Ind. Eng. Chem. Res. 1993,32, 1658-1666
1668
Design of Identification Experiments for Robust Control. A Geometric Approach for Bivariate Processes ChingWei Koungt and John F. MacGregor’ Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
A new approach to the design of experiments is proposed to identify linear multiple-input-multipleoutput (MIMO) models that will provide robust control. The experimental designs for identification are based on minimizing uncertainties in the structure of the multivariate model rather than simply the magnitude of identification error. The experimental designs are derived for steady-state robust stability of 2 X 2 systems using a geometric approach. This approach leads to a simple and unified design approach based on the singular value decomposition (SVD) of the process gain matrix. These robust or control-relevant identification designs for MIMO systems differ considerably from traditional designs developed for single-output systems. Typically in these new multivariate designs, the inputs are correlated, they are not binary sequences, and the magnitudes of the perturbations in low-gain directions are much larger than those in high-gain directions. The results are extended to identification under closed-loop conditions. Dual composition control of distillation processes is used t o illustrate the physical interpretations and the effectiveness of the SVD-based design. 1. Introduction
In multivariable model-based control systems, the effect of model mismatch on the stability and performance of the closed-loop systems is of utmost importance. The thrust of recent robust controller design research has been to incorporate a description of the model uncertainties explicitly into the design methodology in order to enhance the robustness of the control system. The difficulty in obtaining a realistic uncertainty characterization usually leads to conservative designs. However, the ultimate limitation in controller performance and stability is not the uncertainty characterization, but the actual model mismatch itself. Even if we could characterize exactly the model mismatch and use a very sophisticated robust controller design, the control performance would never be better than the one from the same design with a better model. Therefore, this paper focuses on the problem of identifying process models which will lead to more robust control systems. Model mismatch arising from identification errors can be classified into two categories: mismatch due to noise and the finite amount of data (so-called variance errors) and mismatch due to the use of approximate models (socalled bias errors). Included in the latter is the problem of identifying linear models to approximate nonlinear processes, a particularly important problem in process control. A robust model is qualitatively defined here as one which, when used in the design of a control system, will result in a closed-loop system that will remain stable in spite of identification errors in either of the above categories. As a first investigation into this difficult problem of identifying robust multiple-input-multipleoutput (MIMO) models, this paper focuses on a particular robust stability criterion (“integral stabilizability”) where only the steady-state uncertainties need to be considered. However,this condition must be satsified for any dynamic MIMO model used for model-based control with integral action. The methods developed and the insight gained in this limited study will clearly illustrate the issues involved and will showthe important characteristics of experimental designs aimed at achieving more robust models.
* To whom all correspondence should be addressed.
+ Current address: The M. W. Kellogg Co., Houston,TX 77210-
4557.
E-mail:
[email protected](Ching-Wei Koung).
0888-588519312632-1658$04.00/0
A robust model is somehow “close”to the true process. Most past research usually quantifies this closeness with some measures of mismatch magnitude. However,a small enough mismatch magnitude is only a sufficient condition for robust models rather than a necessary one. Koung and MacGregor (1991, 1992) have shown that when the model is intended for model-based control, the closeness of a multivariable model to its true process is much better represented by some characteristics of its multivariate structure rather than its mismatch magnitude. How to obtain a robust model depends on a number of factors. Once the data have been collected, there is little one can do but try to minimize bias errors through the proper selection of data prefiiters and a noise model (Ljung, 19871, and/or iterating on model structures (e.g. Box and Jenkin’s work (1976)). On the other hand, if the data have not yet been collected, it is the design of the identification experiments that will have the greatest impact on the bias and variance errors and on the robustness of the identified models. Through proper designs of experiments, one can m a n i p u l a t e the uncertainty or control its structure (examples are given later in this paper) so as to greatly improve the control robustness of the identified model for use in subsequent controller designs. Therefore this paper focuses on the design of experiments to achieve robust models. For single-input-single-output(SISO)systems,two-level input perturbation designs such as the pseduorandom binary sequence (PRBS)are commonly used and can provide very efficient estimates of linear model parameters. When applied to multivariable processes, traditionally the two-level design is applied to every input variable and employed in such a manner that the inputs are perturbed simultaneously and independently, or one at a time. In the case of multiple-input-single-output(MISO) model identification, these traditional designs still provide efficient estimates of process models. However, for multipleoutput systems designs such as these will not in general provide models that lead to robust multivariable controllers. There are two reasons for this: (1)in these designs the modeling errors in the MIMO model have not been considered jointly, and therefore the robustness properties of the MIMO model have not been addressed, and (2) an independent design for each output does not consider the directionality of a multivariable process. These issues are 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1659 row vectors as shown in Figure lb. This is referred to as the uncertainty region. Due to the stochastic nature of identification errors, we shall assume that not a single point in either uncertainty region can be excluded from the identification results and that the errore in the two outputs are uncorrelated. Under these two assumptions, the following geometric necessary and sufficientcondition can be easily derived from KOung and MacGregor (1991) that the condition in (1)is true, if and only if there exists no h intersecting both uncertaintyregions (2) Figure 1. Geometric analysis: (a, left) geometric interpretationsof model mismatch; (b,right)geometric conditionfor the etabilizability.
