Ind. Eng. Chem. Res. 1988,27, 1863-1872
1863
Multivariable Adaptive Inferential Control Gwo-Chyau Shent and Won-Kyoo Lee* Department of Chemical Engineering, The Ohio State University, Columbus, Ohio 43210
The technique of adaptive inverse modeling based on the deconvolution model has been extended from single-variable adaptive inferential control to multivariable adaptive inferential control. In addition, the adaptive multivariable pole-zero placement controller by Vogel and Edgar is analyzed from the viewpoint of inferential control to facilitate control calculations. Also, an interaction analysis of the adaptive inferential control based on the Vogel-Edgar type inverse model is performed, and several modifications are proposed to reduce the dynamic interaction effect. Simulation results show that each version of the multivariable adaptive inferential control schemes has its own advantages and disadvantages. Therefore, a control system should be designed based on the desired performance criteria.
I. Introduction Most industrial chemical processes are multiple-input/multiple-output (MIMO) systems. Control of MIMO systems represent a class of very important control problems. A conventional approach to MIMO control problems is to use a single-input/single-output(SISO)approach with appropriate pairings. In many cases, one controlled variable is affected by more than one manipulated variable. In such multiple single-loop control, coupling or interaction is said to exist. When interaction is likely to cause a serious control problem, a decoupler is usually used to reduce the interaction between the control loops. However, it has been found that the application of SISO approach with pairing and decoupler is only partially successful because most chemical processes are complicated by many factors such as multiple time delays. It is believed that the use of multivariable controllers can greatly improve the situation because they treat the MIMO process as a single system instead of many individual subsystems. However, successful implementation of the multivariable controllers relies more heavily on good process models. The difference between these two approaches is demonstrated using a 2 X 2 system in Figure 1, parts a and b. Recently, there has been extensive interest in the development of multivariable adaptive controllers. These controllers are attractive because they can adjust the controller settings automatically to compensate for changing process dynamics. Borison (1979) extended the SISO self-tuning regulator (Astrom and Wittenmark, 1973) to a multivariable self-tuning regulator (MSTR). This MSTR was applied to a distillation column (Dahlquist, 1981). A modified MSTR was also applied to a cement raw material blending process (Keviczky et al., 1978). Koivo (1980) developed a multivariable self-tuning controller from Clarke and Gawthrop’s (1975) self-tuning controller. On the basis of the concept of pole assignment, Prager and Wellstead (1980) devised a different type of multivariable self-tuning regulator to allow for different dead times and unknown or varying dead times without requiring an explicit estimate of the dead time. Later, Vogel and Edgar (1982) briefly discussed the disadvantages and advantages of the previously mentioned adaptive multivariable controllers and proposed a new adaptive multivariable pole-zero placement controller
* To whom all correspondence concerning this paper should be addressed. Present address: Inland Steel Company, Research Laboratories, East Chicago, IN 46312. 0888-5885/88/2627-1863$01.50/0
which can handle different and variable dead times. To cope with open-loop unstable processes, a decoupling pole-placement self-tuning controller for MIMO processes was proposed by McDermott and Mellichamp (1984). Further simulation and experimental results were also reported (McDermott et al., 1986). Another multivariable self-tuning control with time delay compensator was reported and experimentally applied to a multicomponent distillation column (Chien et al., 1985a,b). All of the above approaches can be classified into two categories: linear quadratic optimal.type and pole-zero placement type. Although they both have the potential for better performance, some of the tunings have to be done off-line and for most of them no direct adjustments by the operating personnel are possible. Moreover, tuning is usually not easy, causing some difficulties in practice, for example, determination of the weighting matrices to be used in the performance indexes or the location of the desired zeros and poles. On the other hand, the very simple tuning method of inferential (internal model) control leads to a transparent design method and provides an incentive of developing