Mn04-
+ 2Mn2++ 2Hz0
-
Znd. Eng. Chem. Res. 1987, 26, 2151-2157
2MnO(OH)z + Mn3+
+ + 2Mn3+
Mn2+ Mn4+
MKI(OH)~ MnO(OH)z
HzO
MnOz + 2Hz0
This yields an initial zero-order decrease of Mn04- and a concomitant formation of Mn2+. As the reaction proceeds in acidic solution, the formation of solid MnOz starts slowly, initially at the expense of MnO(OH)z,and accelerates later on. We believe that this sol (solid MnOJ formation is the reason why there is an apparent increase in the MnO, peak, as well as in all other peaks in the range X = 200-570 nm for the acidic solutions. The scattered radiation due to Tyndall effects gives the appearance of increasing concentrations for all absorbing species. The above description of the Mn04- reaction is a rather general one. However, it does explain the rather complicated assortment of reaction products in neutral, basic, and acidic solutions. Furthermore, it is consistent with many previously published results concerning the reaction
2151
of Mn04- ion and provides a starting point for the interpretation of experimental results within a broad spectrum of starting conditions. Registry No. Mn04-, 14333-13-2; H202,7722-84-1.
Literature Cited Hamilton, J. J. J . Am. Water Works Assoc. 1974, 66, 734. McDonald, H. 0. Diss Abstr. 1961, 21, 454. Obuchi, A.; Okuwaki, A.; Okabe, T. Nippon Kagaku Kaishi 1974, 1425. Prosselt, H. S.; Reidies, A. H. Znd. Eng. Chen. Prod. Res. Deu. 1965, 4 , 48. Shafirovich, V. Ya; Shilov, A. E. Kinet. Katal. 1978,19(4),877-883. Shafirovich, V. Ya; Khannamov, N. K.; Shilov, A. E. Xinet. Katal. 1978,19(6) 1498-1501. Symons, M. C . R. J. Chem. SOC.1953,3956. Wiberg, K. B.; Gear, R. D. J. Am. Chem. Soc. 1966, 88(24), 5827. Zimmerman, G. J. Chem. Phys. 1955,23(5), 825. Received f o r review February 10, 1986 Revised manuscript received June 29, 1987 Accepted July 27, 1987
Lumping Strategy. 2. A System Theoretic Approach Pamela G. Coxson+ Department of Mathematics, The Ohio S t a t e University, Columbus, Ohio 43210
Kenneth B. Bischoff* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Lumping analysis is placed in the context of linear systems theory, providing simpler and new ways of looking a t the problems, as well as justification to the strategies used in part 1. In particular, the relationships between lumpability of a kinetic scheme and the concepts of observability, controllability, and minimum realization of a system are developed. This also leads to the use of a generalized inverse to find the new lumped rate coefficient matrix from the original full matrix for those applications where this is known and a reduced order representation is desired. Finally, the theorems provide useful guides in identifying appropriate lumping schemes. 1. Introduction It is common practice to view similar chemical species as a single lump or pseudospecies for the purpose of formulating a relatively simple model of a complex reaction network. This approach has worked well in industry (Weekman, 1979) and is supported, in the case of monomolecular reactions, by the theoretical works of Wei and Kuo (1969), Kuo and Wei (1969), Hutchinson and Luss (19701, and Ozawa (1973) and reviewed by Bischoff and Coxson (1987). In part 1 (Coxson and Bischoff, 1987), we developed a systematic method for determining appropriate lumping schemes from experimental data. In this paper, we place lumping analysis in the context of linear systems theory. This approach provides considerable simplification of many known results and fresh insight into some old problems, and it permits a variety of diverse results to be housed under the same roof. Finally, systems theory provides motivation and justification for the lumping strategies presented in part 1. A summary of results from linear systems theory is given in section 2 and those of lumping analysis in section 3. In section 4, the two are combined. The results which emerge are applied to the practical problems of lumping in the remainder of the paper. 'Current address: Member of Technical Staff a t the Aerospace Corporation, Los Angeles, CA 90009-2957.
