I n d . Eng. Chem. Res. 1992,31,253&2546
2638
tems, and NSF Grant IRI-880-7061. Literature Cited Aelion, V.; Powers, G. J. A Unified Strategy for the Retrofit Synthesis of Flowsheet Structures for Attaining or Improving Operating Procedures. Comput. Chem. Eng. 1991, 15 (5),349-360. Barmish, B.R. A Generalization of Kharitonov's Four-Polynomial Concept for Robust Stability Problems with Linearly Dependent CoefficientPerturbations. ZEEE Trans. Autom. Control 1989,34 (2),157-165. de Kleer, J.; Brown, J. S. Qualitative Physics based on Confluences. Artif. Zntell. 1984,24,7-84. Fusillo, R. H.; Powers, G. J. A Synthesis Method for Chemical Plant Operating Procedures. Comput. Chem. Eng. 1987,11,369-382. Fusillo, R. H.; Powers, G. J. Computer-Aided Planning of Purge Operations. AZChE J. 1988a,34,558-566. Fusillo, R. H.; Powers, G. J. Operating Procedure Synthesis using Local Modela and Distributed Goals. Comput. Chem. Eng. 1988b, 12,1023-1034. Gantmacher, F. R. The Theory of Matrices; Chelsea Publishing Company: New York, NY, 1964;Vol. 11. Ishida, Y.; Adachi, N.; Tokumaru, H. Some Results on the Qualitative Theory of Matrix. Trans. SOC.Znstrum. Control Eng. 1981, 17 (l), 49-55. Jeffries, C.Qualitative Stability and Digraphs in Model Ecosystem. Ecology 1977,55,1415-1419. Kalagnanam, J. Qualitative Analysis of System Behavior. P b D . Dissertation, Department of Engineering and Public Policy, Carnegie Mellon University, 1991. Kharitonov, V. L. Asymptotic Stability of an Equilibrium Position of a Family of systems of Linear Differential Equations. Differ. Equations 1979,14,1483-1485.
Kuipers, B. Commonsense Reasoning about Causality: Deriving Structure from Behavior. Artif. Zntell. 1984,24,169-204. Kuipers, B.Qualitative Reasoning: Modeling and Simulation with Incomplete Knowledge. Automatica 1989,24 (4),571-585. Lakshmanan, R.;Stephanopoulos, G. Synthesis of Operating Procedures for Comp1:te Chemical Plants. Part I Hierarchical, Structured Modeling for Nonlinear Planning. Comput. Chem. Eng. 1988a, 12,985-1002. Lakshmanan, R.;Stephanopoulos, G. Synthesis of Operating Procedures for Complete Chemical Plants. Part I1 A Nonlinear Planning Methodology. Comput. Chem. Eng. 198813, 12, 1003-1021. Lakshmanan, R.;Stephanopoulos, G. Synthesis of Operating Procedures for Complete Chemical Plants. Part 111: Planning in the Preeence of Qualitative Miring Constraints. Comput. Chem. Eng. 1990,14,301-317. Mavrovouniotis, M. L.; Stephanopoulos, G. Formal Order-of-Magnitude Reasoning in Process Engineering. Comput. Chem. Eng. 1988,12,867-880. Quirk, J. The Correspondence Principle, a Macroeconomic Application. Znt. Econ. Rev. 1968,9,294-306. Raiman, 0. Order of Magnitude Reasoning. Proceedings of the Sixth National Conference on artificial Intelligence, AAAZ-86, Philadelphia, PA; Morgan Kaufmann Publishers, Inc.: San Matille, CA, 1986;pp 100-104. Samuelson, P. A. Foundations of Economic Analysis; Harvard University Press: Cambridge, MA, 1983. Zadeh, L. A,; Desoer, C. A. Linear System Theory; McGraw-Hill: New York, NY, 1963. Received for review March 9,1992 Revised manuscript received July 2, 1992 Accepted July 23, 1992
Eigenvalue Inclusion for Model Approximations to Distributed Parameter Systems Eric M.Hanczyc and Ahmet Palazoglu* Department of Chemical Engineering, Unioersity of California, Davis, California 95616
The impact of model order reduction on the dynamics of open-loop and closed-loop processes is studied. The concept of eigenvalue inclusion regions (EIR) is utilized to assess the practicality of finite-order models for distributed parameter systems (DPS) such as tubular reactors. It is demonstrated that the EIRs account for the neglected modes of the process and give indications as to the dominance of the models retained in the finite-order model. A simulation study illustrates the computation of EIRs and investigates the significance and the restrictions of the methodology. Introduction Many processes in chemical engineering are typically modeled as distributed parameter systems (DPS) when more than one independent variable exists. In many instances, the governing equations of heat, mass, and momentum transport will contain spatial gradients as well as temporal dependence, and solving this system of partial differential equations (PDEs) may be nontrivial. In general, some approximations are made to obtain a numerical solution that in some limit yields the exact solution, such as the case of finite difference methods. For process control applications, a finite dimensional (lumped) model is generally required to design the feedback controller for general DPS. Since the control action would be calculated with the knowledge of this approximate model, one should answer the question: What is the relationship between the approximate model and the original model? In other words, within the robustness vernacular,how much uncertainty is "produced" by employing
* T o whom correspondence should be addressed.