an adaptive version for both the single-variable and multivariable inferential controllers. Recently, a SISO adaptive inferential control system based on deconvolution controller was proposed to improve the robustness of the inferential control to changing process dynamics (Shen and Lee, 1985,1987). This adaptive deconvolution controller eliminates a factorization required in the design of an inferential controller when unstable or poorly placed zeros are present in the model. In addition, Vogel and Edgar’s (1980) adaptive dead time compensator was analyzed and considered to be a type of adaptive inferential control. In the present study, the adaptive inferential control system has been extended to a multivariable controller for MIMO systems. Moreover, the MIMO version of Vogel and Edgar’s (1982) adaptive dead time compensator is analyzed from the viewpoint of the inferential control and modified to reduce dynamic interactions between input-output pairs. Simulation results are also presented to demonstrate their performances. 11. Process Model A system with p inputs and p outputs can be represented by the following MIMO discrete-time transfer function model: A ( z ) Y ( z )= B(z)U(z) (1) where A ( z ) and B(z) are p X p matrices, Y ( z )is a p X 1 vector of outputs, and V(z)is a p x 1vector of inputs. The 0 1988 American Chemical Society
1864 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988
elements of A(z) and the overparameterized B(z) are defined as - 1a2ijz-2 - .., - aniyz-nlj A i j ( z ) 6, - ~ ~ i j z -
Bij(z)=
(bliJZ-l
+ b2;Jz2 + ... + b(nij+k,.)ijZ-(nij+kij))Z-dij
(2)
SETPOINT NO. 1
CONTROLLER
(3)
where ni.= order of the Ai&) polynomial, 6 , = 1 if i = j (Kronecker delta) and 0 if i = j , dij = minimum expected dead time for the ijth element, and d , + kij = maximum expected dead time for the ijth element. Thus, it is clear from this representation that a process output at present time is related to all the other past outputs and all the past inputs. Equations 1,2, and 3 can be simplified by neglecting the effects of the outputs on each other. This corresponds to setting all the off-diagonal elements of the A(z) matrix to zero. Thus, for a 2 X 2 system, the pulse transfer function model between the input-output pairs becomes
SINGLEVARIABLE FEEDBACK CONTROLLER
SETPOINT
+N
-I -
>
MULTIVARIABLE FEEOBACK CONTROLLER
SETPOINT NO. 1
NO. 2
OUTPUT NO. 1
>
BII(Z)/AII(Z)B ~ z ( z ) / A ~ ~ ( z ) B21(~)/A22(~)
NO. 1
LOAD DISTURBANCE
Bzz(z)/ A 2 2 ( ~ )
I
Actually, this is a MIMO model where the time domain model for each output is in the form of a multiple-input/single-output (MISO) system, since the effect of the outputs on each other is neglected and is accounted for by changes in the B(z)polynomials. It has been noticed that in the absence of a perturbation signal, a recursive identification of an MISO or MIMO model is much more difficult than that of a SISO model (Seborg et al., 1986). Therefore, perturbation signals like PRBS must be cautiously added to all the process inputs. Futhermore, to ensure reliable parameter estimates, all the perturbation signals must be uncorrelated with one another.
111. Multivariable Inferential Control The properties and design methods of the nonadaptive multivariable inferential control have been explored to a great extent by Chen and Brosilow (1984) and Garcia and Morari (1985). Tcq)of the most important properties of the multivariable inferential control scheme are briefly mentioned based on their works: 1. If a process model can exactly describe the process, then the overall system is stable if both the process and the controller are stable. Furthermore, the stability problem can be. addressed by using a filter (which is usually diagonal) if the model mismatch endangers the stability of the overall system. 2. Zero offset can be achieved if the overall system is stable and if the steady-state gain of the controller is the reciprocal of the steady-state gain of the process model. Although the works of two previously mentioned research groups provide very systematic and effective design methods, they are primarily off-line procedures. It is not clear how their design methods could be put into an adaptive system which requires a synthesis of the controller in a limited time period based on on-line identification. Therefore, the emphasis of this study is to obtain an online design method which is appropriate for the practice of adaptive control.
DISTURBANCE MODEL t
‘
I
1
I ,
SETPOINT PROCESS
CONTROLLER
-I T I I@-
I/ ESTIMATION
CONVOLUTION PARAMETER ESTIMATION I
,
Figure 1. Multivariable control systems: (a, top) multiple singleloop feedback control with decoupler; (b, middle) multivariable feedback control; (c, bottom) inferential control.
