Ind. Eng. Chem. Res. 1987,26, 1043-1045 - = limit from lower values
r = liquid molar fraction y = vapor molar fraction
* = equilibrium
' = dimensioned variable
Greek Symbols a = relative volatility t = vapor point efficiency
e = new independent variable; see (7) K
1043
= part of liquid staying on tray
Subscripts b = bottom = reboiler c = condenser f = feed N = top tray 0 = initial condition
Literature Cited Baron, G.; Wajc, S.; Lavie, R. Chem. Eng. Sci. 1981,36, 1819-1827. Cannon, M. R. Ind. Eng. Chem. 1961,53, 629. Lewis, W. K. Znd. Eng. Chem. 1936,28, 399-402. Rivas, 0. R. Ind. Eng. Chem. Process Des. Deu. 1977,16,400-405. Sommerfeld, J. T.; Schrodt, V. N.; Parisot, P. E.; Chien, H. H. Sep. Sei. 1966, 1, 245-279. ToftegArd, B.;Jerrgensen, S. B. submitted for publication in Comp. Chem. Eng. 1987a. Tofteghrd, B.; Jmgensen, S. B. submitted for publication in Ind. Eng. Chem. Res. 198713.
Superscripts + = limit from higher values
Receiued for review March 18, 1985 Accepted February 13, 1987
COMMUNICATIONS Robustness with Respect to Integral Controllability A multivariable closed-loop process may become unstable for small changes in steady-state gains if integral action is used. The results of this work show that the Relative Gain Array (RGA) is related quantitatively to the amount of change (or error) allowed in each individual steady-state gain before the system becomes unstable. The derived criterion is applicable for any type of feedback control system as long as integral action is used. Controllers with integral action are extensively used in chemical process control. As pointed out by several authors (Koppel, 1985; Grosdidier et al., 1985), the integral controllability of a multiple-input-multiple-output (MIMO) process depends largely on the steady-state information of the process. If a process is not integrally controllable, it will eventually go unstable no matter what the controller settings are (usually evidenced by a slow drift into instability). Due to the complexity and strong nonlinearity of many processes, the model used in controller design is usually only an approximation of reality. The ability to remain integrally controllable (stable) in the face of plant/model mismatches is called Robustness with respect t o integral controllability. This is a unique feature of MIMO processes. In a single-inputsingle-output (SISO) process, the system will lose integral controllability only when the steady-state gain changes sign (since a system with positive feedback is not integrally controllable). However, in a MIMO process, positive feedback may arise for small changes in process gains. Robustness for integral controllability is an inherent property of the system, and it can be determined from the steady-state gains, G(0). Grosdidier et al. (1985) related the relative gain array, RGA (Bristol, 1966), to the optimally scaled condition number of G(O),which gives an approximate bound for the allowable modeling error. Their approach dealt with simultaneous perturbations in all the giis. Furthermore, they also pointed out that the RGA is a measure of system sensitivity, i.e., the bigger the ijth element of the RGA, the more sensitive the gij will be to errors. 0888-5885/87/2626-1043$01.50/0
The purpose of this work is to show that each RGA element (Pij) is related quantitatively to the amount of perturbation (Ag,) allowed in each individual element of the steady-state gain matrix before the system loses integral controllability. Integral Controllability Integral controllability has been studied by Morari and co-workers (Grosdidier et al., 1985; Morari, 1985). If multiloop SISO controllers are used, integral controllability is an important criterion in variable pairing. Yu and Luyben (1986) eliminated pairings with negative Morari indexes of integral controllability (MIC) to ensure integral controllability. The MIC's are the eigenvalues of the G'(0) matrix (the plant steady-state gain matrix with the signs adjusted so that all diagonal elements have positive signs). If all of the individual loops are integrally controllable, a negative value of any of the eigenvalues of G+(O)means that the variable pairing will produce an unstable closedloop system if each loop is detuned at an arbitrary rate. It should be noticed that for 3 X 3 or higher order systems, there are instances for which no variable pairing will give MIC's that are all positive.
