I n d . E n g . Chem. Res. 1987, 26, 830-839
830
Co., Elf Aquitaine, Exxon Minerals Co., Everest Minerals Corp., Getty Oil Co., Halliburton Services, IEC Corp., Marathon Resources, Inc., Mobil Foundation, Phillips Petroleum Co., Rocky Mountain Energy Co., and Western Nuclear. Registry No. U, 7440-61-1; (NH4)&O3, 506-87-6; K2C03, 584-08-7; Na2C03,497-19-8; pyrite, 1309-36-0.
Literature Cited Aplan, F. F., McKinney, W. A., Pernichele, A. D., Ed. Solution Mining Symposium; American Institute of Mining- Engineers: New York, 1974. Goddard. J. E.: Brosnahan. D. R. Min. Eng. 1982. 34(11). 1589. Guilinger. T. R. Ph.D. Dissertation, The UnGersity of Texas, Austin, 1983. Kosikov, E. M.; Kokovskii, I. A.; Vershinin, E. A. Obogashch. Rud. (Irkutsk) 1974, 4 , 34.
Liddell, K. C. Ph.D. Dissertation, Iowa State University, Ames, 1979. Marshall, C. E. T h e Physical Chemistry and Mineralogy of Soils; Wiley: New York, 1964. Mishra, R. K. Pd.D. Dissertation, University of Utah, Salt Lake City, 1973. Nelson, M. B. Ph.D. Dissertation, Stanford University, Stanford, CA, 1978. Sato, M. Econ. Geo. 1960,55, 1202. Schlitt, W. tJ., Hiskey, J. B., Ed. Solution Mining Symposium; American Institute of Mining Engineers: New York, 1982. Smith, E. E.; Shumate, K. S. Program 14010 in the Water Pollution Control Research Series, Federal Water Pollution Control Administration, Ohio State University, 1970. Steger, H. F.; Desjardins, L. E. Chem. Geol. 1978, 23, 225. Stenhouse, J. F.; Armstrong, W. M. Trans.-Can. Inst. Min. Metall. 1952, 55, 38. Weiser, H. B.; Milligan, W. 0. Chem. Reu. 1939, 25, 1.
Received for review October 27, 1986 Accepted November 24, 1986
Design and Performance Assessment of Control Systems Using Singular-Value Analysis Richard D. Johnston* School of Chemical Engineering and Industrial Chemistry, T h e University of N e w S o u t h Wales, N e w S o u t h Wales 2033, Australia
Geoffrey W. Barton Department of Chemical Engineering, T h e University of S y d n e y , N e w S o u t h Wales 2006, Australia
Singular-value analysis was used to assess the performance and sensitivity properties of commonly employed industrial multivariable control strategies. The methodology developed was applied to a double-effect evaporator for the design and assessment of conventional multiloop control schemes together with enhancements such as loop decouplers, feedforward compensators, and internal control loops. The most significant improvements, relative to a conventional multiloop control scheme, were achieved by employing an internal control loop which manipulated the steam flow rate to the evaporator feed preheater. This internal control loop not only effectively decoupled the composition and pressure control loops but also reduced the sensitivity of the entire control system to model uncertainties. A scaling policy based on physical arguments was also developed. 1. Introduction
Dynamic simulations are often employed to compare alternative control schemes, to verify that proposed designs are adequate, or to examine the effect of changes to an existing control system or plant design. The process models used for these simulations typically consist of large mixed sets of differential and algebraic equations. While techniques for solving such models are available (Gear, 1971) and are steadily improving in speed and accuracy (Gallun and Holland, 1982), the large number of alternative control schemes, sets of controller parameters, and/or disturbance conditions which may have to be examined tend to make dynamic simulation a time-consuming task, even for moderately small problems. Singular-value analysis (SVA) is a recently developed technique (Doyle and Stein, 1981) that potentially can provide substantially more information about the performance of a control scheme than can dynamic simulation-at a fraction of the computational cost. However, the applications of SVA to chemical process control system synthesis reported to date have been very limited. Arkun et al. (1984) described an analysis procedure, based on singular values, to assess the robustness
* Author to whom correspondence should be addressed.
properties of process control systems. Decoupling control in distillation was used as an example. Lau et al. (1985) used singular-value analysis to quantify interaction in multiloop control schemes, while Grossmann and Morari (1983) briefly demonstrated the use of SVA in assessing the sensitivity of the performance and stability of thermally coupled distillation columns to modeling errors. The vast majority of industrial regulatory control schemes consist of a multiloop structure of proportionalintegral (PI) or proportional-integral-derivative (PID) controllers with, perhaps, some enhancements such as partial or total loop decoupling and/or feedforward compensation. While major economic returns may be possible by employing supervisory control, it is imperative that there is a well-functioning regulatory control structure to implement the supervisory control policy. Thus, due to their industrial importance, this paper will concentrate on the potential uses of SVA in the design and assessment of multiloop PI feedback control systems or enhancements of these systems. 2. Singular-Value Analysis 2.1. Important SVA Results. SVA is a frequency domain technique, the major potential of which lies in the analysis of multivariable or multiinput multioutput (MIMO) systems (see Figure 1) rather than single-input
0888-5885/87/2626-0830$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 831
Figure 1. Block diagram of a multivariable system with a feedback control scheme.