0888-5885/87/2626-2151$01.50/0
2. Linear Systems A brief introduction to the relevant results of linear systems theory is given below. For further details, there are numerous sources varying in level and emphasis (see, for example, Luenberger (1979), Brockett (1970), or Ray (1981)). The reader familiar with the concepts of controllability, observability, and realization might omit this section. A linear dynamic system is given by a system of linear difference (for a discrete system-DS) or differential (for a continuous system-CS) equations forced by inputs u and observed via the measured values y = Mx: x(h + 1) = Gx(h) + c u ( h ) x(hJ = xo y(h) = Mx(h) k(t) = Kx(t) + Lu(t) x(to) =
(DS) XO
y(t) = Mx(t) (CS) where the value of x at a particular time lies in the state space X ( = Y P ) ,u takes values in an input space U (=Yl'), and y lies in the space of measured values M X . If (DS) is a discretized (sample) version of (CS) with x ( h ) = x(to h7), then G and K are related by G = eK7.The systems (DS) and (CS) aboye are completely determined by the ordered triples ^(G,L,M)and (K,L,M),respectively. A system (G,L,M) or (K,L,M) is n-dimensional if X i s an n-dimensional vector space and G (respectively, K), L,
+
0 1987 American Chemical Society
2152
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
and M are matrices. It is time invariant if the matrices are constant. For an n-dimensional time invariant system, the outputs y are related to the inputs u by h-1
y(h) = MGhx0+ CMGh-'-'Lu(i) (with ho = 0) i=O
in the discrete time case (DS) or
+ LtMeK(t-s)Lu(s)ds
y(t) = MeKtxo
(with t o = 0)
in the continuous system (CS). The expressions above suggest that the crucial factor in the input/output relationship is the sequence (MGhL\E,o. For the discrete system, this sequence represents the measured impulse response of the system. The ( i d element of MGhL is the ith component of the output y ( h + 1)if the system, starting at x(0) = 0, is subjected to a unit input in the j t h component of u , i.e., u(0) = (0,...,0,1,0,...,0 ) T j
In a continuous time system, the impulse response is given by MeKtL,which is related to the sequence {MKhL]by the series expansion of eK. We will want to recall later that the response sequence fMGhL}or (MKhLJrepresents data which could be obtained from experiments, independent of knowledge of M, G or K, and L. For much of our subsequent analysis, it will be irrelevant to know whether a triple (A,B,C)represents a discrete time or a continuous time system. Often results will be easier to interpret if viewed in terms of the discrete formulation, so we will favor the discrete case at such times. Whenever it is important to distinguish this, we will indicate why. A major focus of systems theory is the identification of minimal system representations. There are many triples (A,B,C) which are equivalent to a particular system (G,L,M) in the sense tha; they have the same response sequence, (CAhB)= MGhLfor each h. To the investigator who sees only the inputs and outputs, equivalent systems are indistinguishable. A minimal system realization for a given finite dimensional system is a triple (A,B,C) which has the smallest dimension among all of the equivalent representations. All minimal realizations of a system are the same, up to a change of basis (e.g., reordering or scaling of the state components). A minimal system may seem less interesting than a system representation where the state space is chosen to correspond to identifiable physical entities, but it can be valuable to know when a lower order representation exists. If the original representation is not minimal, then part of the system model is not contributing to the input/output (transfer function) relationship and is therefore superfluous in the model. Since the state is not observed directly, but only through the measured values y = M x , it is possible that part of the system is superfluous b_ecause it is never seen. Consider a known system (G,L,M), where X is a space of compositions of n species and M = (l,O,O, ...,0). In this case, only the first species of the composition is measured. It is possible but does not necessarily follow that the remaining species are not "observed". Suppose we began with an unknown composition xo and did not add any material to the system (i.e., u (h) = 0 for all h). Our first n measurements are given by ~ ( 0=) Mx,
~ ( 1 =) MGx,
Because we have measured only one species, the equations above are scalar equations. There are n linear equations and n unknowns (the n components of xo).M and G are known, and the left-hand side is given by measurements. From linear algebra, we know that the unknown values can be uniquely determined if and only if the n X n matrix of coefficients has rank n. Here the matrix of coefficients is
[ G I MG"-'
Even though only one species is measured, if the coefficient matrix has full rank we can eventually (after n measurements) determine or "observe" the values of all species. If the coefficient matrix does not have rank n, then some initial states are not observed. Example 1. Consider the reaction AI A2 with kinetics
- -
where x(t)= (x,(t), ~ , ( t ) xi(t) ) ~ , = mass of species Ai, and with measurements y ( t ) = (1 O)x(t),(Le., y(t) = x l ( t ) ) . If x , and u1 denote the first components of x and u , then y ( t )= e+xl(0) + ~ t e s - t u l ( ds s) 0
The second component, x z ( t ) , has no effect on this measurement. The external observer cannot know that there is a second state component. This suggests that xz can be omitted from the model, leaving a one-dimensionalsystem: Z(t)
= -z(t)
+ (1 O)u(t)
Y(t) = z(t)
