The assistance of the Washington LTIiiversity Computing Facility through National Science Foundation Grant G22296 is gratefully acknowledged. Nomenclature
See the previous paper by Yuan, et al. (1972). literature Cited
Brown, I., Aust. J . Sci. Res., Ser. A 5, 530 (1952). Leland, T. W., Chappelear, P. S., Gamson, B. W., A.I.Ch.E. J . 8 , 482 (1962).
Null, H. It., “Phase Equilibrium in Process Design,” Wiley, New York, N. Y., 1970. palmer, D, A., D.+. ~ i ~ ~ Washington ~ ~ t ~ University, t i ~ ~st., Louis, Missouri, 1971. Renon. H.. Prausnitz. J. R‘I.. A.I.Ch.E. J . 14, 135 (1968). Stookey, D. J., Smith, B. D., paper presenied at‘the Houston meeting of the A.1.Ch.E. in March 1971. Thompson, W. H., Ph.D. Dissertation, Pennsylvania State University, University Park, Pa., 1966. Werner, G., Schuberth, H., J . Prakt. Chem. 31(5-6), 225 (1966). Wheeler. J. D.. Smith. R. D.. A.I.Ch.E. J . 13. 303 (1967). Yuan, W. LC:, Palmer, D. A., Smith, B. D.,‘IND.ENQ.’CHEM., FUNDAM. 11, 387 (1972). RECEIVED for Review October 1, 1971 ACCEPTED June 2, 1972
A Root-Locus interpretation of Modal Control B. R. Howarth,’ E. A. Grens 11,” and A, S. Foss Department oj Chemical Engineering, University of California, Berkeley, Calij. 94730
Modal control for single modes of linear, lumped-parameter processes i s examined from the point of view of root-locus methods. Such control is equivalent to exact pole-zero cancellation. Sensitivity of the controlled process to parameter variations may therefore be high. However, this i s a significant effect only when the gain i s chosen such that ihe closed-loop eigenvalue of the controlled mode i s nearly equal to that of some other mode. At other gain values the behavior of the single mode controller i s quite insensitive to large perturbations in controller parameters. Such observations suggest that control systems with restricted measurement and manipulation may achieve results for control of a single mode approaching those available with an ideal modal controller.
T h e concept of modal control for chemical processes seems to have been first advanced by Rosenbrock (1962). In its original form, this approach is very attractive from a theoretical viewpoint. The reduction of overall process response times by action on the decay times of the modes and the noninteraction of the control loops, together with the simplicity of the control law, have tended to overshadow the practical implications of the large number of measured and manipulated variables required for implementation. While these requirements for measurement and manipulation restrict the possibilities of application of modal control, nevertheless a better understanding of the effects of modal control may assist the design of more practicable controllers based on the principles of modal analysis. By consideration of the special case of control of only one mode, we have found that the behavior of modal controllers can be interpreted in terms of root-locus theory. This interpretation derives from an analysis of the structure of the characteristic equation of the closed-loop system; it is based on the observation that modal controllers achieve their effect through pole-zero cancellation. Thus a high sensitivity of these controllers to variations or uncertainties in their parameters is to be expected and can be investigated through root-locus techniques. It is possible to extend this treatment of single-mode control to the control of several modes in a qualitative way, but the conclusions that can be drawn from the more general case are not so clear. Present address, Honeywell Pty. Limited, Sydney, Australia.