single-output (SISO) systems. In MIMO systems, it is the performance of the overall control system which is of prime importance, whereas SISO analysis examines the performance of individual control loops (which may well be part of a multiloop system). From the block diagram of the multivariable system, various matrix quantities related to the control system performance may be obtained. For example, the closedloop relationship (in the Laplace or frequency domain) describing the dynamic response of the output vector y to the external disturbance vector w is y = (I
+ GK)-’GL w
(1)
where G(s) is the process transfer function matrix, K(s) is the matrix of controller transfer functions, and GL(s) is the load transfer function matrix. For high-quality regulatory control, all outputs should remain close to their setpoints. Since the vectors in Figure 1 represent perturbation variables (i.e., deviations from the steady-state operating condition), the objective of regulatory control is to keep all the outputs as close to zero as possible. In other words, any measure of the size of the vector y , denoted by IIyII, should remain small under the influence of external disturbances. If, in the frequency ( w ) domain, the size of vectors is measured by the Euclidean norm (Stewart, 1973), it can be shown from eq 1 that
IWl/llWll 5 u*((I + GK)-’Gd
(2)
where u* denotes the maximum singular value of a matrix quantity. The singular values (Klema and Laub, 1980) of an arbitrary matrix P(iw) are the positive square roots of the eigenvalues of P*Pwhere P*is the complex conjugate transpose of P. P*Pis a positive semidefinite Hermitian matrix, the eigenvalues of which are real and greater than or equal to zero. Hence, there is one singular value for every eigenvalue of a matrix. The maximum singular value is obtained from the largest eigenvalue of P*P and the minimum singular value, denoted by u*, from the smallest eigenvalue of this Hermitian matrix. In practical terms, u*((I + GK)-’GL) may be thought of as the “size” of the matrix (I + GK)-lGL which provides an upper bound on the ratio of the size of the response of the vector y to the size of a sinusoidal change in the vector w . Each component of y and w will oscillate with a different amplitude. Taking Euclidean norms produces scalar quantities which are essentially the average of the contributions of the various components of each vector. It is assumed that all components of the vectors y and w oscillate with the same frequency. However, these oscillations need not be in-phase, and in fact, the size of the response in the vector y will depend upon the phase difference between the disturbance variables. In summary, for high-quality regulatory control, the process G(s) and the control system K(s) should be designed such that while preserving closed-loop stability. This last point is
important since an underlying assumption of SVA is that the system is closed-loop stable. Where a system can be described by a state-space model, closed-loop stability is most conveniently checked by the method of Johnston and Barton (1985a). In the more general case, particularly in cases where significant dead times are involved, the multivariable Nyquist stability criterion (MacFarlane, 1976) may be used. Apart from regulatory control quality, SVA can be employed to quantify many other aspects of control system performance (Johnston and Barton, 1985a). Changing set points are closely tracked if
u*(GK(I
+ GK)-’) = u,(GK(I + GK)-’j = 1
(4)
while from the closed-loop relationship
u = -K(I
+ GK)-’GLw
(5)
u*(K(I + GK)-’GLJis seen to provide a direct measure of the amount of regulatory control action required as a result of external disturbances. If u*[K(I GK)-’GLJis small, there will be less likelihood of the manipulated variables reaching their physical constraints or “saturating”. To this point, it has been assumed that the model G(s) is an accurate representation of the real process. However, if G(s) deviates from the real process, the regulatory control quality achieved may deteriorate from that measured by criterion 3 or the underlying assumption of closed-loop stability may be violated. The effect of a mismatch between the process and the process model (that is, model uncertainty) on regulatory control quality and stability will be referred to as performance sensitivity and stability sensitivity, respectively. Using the multiplicative uncertainty representation (Doyle and Stein, 1981), it may be shown that, if the nominal system is closed-loop stable, stability in the face of model uncertainties at either the inputs or outputs is guaranteed if
+
6 = u*(K(I
+ GK)-’Ju*(GJC l/p,(w)
(6)
where p,(w) is the uncertainty radius. 6 will be referred to as the “stability sensitivity number” of the system, which should be small over as wide a frequency range as possible if stability in the face of model uncertainty is to be guaranteed. If the process model deviates from the real system, then the regulatory control quality may deteriorate from the level measured on the basis of the process model. Johnston and Barton (1987) have shown that the performance sensitivity is low in closed-loop systems where
u*(GJu*(K] is small
(7)
2.2. Relative Importance of SVA Criteria. Rarely can all the previous requirements for good control system performance be satisfied simultaneously and tradeoffs will normally be necessary, just as is the case with SISO systems. As a simple example, although high controller gains may lead to high-quality control (as measured by criteria 3 and 4), the stability of the resulting system will usually be quite sensitive to model uncertainties (i.e., large values of 6). Thus, lower gains would have to be used. While a number of SVA results have been presented above, experience has shown that two stand out as the most useful in assessing plant controllability (Johnston, 1985). The regulatory control quality, measured by u*{(I + GK)-’GLJ,and the stability sensitivity number (6) provide reliable indications of the overall control system performance. It is the establishment of a high-performance regulatory control structure which remains stable in the
832 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