particularly important for ill-conditionedprocesses where very small mismatch magnitudes can lead to severe problems in closed-loop stability and performance. In the following, we will present a new approach to experimental design in which the design criterion is to minimize the mismatch in multivariate model structure, rather than to minimize mismatch magnitude as done traditionally. 2. Preliminaries 2.1. Stabilizability. In general, robust control properties are determined not only by the identified model but also by the controller design. The only exception is where robustness properties involve only the steady-state conditions of the closed-loop system because the steady-state conditions are the same for all controller designs with integral action (Garcia and Morari, 1982) (e.g. the steadystate gain of internal model control (IMC) controller must be the exact inverse of the identified model). In this situation, Garcia and Morari (1985) have shown that any control system with integral action is controllable if and only if
Re(hi(G&’))> 0 V i (1) where hi is the ith eigenvalue of a matrix, while G and are the steady-state gain matrices of the true and identified processes, respectively. This condition is often referred as “integral controllability” (Morari and Zafhiou, 1989) or “stabilizability”(Koung and MacGregor, 1991). If the stabilizabilitycondition is violated, no model-based control system with integral action would ever work regardless of the controller design and tuning. In SISO systems, (1) simply corresponds to the well-known uncontrollable situation where a process and its model have opposite signs in their steady-state gains. In MIMO processes, (1) requires that all the eigenvalues of GG-l lie in the right half of the complex plane. Koung and MacGregor (1991) have derived several conditions that relate the geometric interpretations of model mismatches to the locations of these eigenvalues for 2 X 2 systems, and that work forms the basis of the following section. 2.2. A Geometric Condition for Identification Errors. Koung and MacGregor (1991) interpreted the differencesbetween the true and mismatched gain matrices of 2 X 2 systems geometrically as the rotation and length change of a true row/column vector to become the mismatched row/column vector (Figure la). The geometric differences between the true and mismatched row/ column vectors were then related to the locations of the two eigenvalues of GG-1 in the complex plane. We can draw the two row vectors of the true gain matrix on a plane and then define a “singularity line” (h) as a straight line passing through the intersection point of the two row vectors (Figure lb). When the mismatched gain matrix is a result of identification,we can draw a boundary around each true row vector to enclose possible identified
e:
One should note the assumption of uncorrelated errors. If the identification errors in the rows (G1,G2) of the gain matrix are correlated, as might be the case if they arose partially from structured nonlinearities, then the above condition would be conservative.