State Matrix for the Closed-Loop System
For small deviations from some steady state, the response of many chemical processes can be represented by linear, lumped-parameter models. Then the process dynamics can be represented by the so-called state equations
X=A.X+Bu
y
=
cx
(1)
(2)
where A is the constant state matrix for the uncontrolled system, B is the constant control-effect matrix, C is the output matrix, u is the control vector (dimension m), x is the state vector dimension n), and y is the output vector (dimension T ) . Here the state matrix A is assumed to have distinct eigenvalues XI, X2, . , ,, A,, with corresponding right eigenvectors wl,w2,. . . , w,, and left eigenvectors V I , ut, . . . , u,. If W is the matrix with columns wI, j = 1, . . . , n, V the matrix with columns u,,j = 1, . . . , n, and A the diagonal matrix of eigenvalues, then
A
= WAVT
(3)
For a system of this type, the modes are defined to be the components of the dynamic response of the system referred to the right eigenvectors as basis vectors; they have decay times equal to the reciprocals of the real parts of corresponding eigenvalues. Thus, the free response of the Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
403
system can be represented by the superposition of the activations of these modes (Zadeh and Desoer, 1963) n
,EeX'twiviTx(0)
x(t) =
(4)
2=1
I n the direct application of modal control it is necessary that both matrices B and C be square and of rank n, that is, that there be as many measured variables and as many manipulated variables as there are state variables. The reduction in response times of the modes is achieved through increases in the magnitudes of the eigenvalues (which have negative signs if the process is stable), obtained with the feedback control law
u
=
B-'WKVTC-'g
(5)
where K is a diagonal matrix. Substitution of (5) and (3) into (1) leads to X = W(A
+ K)VTx
(6)
Thus, if the diagonal elements of K are large and negative, system stability is preserved, and response time is reduced for the individual modes and hence for the system as a whole. For the special case of control of the single mode i, only one element of the matrix E; is nonzero. The control of a single mode implies that the eigenvalue of this controlled mode is real. The state equation for the closed-loop system can then be expressed
k
=
(A
+ k,F&
(7)
wtviT
(8)
where
FI
=
is a dyadic product matrix. Because the columns of such a matrix are scalar multiples of the single vector W I , the dyadic matrix has rank 1. Characteristic Equation of the Closed-loop System
Po(X) is the collection of terms in the expansion that are independent of k ; it must be the characteristic polynomial for the matrix A, and therefore its roots must be the eigenvalues of the open-loop system. The highest power of X that can occur in Pl(X)is n - 1, since all terms in the expansion of P(k,X) that have the form k$(X) involve only (n - 1) elements of the matrix (A - XI); thus Pl(X)can have at most (n - 1) roots. The values of the roots of Pl(X)can be deduced from consideration of (6). When only one element of K is nonzero, that is when only one mode is controlled, then all but one of the eigenvalues remain constant a t values A, for all values of k . Thus P(k,Xj) = Po@,)
+ kPI(X1) = 0
(for all k , for a l l j # i)
(12) where i is the controlled mode, but PO(hj) = 0
( j = 1, 2, . . ., n)
(13)
and therefore Pl(Xj) = 0 (for a l l j # i)
(14)
Thus the roots of P I @ )are the same XI, excluding Xi, that are the eigenvalues of the uncontrolled modes. There are n - 1 of these, so that the polynomial is completely determined, with the roots of Pl(X) identical with n - 1 roots of P&). Root-locus Interpretation
For single-mode control, there is only one term subject to adjustment in the closed loop system, the controller gain. Only one eigenvalue is affected, while those for the uncontrolled modes remain unaltered. Thus interpretation of this case of modal control can be readily accomplished by examination of the effect of variations in a single parameter, the gain, on the roots of the characteristic polynomial of the closed-loop system. It is precisely for this type of problem that root-locus techniques were developed. The polynomial P(k,X) is a function of A with one scalar parameter k. It can be rearranged
The eigenvalues of the controlled system are the roots of the characteristic polynomial in X. For the control of a single mode (mode i) as expressed in (7) det(A
+ kF - XI) = 0
(9)
in which it is understood that k and F refer to the controlled mode. This determinant, which has two scalar variables X and k , can be difficult to evaluate even for the systems of small dimension. However, from consideration of the rules for expansion of determinants it is clear that, in its most general form, the characteristic polynomial can be arranged as a sequence in powers of k , each multiplying a polynomial in X
P(k,X)
=
Po(X)
+ kPl(X) + kZPz(X) + . . . + k"P,(X)
(10)
Consider the terms in the expansion that contain kZ. For each term of the form kZjiJpg$(X)there must be a term -k2fipfpj$(X), since the latter term is different from the first by one permutation. Clearly the sum of these two terms contains a minor of F of order two, the term (fl,fp, - fi4fp,). However, since F is of rank 1 all such minors must be zero, and the polynomial Pz(X) vanishes for the case of single-mode control. A similar argument can be applied to all terms in the sequence that involve higher powers of k . Thus, for this special case, the characteristic polynomial takes the simple form