where z ( t ) = x l ( t ) . The response sequence, {MKZhL), is l(-l)h(l0) or ((-l)h0). We compare this to the response sequence of the two-dimensional system,
which is just ((-lIh 0), the same as the smaller system. Thus, the two systems are equivalent. The measured values, y , corresponding to a given input function, u , are identical for the two systems. The new state space, 2,is the space of values of species one. In example 1, the measured value, y = x,, does not reflect changes in the rest of the system. If we had measured the second species, that is if M = (0 l),then the measurement would reflect changes in species one since x 2 ( t ) = e%JO) + te+xl(0) (for u ( t ) = 0). The amount of A, can be inferred from changes in the amount of A2: x l ( t ) = (xz(t) - etxz(0))/t. In this case, it is not possible to find a smaller equivalent system representation (see the discussion following theorem 1). A state, x(O),which does not affect the measured values, y , at any time is said to be an unobservable state. Since
cs:
y ( t ) = MeKtx(0)
(with u(t) = 0)
DS:
y ( h ) = MGhx(0)
(with u(h) = 0)
x ( 0 ) is unobservable if and only if MGhx(0) = 0 (respectively, MKhx(0) = 0) for each h 1 0. In example 1,we saw that x = (0 a)T was an unobservable state for any a. The unobservable states form a subspace of X . If the system
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2153
(K,L,M) has no non-zero states which are unobservable, it is said to be completely observable. A necessary and sufficient criterion for complete observability is given by the following well-known result (see Luenberger (1979) for example). We denote the state matrix by F, representing either a continuous time rate matrix, K,or a discrete time system matrix, G. Theorem 1. The n-dimensional time invariant linear system (F,L,M) is completely observable if and only if the observability matrix, 0,has rank n, where
consider all of the species independently. An alternative is to group similar species together in a lump, as discussed in part 1. The composition, z , of the new lumped pseudospecies is related to the original species, x,by a linear transformation, z = M x . The composition, z , is a proper lumping of x if M has full rank and each column is a canonical unit vector (e.g., (0 1 O)T). Proper lumping implies that each species is assigned to one and only one lump. A reaction system with system matrix G is exactly lumpable by M if the lumped variables M x also satisfy a linear reaction scheme, i.e., if there exists G, such that z(h + 1) = G,z(h) z(h) = Mx(h)
+
This is a purely algebraic criterion depending only on the two matrices, M and F. For the two-dimensional system of example, 1, we note that
.=[' -1
O] 0
has rank one. If M = (0 l ) , the observability matrix is
-:I which has rank two. Thus, measurement of the second species alone results in a system which is completely observable, as we showed. Analogous results can be given for controllability, reflecting the ability of the system inputs, u , to alter the state. An n-dimensional system, (F,L,M), is completely controllable if the controllability matrix, 6 = (L,FL,F2L,...,Fn-lL), has rank n. For the purpose of this discussion, we will be interested primarily in systems for which L has rank n (e.g., L = In),so that controllability is immediate. A system can fail to be minimal because of a deficiency of observability or of controllability. In fact, minimal systems admit the characterization in Theorem 2, e.g., Brockett (1970). Theorem 2. An n-dimensional time invariant linear system, (F,L,M),is minimal if and only if it is completely observable and completely controllable. Algorithms for constructing minimal order realizations of a system, (F,L,M),have been proposed by Kalman and Ho (Kalman et al., 1969), Silverman (1971), and others. These algorithms utilize the semiinfinite block Hankel matrix:
F; Ho
% =
Hi
By use of the kinetics of x,z ( h + 1) = Mx(h 1) = MGx(h), and by exact lumpability z(h + 1) = G,z(h) = G,Mx (h). It follows that the system is exactly lumpable by M if and only if there exists G, such that MG = G,M. Although we have focused on the discrete form for the kinetic equation, the criterion for exact lumpability of continuous time systems is the same, MK = K,M. For our purposes, the following characterization (Coxson, 1984) will be useful. Theorem 3. A finite n-dimensional, time-invariant, linear kinetic system with system matrix F (discrete or continuous) is exactly lumpable by the lumping matrix, M, if and only if rank
where H k = MFkL. We have noted previously that the matrices Hk represent measured values of the system output which can be determined experimentally without knowledge of M, F, and L. 3. Lumping Analysis: Exact Lumpability Consider the reaction system given (in discrete form) by x(h + 1) = Gx(h) x(0) = x,,
When the number of species in the composition vector, x, is very large, it may be neither practical nor desirable to
= rank M
Proof. The system is exactly lumpable if and only if there exists F, such that MF = F,M. This equation expresses each row of MF as a linear combination of the rows of M, the combination given by the corresponding row of F,. Therefore, appending rows of MF to M does not increase the rank. Criterion 3-1 provides a test for exact lumpability which is particularly simple to evaluate for proper lumping matrices, M. We need only observe whether or not the rows of MF are linear combinations of the rows of M. To illustrate its simplicity, consider the following examples. The first two are systems studied by Wei and Kuo (1969) and Ozawa (1973), showing one lumping scheme that is exact and one that is not. The third illustrates the principle for a discrete system. Example 2.