3. Design of Experiments for Stabilizability Traditional criteria for the optimal design of identification experiments have always involved minimizing some overall measures of parameter uncertainties associated with one output only (i.e. for the MIS0 model). Nevertheless, the geometric condition in (2) implies that we should design the experiments such that the parameter uncertainties associated with both outputs should be considered. In this section, we first take a geometric approach to shape the uncertainties such that the geometric condition in (2) will be satisfied. Second, we show that the geometric approach can be generalized in a very simple fashion by using transformed input variables defined by singular value decomposition (SVD). Then, on the basis of the geometric derivations for open-loop designs, it is shown that identification under closed loop can help identify more robust models with less prior knowledge on the process. Finally, a special class of nonlinear processes is introduced, and it is shown that the same design approach can be used to obtain linear models that are robust to both variance and bias errors for these processes. 3.1. GeometricAnalysis. Under the assumption that the errors in the kth output are normally distributed with a mean 0 and variance G e t , it is well-known that the least squares estimates of the parameters in the kth row of a gain matrix has a joint confidence region defined by the following (Beck and Arnold, 1977):
(G,
- ok)(MTM)(G,-
= c2
(3) where G, = [ g k l gk21 is the kth row vector of G, M is the design matrix of manipulated input settings, and c2 is a constant determined by the number of experiments and u e t . For 2 X 2 systems, the design matrix is given by
LI.l
i.1
J
and the joint confidence region is an ellipse: n
n
This joint confidence region is the uncertainty region of
1660 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
Igb2
Figure 2. Uncertainties from different input designs: (a, left) au arbitrary uncorrelated design; (b,middle) a modified design of (a) with increased magnitudes or number of experimenta; (c, right) the new design of correlated inputa.
the kth row vector defined earlier in Figure lb. Note that, aside from their locations, the uncertainty ellipses for the two row vectors are the same because the W M matrix appears in (3) for both vectors. With the additional assumption that the errors in the two outputs are independent, we can apply the geometrically necessary and sufficient condition in (2) to examine how different designs affect the stabilizability of the closed-loop system using the identified models. The uncertainty regions can be classified as follows: (1)ellipses with a horizontal or vertical major axis when the cross-product term is 0 (Le. the two inputs are designed to be uncorrelated) or (2) ellipses with inclined axes when the two inputs are correlated. The most common practice in identification is to design the inputs to be uncorrelated. This can be achieved by either perturbing only one input at a time (Le. each mlim2i is 0), or all inputs simultaneously but independently (Le. &'~lm,im, is close to 0). A well-known example for the latter is a two-leveldesign in which each input is perturbed independently at its upper and lower levels. It can be shown that the latter two-level design is also able to minimize the area of the joint confidence region for all the parameters associated with one output in (3). This is the so-called "D-optimality" criterion, in which the determinant of MTM in (4) is maximized by the design. For dynamic systems, independent PRBS's with well-chosen switching intervals are a good approximation to the D-optimality design. Figure 2a illustrates the joint confidence regions of the two row vectors that would be obtained from applying independent inputs to an ill-conditionedprocess. We can see from this figure that because of the collinearity of the true gain vectors coupled with the orientation of the major axis of the uncertainty regions, the necessary and sufficient condition for the stabilizability in (2) cannot be satisfied by this uncorrelated design. In order to meet this stabilizability condition, the uncorrelated design would have to be modified to have more experiments or to use input perturbations with larger magnitudes such that the areas of the ellipses would be reduced sufficiently (Figure 2b) so that there exists no singularity line (h) that can intersect both regions. However, for ill-conditioned processes the required number of experiments, or the increased variations in the output responses as a result of increasing the input magnitudes, is often unacceptable. Geometrically,an obvious method to meet the condition in (2) is to rotate the ellipse in Figure 2a such that the major axis is parallel to the row vectors, and to simultaneously decrease the ratio of minor to major axis length such that the major uncertainties are only in the lengths of the two vectors (Figure 212). Of course, this design strategy requires some prior knowledge about the direction of the row vectors to be identified. If this is not available, it can be achieved by using a sequential design approach in which such information can be obtained gradually. The most significantresult from Figure 2c is that proper designs for stabilizability of this type of ill-conditioned process
U matrix
Scaling
p
Row vectors of t,nnsformed
a gair matrix
+=,
Figure 3. Formation of a transformed gain matrix.