face of process variations which is a prime objective in control system design. 2.3. Scaling. SVA results are dependent upon the scaling of the variables constituting the system model. The problem of how a system model should be scaled so as to produce usable SVA criteria has received increased attention recently (Lau and Jensen, 1985; Perkins and Wong, 1985). All work to date has only considered the effects of scaling on the inherent process characteristics &e., a*(G) and the condition number y = a*(G)/ar{G)).Whatever scaling policy is adopted, it must embody the basic objective of scaling which is to avoid domination of the results by some subset of the output, manipulated, or disturbance variables. The scaling policy employed in this study is outlined below. (a) Output Scaling. The outputs should be scaled such that a change of given magnitude has equal significance for all outputs from a controllability point of view. This is best explained through a simple example. Two outputs of the double-effect evaporator to be examined in section 4 are the product composition (steady-state value = 0.15 mass fraction) and the pressure in the second effect (steady-state value = 60.8 kPa). If the model was written in terms of mass fractions and kilopascals, then a change of magnitude 0.1 represents a dramatic departure of the composition from its steady-state value but only a negligible change in pressure. The problem is clearly improperly scaled since all SVA results would be dominated by the composition entry in the output vector. However, the model was written such that both composition and pressure were represented as percentage changes from their steady-state values. Then, a change of given magnitude, say 10 (Le., a 10% change in this case), in both outputs was considered to be of equal significance from a controllability and operability point of view, and the problem was properly scaled. That is to say, a 10% change in product composition was considered to be as undesirable 89 a 10% change in pressure with regard to operability and performance. Although it was the case in the above example, normalization of the outputs by dividing by their steady-state values should not be seen as a general approach to output scaling. A 10% change in a distillation column temperature (e.g., steady-state value = 400 K) is clearly much more critical than a 10% change in the level of liquid in a reflux accumulator (e.g., steady-state value = 1m). If a 10-deg temperature change was considered to be as undesirable as a 0.5-m liquid level change, then a scaling factor of 0.05 may be applied to the temperature output. In this way, a change of given magnitude in the level or the scaled temperature would be equally undesirable. (b) Manipulated Variable Scaling. Manipulated variables should be scaled such that a change of given magnitude in all manipulated variables represents an equivalent amount of control action or, in other words, equal valve movement. It is usually adequate to represent all manipulated variables as fractional or percentage changes from their steady-state values. If all valves are at the midpoint of their operating range at steady state, a 100% change in any manipulated variable would indicate either complete opening or closure of a valve. (c) Disturbance Scaling. Disturbances can be scaled according to the expected magnitude an likelihood of each disturbance. As an example, consider a rocess where the major disturbances are changes in the feed temperature and composition. If a feed temperature change of magnitude 1 (say 1K) was considered to be as likely to occur as a feed composition change of magnitude 1 (say l % ) ,
$
Table I. Outputs, Manipulated Variables, and Disturbances for the Double-Effect Evaporator variable comments outputs final product composition (state zz) Y1 vapor pressure in second effect (state x 8 ) Y2 liquid volume in first effect (state rl) Y3 liquid volume in second effect (state x 5 ) Y4 manipulated variables flow rate of steam to first effect u1 flow rate of cooling water to condenser u2 flow rate of final product stream (leaving u3 first effect) flow rate of liquid stream leaving second UP effect flow rate of steam to preheater U5 disturbances feed flow rate (expected range *20% of W1 the steady-state value) feed composition (expected range *20% WZ of the steady-state value) temperature of feed to preheater w3 (expected range A10 K = f3.4% of the steady-state value)
then a model with feed temperature expressed as degrees kelvin and feed composition as a percentage would be properly scaled for SVA. However, if the feed composition was written as a mass fraction, then clearly a change of magnitude 1in the feed composition would be much less likely than a change of 1K in the feed temperature. Such a system would be improperly scaled. An alternative approach is to weight the disturbances with respect to their expected size such that changes of equal magnitude in each scaled disturbance may be expected. For example, in the double-effect evaporator to be considered in section 4, all variables were initially normalized about their steady-state values. Table I, however, indicates that the expected fractional or percentage changes in the disturbance variables are not all equal. The disturbance variables were thus weighted according to their expected ranges, as follows: disturbance
weighting
feed flow rate (wl) feed composition (wz) feed temperature (w3)
1.0 1.0 0.17
This means that the third column of the load transfer function matrix (GL) was multiplied by a factor of 0.17. If this was not done, the singular values of any matrix quantity involving GLwould be unrealistically dominated by the effects of the feed temperature. In other words, w3 was replaced by the scaled disturbance w3/, where w3 = 0 . 1 7 ~ 4 .For example, if the original closed-loop relationship between the output y1 and w3 was represented by yl(s) =
ql3(s)w3(s)
(8)
then in terms of the scaled disturbance (w3’), this relationship would become As a result, to achieve the same response in each output as would be caused by a 3.4% change in w3, a 20% change in w3/ would need to be applied. With the above weightings, a 3.4% change in w3 is now “equivalent” to a 20% change in wl, w2, and w3’. In the singular value analysis, the disturbance vector is now (wl, w2,w ~ ’ )with ~ , equal changes (of up to 20%) expected in each component of this vector. An important feature of this scaling policy is the inclusion of information (often subjective in nature) relating
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 833
Figure 2. Feedback control system with input-output decouplers.
?m
Figure 4. Double-effect evaporator.
evaluated for s = 0) often proving the most convenient. With a feedforward controller installed, the closed-loop relationship between the outputs and the set points, measurement noise, and disturbances becomes
I
1
Figure 3. Block diagram representation of combined feedback and feedforward control.
to the importance of each output and the size of the disturbances a control system will have to cope with. Previous scaling policies have not taken such considerations into account but have been based on purely mathematical approaches (Lau and Jensen, 1985; Perkins and Wong, 1985).