P(k,X) = det(A
+ k F - XI) = Po(X)+ kPl(X)
404 Ind. Eng. Cham. Fundam., Vol. 1 1 , No. 3, 1972
(11)
In root-locus terminology the roots of Po(A)are the poles of the system while the roots of P I @ ) are the zeros. As k is varied from zero to infinity the roots of P(k,X) shift from their initial coincidence with the roots of P&) to become the roots of Pl(X),with the exception of one root which becomes infinite. However, the roots of Pl(X)are the same as n - 1 roots of P&), so that the closed-loop eigenvalues corresponding to these roots do not shift; in root-locus terms there is pole-zero cancellation. This cancellation effect can form the basis for an interpretation of modal control. Zadeh and Desoer (1963) give an interesting discussion of the relation between pole-zero cancellation, controllabilityobservability, and modal decomposition. They show that pole-zero cancellation occurs when a mode is either uncontrollable or unobservable. When, as in the present case, only a single mode is controlled, it can be easily verified that all modes except the controlled mode are both uncontrolled and unobserved. Effect of System Perturbations
Pole-zero cancellation is generally considered unfavorable in control system design, because it makes the controlled
system sensitive to parameter variations in the controller and controlled process. This is more particularly the case if one of the “cmceled” poles is unstable. If exact cancellation occurs for an unstable pole, then the process output appears stable. However, even a small parameter variation can destroy the cancellation, with the output reflecting the instability of the system. This sensitivity is encountered in modal control, and the effects are accentuated by inaccuracies inherent in application of the technique. The state matrix for any real system cannot be identified with perfect accuracy, even if the process should be truly linear. Moreover, the eigenvectors used to define matrix F can be determined only with limited precision, so that while the feedback matrix F used in the controller may be nearly correct, it cannot be expected to have the elements required to give exact pole-zero cancellation for the actual process. An inability to measure or manipulate certain variables would also be equivalent to a mjaor perturbation of the ideal controller. The effect of small inaccuracies in the vectors used to generate the controller feed-back matrix can best be seen by consideration of the variation of eigenvalues with gain. In the case studied as an example, which is based on a binary distillation column, it is assumed that the process is invariant and that small perturbations are applied to elements of the vectors defining F in the controller. This matrix retains its unity rank, but the zeros of the characteristic equation shift relative to the poles, which, being defined by the open-loop system, are fixed. The eigenvalue-gain relation in the ideal case ( t e . , exact cancellation) is shown in Figure l a . The controlled mode eigenvalue appears to cross the constant eigenvalues of the uncontrolled modes as gain is made larger, and the slope of this curve is unity. Small perturbations in elements of the feed-back matrix produce attendant perturbations in the magnitude of the system zeros. For the simple case of two zeros considered in Figure la, they may each increase or decrease in magnitude independently. Thus for this case there are four different situations: both zeros increase, the first increases while the second decreases, the second increases while the first decreases, and both decrease. The situations where the zeros are perturbed in the same direction are basically different from those where they change in opposite directions. The situation where both zeros have decreased in magnitude as the result of perturbations is represented in Figure lb. It can be seen that the eigenvalues remain real for all values of gain but that the controlled-mode eigenvalue no longer appears to cross those of the uncontrolled modes. Instead, a characteristic pattern of “approach and divergence” can be observed. This behavior concentrates the effects of the zero perturbations to regions of gain values near those which produce a crossing with exact cancellation. The situation where the controller perturbations induce one zero to increase in magnitude and the other to decrease is considered in Figure IC.The eigenvalue-gain curves are again changed in character, but now rather than an “approach and divergence” pattern there is the formation of a complex conjugate pair of eigenvalues over a limited range of values of gain. This range of gains includes the location of a crossing point for the ideal case. In these figures it can be observed that perturbations have their most significant effect on the eigenvalue-gain relationship for gain values in the region of those a t which crossing points are found in the ideal case. For gain values well removed from these regions there is little difference in eigenvalue
L 0
I
G A I N (-k)
-x COMPLEX EIGENVALUE
X=POLE
r
J
GAIN (-k)
0 =ZERO
Figure 1 . Effects of perturbations upon the eigenvalue-gain behavior of a single mode controller: a, ideal case; b, two zeros perturbed in direction of controlled mode pole; c, two zeros perturbed in opposite directions
-A
0