RGA and Robustness The RGA gives some information on variable pairing. Grosdidier et al. (1985) gave an updated summary of the uses of the RGA. Each element of the RGA is defined as
p., v = g$.. 1 11
(1)
where pi, = the ijth element of RGA, gij = the ijth element 0 1987 American Chemical Society
1044 Ind. Eng. Chem. Res., Vol. 26, No. 5 , 1987 Table I. A Deisobutanizer Column Example (Yu and Luyben, 1985) R- V D-V G(0)
RGA IRA
16.3 1-26.2
I -:;:: I 0.012 -0.011
II II
-18.0 -2.3 28.63 3.1 87.1 0.56 -86.1 0.44 -0.0111 (-1.786 0.012 -2.272
:I” I 1-6:
I I1
Table 11. Sidestream Distillation Column (Oaunnaike et al.. 1983)
-::::1
0.44 0.56 1-0.48 1.48 -0*48 -2.272 -0.676 -1.786 2.083 -0.676 2.083
I I
of G(O),and gji = the jith element of [G(O)]-’.Grosdidier et al. (1985) related the sensitivity of gij to Pij by dPij
-
(1 - Pij)Pi;
dgij
1 1 1
RR-V G(0) RGA IRA
(See the Appendix or Grosdidier et al. (1985) for the derivation.) Since Pij = is the limit of integral controllability (or stability), the allowable change in gij (Agi,) can be obtained by integrating eq 2. gij
(3)
For the case of Pij > 1, the left-hand side (LHS) of eq 3 becomes
LHS =
IRA
J pij J,, dpij
= Iln
&
Pij
dpij
-
- In (Pij - l)l;,J
= Iln [Pij/(Pij - 1)11i, = In (1 - l/Pij)
(4)
The right-hand side (RHS) of eq 3 is RHS = In [kij + &ij)/gij] = In (1 + Agij/gij)
(5)
The result of the integration is
Notice that the same result can be obtained for PI, 1. Equation 6 shows that -l/& is a measure of the relative change allowed in g,]. For a system with a big P,,, a small change in the g element will produce instability. It should be emphasize8 that if a fractional change in g,, exceeds -1/PV, the closed-loopsystem will become unstable for any controller with integral action (e.g., multiloop SISO controllers, Internal Model Control, Dynamic Matrix Control, etc.). It is convenient to define a matrix of robustness measures called the “Integral Robustness Array” (IRA). The elements of IRA (a,,) are ff,, = -1/& (7) all is the fractional allowable change in g, (Ag,/g,). The IRA gives control engineers a quantitative measure of a system’s robustness to integral controllability. It should be emphasized that the IRA gives the maximum change permitted to maintain integral controllability when only one element is changed. The analysis of Grosdidier et al. is applicable when all elements are changed simultaneously.
Examples Three distillation column examples are presented below:
1.11
-34.68 2.01 -0.72 -0.29 -0.50 1.39 3.50
-0.61 -2.36 46.2 -0.65 1.83 -0.18 1.55 -0.55 5.59
-0.0049 -0.01 0.87 0.355 -0.10 1.47 10.00 -0.68
Table 111. Complex Distillation Example (Elaahi and Luvben. 1985)
(2)
gij
0.66
I
13.4 9.2 -5.3 -11.2 13.4 9.2 -5.3 -11.2 -6.6 -58.1 70.7 -4.3 0.512 0.017 -0.014 0.233
-1.4 3.6 -1.8 -4.5 -1.4 3.6 -1.8 -4.5 0.13 3.11 -2.41 1.71 -7.692 -0.323 0.417 -5.882
-21.6 -14.1 8.1 19.0 21.6 14.1 -8.1 -19.0 7.38 55.26 -66.30 4.60 -0.136 -0.018 0.015 -0.215
0.76 0.47 -0.11 4.90 0.76 0.47 -0.11 4.90 0.086 0.731 -0.305 0.489 -12.50 -1.37 3.30 -2.041
A. Example 1: Three-Component, 32-Tray Column (Yu and Luyben, 1985). This is a high-reflux-ratio column. Three 2 x 2 control structures were studied: (a) the conventional R-V (reflux and vapor-boilup) structure, (b) the D-V (distillate and vapor-boilup) structure, and (c) the RR-V (reflux ratio and vapor-boilup) structure. Table I gives the G(0) matrix, the RGA, and the IRA matrices for all three control structures. The R-V structure is very sensitive to parameter change in any of the gij’s. A 1.2% change in any one of the gij’s causes the system to become unstable. Also notice that the conventional pairing produces negative relative gains in the diagonal elements. The D-V structure is very robust. More than 100% changes in gij are allowed. For example, if g,, decreases by 180%,the system will become unstable. However, for a 100% change in g22,the system will lose integrity (e.g., loop 2 will be unstable by itself). The robustness of the RR-V structure falls between those of the D-V and R-V structures. A 68% deviation in gll or gZ2can be tolerated. The D-V structure is the control scheme recommended by Yu and Luyben. B. Example 2: Sidestream Distillation Column (Ogunnaike et al., 1983). This 3 x 3 distillation process uses reflux flow, sidestream flow, and vapor boilup to control two compositions and one tray temperature. The steady-state gains and RGA and IRA matrices are given in Table 11. The largest relative gain (fill) is 2.01. A 50% decrease in g,, will cause the closed-loop system to become unstable. The off-diagonal elements can tolerate more than 100% changes in steady-state gains. Therefore, this 3 X 3 process has a fairly robust cbntrol structure. C. Example 3: Complex Distillation Column (Elaahi and Luyben, 1985). This column has three feeds and two sidestreams and represents a 4 X 4 multivariable process. Severe nonlinearity was also observed in this column. The G(0)matrix obtained from pulse testing the dynamic model is given in Table 111. Notice that there is no reasonable pairing which gives all positive MIC’s. The eigenvalues of G+(O)are 21.7, 5.65, 2.76, and -0.12.