IV. Multivariable Adaptive Inferential Control Based on Deconvolution Model The idea of SISO adaptive inferential control (Shen and Lee, 1985, 1988) has been extended here to the MIMO case. Figure ICillustrates the structure of the inferential control system. The SISO adaptive inverse modeling has the process model output as an input to the inverse convolution (deconvolution) model, and coefficients of the deconvolutionmodel are adapted to cause its output to be a best fit to the delayed process input. By the same token, for the multivariable adaptive inferential model, all the process model outputs are taken as inputs to the deconvolution model, and the deconvolution model coefficients are updated to have its output to be a best fit to a certain delayed process input. This can be illustrated using a 2 X 2 system as follows: For a 2 X 2 system, a simplified discrete transfer function model is
On the basis of the model outputs Yl and Yz,a recursive parameter estimation method is used to update the de-
Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 1865 convolution model coefficients (eq 7) so that the deconvolution model output is a best fit to a delayed process input. For example, if the recursive least-squares (RLS) method is used for on-line parameter estimation, the deconvolution model obtained is a best fit to a delayed process input in the sense of least squares of residual. Thus, the deconvolution model has the following form:
Hence, the deconvolution model is modified to be
where
EDi: where
In matrix form, eq 6 becomes
= lim DijE
(14)
2-1
Thus, zero offset is guaranteed. It should be noticed that, with the deconvolution controller, the process model can be either the transfer function model or the convolution model. However, relatively intensive calculations may be required when a convolution model is used for a MIMO system.
V. Multivariable Adaptive Inferential Control Based on Vogel and Edgar's Inverse Model Vogel and Edgar (1980) developed the SISO adaptive dead time compensator using Dahlin's controller in the Smith predictor structure. Later, they derived the same adaptive dead time compensation from the viewpoint of pole-zero placement and extended this design method to a multivariable pole-zero placement controller/dead time compensator (Vogel and Edgar, 1982). In this section, the Vogel-Edgar's multivariable controller with the structure of Figure l b is first analyzed from the viewpoint of the inferential control. Then, an effort is made to analyze and to reduce the interaction between the input and output pairs by modifying the original Vogel-Edgar's controller. If the Vogel-Edgar's controller is used in a feedback closed loop, the closed-loop response is given by
Y ( z ) = [I + G(~)GvE(~)I-'G(~)GvE(~)R(~) (15) Thus, it can be seen that the product of the deconvolution controller and the process model is approximately equal to a delayed unit matrix, i.e.,
Generally, the goodness of deconvolution model depends on the magnitude of the residual sequence. A good deconvolution model (with small residual sequence) can only be obtained by a careful selection of the delay y and the number of terms in the deconvolution model. A trialand-error procedure is usually necessary in this regard. Since the deconvolution model is obtained as an inverse model in the sense of least squares of residual when the RLS is used, there is no guarantee that the steady-state gain of the deconvolutionmodel will be exactly the inverse of that of the process model. Thus, the deconvolution model must be modified to have the correct steady-state gain. Let
where Y is the process output, G is the process, GW is the Vogel-Edgar's controller, and R is the set point. Extended directly from the derivation of SISO VogelEdgar's controller, the desired closed-loop response for the MIMO system is
Y(z)= G(z)Q-'(z)F(z)R(z)
(16)
Q ( z ) = A-'(z)CB(z)
(17)
where and the elements of CB(z) are defined as
Hence, Q ( z ) can be viewed as a simplified predictive model. F(z) is a diagonal matrix with its diagonal elements being first-order digital filter; i.e., [I - exp(-T/hi)]z-'
Fii(z) =
1- exp(-T/ Xi)z-'
(19)
where T is the sampling period, and hi is the time constant of the filter. If eq 15 and 16 are equated the Vogel-Edgar's controller has the form then
GVE= 1
is the desired steady-state gain for deconvolution model.