4
-122
K = 10
M=
A[ : r]
10 -10
MK =[-lo 10
HZ
": ": i:]
kF]
-10 17 10 -10
The system is exactly lumpable since the rows of MK are clearly linear combinations of the rows of M; i.e., they have the format (u u u). Example 3. K =
[:a "1 -1; 10 -10
= [ 01 10 11 0
Here, the rows of MK clearly cannot be made from linear combinations of the rows of M since they do not have the form (u u u). Therefore this lumping is not exact. Example 4. The previous two examples used rate matrices from continuous time reaction systems. In this example, we consider the discrete time system, made from
2154 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
the K matrix by G = e K r ,7 = 0.05. G=
(i) M = (iii M =
I
I
0.55700 0.08463 0.12642 0.12694 0.59931 0.i8964 0.31606 0.931606 0.68394
[
[AI, y] [i " 1'
1 1
0.68394 0.68394 0.31606 M G = 0.31606 0.31606 0.68394
M G = [ 0.55700 0.08463 0.12642 0.44300 0.91537 0.87358
(,
The system is exactly lumpable by scheme i since M and MG are both of the form (u u u ) , but in scheme ii this is not the case. This is expected here, of course, since the continuous and discrete representations are entirely equivalent. Another reason for emphasizing criterion 3-1 is that it serves as a link to linear systems theory, which will be explored in the next section. 4. A System Theoretic Context The n-species reaction system can be embedded in a linear system in a very simple and natural way. Rather than thinking of the lumped variables as new pseudospecies (state variables), we first view them as measurements of the original species. This viewpoint is consistent with approaches suggested by Liu and Lapidus (1973) and Aoki (1968). It is also intuitively satisfying, since by lumping we are indeed deciding how the species of the system will be measured. The system we consider is (F,In,M),where F is the underlying system matrix, M is an m X n lumping matrix, and I, is the n X n identity matrix: x(h + 1) = Fx(h) I,u(h)
+
~ ( h=)Mx(h) The lumpability criterion 3-1 and the criterion for observability of the system (F,In,M) have a relationship which is made explicit in the following theorem (see also Coxson (1984)). Theorem 4. An n-species linear reaction system is exactly lumpable by M if and only if the rank of the observability matrix for (F,In,M)is equal to the rank of M. Proof. (a) If rank 0 = rank M, then rank
EF]
= rank
M
a fortiori, and the system is exactly lumpable by criterion 3-1. (b) Let E F
1
It is tedious but straightforward to show that if rank Oj = rank Oj+l,then rank Oj = rank Ob for all k 1 j (Luenberger (1979), p 278). Criterion 3-1 requires that rank 0, = rank 02.Therefore, by induction, exact lumpability implies rank M = rank 0. Corollary 1. If rank M is less than n, then the n-dimensional system (F,In,M)cannot be both exactly lumpable and completely observable. Since rank 0 1 rank M for any system (F,I,,,M),theorem 4 implies that exact lumpability is equivalent to minimal observability. This is not suprising since the goals of lumping and observing the system are in opposition. Observability is a measure of the information about state
x which can be determined from the measurements of Mx. We already have full information about M x , since it is measured directly. If rank 0 > rank M, additional information can be recovered. In this paper, our purpose in lumping states together is to simplify the mathematical model and to avoid accumulating information about the individual species. We reduce observability by careful choice of the structure of system measurements. A lumping scheme is exact precisely when the lumped measurements dc not provide any information about the composition within lumps. So far, we have seen that systems theory provides a convenient context and terminology for discussing lumpability. We will show that known results and recent advances in systems theory can be exploited to deal with several important issues in the lumping problem. 5. The Lumped State Realization