Ind. Eng. Chem. Res. 1987,26, 1045-1048 One way to achieve integral controllability is to sacrifice the single-loop performance of loop 3 (Le., loop 3 cannot be stabilized by itself). In this case G+md is used instead of G+(O). The G+md matrix is G+(O) with one negative diagonal element (g33).The resulting MIC's become 4.46 f 7.17i, 4.77, and 0.12. Table I11 also gives RGA and IRA values. Some giis are very sensitive to modeling error. For example, a 2% change in gZ1,g23, g31,or g,, causes the closed-loop system to become unstable. Unfortunately, as pointed out by Elaahi and Luyben (1985), strong nonlinearity may easily result in more than 100% changes in some of the steadystate gains (i.e., g,, changes sign at different operating conditions).
Conclusion The RGA provides a useful measure of robustness with respect to integral controllability. A system with a large relative gain element (&) will become unstable for a small change in that particular steady-state gain (gi,). Furthermore, the fractional change in gij cannot exceed -1/& in order to preserve closed-loop stability. The result of this work (eq 6, IRA) gives process control engineers a quantitative measure of how much change (or error) in the steady-state gain is allowed in a multivariable process. This criterion is independent of controller tuning and is applicable to any type of feedback control system (multivariable or multiloop SISO controllers) as long as integral action is used. Nomenclature
1045
Appendix The derivation of eq 2 follows directly from the definition of RGA. (-41) pij = g.& 1 I1
where det G = the determinant of G and Gij = the matrix G with ith row and jth column removed. Differentiating eq A2 with respect to gij gives d&j -1i+j det Gij d[l/det GI _gij(-l)i+jdet Gij dgij det G dgij
+
Substituting eq A2 into eq A3 gives -dptj = - - -Pij =
Pi?
gij
gij
dgij
Pij(1
- Pij)
gij
(A4)
Literature Cited Bristol, E. H. IEEE Trans. Autom. Control 1966, AC-11, 133. Elaahi, A.; Luyben, W. L. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 368. Grosdidier, P.: Morari, M.; Holt, B. R. Znd. Eng. Chem. Fundam. 1985, 24, 221.
D = distillate flow rate G = plant transfer function matrix G+ = G with positive diagonal elements G+md..= G+ with at least one negative diagonal element gij = ilth element of G gji = jith element of G-' Agij = deviation of gij from its nominal value R = reflux flow rate RR = reflux ratio V = vapor boilup
Koppel, L. B. AZChE J. 1985,31, 70. Morari, M. IEEE Trans. Autom. Control 1985, AC-30, 574. Ogunnaike, B. A,; Lemaire, J. P.; Morari, M.; Ray W. H. AIChE J . 1983, 29, 632.
Yu, C. C.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 498.
Cheng-Ching Yu, William L. Luyben* Process Modeling and Control Center Department of Chemical Engineering Lehigh University Bethlehem, Pennsylvania 18015
Greek Symbols aij = ijth element of IRA
Received f o r review October 7, 1985 Accepted December 29, 1986
Oij = ijth element of RGA
Dissociation Extractive Crystallization The principle of dissociation extraction has been adopted to create a crystalline phase to realize separations in difficult systems. This somewhat novel strategy is illustrated with a study of separation of close boiling point mixtures of substituted anilines, N-substituted anilines, and substituted piperazines by selective crystallization with aromatic sulfonic acids. Very high values of separation factor were also obtained for the separation of p-cresol/2,6-xylenol by crystallization with piperazine. The separation of close boiling, acidic/ basic, isomeric/nonisomeric, organic compounds by dissociation extraction has received considerable attention in the last 15 years (Anwar et al., 1971a,b, 1973,1974,1979; Gaikar and Sharma, 1985a-c; Jagirdar and Sharma, 1980, 1981a,b; Wadekar and Sharma, 1981a-c). This two-phase method exploits the difference between the dissociation constants and the difference between the distribution coefficients of the components of the mixture. A single-stage dissociation extraction involves equilibrating a mixture of organic acids (or bases) dissolved in a suitable water-immiscible solvent with an aqueous phase containing the
neutralizing agent in stoichiometric deficient amount; i.e., the amount of neutralizing agent is just sufficient to neutralize the stronger component of the mixture. The competition for the neutralizing agent between components of the mixture in the aqueous phase leads to preferential neutralization of the stronger component. This results in enrichment of the aqueous phase by the stronger acid (or base) in the form of the pertinent salt while the organic phase gets enriched in the weaker acid (or base). Jagirdar and Sharma (1981b) have extended this method to the gas-liquid mode to separate substituted anilines, where anhydrous HC1 gas was directly introduced into a
0888-5885/87/ 2626-1045$01.50/0 0 1987 American Chemical Society