the lumped model in place of the original system of PDEs? In control system design, this becomes critical since any controller based on an approximate model may result in the loss of desired performance characteristics and even stability when implemented on the actual process. In the process control literature, the modeling and control of distributed parameter systems have been addressed by various researchers (Ray, 1981;Fose et al., 1980; Bonvin et al., 1983). The most common approach to modeling and subsequent control design is based on lumping procedures using orthogonal collocation. The set of PDEh is transformed into a set of ordinary differential equations (ODES) by collocation, and typically the resulting lumped model is of very high order. Thus, further order reduction (Bonvin and Mellichamp, 1982) is necessary to use available control design methodologies. These steps unavoidably result in a mismatch between the simple, low-order model and the original system, thus raising questions about the feasibility of the implemented feedback control system. Despite its importance,however, the issue of model/plant mismatch for DPS is seldom addressed (Palazoglu and Owens, 1987).
0888-5885/92/2631-2538$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2639 This paper focuses on linear (or linearized) DPS and is concerned with how the eigenvalues of a lumped reduced-order model are related to thoee of the original DPS. This leads to a method for quantifying the dynamic information lost in reducing the model order, and provide us with valuable insight toward the trade-off between model complexity and accuracy. Within the context of feedback control, this method yields a sufficient condition for the stability of the closed-loop system, using available information about the approximate model, thus making it possible to robustly control DPS. This concept has been discussed previously by Franke (1986,1986). Starting with well-known Gershgorin theorems for finite dimensional matrices, Franke extended these to the infinite dimensional case, where there exist infinitely many discrete eigenvalues, making it possible to estimate the eigenvalue locations in the complex plane. Using mathematical examples, Franke showed that stability of the original system could be guaranteed by analyzing the finite-order model. The finiteorder model is constructed by the method of weighted residuals (MWR). Specifically, we focus on Galerkin's method, since it results in a series truncation of the infinite dimensional DPS. As the order of approximation tends to infinity, the number of eigenvalues goes to infinity and the estimate converges to the original DPS (Finlayson, 1972). This paper is structured as follows. First the DPS problem is formulated,and the MWR approach is reviewed briefly. Then, we present four theorems for estimating the bounds on each of the infinite set of eigenvalues. A similarity transformation in the state space is introduced for determining smaller eigenvalue enclosures. This culminates in the main result, a sufficient stability condition for the closed-loop system when a finite-order model is used. Finally the methodology is illustrated with two simulation studies and the results are discussed.
Problem Formulation To demonstrate our approach, we shall start with the following linear, time-invariant plant description:
The boundary conditions are given as w(t,O=)Rlx(t,l) = 0 (2) where x(t,z) E L,[O,l] is the state variable, u(t) E Rp is the vector of manipulated variables, and D,R,,, and R1 are spatial differential operators, comprising an eigenvalue problem with a discrete spectrum. The vector b(z) spatially distributes the control action, and the output vector is given by the integral (3)
which leads to the feedback control law, ~ ( t= ) -Ky(t) (4) where K is assumed to be a constant gain matrix for simplicity. A typical approach would be to attack this problem using the MWR procedure (Finlayson, 1972). This class of model reduction methods approximatesthe state x(t,z) as follows: n
X(")(Z,t)
= E:zi(t)4 i ( Z ) = 4%) x(t) i=l
(5)
where the functions &(z) (i = 1, ...,n) are the elements of a smooth, complete, orthonormal set in [0,1]. The boundary conditions become R&(O) = R1&(l) = 0, i = 1, 2, 3, (6) Substituting x(")(z,t) for x(z,t) in eq 1 creates a residual: Res(t,z) = 4T(z)t ( t ) - D$J~(z) x(t) + bT(z)u(t) (7)
...