3. Singular-Value Analysis of Enhanced Control Systems There are various means of enhancing the performance of conventional multiloop control systems. Examination of these enhancements by SVA has not been previously addressed. How such enhancements may be incorporated into SVA will now be outlined. 3.1. Input-Output Decouplers. Interaction between control loops can often be reduced through the introduction of partial or total input-output decouplers. For a two-input two-output process model, G(s), the inputoutput decoupler matrix Dio(s)in Figure 2 is
y = GK(1
+ GK)-'(r - n) + (I + GK)-'(GF, + GL)w (12)
Thus, to examine feedforward controllers using SVA, GL should be replaced by (GF, + GL) wherever the former appeared in the feedback control system analysis. 3.3. Internal Control Loops. In many chemical processes, there are often manipulated variables available in excess of the number required to form a multiloop control system. Johnston and Barton (1984) have previously shown how it may be possible to employ these excess manipulated variables in internal control loops (ICLs) in a way that significantly improves the process characteristics, y and o,(G]. ICLs may be similarly designed to enhance closed-loop control properties. They are incorporated into the singular-value analysis as extra state equations in the statespace representation of the system. 4. Control of a Double-Effect Evaporator SVA will now be used to assess various control schemes for the double-effect evaporator (DEE) illustrated in Figure 4. A state-space model of this evaporator was developed by Johnston (1985) in the form d = A x Bu Ew (13a)
+
where
f12(s) = -gds)/g11(s) and f2l(S)
= -g21(s)/g22(s)
gij(s)is the transfer function relating the ith output to the j t h input. The overall control system is composed of the multiloop control scheme plus the input-output decoupler, Dio(s). Thus, if a multiloop control system with input-output decouplers is to be examined by SVA, this is achieved by replacing the matrix K(s) by Dio(s)K(s)wherever the former appeared in the basic feedback analysis. 3.2. Feedforward Control. A feedforward controller, F,(s) in Figure 3, is rarely used on its own but rather in conjunctionwith a feedback control scheme. The objective of feedforward control is to compensate for the disturbances (w) before these disturbances have a major effect on the system outputs. In the absence of model uncertainty, perfect feedforward control (Owens, 1978) may be achieved by specifying F,(s) = -G(s)-'GL(s)
(11)
However, F,(s) as defined by eq 11 is usually too complicated to implement (it may also be physically unrealizable), and the models G(s) and GL(s) will always be subject to some uncertainty for real systems. In practice, a stable, realizable approximation to -G(s)-lGL(s) is used for F,(s), with a steady-state feedforward controller (i.e.,
+
y=cx (13b) where x is the vector of state variables, A is the state matrix, B is the input matrix, C is the output matrix, and E is the disturbance matrix. Details of this model are given in the Appendix. 4.1. Basic Multiloop Control Scheme. The principal objective of the double-effect evaporator is production of a 15% w/w caustic soda solution from a feed stock which averages 4% w/w caustic soda. The average feed rate is 3.333 kg/s. The original dynamic model of the DEE was linearized, and all variables were normalized by dividing by the appropriate steady-state values. Therefore, all variables are in a normalized, perturbation form, ( x x s s ) / x s s .The resultant 17th-order state-space model was subsequentlyreduced to a 9th-order model by replacement of faster dynamic relationships with their equivalent steady-state form (Johnston, 1985). The potential manipulated variables and outputs for this system are listed in Table I together with the major disturbances and the expected disturbance ranges. With the variables normalized about their steady-state values, no further scaling was considered necessary for the outputs and manipulated variables. However, weightings were applied to the disturbances as previously outlined. Conventional industrial control schemes for a DEE indicated that the output-manipulated variable pairings yl-ul, y2-u2, y3-u3, and y4-u4 should form an acceptable basic multiloop control system. These pairings were also supported by relative gain analysis results (Bristol, 1966).
834 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
In the present DEE model, each liquid level was modeled as a pure integrator. While this is the simplest modeling approach, such a representation can lead to problems with the maximum singular value of G(s) and consequently the process condition number (y) going off to infinity at low frequencies (Johnston, 1985). This gives an unrealistic picture of the dynamic characteristics of this process. One solution to this problem is to make the two level control loops an integral part of the "open-loop" DEE model. Pairing the liquid volumes y3 and y4 with the manipulated variables u3and u4,respectively, via P I controllers, produces two extra differential equations which describe the time rate of change of u3 and u4. These differential equations are each a function of state, manipulated, and disturbance variables. Thus, u3 and u4 behave like two extra state variables and as such may be absorbed into the state matrix (A). Setting proportional gains of -5 and reset times of 300 s in these level control loops (these parameters were selected to give relatively fast response in these loops, thereby ensuring that the steam chests remained covered with liquor), an 11th order state-space model was produced after absorption of u3and u4into the state matrix (see Appendix, eq A.5). Note that the proportional gains are dimensionless since all variables were represented as fractional changes. The input matrix (B) now relates to three manipulated variables, ul,u2,and u5. However, u5played no part in the basic multiloop control scheme to be considered initially. Hence, u5was assumed to be constant a t its steady-state value. The third column of B shown in eq A.5 was therefore omitted, giving a matrix of dimensions 11 x 2. The output matrix (C) has dimensions 2 X 11 and relates only to y1 and yz. It is this 11th-order state-space model on which the following analysis was based. Absorption of the level control loops into the state matrix in the above manner means that the performance of these loops will be implicit within the input-output model, G(s). That is, the singular-value analysis will not explicitly reveal information on the level control loops. Rather, all analysis will reveal information on the relationship of the outputs y1 and y 2 to the manipulated variables u1 and u2 and the three disturbances. This does not represent a severe limitation, however, since it is these two outputs which are most critical from a control system performance viewpoint. The caustic concentration in the final product (yl) is, economically, the most important output, while satisfactory control of the pressure (yz) in a DEE is critical for smooth operation. (The effect of the pressure control loop on closed-loop stability is discussed below.) The proportional gains, K,, and reset times, Ti,for the control loops yl-ul and yz-u2 were selected to give a reasonable compromise between the control quality, the amount of control action required, and the performance and stability sensitivity: Composition loop Yl-ul K , = 20 Ti = 480 s Pressure loop
Y2-%
K , = -20 Ti = 60 s With higher proportional gains or smaller reset times, the amount of control action and the performance and stability sensitivity increased markedly. Closed-loop stability of the system with these controller parameters was checked by using the method of Johnston and Barton (1985a). Note that the system was found to require tight
KEY
.