Figure 4. Uncertainties from two uncorrelated designs of experimenta: (a,left)the same perturbationmagnitudein two rotated inpute, (b, right) the second rotated input with a larger magnitude than the f i t .
will have correlated input perturbations (Le. the ellipse has an inclined major axis). This is contraryto the general belief that we should use independent perturbations in the two inputs. The latter would provide uncorrelatd parameter estimates but less robust models. Another important advantage of taking this approach to design is that the robust identification can be accomplished with many fewer experiments or with input variations that cause much smaller output variations than the case in Figure 2b. More details on the latter will be discussed later. In summary, we have established a geometric basis for experimental design that considers the orientation of the uncertainty regions for the row vectors in the gain matrix of 2 X 2 systems. The orientation of the ellipses as well as the required ratio of the minor to major axis lengths depend on the particular MIMO process being considered. Therefore, in the following, we provide a unifying basis for the designs in a transformed input space. 3.2. Unified Design in Termsof the Rotated Inputs. Consider the following input transformation based on the SVD of a gain matrix:
G*M= (U2)(VT*A4) A 0.4 (6) where 4 is a result of multiplying the input vector by the input rotation matrix and therefore will be referred to here as a rotated or transformed input vector. Design of experiments for the settings of 4 will determine the uncertainties in the elements of the transformed gain matrix Q,which can be interpreted as the output rotation matrix being rescaled columnwise by the singular values. Figure 3 provides a geometric interpretation for a transformed gain matrix. Figure 4a shows the uncertainty regions of the transformed gain vectors from an uncorrelated design with the same perturbation magnitude in both rotated inputs.
Ind. Eng. Chem. Res., Vol. 32,No.8,1993 1661 According to (2), the design in Figure 4a does not lead to robust identification. It is clear from this figure that the major problem with ill-conditioned processes is the very small minimum singular value, which scales the vertical components of row vectors to be so small that both uncertainty regions can easily have their vertical components being positive or negative. Then the horizontal axis itself is a singularity line intersecting both uncertainty regions. A proper design for the transformed process should yield uncertainty regions similar to Figure 4b. This design would still be an uncorrelated design in the rotated inputs but would have a larger magnitude in the rotated input corresponding to the smaller singular value and hence provide more information in this direction. When the input magnitude in this direction is large enough that neither uncertainty region has the possibility of a sign mismatch, then stabilizability is guaranteed because then no singularity line can intersect both regions. Therefore, we conclude that an uncorrelated design in the rotated inputs with a sufficiently larger magnitude perturbation in the second rotated input (corresponding to the smaller singular value) will be efficient for identifying general transformed processes that satisfy the stabilizability condition. Given an estimate of the input rotation matrix, the design of rotated inputs discussed above can be easily translated into the design of the physical inputs by a simple inverse calculation, M = Vt. Consider as an example the ill-conditionedprocess with nearly collinear column vectors discussed in Figure 4. This process has its true rotated input vector as follows:
'=[
[
0.71 -0.71 mi = 0.711711 -0.71Q -0.71 -0.71][ m,] -0.71rnl -0.71m:
1
(7)
Suppose we design the rotated inputs using an independent two-level design where the second rotated input has a 10 times larger perturbation than the first. Then the four components of both the rotated and the corresponding physical input perturbations are given as follows:
Note that the two physical inputs always change in the same direction (Le. either increase or decrease simultaneously) and therefore are highly correlated. This result is the same as that from the earlier approach shown in Figure 2c which directly designs the physical inputs based on the geometric analysis of the original gain matrix. It should be emphasized that this correlated design yields approximately the same output range of Yj = [*l *lIT as the traditional uncorrelated design of Mi = [ f l *UT (the gain matrix and its SVD matrices are given later in the example). In other words, the proposed design approach can reduce the parameter uncertainty without the cost of a wider range of output variations. More discussions on the output variations are presented in the next section. If the input rotation matrix could be estimated exactly, then applying the designed physical inputs would yield a physical model with stabilizability because
G&-'= QVT(&p)-'= Q&-' (9) However, the true V matrix is never known exactly, and
y2
y2
Figure 5. Output space interpretations for two different deaigne: (a, left) the same magnitude of the two rotated inputs; (b,right) the optimal relative magnitudes according to (10).
