pairing is superior at T = 35 O C . (Dottedline, setpoint;solid line, temperature,cold water flow rate; dashed line, level, hot water flow rate; daah-dot line, disturbance flow rate.)
were used as control variables (controller outputs), the steady state of the system would become linear and decoupled with respect to these variables, that is,
When the control system is implemented, it is still necessary, of course, to determine the flow rates mH and mc and ultimately the inputs UH and uc to the stepping motors. The flow rates are obtained from eqs 13 as
where U hand uTare controller outputs. The other variables on the right-hand sides have to be measured, estimated, or replaced by nominal steady-state values. The inputs to the stepping motors can then be determined by means of eqs 3. Equations 15were derived using steady-state arguments. Therefore, it is important to check the dynamic behavior of the process when uh and U T are used as input variables. Assuming that all variables on the right-hand sides of eqs 15 are known, elimination of mH and mc from eqs 1 by means of eqs 15 gives
Figure 7 shows an experiment (MBS) with the modelbased control system described above. All disturbance variables on the right-hand sides of eqs 15 are measured. This results in a nonlinear feedforward/feedback control system. It is thus no surprise that the control performance is very good compared to the SISO experiments. The closed-loop time constants were chosen as T h p = TT,, = 0.5 min, which gave the controller gains kh,, = 13 and kT,c = 1(see Table VII). The integration time for the temperature control loop was chosen according to eq 18b, that is, T T , ~= 0.5 min, but for the level control loop Sh,i = 2 min was chosen, because the choice according to eq 18a ( 7 h J = 6.4 min) resulted in a very sluggish level control. The same process input constraints were used as in the SISO experiments (1V I UH I 9.9 V, 1 V I uc I 9.9 V), but for simplicity they were transformed to corresponding constraints on mH and mc according to eqs 3. If the controller outputs U h and UT would have resulted in a process input (i.e., mH and mc) outside this range, the input was left at the constraint. In order to avoid integral windup, the values of Uh and U T corresponding to the realized inputs were recalculated accordingto eqs 13.These updated controller outputs were then used in the incremental PI algorithm at the following sampling instant. Because all disturbances were measured, the comparison with SISO control is not fair, of course. However, the control algorithm can easily be modified to a pure feedback algorithm using the same measurements as in the SISO case. The temperatures THand Tc of the inlet streams are then replaced in eqs 15 by the nominal steady-state values THand T'c. The disturbance stream mD is also replaced by the nominal value m~ = 0 kg/min. This will
Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993 2659 MBS
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Figure 7. Model-based steady-state design. (Dotted line, setpoint; solid line, temperature, cold water flow rate; dashed line, level, hot water flow rate; dash-dot line, disturbance flow rate.)
Table VII. Controller Parameters for Model-Based Control experiment kh,c rhj/min BtJmin kT,c TTjlrnin BTlmin 1.00 0.50 13.0 2.00 MBS 0.50 0.50 2.00 6.4 1.00 MBDT 13.0 0.25 0.25 1.00 3.2 0.50 MBDR 6.5
degrade the control quality somewhat for disturbances in mD, but hardly at all for setpoint changes (Hiiggblom, 1991). 4.2. Dynamic Design for Setpoint Tracking. For a dynamic system the steady-state design described above is not optimal, of course. It would be more satisfactory if such control variables Uh and U T could be found that would make the open-loop system behave, for example, like the linear, decoupled, first-order system
Such control variables can be derived as follows. Elimination of the time derivatives from eqs 19 by eqs 1 yields the equations = h + -(mH
PA
+ mc + m D - m)
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uh
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Figure 8. Model-based dynamic design for setpoint tracking. (Dotted line, setpoint; solid line, temperature, cold water flow rate; dashed line, level, hot water flow rate; daeh-dot l i e , disturbance flow rate.)
which define the control variables that would make the system behave as eqs 19. The process still has to be manipulated through mH and mc (and ultimately, UH and uc), which can be determined from eqs 20 as
mc = [d,-'pA(TH - T)(uh- h) - BT-'pAh(UT - r ) + m(TH - r ) - mD(TH - TD)I/(TH - Tc) (2lb) These transformations from the controller outputs Uh and UT to the process inputs mH and mc are different from the corresponding transformations in the steady-state design, eqs 15. In eqs 21, the controller outputs affect the parts following from the inclusion of dynamics in the design, not the steady-state part as in eqs 15. At steadystate, however, where h = U h and T = U T , eqs 21 reduce to eqs 15 (note that m is given by eq 2). Figure 8 shows an experiment (MBDT), where this control algorithm is used. The time constants of the transformed system, Oh and 8T,were chosen equal to the linearized time constants 7 h and TT (see eq 5). The same controller parameters as in the steady-state case, eqs 18, are therefore used. In order not to use more measurements than in the SISO case, TH= TH,Tc = Tc, and mD = A D = 0 kg/min are also used in eqs 21. The full nonlinear controller is thus a pure feedback controller. Because the level is the output more difficult to control (see Figure 7; note also that 7 h is an order of a magnitude , 111, the constraint handling scheme larger than 7 ~Table used in the steady-state case was modified as follows. The
2660 Ind. Eng. Chem. Res., Vol. 32, No. 11, 1993