\
0.011 0,001
0.01
1
1
01
FREQUENCY (rad/s)
Figure 5. DEE base case. Regulatory control quality and set-point tracking.
31100 Y
5 ,o a LT
I
. .
0
_______
2 2
:>.
'-71
zin
0 001
001 01 FREQUENCY ( r a d / s i
Figure 6. DEE base case. Amount of control action, stability sensitivity, and performance sensitivity.
pressure control to remain closed-loop stable. If the pressure control loop yz-uz was detuned, for example, by setting K , = -2 and Ti = 180 s, the resulting system was closed-loop unstable. With the above control system installed, a complete singular-value analysis was performed to produce the results shown in Figures 5 and 6. All SVA criteria showed the DEE to be most susceptible to disturbance frequencies in the vicinity of 0.01 rad/s. At these frequencies, disturbances entering the system are not greatly attenuated and the rate of oscillation is sufficiently great to cause problems in the response of the control system. The amount of control action, quantified by u*(K(I + GK)-'GL}a t frequencies around 0.01 rad/s, would, in all probability, result in manipulated variable saturation. For example, if a*(K(I + GK)-'GL) = 5, 20% changes in the disturbance variables could translate into 100% changes in the manipulated variables since this SVA quantity provides the upper limit to disturbance amplification. If the manipulated variables were at the midpoint of their operating range in the steady state, a 100% change would imply complete closure or complete opening of the valves. As values of o*(K(I GK)-lGj significantly greater than 5 are obtained a t frequencies around 0.01 rad/s, manipulated variable saturation is highly likely. Under the influence of disturbances around 0.01 rad/s, closed-loop stability problems are likely in the event of significant model/plant mismatch. This may be seen from the sharp peak in the stability sensitivity number, a*(K(I + GK)-'}a*(G],in Figure 6. Regulatory control quality (measured by a*((I + GK)-'GL) in Figure 5) is also very poor at frequencies in the vicinity of 0.01 rad/s. The control quality and amount of control action predicted by SVA may be tested by dynamic simulation of the 11th-order linear model. The response of the outputs y1 and y 2 to simultaneous step changes in the three possible disturbances is shown in Figure 7. The step changes used for this and all subsequent dynamic simulations (i.e.,those where an enhanced control system was used) were selected to represent a worst-case situation as follows: Feed flow rate (wl) +20% Feed composition (w,) -20% Feed temperature (w3) -3.4% (that is, w3'= -20%)
+
~
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 835
I
0.04
0-0.6-
0
I
300
600 900 TIME I SI
1200 1500
0.001
0.1
0.01
FREQUENCY (rad/s)
Figure 8. DEE regulatory control quality before and after installation of input-output decouplers.
BASE C A S E
eu -2
0
600 900 T I M E is)
300
1200
1530
Y
Figure 7. DEE base case. Dynamic simulation of step response; output ISElW 0.571.
The combined integral squared error (ISE) of the two outputs was also computed for the time period 0-1500 s (ISE,, = 0.571). The oscillatory nature of the response may be attributed to the following factors: (i) outputmanipulated variable interactions (see section 4.2.3) and, in particular, the effect of the steam flow rate to the first effect (uJ on both the final product composition and the second effect pressure; (ii) the combination and type of disturbances entering the system are the worst possible; (iii) a high proportional gain and small reset time were required in the pressure control loop to maintain stability. This response is also typical of many industrial double-effect evaporators (Pitblado et al., 1986). Despite the oscillatory behavior, however, both outputs remain within *3% of their set points. In the frequency domain, a step response may be considered as equivalent to the response obtained from a series of sine waves, dominated by low-frequency components. “Low” frequency in the case of the DEE may be taken as less than about 0.01-0.02 rad/s. Therefore, from a SVA point of view, it is these frequencies which are of most interest. In all following studies of enhanced control schemes, the basic multiloop PI control system with the proportional gains and reset times given in this section will remain intact. Therefore, Figures 5-7 will provide the base case against which other control schemes can be judged. 4.2. Control System Enhancements. 4.2.1. InputOutput Decouplers. Steady-state input-output decouplers are usually the easiest to implement. The matrix of steady-state gains between the outputs and manipulated variables of the DEE was obtained from the state-space model, eq A.5, as r
u1
UZ 1
G( 0)=
from which, using eq 10,
-
Dido) =
1
4.29 L
2 1
-
-0.0058 1
I
After this steady-state input-output decoupler was added to the basic multiloop control scheme, the resultant system remained closed-loop stable. Singular-value analysis of the DEE with this input-output decoupler installed produced the results shown in Figures 8-11. When compared with the base case, there appears to be
’b
0,001
0.01 FREQUENCY irad/si
0.1
Figure 9. Amount of control action required in the DEE before and after installation of input-output decouplers.