therefore the design of experiments in terms of the rotated inputs will only asymptotically ensure the robustness of physical models. Therefore, it is crucial to also obtain a precise estimate of the V matrix for this generalized design approach to work. Fortunately, Koung (1991) has proven that, for 2 X 2 systems, the above robust design procedure of using uncorrelated input sequences in the rotated inpute with a greater variation in the direction of the second rotated input is also a D-optimal design for estimating the input rotation matrix V. The optimal ratio of the magnitudes of the perturbations in the two rotated inputs was also shown to be
g2/g1 =K (10) where K is the condition number of the gain matrix. Note that, however, for ill-conditioned processes the above ratio may not be implementable due to physical constraints on the inputs. Therefore, faced with the dilemma that the V matrix may not yet be well identified, a sequential design procedure should be used. On the basis of a current estimate of G and V,a designed experiment is run and updated estimates of G and V are obtained. The new estimate of V is then used to design a second experiment, etc. In summary, we can conclude from this section that an identification design consisting of uncorrelated two-level perturbations in the rotated inputs (with a larger magnitude in the direction of the smaller singular value) achieves two objectives simultaneously: (i) On the basis of the current estimate of the V matrix, it minimizes the impact on the stabilizability of any integral controller (i.e. it provides a robust model), and (ii) it provides an optimal estimate of the V matrix. 3.3. Implications of Closed-Loop Identification. The presence of output directionality distinguishes MIMO from SISO linear processes. In terms of the transformed gain matrix, the output vector is a linear combination of the two column vectors of the U matrix weighted by different singular values Y = ul*ul~l + UfU&
(11) where Ujis the jth column vector of the U matrix. When we perturb the rotated inputs one at a time with the same magnitude, the responding output vector will have different magnitudes in different directions. The maximum magnitude of output vector measured in the 2-norm is the maximum singular value in the direction of U1,while the minimum magnitude is the minimum singular value in the direction of UZ. Ill-conditioned processes with very different magnitudes in the singular values must have strong directionality. Figure 5a provides an example of a 2 X 2 ill-conditioned process in which the strong directionality also makes the two outputs highly correlated (i.e.yl= yz). The proposed design strategy for robust identification is to have larger perturbation magnitudes in the rotated inputs corresponding to the smaller singular values. If we do not
L
I
7
..........................
i
........I .......
b\J
P
r,
ce
_I__-
YD
.I
092
0
Figure 7. Nonlinearities in distillation columns.
Y = LU1&U1
+ (.
'3)U2
where 1= f l . Figure 5b showsthegeometric interpretation of (12). Therefore, a simple interpretation of the proposed design strategy in terms of output variations is to perturb the outputs in such a fashion that the output correlation is minimized. This design of the output perturbations may be accomplished very simply by using controllers, i.e. conducting the identification experiments under closed loop. For identification of a process model using closed-loop data, independent input perturbations or "dither signals" can be added to the closed-loop system (e.g. see Box and MacGregor (1974)). The dithers can be added into the set points or equivalently into the manipulated inputs (Figure 6). Since the closed-loop robust identification ideas discussed above set requirements on the outputs, we will consider only the set-point dithers here. A simple design for the set-point dithers is to introduce independent two-level perturbations in each set point. If the control were perfect and no constraints were hit, the design of Figure 5b would be achieved immediately. However, even when the controlis not tight, the controllers will adjust the physical inputs such that the rotated inputs corresponding to the small singular values are increased more whenever a set point change is made in the direction which is orthogonal to that of the process output correlation. In other words, the required characteristics of the physical inputs from the open-loop identification can be approximated automatically without any knowledge of the input rotation matrix and the process singular values. This shows a unique advantage to the closed-loop identification of MIMO systems; it inherently produces input variations with the correct correlations and relative magnitudes which will lead to the identification of more robust models. Since tight control of ill-conditioned processes is not possible before we identify arobust model, the set-point changes could be held for a long enough duration so that the process outputs can approach the steady-state points as shown in Figure 5b. These results are consistent with the findings of Andersen et al. (1989,1991), who reported that a closedloop identification scheme perturbing the set points of two proportional controllers on the compositions of a binary distillation column yielded a good estimate of the minimum singular value at low frequency. 3.4. Designs for a Special Class of Nonlinear Processes. So far we have considered the situation where an identified linear, steady-state model is subject to variance errors only. When a linear model is used to control a nonlinear process over a wide region, the identification