IQk)- F ( Z ) G ( ~ ) I - ~ F ( ~ )
(20)
The designer only explicitly specifies the poles of the closed loop through eq 19, and the closed-loop zeros are determined through the zeros of the process model as shown in eq 16.
1866 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988
g
8
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0
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Figure 3. Comparison of model and process outputs in multivariable adaptive inferential control system with transfer function model: (a, top) first model and process outputs; (b, bottom) second model and process outputs.
Replacing GI with Q-' as in eq 17, we have 2
I -
G, = [I - Q-~FGM]-~Q-'F = {&[I - Q-~FGM]}-'F = [Q - FGM]-'F
I
1.03
r4
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Figure 2. Performance of multivariable adaptive inferential control with transfer function model: (a, top) first process output; (b, middle) second process output; (c, bottom) controller actions.
The Vogel-Edgar's controller for SISO systems has been analyzed from the viewpoint of an adaptive inferential control (Shen and Lee, 1988) and shown to be equivalent to each other. This analysis is extended to the VogelEdgar's controller for the MIMO case. In the inferential control structure as shown in Figure IC,a process model and an inverse model, GI, as part of the controller are required. If the process model and the inverse controller are combined in the simple feedback form, the simple feedback controller becomes G, = [I - G~FGM]-'GIF
(21)
where GM is the process model and GI is the inverse model.
(22)
which is exactly the same as eq 20, the Vogel-Edgar's controller. Thus, it is clear that the multivariable Vogel-Edgar's controller is equivalent to an adaptive inferential controller with a special inverse model. Moreover, because of the simplified dynamics of Q(z) in eq 17, the resulting inverse controller does not have the potential problem of instability due to inversion of the process model as might be faced in the design of inferential controller. As a result, this special attribute lends itself to a fairly attractive on-line design method computationally for multivariable adaptive inferential controller. However, this inverse controller is by no means best or optimal in any other sense. The issues of interaction and decoupling in multivariable process control are very important. Various forms of decoupling can always be achieved without much difficulty. However, it should be bore in mind that possible disadvantages may arise by imposing certain structural constraints on the system response (Garcia and Morari, 1985). Consequently, decision has to be made in advantage of a certain characteristic at the price of sacrificing the others. This point will be seen very clearly in the analysis of the adaptive inferential controller based on the Vogel-Edgar-type inverse model using a 2 x 2 system.
Table I. Parameters Used in the Simulation Study sampling period length of PRBS magnitudes of PRBS open loop closed loop PRBS signal interval initial diagonal elements of covariance matrix set-point filter constants controller output limits no. of terms in process model 1st model 2nd model
2 time units 1023 0.003 0.01 (added to set point) 6 time units 10 000.0
0.1, 0.1 -0.5, 0.5
Bz2CB11- B21CBi2
(29)
Thus, the degree of interaction with this control system can be measured from the off-diagonal elements of eq 23 and 24. In the Vogel-Edgar’s controller where a diagonal filter matrix is used, i.e., F12 = Fzl = 0, eq 24 reduces to
2 denominator; 3, 4 numerator 2 denominator; 6, 4 numerator
Table 11. Parameters Used for Runs with Vogel-Edgar-Type Inverse Model and Its Modified Versions tuning parameters of variable forgetting factors for mode1ing lower limit for forgetting factor internal filter constants
0.001
0.5 0.2, 0.2
Table 111. Parameters Used for Runs with Inverse Model Based on Deconvolution Model no. of terms in deconvolution model time delay used for deconvolution model initial diagnoal elements of covariance matrix for inverse modeling tuning parameters of variable forgetting factors for modeling tuning parameters of variable forgetting factors for inverse modeling lower limit forgetting factor internal filter constants
4, 4; 4, 4 2 time units 10000.0
0.001 1.0
0.5 0.5, 0.5