Given that an exact lumping matrix, M, has been identified, we show how to find the lumped state realization without knowledge of the original system matrix, F. This is important, since one of the reasons to lump variables is to avoid having to identify the system matrix, F. At this point, we want to recall the observation in section 2 that the sequence (MFhLJrepresents measurements of the system (F,L,M)which can be determined experimentally. When L = I,, this sequence is simply (MFh).Thus, the entries of MFh,h = 0, 1,2, etc., can be measured even if the system matrix, F, is unknown. With this observation, criterion 3-1 and the extended criterion of theorem 4 relate the abstract notion of exact lumpability to experimental measurements. For a proper lumping scheme, theorem 4 says simply that the system is exactly lumpable if and only if separate experiments starting with different pure species from the same lump result in the same measured response of the lumped variables. This principle of invariant response was identified by Wei and Kuo (1969) and was used to establish the 10-lump model for catalytic cracking, as described in Weekman (1979). Although both MKh and MGh can in theory be measured without knowledge of the rate matrix K, in practice MGh is relatively easy to measure and MKh is relatively difficult. Note that x ( t ) = Kx(t), so the hth derivative is ~ ( ~ )=(Khx(t), t ) and thus for z = Mx dh)(t)= MKhx(t) In other words, at each time t , MKh represents the hth derivative of the lumped process. Even the first derivative would be difficult to estimate from measurements. Higher order derivatives are unlikely to be measureable in any practical sense. MGh,on the other hand, represents direct measurements of the lumped species composition, sampled at equal intervals over time. Therefore, although the theoretical results for the continuous and discrete representations are essentially parallel, the discrete representation was favored in our discussion of lumping strategies in Coxson and Bischoff (1987) and below in section 6. If the system (F,In,M)is exactly lumpable, then it is not observable (corollary 1) and therefore is not minimal (theorem 2). A lumped state realization is determined by the identity MF = F,M as follows. Theorem 5. If the m x n matrix, M, of full rank m and the n X n matrix, F , satisfy MF = F,M, then the m-dimensional system (Fz,M,Im) is a minimal realization of the n-dimensional system (F,In,M). Proof. The response sequence of (Fz,M,Im)is (I,F,hM} = (F,hM). Since F,M = MF, FZhM= F,h-l(F,M) = F,h-lMF, and hence (F,hM)= (MFh},which is the response sequence for (F,I,,M). Thus, the two systems are equiv-
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2155 alent. The m X nm controllability matrix for (F,,M,I,) is (M,F,M,...,FZm-'M),which has full rank equal to m since M does. The nm X m observability matrix is
TI,
1
which has rank m since I, has rank m. By theorem 2, (Fz,M,Im)is a minimal realization. If the m X n (m 5 n) matrix, M, has full rank, then MF = F,M implies F, = MFM+, where M+ is the MoorePenrose pseudoinverse of M. Briefly, the Moore-Penrose pseudoinverse is the matrix M+ w'hich satisfies (i) MM+M = M, (ii) M+MM+ = M+, (iii) (MM+)T= MM+, and (iv) (M+M)T= M+M. In general, the numerical problems involved in computing the Moore-Penrose inverse are similar to those encountered in computing an inverse. Indeed, when M is a square, nonsingular matrix, M+ = M-l. For an m X n matrix, m < n, of full rank, M+ = MT(MMT)-l,again involving an inverse. However, if M is a proper lumping matrix, M+ has the following explicit representation: M+ = MT diag [1/n1,1/n2, ...,l/n,] (5-1) where n; is the number of species in the ith lump. This follows from the observation that, for a proper lumping matrix, MMT = diag [n1,n2,...,n,]. Example 5. M=