is also subject to bias errors. These result from process nonlinearity effects which can make the locally linearized process at various operating points very different from the nominal model. It might be possible to analyze the (structured) uncertainties arising from nonlinearities of any individual process and then design experiments to minimize the bias impacts on control applications. However, since there is no way to generalize process nonlinearities, it is impossible to conclude a general design approach for robust model identification against bias errors resulting from nonlinearities. However, Koung and MacGregor (1991, 1992) have discussed a special class of nonlinear processes in common existence which can be easily analyzed. This class of processes consists of those in which the nonlinearities have very little effect on the angles between the column vectors of the locally linearized process gain matrix at different operating points within a region. For 2 X 2 processes with such a characteristic, Koung and MacGregor (1991)proved that stabilizability is ensured within that region no matter how large the mismatch magnitude. Dual composition control of a high-purity distillation column is one physical example where nonlinearities lead mainly to such a equalangle rotation in both column vectors of the gain matrix as a high-purity specification is shifted from the top to the bottom or vice versa (Figure 7). For this class of nonlinear processes it is critical that this angle between the gain vectors be precisely estimated. Therefore, in order to identify a linear model that would lead to robust control of these nonlinear processes, the objective of the designed experiments should be to minimize the mismatch in the angle between the two column (row) vectors. Note that in the geometric analysis of ill-conditioned systems in Figure 2 the uncertainties in the best design were mainly in the vector lengths and not in the vector angles (Figure 2c). Therefore, designs for robust linear models which guard against variance errors from linear ill-conditioned processes will be the same as these which guard against bias errors when the identified model is used to control this special class of nonlinear systems. Alternatively Koung and MacGregor (1992) proposed a different characterization of structured uncertainties, in which the above special class of nonlinear processes are described as locally linearized processes which have the V matrices in the SVD of their gain matrices unaffected by the nonlinearities. Under this situation, it has been shown that, the more accurately the fixed V matrix is identified, the larger the operating region over which the linear model can be used to control nonlinear process with stabilizability. Therefore, in terms of the SVD mismatch descriptions, the experimental design objective for robust models which guard against bias errors is to maximize the information content on the V matrix. As shown in Koung (1991), a D-optimal design for the objective is once again
Ind. Eng. Chem. Res., Vol. 32, No. 8,1993 1663 where the reflux ratio (LID) is large (usually over lo), then the second rotated input of LV control approximately equals 1.4 times that of the DV control: L I
D
T
feed
(f= 0.707(L + V)
=
0.707(2V) when L >> D
(15a)
~
(15b)
= 2112g + d
I-.z-qpj, 5
sxFigure 8. Distillation column.
the same as the designs discussed earlier that minimize the impact of variance errors on the stabilizability. In conclusion, the special class of nonlinear processes (more physical examples are given in Koung and MacGregor (1992))in which the nonlinearities lead to an equalangle gain vector rotation, or equivalently an unchanged V matrix, can be controlled using linear models over a wide region, if these robust designs are employed during the collection of data for the identification. 4. Experimental Designs for Distillation Columns 4.1. Process Descriptions. The dual-composition control of high-purity distillation columns has been thoroughly studied in recent years because it presents unique challenges arising from severe process nonlinearities and from highly ill-conditioned nominal steady-state gain matrices (Skogestad and Morari, 1988;Skogestad et al., 1988). Two control strategies, LV and DV control, particularly have been the focus of attention. LV control of a column refers to controlling the overhead and bottom compositions with reflux flow (L)and boil-up flow (V), while DV control refers to controlling the dual compositions with overhead product flow ( D )and boil-up flow (V) (Figure 8). Both the reflux flow and boil-up flow circulate within the distillation column, and therefore if they are increased/decreased simultaneously, the changes are in the internal flows. The distillate flow is the product flowing out of the column and is therefore called the external flow (Skogestad et al., 1988). The input rotation matrices of LV and DV control are approximately the following for most binary columns (Koung and MacGregor, 1992):
where superscripts E and M denote LV and DV control, respectively, and L, V, D are all deviation flow rates. Since the steady-state relationship between LV and DV control is just a linear transformation due to material balances
[:]=[a’
:I[;]
the directions of the rotated inputs for the LV and DV configurations will be the same. The first rotated input is the overhead product flow (i.e. D = -(L V),an external flow in both configurations. For high-purity distillation
-