1coo(
I
Y
‘b
0.001
0.01
01
FREQUENCY Irad/s)
Figure 10. Stability sensitivity of the DEE before and after installation of input-output decouplers.
0001
0 01
01
FREQUENCY ( r o d / s )
Figure 11. Performance sensitivity of the DEE before and after installation of input-output decouplers.
little benefit gained from the installation of steady-state input-output decouplers from either a control quality or sensitivity point of view. Any small improvements in the regulatory control quality, the control action required, or the stability sensitivity must be weighed against the significant increase in the performance sensitivity caused by the input-output decouplers. Of Course, the gains calculated for the steady-state input-output decoupler (see eq 15) may not represent the optimum gains from an overall performance point of view. An optimization strategy may be formulated to determine whether “better” values of the gains do exist, though this will not be pursued here. Since step changes cannot be associated with disturbances of specific frequencies, the SVA results should be treated in a semiquantitative manner when analyzing the step response of a system. The dynamic simulation of the DEE’S response to the set of step disturbances, after installation of input-output decouplers (see Figure 12), supported the SVA prediction of minor improvements in regulatory control quality and the amount of control action
836 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
",:y --006
O
0
o 4 600 900 1200 TIME I s 1
300
L 1500
0 001
001
01
FREOUEN CY irad/s)
Figure 14. Amount of control action in the DEE before and after installation of feedforward compensator.
-2L 0
900
600
300
TIME
1200
1500
(51
I
- 0 oh{
Figure 12. Dynamic simulation of step response of the DEE after installation of input-output decouplers; output ISEISoo= 0.456.
0
300
600 900 1200 TIME (5)
21
600 1
i
$ 0001
001
-11
"r2/ 0 300 ,
01
FREQUENCY lrad/sl
I
,
600
900
I
1200
Ij
1500
TIME is1
Figure 13. Regulatory control quality of the DEE before and after installation of feedforward compensator.
Figure 15. Dynamic simulation of step response of the DEE with feedforward compensation; output ISE,,oo = 0.0691.
required (combined output ISE,, = 0.571 for the base case; ISE,,, = 0.456 with input-output decouplers installed). 4.2.2. Feedforward Control. T o construct a steadystate feedforward controller, F,(O),the steady-state gains between the outputs and manipulated variables and the steady-state gains between the outputs and disturbances are required. The former are given in eq 14, while the latter may be obtained from the state-space model, eq A.5, as
of 3-5 in the low-frequency range. Thus, the norm of the output oscillations should be reduced by an equivalent factor. This was confirmed by the dynamic simulation results shown in Figure 15 where the combined output integral squared error, ISE1500,was reduced by an order of magnitude on the base case. Figure 14 indicates that significantly less control action is required, and this was borne out by the simulation results in Figure 15. 4.2.3. An Internal Control Loop (ICL) for the Double-Effect Evaporator. The obvious use for the excess manipulated variable (i.e., additional to the requirements of a basic multiloop control scheme), the steam flow rate to the preheater (us),is for control of the temperature of the feed to the second effect. However, both SVA and dynamic simulation showed such a control loop to be of little value regardless of the controller parameters used. Greater benefits may be obtained by pairing it with some other measurement or combination of measurements. It will now be shown how this excess manipulated variable may be employed in an internal control loop (Johnston and Barton, 1984) which brings major operability and controllability benefits. Previously it was shown that, with a multiloop control scheme, several SVA quantities exhibited a peak in the vicinity of 0.01 rad/s. Thus, incorporation of an ICL to reduce the stability sensitivity number (6) at frequencies in this vicinity would be one highly desirable option. Calculation of 6 requires significantly more computation than calculation of the condition number ( 7 ) . However, as minimization of y generally also brings about a significant reduction in 6, the former was used as the more convenient objective function for minimization. It was assumed that measurements of the following were available for inclusion in the ICL: q,temperature of the liquor in the first effect; x6, concentration of caustic soda
w1
*I
*3
r
By use of eq 11 r
F,( 0 ) =I -G( 0 ) - 'GL(0 ) =
1
T
1.230 -0.291 -1.103 2.623 -1.248 3.392
Singular-value analysis of the DEE with the basic multiloop control scheme plus the above feedforward controller produced the results shown in Figures 13 and 14. Note that as the performance and stability sensitivity are unaffected by the addition of a feedforward controller, the SVA results pertaining to these quantities have not been included. Since eq 17 is a steady-state controller, the major benefit from its installation would be expected at low frequencies. This is clearly revealed in Figures 13 and 14. At frequencies greater than about 0.02 rad/s, the control quality deteriorates and the amount of control action required increases after installation of the steadystate feedforward controller. Nevertheless, in response to step changes in the disturbances, significant benefits should be gained through the inclusion of the feedforward controller. a*((I + GK)-lGL)has been reduced by a factor
Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987 837
BASE CASE WITH
ICL
--. '.'.
z
0 U
0.0001
0001
I
I
0.01
03
0.001
FREQUENCY Irad/sl
Figure 16. Condition number of the DEE before and after installation of an internal control loop.
0.01 FREQUENCY l r a d / s l
Figure 19. Amount of control action in the DEE before and after installation of the ICL.
niin
-Y
Ool
0 01 FREQUENCY(rad/sl
0.0 1 FREQUENCY (rod/sl
10
0 01
0 001
0.1
03
FREQUENCY Irad/sl
Figure 17. Regulatory control quality of the DEE before and after installation of the ICL.