1 1 0 0 11
[o
For K of examples 2 and 3,
This is the same result found in part 1 by another route. Theorem 5 and the remarks above show that the lumped realization, F,, of a system which is exactly lumpable by a given proper lumping matrix, M, can be easily detercv mined from perfect measurements, MF = MF. When the rv
measurements, MF of MF, are subject to error, the measured values are given by rv MF=MF+N where N = [nij] is the error matrix. Assuming that the entries in N are random variables with zero mean (E(N) =
O),
rv
MFM+ = MFM+
+ NM+
= E(MFM+ + NM+) = MFM+ E ( x ) denotes the expected or mean value of x . If the covariance matrix of each column, Nj of N, is the identity, E(NjNjT) = I,, indicating that the measurement errors are not correlated, then the entries of the measured rv MFM+ are minimum variance unbiased estimates of the desired values of the true MFM+ (see Luenberger (1969), Chapter 4). This is not surprising in the case of proper lumping, since postmultiplying by M+ simply takes the
average of the columns of MF which should be equal by invariant response. Conservation of Mass. The lumped system matrix, F,, inherits eigenvalues and eigenvectors from F , in the following sense. If vi is an eigenvector of F with associated eigenvalue A, then either Mvi is zero or Mvi is an eigenvector of F, associated with eigenvalue A,. If F is diagonalizable and M has full rank, then every eigenvalue-eigenvector pair of F, is obtained in this fashion. If M is a proper lumping matrix, then the lumped system also inherits the property which corresponds to conservation of mass. That is, 1 .K = 0 for (CS); 1 .G = 1 for (DS), where 1 denotes the appropriate size row vector (1,1,...,1). Theorem 6. Let M be an m X n proper lumping matrix. If the n X n matrix, F, satisfies 1 .F = 0 or 1 OF= 1 , then 1 .MFM+ = 0 or 1 .MFM+ = 1 , respectively. Proof. From the definition of a proper lumping matrix, 1 -M = 1 . The construction of M+ (eq 5-1) shows clearly that 1 .M+ = 1. The result follows directly when these two relations are utilized in the evaluation of 1 -MFM+: continuous case: 1.K = 0
l*MKM+= 1*KM+ = O.M+
=o
discrete case:
1-G = 1 l.MGM+ = 1*GM+ = 1.M+ =1 6. Finding an Appropriate Lumping Scheme The lumped state realization described in section 5 assumed that a proper lumping matrix, M, was given. In practice, finding an appropriate lumping scheme and its corresponding matrix, M, is the objective of lumping analysis. In part 1,we outlined a systematic procedure for selecting a lumping scheme based on discrete time (sampled) experimental or simulated data. The procedure is summarized below. (1)From basic chemical principles and the process objectives, determine the coarsest lumping scheme, M,, that could possibly describe the essential features of the system. For the catalytic cracking model of part 1, this was the division of species into feedstock (gas oil), desired products (gasoline-range species), and undesired products (the Clump). There is no reason to expect that such a coarse scheme will be exact or even nearly so. (2) From experimental data for the system (G,In,Mc), construct the impulse response matrix: E:G
1
E($FM+)
The jth column of M,Gh contains the coarsely lumped measurements following an initial composition of pure species j, measured at the end of the hth sampling interval. (3) An exact (or nearly exact) proper lumping matrix, M, is determined by lumping together species which correspond to identical (or nearly identical) columns of R(c). Justification of step 3 is provided by theorems 7 and 8 below. The first result shows that any exact lumping
2156
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
Chart I 1.0000 0.0000
1.0000 0.0000
1.0000 0.0000