In the inferential control, a perfect inverse process model is most desired for the controller. Ideally, the combination of the inverse controller and the process yields the identity matrix. However, this is not always possible due to the problem of realizability of the exact inverse model. Consequently, special care is always required for the design of the inverse controller. The design of the adaptive inferential control is more difficult than off-line design methods because of the additional requirement of on-line computation of controller parameters. With the VogelEdgar-type inverse model, the product of the inverse controller (including a filter) and the process becomes, assuming correct process model, GGlF = GQ-lF
L
With use of this diagonal filter, dynamic interaction will disappear only when M and N are zero. For example, this is possible when each of the four transfer functions is first-order without dead time, i.e., B , = CBi,. On the basis of the assumption that a process model is a fairly good representation of the process, the dynamic interaction can be reduced to any desired extent by designing different controller filters. For example, if complete decoupling is desired, a filter can be designed as follows: Let HF12 + JF22 = 0 (31) and
KF11
+ LF21 = 0
(32)
we have
F12 = -JF,,/H
(33)
F21 = - K F , , / L
(34)
and After some mathematical manipulations with eq 33 and 34, eq 24 becomes
Two problems arise with this filter design. First, a comparison of eq 35 with eq 24 or eq 30 shows that longer dead time will exist in the diagonal elements if all the Bi;s contain time delays. In other words, the elimination of interaction is obtained at the price of slower response of each process output to control action. In addition, instability will arise if H and L have unstable zeros, as seen in eq 33,34, and 35. To solve this problem, F12and FI2can be modified to be F12 = -JF22/A (36) and
F21 = - K F i , / A
(37)
The resulting combination of the controller and the process, eq 24, becomes
Compared with eq 30, eq 38 shows a reduced interaction but a slower response speed. On the contrary, eq 38 is better than eq 35 in the system’s response speed, but more interaction can be observed. Clearly, this is a trade-off between the system’s response speed and interaction. This analysis supports the point that design has to be made to
1868 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 o
o
MAICtDC
o
o
MAIC+DC
1' I;
Figure 4. Steady-state gains of models in multivariable adaptive inferential control system with transfer function: (a, top) first model; (b, bottom) second model.
the advantage of a certain characteristic a t the price of sacrificing the others. Control filter in the adaptive inferential control system should be designed based on the knowledge of the specific process to be controlled and the desired performance of the overall control system. It should be noticed that the steady-state gains of all the above controller filters are kept as the identity matrix to ensure zero offset of the closed-loop system. An alternative way of reducing the interaction is to modify the inverse model itself instead of the controller filter matrix. The inverse of the original predictive model is
2
8 MAICtUE. NO DECOUPLING
E1 21
c)
0
LOO
1. 14
1.29
3:,1
,
1.57
1.71
1.86
2.00
,103
Figure 5. Performance of multivariable Vogel-Edgar's controller: (a, top) first process output; (b, middle) second process output; (c, bottom) controller actions.
A close examination of eq 23 reveals the fact that the interaction arises as a result of the simplified dynamics of the predictive model. Therefore, to reduce the interaction, eq 39 can be modified to be
With this inverse model and a diagonal filter, the combination of the controller and the process becomes
Once again, smaller interaction is obtained at the price of sluggish closed-loop response. The performance of the partially decoupling filter (eq 36, 37, and 38) and the modified inverse controller (eq 40 and 41) will be shown in the next section using computer simulation. VI. Simulation Results The simulated process used in the present study is a modified version of a 2 X 2 distillation column model developed by Wood and Berry (1973). The original model is modified from first order to second order to test the
Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 1869 8
8 MAIC+UE. PARTIALLY DECOUPLING FILTER
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8
MAIC+UE. DECOUPLING CONTROLLER
AI
1'
8
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0
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8 MAICeUE. DECOUPLING CDNTROLLER
1' 1'
1'
-a1
Z?
00
n c m w
s 0
' R
131
0
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Figure 6. Performance of modified multivariable Vogel-Edgar's controller with partially decoupling filter: (a, top) first process output; (b, bottom) second process output.