0.011 0 001
ICL
+
=e t 0.001
01
Figure 20. Stability sensitivity of the DEE before and after installation of the ICL.
Y
0.1 FREQUENCYIrod/sl
Figure 18. Set-point tracking ability of the DEE before and after installation of the ICL.
Figure 21. Performance sensitivity of the DEE before and after installation of the ICL.
in the second effect liquor; x7, temperature of the liquor in the second effect; xg,temperature of caustic soda leaving the feed preheater; and u5, steam flow rate to the preheater. Incorporation of ICLs in a variety of process models has shown that inclusion in the ICL of meaqurements already being used in the principal control structure rarely brings any additional benefits. These measurements could, of course, be included-at the cost of increasing the number of variables in the optimization problem. However, in the present example, the measurements were restricted to the above five. The 11th-order DEE model, eq A.5, with implicit level controllers, provided the basis for the following optimization problem: Find the coefficients in the ICL equation
showed improvement in some form as a result of the inclusion of this ICL. The stability sensitivity number (6) was dramatically reduced at all frequencies and particularly in the vicinity of 0.01 rad/s (see Figure 20). Similarly, over the majority of the frequency range shown, control quality was significantly improved and the sensitivity of control quality to model uncertainties reduced. For example, while the maximum value of a*((I + GK)-lG+),a measure of the regulatory control quality, in the original multiloop system was 1.66, the corresponding maximum for the system with the ICL was only 0.137. Finally, the ICL also gave the system greater ability to track changing setpoints at most frequencies, and the amount of control action required was reduced, particularly at frequencies near 0.01 rad/s. The SVA results indicating improved control of the DEE by inclusion of an ICL were confirmed by the dynamic simulation results shown in Figure 22. The much smoother response suggests that the ICL significantly reduced control loop interaction. This was confirmed by calculating the dynamic direct gain interaction matrix, DGM(w) (Johnston and Barton, 1985b). Elements in the DGM(w) matrix approaching unity indicate that the corresponding output-manipulated variable pairing would be noninteractive, while elements approaching zero indicate strong interaction. For the basic multiloop control system,
ri5
= u12.3x3
+ u12,6x6 + a12,7x7 + u12,@9 + u12,12u5
(la)
which minimizes the process condition number, y, at a frequency of w = 0.01 rad/s. Solving this problem, the ICL was found to be
which, when included in the state-space model, produced a modified process G(s) with y = 2.466 a t w = 0.01 rad/s. Not only was y significantly reduced at w = 0.01 rad/s but, as Figure 16 shows, the improvement extended over the entire frequency range. Employing the original controller parameters in the basic multiloop control scheme, the system with the above ICL included remained closed-loop stable and SVA produced the results shown in Figures 17-21. All SVA results
r
DGM(w = 0.01)=
Ul
(20)
showing that the steam flow rate to the first effect (uJ has a major effect on the pressure (y2),which is a major cause
838 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0
0
0
1
0
0
0 0
0 0 0
-0.7957e-03
0.1372e-02
0
0 0 0
-0.7085e-00
0
0 0
0.1267e-01
-0.1266e-01
0
0
0 0
O.lllOe-O1
of the oscillatory response observed in Figure 7. After installation of the ICL, r
DGM(w = 0.01)=
u1
u2-
(21)
which shows that the two loops are almost decoupled. In fact, the loop interaction was significantly reduced by the ICL for all frequencies less than about 0.05 rad/s.
Conclusions To achieve a closer integration of chemical plant and control system designs, techniques considerably faster than dynamic simulation are needed to rapidly assess the performance of controlled processes. Singular-value analysis is such a technique. In this paper, it has been shown how singular-value analysis can be used to quantify the performance and the sensivitity to modeling errors, of commonly employed industrial control schemes, with a double-effect evaporator being used as an example.
0 0
-0.4465e-03
Conventional steady-state input-output decouplers brought few benefits to the evaporator from the point of view of regulatory or servo control quality or the sensitivity to model uncertainty. The benefits of steady-state feedforward control were seen through the improved control quality, but the sensitivity to model uncertainty was unchanged from that of the original multiloop feedback control scheme. On the other hand, an internal control loop, which was designed principally to reduce the stability sensitivity of the evaporator, brought major benefits in all measures of the control system performance. Loop interaction analysis showed that inclusion of the ICL had effectively decoupled the composition and pressure control loops, resulting in a control system whose response was much smoother and more rapid. Singular-value analysis has been shown to be a highly efficient technique for assessing many aspects of the dynamic performance of controlled processes. A much closer integration of chemical plant and control system designs should be possible through the use of singular-value
Ind. Eng. Chem. Res., Vol. 26, No. 4,1987 839 Table 111. DEE Flow Rates and Disturbances
steadyState