1.0000 0.0000
1.0000 0.0000
1.0000 0.0000
1.OW0 0.0000
1.moo 0.0000
0.6197 0.380.'1
0.4843 0.5157
0.6915 0.3085
1.000@ 0.0000
0.8018 0.1982
0.3156 0.6844
0.6331 0.3669
1.0000 0.0000
0.6682 0.3318
0.8378 0.1622 0.7147 0.2853
1.0000 0.0000
0.4480 0.5520 0.3736 0.6264
0.2703 0.7297
0.6248 0.3752
1.0000 0.0000
0.5806 0.4194
0.5446 0.4554 0.3477 0.6523 0.2728 0.7272
0.6228 0.3772
0.3462 0.6538
0.2709 0.7291
0.6280 0.3720
1.0000 0.0000
0.5257 0.4743
0.2551 0.7449
0.3422 0.6578
0.2890 0.7110
0.6348 0.3652
1.0000 0.0000
0.4937 0.5063
0.3505 0.6495
0.3138 0.6862
0.6434 0.3566
0.3654 0.6346 0.3839 0.6161
0.3406 0.6594
0.6530 0.3470
1.0000 0.0000 1.0000 0.0000
0.3678 0.6322
0.6634 0.3366
0.4043 0.3957
0.3944 0.6056
0.6743 0.3257
omoo
1.0000 0.0000 0.0430 0.9570
1.0000 0.0000 1.0000 0.0000
1.0000 0.0000
1.0000 0.0000
0.0842 0.9158 0.1237 0.8763
1.0000 0.0000
0.5555 0.4445
1.0000 0.0000
0.1614 0.8386
1.0000 0.0000
0.5077 0.4923 0.4752 0.5248
1.0000 0.0000
0.1975 0.8025
1.0000 0.0000
0.4780 0.5220
0.2639 0.7361 0.2845 0.7155
1.0000 0.0000
0.2320 0.7680
1.0000 0.0000
0.4736 0.5264
0.3100 0.6900
0.4548 0.5452
i.0000 0.0000
0.2651 0.7349
1.0000 0.0000
1.0000 0.0000
0.4770 0.5230
0.3372 0.6628
0.4438 0.5562
1.0000 0.0000
0.2967 0.7033
1.0000 0.0000
1.u000
0.4857 0.5143
0.3645 0.6355
0.4402 0.5598
1.0000 0.0000
0.3270 0.6730
1.0000 0.0000
scheme which is a refinement of the coarse scheme, M,, will leave a pattern in the coarse system response matrix, R(c). Specifically, species which can be lumped exactly will have identical columns in R(c). The second result is essentially the converse. It states that sets of identical columns in R(c) can indicate the existence of an exact lumping scheme which lumps the species corresponding to identical columns. Theorem 7. If the linear system with n X n system matrix, F, is exactly lumpable by the m X n proper lumping matrix, M, with response matrix, R, and if the m, X n proper lumping matrix, M,, is a proper lumping of the lumps in M, then the impulse response matrix, R(c), for (F,In,Mc)will have identical columns corresponding to each lump of M. Proof. Consider v with 1 in the ith position, -1 in the jth postion, and 0 elsewhere for any i and j associated with the same lump of M. For such v , R-v = 0 because columns i and j are identical by exact lumpability and therefore they cancel each other in the matrix product. We will show that R.v = 0 implies R(c).v = 0, which says columns i and j of R(c) also are identical. This is the conclusion of theorem 7. The result follows from the observation that M, = PM (with P being a proper lumping matrix), since MFk.v = 0 implies PMFk.v = 0. Thus, if R-v = 0, then R(c)-v= 0, where r M
-I
TM,
1
The statement of theorem 8 makes use of the following definition. Definition. Two subspaces of 93" are said to be linearly independent if their intersection has only the zero vector. Theorem 8. Suppose the response matrix, R(c), resulting from the m, x n lumping scheme, M,, has a set, S , of identical columns. Let R(s) denote the response to the lumping scheme, M,, where M, places the columns of S in a separate lump and is otherwise identical to M,. If the columns of S span a subspace of Bmgwhich is linearly independent of the subspace spanned by the remaining columns, then every pair of columns which have the same
1.0000 0.0000
values in R(c) will also be identical in R(s). Proof. The proof of theorem 8 is given in Coxson (1985), theorem 2 . Corollary. If R(c) is partitioned into m sets, Si,or identical columns and if each set is independent of the subspace spanned by the remaining columns, then there exists an m-dimensional exact proper lumping of the system which places all of the states associated with S , in the same lump. The condition of linear independence is important. Example 9 shows why it cannot be ignored. Example 9.