Figure 7. Performance of modified multivariable Vogel-Edgar's controller with decoupling controller: (a, top) first process output; (b, bottom) second process output.
performance of the multivariable adaptive inferential control schemes. This model is given by
a SISO process. Uncorrelated perturbations are always required. Furthermore, it has been found in the present study that, even with the aid of uncorrelated PRBS signals, process model parameters still experience unexpected variation upon changes in the process input signals. This problem is especially serious when a sudden set-point change is imposed on the process. Although the controller filter can be used to slow down the rate of control signal variations, controller detuning still occurs. This phenomenon was also observed by others, e.g., Prager and Wellstead (1980). They found that rate limiting is a useful technique to reduce controller detuning due to sudden set-point changes. In this study, two additional fmt-order digital filters of the type of eq 19 are used for the set-point change before the signals enter the closed loop. With these two set-point filters, controller detuning becomes less severe and control performance is then acceptable. The performance of the multivariable adaptive inferential control schemes to sudden changes in load disturbances is investigated for the autotuning case only. Because a sudden change in load disturbance strongly affects on-line parameter estimation, adaptive control under this condition may suffer greatly from inaccurate process models. Although random walk or covariance resetting (Wellstead and Zanker, 1982; Vogel and Edgar, 1982; Seborg et al., 1986) may be used to resolve this problem to some extent, it has been found by the authors that its performance is very unreliable and highly sensitive to how the random walk is invoked. For the MIMO case, tuning of the random walk is even more difficult and extremely case dependent. The use of a new on-line pa-
G(s) =
12.8 exp(-1.0s) (16.7s + 1)(2s + 1) 6.6 exp(-7.0s) (10.9s 1)(4s 1)
1
+
+
-18.9 exp(-3.0s) (21.0s l)(s 1) -19.4 exp(-3.0s) (14.4s 1)(3s 1)
+
+
+
+
]
(42)
To investigate the ability of the multivariable adaptive inferential control algorithm for adjustment to changing process dynamics, the gains and the dead times of the simulated process are changed at t = 1500.0 from eq 42 to the following equation (Vogel and Edgar, 1982):
G(s) =
I
6.1 (16.7s + 1)(2s 1) 3.5 exp(-l.os)
+
(10.9s +-1)(4s + 1)
-8.4 exp(-2.0s) (21.0s l)(s 1) -8.8 exp(-1.0s) (14.4s + 1)(3s + 1)
+
+
1
The set-point changes are introduced according to the following sequence: time 1000.0 1250.0 1400.0 1700.0
1850.0
set point 1
set point 2
0.0
0.0 0.0 0.0
0.5 0.0 0.0 0.0
0.5 0.0
It has been mentioned previously that on-line parameter estimation of a MIMO or MIS0 model for a MIMO process is much more difficult than that of a SISO model for
1870 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 g
g MAIC+DC.
-, -,
SELF-TUNING
g,
g, MAICrUE. NO OECOUPLING
SELF-TUNING
A
T
-103
o
9,
DECOUPLING z, MAICrUE. SELF-TUNING P
NO
n
Figure 8. Performance of autotuning control using multivariable adaptive inferential control: (a, top) first process output; (b, bottom) second process output.
Figure 9. Performance of autotuning control using multivariable Vogel-Edgar's controller: (a, top) first process output; (b, bottom) second process output.