controller outputs U h and U T were constrained between the minimum and maximum values that can be determined from eqs 20 when mH and mc are replaced by their minimum and maximumvalues, respectively. In practice, this had the effect that U h was never constrained (because the sum mH + mc demanded by U h could always be realized), only U T . This meant that the level control was not disturbed by an input constraint, only the temperature. As can be seen from Figure 8,the level control is superior to the level control in the previous cases, and the temperature control is also good, although slightly slower than in the steady-state case (where also T H ,Tc, TD,and mD were measured). By careful tuning of the controllers, it would be possible to improve the control performance somewhat, but not very much because the extreme temperature setpoints (45 and 25 "C)are such that the flow rates are close to their minimum and maximum values. Other limiting factors are the slowness of the stepping motors, the hysteresis, and the rather long sampling period (6 s) used. 4.3. Dynamic Design for Improved Disturbance Rejection. When the disturbances and the temperatures of the inlet streams are not measured, the flow rates in the previous design are determined from
MBDR
50
I
-_0
20
10
30
40
50
60
! 50
60
40
01
0
'
! 10
'
! 20
'
! 30
'
! 40
'
12,
I
10
-
8
. 26 .E
2
mc = [6;'pA(FH - T ) ( u h - h) - 6{'pAh(UT - T ) ~ ( F- HT ) ] / ( F H- Fc) (22b)
0
10
20
30
40
50
60
Time (rnin)
where m is given by eq 2. The resulting open-loop behavior of the system is then (assuming 5'" = T H ,Fc = Tc)
dh + h = U h + *hmD Oh dt PA
From eqs 23 it can be seen that the steady-state gains between the outputs (hand T ) and the disturbances, that is, the disturbance rejection capabilities, can be affected by the choice of oh and OT. The disturbance rejection can, in fact, be affected independently of setpoint tracking. Two PI controllers with the outputs U h and UT and the respective parameters Oh
kh,c
= -9
Th,i
= oh
(24a)
'h,c
give the closed-loopsystem, expressed in transfer function form, 6h(s)= -6 h r ( s ) + 'h,cs
+
6T(s)= -6Tr(s)+ + 'T,cS
Figure 9. Model-baaed dynamic design with improved disturbance rejection. (Dotted line, setpoint; solid line, temperature, cold water flow rate; dashed line, level, hot water flow rate; dash-dot line, disturbance flow rate.)
where hr and Trare the setpoints for h and T ,respectively. As eqs 25 show, the effect of unmeasured disturbances on the outputs h and T can be affected by the choice of 8h and *T without affectingthe setpoint tracking characterized by the time constants Th,c and rT,c-lower values of 6 h and 6~ result in better disturbance rejection. Figure 9 shows an experiment (MBDR) with 6 h and OT reduced to 50% of the values used in MBDT (Figure 8). The controller parameters are changed correspondingly (see Table VII). The experiments show that the disturbance rejection is improved in MBDR, but the setpoint tracking is not as smooth as in MBDT. Since MBDT demands larger and faster changes in the flow rates, this is hardly surprising considering the performance limitations mentioned in the previous section. 5. Conclusions Conventional multiloop SISO and nonlinear "modelbased" MIMO control have been applied to a laboratoryscale mixing tank. Nonlinear MIMO control was clearly superior to multiloop SISO control. The superiority was especially pronounced for large setpoint changes. The reason for this is not only the nonlinearity of the process, but also the interactions between the control loops in multiloop SISO control. The combined effects of interactions and nonlinearity also affect the variable pairing for multiloop SISO control: a variable pairing that is optimal at one operating point may be the wrong one at another operating point. Conventional multiloop SISO control could match, or even outperform, model-based MIMO control only with respect to disturbance rejection capability. This result was obtained after considerable tuning of the PID con-
trollers used in the SISO case, whereas only PI controllers were used in the MIMO designs. Another reason why the model-based MIMO design, where the disturbance rejection was explicitly taken into account (MBDR), did not reject disturbances better than multiloop SISO control is that the multiloop SISO control schemes directly manipulated the inputs to the stepping motors (and were tuned for this), whereas these inputs were calculated from static relationships in the model-based designs. Because the stepping motors used were quite slow, this introduced errors at high frequencies that were not taken into account in the model-based designs. The possibility of using simple controllers that are easy to tune is an important factor in favor of (nonlinear) modelbased control schemes. Furthermore, when disturbances can be measured or inferred from other measurements, it is easy to take advantage of the information in the modelbased schemes as illustrated in the MBS scheme. As shown in the MBDT scheme, there are also good possibilities to handle process constraints intelligently in such control schemes. It is thus quite clear that generally a (nonlinear) “model-based” approach is required to handle a MIMO process adequately.
Acknowledgment Financial support from the Academy of Finland is gratefully acknowledged. Nomenclature A = cross-sectional area of mixing tank h = height of liquid in mixing tank k = gain k,, = gain from u to y m = mass flow rate T = temperature u = controller output v = input to stepping motor Greek Symbols a = parameter for valve and stepping motor, eqs 3 p = parameter for turbulent flow, eq 2 y = parameter for valve and stepping motor, eqs 3 6 = deviation operator (6x = x - 2)
e = time constant for reference model X = relative gain A, = relative gain from u to y
p 7
= density = time constant
Subscripts c=
closed loop, controller C = cold water d = derivative time D = disturbance h = height of liquid H = hot water i = integration time m = mass flow rate o = offset T = temperature Other Notations
- = nominal (steady-state) value
-
Ind. Eng. Chem. Res., Vol. 32, No. 11,1993 2661 = varying controller gain (gain scheduling),eqs 10
’ = dimensionless variable, see below eqs 7 Literature Cited
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Received for review February 9, 1993 Revised manuscript received June 7, 1993 Accepted June 22, 1993. 8
Abstract published in Aduance ACS Abstracts, September
1, 1993.