GArN (-k)
Figure 2. Eigenvalue-gain behavior of a modal controller for a single mode very slightly perturbed from the ideal (pole-zero cancellation) condition
behavior among the three cases. The extent of deviations of the eigenvalue-gain curves from those for the ideal case is related to the magnitude of the perturbations in the controller. The eigenvalue-gain behavior observed for these perturbed cases leads to an interesting view of ideal control of a single mode. Here the apparent crossing of successive uncontrolled eigenvalues by the controlled eigenvalue can also be interpreted as successive shifting of the eigenvalues, each to the value for the next faster mode; for this ideal case, whether there is “crossing” or “shifting” is a moot point. However, in any implementation, small deviations from ideal parameters in the controller will result in behavior as described previously, and a successive shifting of modes, as shown in Figure 2, will occur. Thus this latter interpretation would appear preferable also for the ideal case. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
405
Extension to Multi-Mode Control
The above interpretation has been based on examination of single-mode control. This is the simplest case for study and is the only one to which root-locus techniques apply directly. With multi-mode control the gains for each mode are independent: root-locus techniques only apply where there is only one independent scalar variable. Although such multimode control cannot be subjected to the same detailed analysis that has been presented for the single mode case, the same basic considerations apply in a general way. If two modes are to be controlled, the state equation for the closed-loop system can be written in the same way as (7). It becomes X = (A
+ klFl + k z F 2 ) ~
From this equation, the characteristic equation can be derived
P(h,b,X)
=
Poh)
+ kiPi(X) + kzPz(X) + kikzPiz(X)
(17)
Pl(X) and P2(X) are both polynomials of degree (n - 1); PI2(X)is of degree (n - 2). All of these polynomials have as their roots eigenvalues of the open-loop system. Similar relations apply for control of three or more modes, with increasing complexity in the terms involving products of gains. Conclusions generally similar to those described previously can be drawn concerning the effects of controller perturbations. These effects can be expected to be even more pronounced in these cases because of interactive effects among the control loops.
relation is not particularly sensitive to controller parameter variations. One potentially advantageous parameter variation is that resulting from elimination of measured and manipulated variables, since the requirement for large numbers of such variables is one of the principal obstacles to modal control of chemical processes. Much the same eigenvalue-gain behavior could be expected for controllers with reduced numbers of measured and manipulated variables if the vectors that comprise the dyadic product control matrix F were selected such that zeros be located close to poles. Such modal controllers could be more easily implemented because of their greatly reduced requirements for measurement and manipulation. At least for single-mode controllers they would have the additional advantage that they would be less sensitive to parameter variations than true modal controllers, because of the less stringent requirements on placement of the zeros (Howarth, 1970, 1972). This limited intepretation of the special case of singlemode control by use of root-locus techniques has given some insight into the question of “how modal control works” when applied to systems typical of many chemical processes, and into the effects of parameter variations on system performance. While consideration of only single-mode control might appear to be a serious limitation, it is of great importance in chemical process control where the modes often have widely separated time constants. The analysis indicates that it may be possible to design controllers using only a few measured and manipulated variables which give nearly the same behavior as ideal modal controllers. Acknowledgement
Discussion and Conclusions
Two interesting observations can be made on the basis of this analysis, First, it is clear that with modal control the strongest effects of parameter variations are observed when gain values are such that two or more eigenvalues in the closed-loop system are approximately the same. Thus it would appear unwise to use such gain values in a modal controller; the effect of parameter uncertainties would be such as to make controller performance unpredictable. Secondly, in view of these restrictions on gain settings, it appears that it is not necessary to use true modal controllers to gain useful improvements in process performance. For gains significantly different from those corresponding to eigenvalue “crossing” in the ideal case, the eigenvaluegain
406 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for partial support of this research. literature Cited
Howarth, B. R., Ph.D. Thesis, University of California, Berkeley, Calif., 1970. Howarth. B. R., Foss, A. S., Grens, E. A., IND. ENG.CHEM. F U N D ~inM press . (1972). Rosenbrock, H. H., Chem. Eng. Progr. 58 (9), 43 (1962). Zadeh, L. A., Desoer, C;, A., “Linear Systems Theory: The State Space Approach, McGraw-Hill, New York, N. Y., 1963. RECEIVED for review September 20, 1971 ACCEPTEDApril 27, 1972