rameter estimation scheme proposed by Shen and Lee (1987) is found to be not satisfactory for the MIMO case, either. Therefore, if frequent sudden change in load disturbance is likely to be a problem for adaptive control, control in the form of autotuning should be used instead i.e., controller adaptation is performed at the operator's careful decision. Selection of the sampling period is a very important issue for adaptive control. For a SISO process, a rule of thumb is to choose the sampling period between 1/15 and 1/4 of the 95% settling time of the process (Isermann, 1982). For the multivariable control of a MIMO process, difficulty with choosing the sampling period may arise if the process has several quite different time constants. A further problem is that the selected sampling period using this rule of thumb may be too large if an overparameterized model is used to model the variable dead time, as in the present simulation example. As a result of compromise, the sampling period of two time units is selected for this simulation study. To obtain the initial estimates of the model and controller parameters, two different PRBS signals, each with a magnitude of 0.003, serve as the process inputs in open loop for the first 1000 sampling times. A t t = 1000.0,the loop is closed and the two PRBS signals are switched to the set point, and their magnitude is changed to 0.01. This is done so that the magnitudes of the PRBS do not have to be changed according to the operating conditions. U-D factorization with variable forgetting factor (Fortescue et al., 1981) is used for parameter estimation. To obtain a good process model in the presence of process
changes, the fixed forgetting factor has been found to be more efficient than the variable forgetting factor. However, covariance windup due to a lack of signal excitation would be a more serious problem for control of the MIMO system. Although a larger perturbation signal can keep the magnitude of the covariance matrix at a certain level, it is usually not desirable in practice. For the sake of clearness and brevity, other parameters associated with adaptation of the model and the controller, which are of less importance, are given by Tables 1-111. A. Multivariable Adaptive Inferential Control Based on Deconvolution Model. The multivariable adaptive inferential control based on the deconvolution model (eq 6 and 7) is first applied to the simulated process. The results are shown in Figures 2-4. Parts a and b of Figure 2 are the process outputs compared with the set point. The control signals are presented in Figure 2c. The response of process outputs to the individual set-point change is a little sluggish partly because of the set-point filter mentioned previously. For the set-point change in the first process output, the interaction effect is not very severe. However, because of the more significant model mismatch resulting from the process changes, a more severe interaction effect due to the set-point change in the second process output can be observed in the first output, as shown in Figure 2a. Parts a and b of Figures 3 show the performance of the parameter estimation using the variable forgetting factor. After the process changes at t = 1500.0, the first process model has a problem following the process change. How-
!,
!,
Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988 1871 MAICNE. PARTIALLY DECOUPLING FILTER SELF-TUNING
?
-1 -1
MAIC+UE, PRRTIALLY DECOUPLING FILTER SELF-TUNING
y
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DECOUPLING CONTROLLER 5 .! MAIC+UE. SELF-TUNING 1 1
11.71
li.ffi
ion
Figure 10. Performance of autotuning control using modified multivariable Vogel-Edgar’s controller with partially decoupling filter: (a, top) first process output; (b, bottom) second process output.
Figure 11. Performance of autotuning control using modified multivariable Vogel-Edgar’s controller with decoupling controller: (a, top) first process output; (b, bottom) second process output.
ever, the second process model works well. Since the same process input signals are used for the estimation of both the process model parameters, lack of dynamic information in the second process output seems to be the only reason for such a difference. Variations of the four process steady-state gains are shown in parts a and b of Figure 4. Controller detuning due to changes in the process input and output signals can be seen clearly here. A closer examination of these two figures reveals a very interesting fact: set-point changes in the first (second) process output have a larger effect on the first (second) model parameters than on the second (first) model parameters. Again, the large variation of the first (second) model parameters is probably a result of the large change in the first (second) process output. Hence, it is obvious from this observation that parameters of the MIS0 model used are dependent on the input and output signal levels and, therefore, are not unique even for a process with constant parameters. The phenomenon of variation in the controller parameters as the operating point changes is also reported by Prager and Wellstead ( 1980).