1
0.55700 0.08463 0.12642 G = 0.12694 0.59931 0.18964 as in example 4 0.31606 0.31606 0.68394
[
Take M, = [l 1 I]. Then
All three columns of R(c) are identical. The subspace spanned by any subset of columns is not independent of the remaining column(s). Thus, the separate fact that columns 1 and 3 are identical cannot be used to justify lumping species 1 and 3 without species 2 . This scheme is too coarse to provide any useful information, as is obvious in this simple case where the lump merely represents total mass balance of unity. In more complicated and less obvious cases, the linear independence condition will provide correct results. In example 4,we saw that species 1 and 2 can be lumped, but species 2 and 3 do not form an exact lump. A simple check with
would show that 1 and 3 also cannot be lumped. The lumping scheme M = (1 1 1) is too coarse to identify lumps for a model of the process. The response, R(c), shows only that M s (the total mass) never changes. The number and composition of lumps in a model depend on the purpose for which the model will be used. M, must keep the critical variables, such as desired and undesired products, separate from other variables. Recall that the system of example 9 and example 4 has an exact reduced order model in which species one and two are lumped
Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987 2157 together. This would be irrelevant if the first species were a product one wished to predict. In the 10-lump Mobil model considered in part 1,the coarse lumping scheme M,=
1
1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1
[
kept the desired product gasoline (species 9) and the undesired C-lump (species 10) separate from all other species. If the C-lump is included with the feedstock, this is,
[
1
1 1 1 1 1 1 1 1 0 1 Mz= 0 0 0 0 0 0 0 0 1 0
then the response matrix, R (example 10 below), with blocks, MzGk, shows clearly that species 4 and 8 (the aromatics) and 10 (C-lump) form an exact lump. Example 10. Response, R, for the Mobil model (two course lumps) is given in Chart I. Again, this does not sufficiently differentiate between certain feeds and products. In practice for catalytic cracking, while it is true that the aromatic rings as well as coke and light ends are refractory as regards gasoline formation, it still is of interest to keep them separate. Thus, careful consideration must be given in the coarsest lumping to what are the truly minimum number of lumps required to give useful information-in catalytic cracking it is three.
7. Conclusion We have placed lumping analysis in a system theoretic context. The relationship of lumpability to system observability and minimal realization was exploited to identify appropriate lumping schemes and to obtain the reaction rate matrix for a given scheme. These results provide a theoretical basis for the lumping strategies described in part l. Nomenclature A j = jth species in a reaction 6 = controllability matrix F = system matrix (discrete or continuous time) F, = system matrix for lumped variables z G = discrete system matrix G, = discrete lumped system matrix 7f = Hankel matrix with entries Hk H k= system response (MFkL) I, = n x n identity matrix K = continuous system matrix
K, = continuous lumped system matrix L = input matrix M,M,,M2 = measurement or lumping matrices M, = coarsely lumped measurement matrix M+ = Moore-Penrose pseudoinverse of M N = error matrix 0 = observability matrix O j = partial observability matrix P = another proper lumping matrix Rk= space of real valued k-dimensional vectors R = system response matrices R(c) = coarsely lumped system response S,Si = sets of identical columns of a response matrix u ( t ) = inputs to a continuous system u ( h ) = inputs to a discrete system x ( t ) = state vector for the continuous system x ( h ) = state vector for the discrete system xi = jth component of the state vector y ( t ) = measurements of the continuous system y ( h ) = measurements of the discrete system z ( t ) = continuous system lumped state vector z ( h )= discrete system lumped state vector Greek Symbol T = sampling interval Literature Cited Aoki, M. IEEE Trans. Automat. Control 1968,AC-13, 246-253. Bischoff, K. B.; Coxson, P. G. Proc. Int. Chem. React. Eng. Congr. 1987,in press. Brockett, R. Finite Dimensional Linear Systems; Wiley: New York, 1970. Coxson, P. G. J . Math. Anal. Appl. 1984,99, 435-446. Coxson, P. G . IEEE Trans. Automat. Control 1985,AC-30,478-480. Coxson, P. G.; Bischoff, K. B. Ind. Eng. Chem. Res. 1987, 26, 1239-1248. Hutchinson, P.;Luss, D. Chem. Eng. J . 1970,I , 129-135. Kalman, R.; Falb, P.; Arbib, M. Topics in Mathematical Systems Theory; McGraw-Hill: New York, 1969. Kuo, J. C. W.; Wei, J. Ind. Eng. Chem. Fundam. 1969,8,124-133. Liu, Y.A.; Lapidus, L. AIChE J. 1973,19, 467-473. Luenberger, D. G. Optimization by Vector Space Methods; Wiley: New York, 1969. Luenberger, D. G. Introduction to Dynamic Systems; Wiley: New York, 1979. Ozawa, Y. Ind. Eng. Chem. Fundam. 1973,12, 191-196. Ray, W. H. Advanced Process Control; McGraw-Hill: New York, 1981. Silverman, L. IEEE Trans. Automat. Control 1971,AC-16,554-567. Weekman, V. W. Lumps, Models, and Kinetics in Practice; AIChE Monograph Series; AIChE: New York, 1979; Vol. 75. Wei, J.; Kuo, J. C. W. Ind. Eng. Chem. Fundam. 1969,8,114-123.
Received for reuieu! November 18, 1986 Accepted July 7, 1987