In other words, both control strategies have the internal flow as their second rotated input. The proposed design strategy for robust identification experiments is to perturb the second rotated input with amuch larger magnitude than the first rotated input when the process to be identified is very ill-conditioned. When applied to distillation processes, this experimental design strategy requires much larger changes in the internal flows than the external flows. Such a requirement can almost always be fulfilled in practice because the reflux flow and boil-up flow are indeed able to have much larger variations than the two product flows. However, there are still some physical constraints such as flooding of the columns that will limit the perturbation magnitudesof the internal flows. In terms of the changes in the external flow (fiistrotated input) and internal flow (second rotated input), there is virtually no difference in the design of experiments for LV and DV control. Therefore in the following, we will only examine the LV process. A simple nominal gain matrix = 0.505 -0.495 0.495 -0.505
[
1-
-
0.707 -0.707 0.707 0.707
][ o
1.0
0 O . O I ] [ %7%
-0.707 -0,7071
(16)
which is very similar to those gain matrices of many columns with equal purities in both top and bottom products, will be used to provide all the numerical confirmations for the theoretical developments. 4.2. Experimental Designs. Three experimental designs to identify the distillation column gain matrix will be compared here. Two of them are open-loop designs, and the other is a closed-loopdesign. For a fair comparison, all designs are restricted to yield approximately the same range of variation in each output. From (16),the output vector is dominated by the first rotated input as the following:
where XD and XB are scaled deviation variables for composition. Therefore, if we choose the fiist rotated input magnitude to be 2112,the two outputs will be limited approximately between -1 and +1 when there is no noise. This bound will constrain both physical input variables between -11 and +11. The first design considered here (design A) is a traditional design with uncorrelated physical inputs, in which the physical input perturbations have the components of [A1 All. Note that the component [l -11 or 1-1 13 corresponds to the first rotated input change with the magnitude of 2112,and zero second rotated input change, while the component 1111 or [-1-11 corresponds to zero first rotated input change and the second rotated input change with the magnitude of 2Il2. Therefore, the traditional identification is conducted by perturbing the external and internal flows with the same magnitude in an uncorrelated fashion. Since the two rotated inputs
1664 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 Table I. Summary of the Three Designs design A
one time constant
duration
MTM
design B
design C
one time constant
five time constants
[
PT? e e
19800
20200 19ew
202301 4WwI 0
(400 0
have the same magnitude, the second term on the righthand side of (17)has negligible effects on the output vector. As a result, the output space is virtually along the [l 1IT vector (Le. XD = XB), which indicates a strong directionality of the output space. The proposed design for robust identification is to reduce this directionality of the output space. In open-loop experiments, this can be achieved by having a larger magnitude of change in the internal flow. Therefore in the second design (design B) we choose to increase the magnitude of the internal flow change to be 10times larger than the external flow change. The corresponding physical input settings of design B have been shown in (8),and they are highly correlated. The third design (design C) is for closed-loop identification where dithers in the set points are used. Given the range of output variations (i.e. between -1 and +1for both outputs),we can simply design the set-point dithers to be -1 and +1in both outputs. As discussed earlier, such a design of uncorrelated output perturbations can eliminate the output directionality completely if the controllers have high performance. However, it is impossible to have high-performance control prior to a good identification of the process. Thus we need to make each set-point dither last long enough to allow the loosely tuned controllers to move the outputs in all directions. Moreover, in the following simulation, the set-point dithers are deliberately chosen smaller than the tolerable range of output variations (i.e. +1 and -1) to allow some output overshoots due to the control actions. Table I provides a summary of the three experimental designs used later in simulations. It contains the perturbation components, duration of every perturbation (more details discussed later), and information matrices as well as for the rotated for the physical inputs (MTM) inputs; i.e.
L
J
respectively. Note that the latter matrices for design C must be evaluated from closed-loopsimulations, since they are functions of controller tuning parameters and the duration of a perturbation. 4.3. Simulations. In order to make the simulations more realistic, a common first-order dynamic term, 1 0.6z-l, is brought into the transfer function matrix (i.e. 0.4/(1- 0.62-l)multiplied by the gain matrix in (16)) for dynamic simulations. The three designs discussed earlier will be applied to identify this simple dynamic process. A pseudorandom switching sequence between the binary inputs is used. The switching period of open-loop experiments (Le. designs A and B) is chosen as one time constant for this example, while that of the set-point dither
Noise-Free Output Responses 15
-
dmt