variable steam flow rate to first effect steam chest ( u J , kg/s condenser cooling water flow rate (u2),kg/s first effect caustic product flow rate (us),kg/s second effect caustic product flow rate (u4),kg/s steam flow rate to preheater (us),kg/s evaporation rate in first effect, kg/s evaporation rate in second effect, kg/s feed rate of weak caustic to preheater ( w ~ )kg/s , caustic concn in feed to preheater (wz), mass fraction temp of feed to preheater (tu3),K
value 1.089 33.42 0.889 2.109 0.376 1.220 1.224 3.333 0.04 293
analysis offering, as it does, the potential to rapidly assess a wide range of control system/plant design combinations. Acknowledgment This work was supported by the Chemical Engineering Foundation within the University of Sydney. Nomenclature A = state matrix B = input (manipulated variable) matrix C = output matrix d = vector of the effect of disturbances on the outputs Di,(s) = matrix of input-output decoupler transfer functions E = disturbance matrix e = vector of error signals to the controllers F,(s) = matrix of feedforward controller transfer functions f (s) = individual input-output decoupler transfer function %(s) = process model transfer function matrix GL(s) = load transfer function matrix g,(s! = single process transfer function within G ( s ) I = identity matrix K(s) = matrix of controller transfer functions K , = proportional gain LI = vector of measurement noise r = vector of controller set points s = Laplace variable Ti = reset time u = vector of manipulated variables (inputs) w = vector of disturbances x = vector of state variables y = vector of system outputs 11*11 = Euclidean vector norm Greek Symbols
6 = stability sensitivity number (eq 6) y = process condition number p,(w) = model uncertainty radius
c*(P)= maximum singular value of the matrix P a*{P)= minimum singular value of the matrix P w = frequency, rad/time
Superscript (on a M a t r i x )
* = complex conjugate transposition Appendix. Double-Effect Evaporator Model On the basis of unsteady-state heat and material balances, a 17th-order (nonlinear) dynamic model can be derived for the double-effect evaporator (DEE) illustrated in Figure 4. The state variables in this model are listed in Table I1 together with their steady-state values. Table I11 gives the steady-state values of the manipulated and disturbance variables. As stated in the text of the paper, the original 17th-order nonlinear model was linearized, and all variables were expressed as normalized (against their steady-state values) perturbations. If some of the “faster” (i.e., smaller time constant) differential equations were replaced by steadystate algebraic equations, this 17th-order linear model was
reduced to a 9th-order state-space model. The state variables in Table I1 which have not been identified by the symbol x i correspond to those state equations replaced by steady-state algebraic equations. The manipulated and disturbance variables are, of course, the same for both the 17th- and 9th-order models. As explained in the text, the level control loops y3-u3 and y4-up may be absorbed into the state matrix. From the ninth-order state-space model, 2 1 = a 1 2 X 2 + (1.1323 + U14Xq + (1.16Xg + b13U3 b 1 4 ~ 4 (A.1) and 3 5 = (1.5626 + U5727 + a 5 8 2 8 + b 5 4 ~ 4+ e51W1 e52W2 (A.2)
+
+
where uij, b,, and eu are the coefficients of the state, manipulated, and disturbance variables. The differential equations describing regulatory (i.e., fixed setpoint) PI control of y3 (=xJ and y4 (=x5) with proportional gains of -5 and reset times of 300 s are 0 3 = 5 2 1 + 5X1/3OO (A.3) and Li4
= 5x5
+ 5X5/3OO
(A.4)
Substitution of eq A.l and A.2 into eq A.3 and A.4, respectively, yields linear functions describing the time rate of change of u3 and u4. In the 11th-order state-space model, eq A.5, Chart I, these linear functions produce the 10th and 11th rows of the A, B,and E matrices, and the manipulated variables u3and u4contribute the coefficients in the 10th and 11th columns of the state matrix, A. (The notation e-04 in Chart I stands for value X It is this 11th-order model, with implicit level controllers, which provides the basis of all singular-value analysis of the DEE. Literature Cited Arkun, Y.;Manousiouthakis, B.; Palazoglu, A. Znd. Eng. Chem. Process Des. Deu. 1984, 23, 93-101. Bristol, E. H. ZEEE Trans. Autom. Control 1966,11(1), 133. Doyle, J. C.; Stein, G. ZEEE Trans. Autom. Control 1981, 26(1), 4-16. Gallun, S. E.; Holland, C. D. Comput. Chem. Eng. 1982, 6(3), 231-244. Gear, C. W. Numerical Initial Value Problems in Ordinary Differential Equations; Prentice-Hall: Englewood Cliffs, NJ, 1971. Grossmann, I. E.; Morari, M. Paper presented at the 2nd International Conferenceon the Foundations of Computer-aided Process Design, Snowmass, 1983. Johnston, R. D. Ph.D. Dissertation, University of Sydney, N.S.W., 1985. Johnston, R. D.; Barton, G. W. Znt. J. Control 1984, 40(6), 1051-1063. Johnston, R. D.; Barton, G. W. Paper presented at Process Systems Engineering 85, Cambridge, 1985a. Johnston, R. D.; Barton, G. W. Znt. J. Control 198513, 41(4), 1005-1013. Johnston, R. D.; Barton, G. W. Znt. J. Control 1987,45(2), 641-648. Klema, V. C.; Laub, A. J. ZEEE Trans. Autom. Control 1980,25(2), 164-176. Lau, H.; Alvarez, J.; Jensen, K. F. AZChE J. 1985, 31(3), 427-439. Lau, H.; Jensen, K. F. AZChE J. 1985,31(1), 135-146. MacFarIane, A. G. J. AZChE Symp. Ser. 1976, 159, 126-143. Owens, D. H. Feedback and Multivariable Systems; Peregrinsus: New York, 1978. Perkins, J. D.; Wong, M. P. F. Paper presented at Process Systems Engineering 85, Cambridge, 1985. Pitblado, R. M.; Barton, G. W.; Sjoberg, P. Paper presented at the 3rd Conference on Control Engineering, Sydney, 1986. Stewart, G. W. Introduction to Matrix Computations; Academic: New York, 1973. Received for review June 25, 1985 Revised manuscript received March 27, 1986 Accepted December 6, 1986