first process output produce a larger interaction effect than the one based on the deconvolution model. The major difference between the performance of these two control systems can be seen from Figures 2c and 5c. Even with a more sluggish filter, the control signal of the one based on the Vogel-Edgar-type inverse model exhibits a sharper change upon set-point changes. This difference is believed to result from the difference in the inverse model structure. To reduce the interaction of the system, the partially decoupling filter of eq 36 and 37 based on the identified model is implemented. The results are presented in parts a and b of Figure 6. Indeed, the interaction is greatly reduced. The reduction in response speed is relatively insignificant and cannot be seen in parts a and b of Figure 6. This is believed to be attributed to the very small values of M and N in eq 27, 28, and 38. Also noticed is the oscillation of the control signal for t > 1700.0 and the corresponding oscillation of the process output in Figure 6. Variations in the model parameters based on on-line data are believed to be responsible for this; the use of a slower controller filter should be able to alleviate this problem to some extent. The modified Vogel-Edgar-type inverse model (eq 40) is also applied to the simulated process. The results are shown in parts a and b of Figure 7. Compared with the partially decoupling filter, this modified inverse controller yields even less interaction when the process model is good and a worse performance when the model mismatch becomes serious, as can be seen in Figure 7. Similar to the use of partially decoupling filter, a more sluggish controller
B. Multivariable Adaptive Inferential Control Based on Vogel-Edgar-Type Inverse Model. The performance of the multivariable adaptive inferential control based on the Vogel-Edgar-type inverse model (inverse of eq 17) is represented in parts a and b of Figure 5. A comparison of parts a and b of Figures 5 and 2 shows that this control system yields a better performance for set-point changes. However, the set-point changes in the
1872 Ind. Eng. Chem. Res., Vol. 27, No. 10, 1988
filter is necessary to reduce the oscillation in the control signal when the process model parameters are experiencing some oscillation. C. Autotuning Control Using Multivariable Inferential Control Schemes. As mentioned previously, the multivariable adaptive inferential control is not suitable when frequent unmeasured disturbances exist in the process, because under this condition it is extremely difficult to make the on-line parameter estimation function well. Therefore, it is felt by the author that autotuning is a better approach than “pure“ adaptive control under this condition. In the following simulation runs, a load change of 0.34 is imposed on the process at t = 1500.0. The vector of transfer functions for the load disturbance is 3.8 exp(-8.ls) GL(s) =
14.9s + 1 4.9 exp(-3.4s) 13.2s + 1
1
(44)
1
The on-line parameter estimation with the aid of the PRBS signals is terminated a t t = 1000.0. The control performance of all four types of multivariable adaptive inferential control schemes is compared as presented in Figures 8-11. The following remarks can be made based on this comparison. 1. The control based on the deconvolution model (eq 6 and 7) is the best in terms of the disturbance rejection ability. However, it is the worst in response speed to set-point changes. 2. As far as interaction is concerned, the control based on the deconvolution model is better than that based on the Vogel-Edgar-type inverse model (inverse of eq 7). The two modified versions of the Vogel-Edgar-type inverse model (eq 36 and 37, and eq 40), however, are superior to that based on the deconvolution model. 3. The performance of the two modified versions of the Vogel-Edgar-type inverse model is almost the same. The one with decoupling controller (eq 40) has less severe dynamic interaction. It should be mentioned that the conclusions made above are specific to the numerical example used and are for autotuning only. Therefore, extension of this result to other systems should be done with caution. The controller designer should always choose the control design based on the preferred performance index, e.g., ability of disturbance rejection, response speed to set-point change, or minimum interaction.
VII. Conclusion Four types of multivariable adaptive inferential control schemes were presented, with each having its own desirable features. With the inferential control structure, various modifications can be made for either the inverse controller or the controller filter to obtain the desired control performance. Moreover, under the condition of frequent changes in unmeasured load disturbance, the control in the form of autotuning instead of adpative control is strongly suggested to ensure a good control quality.
Nomenclature A, B = model transfer functions (matrices) Aij = element of A matrix (polynomial) Bij = element of B matrix (polynomial) a, b, c , d = steady-state gains of process model b,” = element of Bij DijE = discrete deconvolution model
dij = minimum expected dead time for ijth element
F = filter matrix Fii = element of filter matrix G = process transfer function GI= inverse process model transfer function GL = load disturbance transfer function GM= process model transfer function Gm = Vogel-Edgar’s controller gkij = discrete deconvolution model coefficient H = design variable (eq 26) I = unit matrix J = design variable (eq 27) K = design variable (eq 28) L = design variable (eq 29) M = design variable (eq 27) N = design variable (eq 28) nij = order of Aij polynomial Q = simplified predictive model R = set point T = sampling period t = time U = process input (vector) Y = process output (vector) Greek L e t t e r s
A = design variable (eq 25) 6ij = Kronecker delta y = delay used in deconvolution model A = time constant of Fii
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Received for reuiew November 10, 1987 Revised manuscript received June 16, 1988 Accepted July 5 , 1988