which can be driven to zero using a variety of different weighting functions. If themselves are used as the weighting functions, the Galerkin version of MWR results. Evaluation of the integral JIRes(t,z) +(z) dz = 0
(8)
yields the finite dimensional representation of the original system t ( t ) = A(")x(t)
(9)
where
A(")=
1
$(z)
D4T(z)dz -
As noted earlier, the system eigenvalues approach thoee of the original system as n tends to infinity. Galerkin's method actually results in a truncation of a sequence; thus it leads itself nicely to comparison of different orders of model approximation as the order increases. This would not have been poasible using orthogonal collocation, since for that method, the model structure (collocation points) change as the order is increased, thereby not allowing a direct assessment of the impact of the additional collocation point and its relevance to the remainder of the structure. The goal is to determine eigenvalue inclusion regions (EIR) that reveal the contribution of neglected eigenvalues on the remaining ones. The four theorems that follow establish conditions for the existence of finite radii for the Gershgorin disks (Rosenbrock, 1974) that bound the EIRa for an n-dimensional real-valued square matrix M = (mij) (e.g., the state matrix A). The eigenvaluescan be enclosed in disks centered at mjj with radii determined by n
rj = CImijl, j = 1, 2, ..., n
(11)
a=1
i#J
-
The following two theorems establish conditions which =. guarantee that rj exists for each j as n Theorem 1 (Franke, 1986). Let M be an n-dimensional sequence for which the following inequalities hold for each integer n:
mjj I-Cl*p, j = 1, 2, ..., n lmijl IC2*i@*jr, i # j , i, J = 1, 2,
(12)
..., n
(13)
where C1and Czare positive constants and a,8, and y are real numbers constrained by a 1 0 and p + y + l < a (14) Then the eigenvalues of A4 as n OJ will be enclosed in Gershgorin disks centered at mjj,j = 1,2,3, ...,with radii rj = C,*jb'Y*[t(S-8) -j-(&@)],j = 1, 2, 3, ... (15)
-
where 6 is an arbitrarily selected real number in the interval
2540 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
/3+1 1 (20) Then the eigenvalues of M as n w will be enclosed in Gershgorin disks centered at mjj = 1,2,3, ...,with radii
-
rj = 2C2*j@+y*n6),j = 1, 2, 3, ...
(21)
The fmt extended Gershgorin theorem has been found to be more useful for cases where the basis functions, &(z), are the eigenfunctions of the operator D. The second, based on different assumptions concerning diagonal dominance, yields better results in the more general Galerkin case. Under feedback, the closed-loop A matrix is the sum of two matrices. The matrix entries often have different dependencies on the row and column indices. The first two theorems are sometimes inadequate to evaluate the closed-loop problem under these circumstances. This has not been fully recognized by Franke (1986). The following two theorems are developed to provide tighter enclosures for closed-loop analysis. The theorems individually bound the open-loop and feedback components of the matrix A with different functional relationships between the offdiagonal elements and the corresponding row and column indices. Theorem 3. Let M be an n-dimensional sequence for which the following inequalities hold for each integer n: mjj 5 -Cl*p, j = 1, 2, ..., n
+
-
and 8 > 1
(16)
(22)
e + v I a ,
(29)
+
where 6 is an arbitrarily selected real number such that @ + 1 < 6 S a - y , 6 - € > 1 , and q + b - 8 S a (31) C4is a positive constant determined by the values of (6 t i and 8;p is a real number given by p = min(6 - e,8) (32)
For proofs of theorems 3 and 4, see the Appendix. These theorems contain the necessary tools for calculating the radii of the EIRa. However, the size of these disks can be further reduced. A similarity transformation is now introduced with adjustable parameters, which allows manipulation of the relative sizes of the dominant and nondominant eigenvalue enclosures. We note that the dominant eigenvalues correspond to the ones retained in the chosen model, by virtue of their dominant impact on the dynamic proeprties of the system. Let A(m)represent the reduced mth order Galerkin model of the original DPS. Al, ...,A,, are the eigenvalues of A("), and V = (uij) is the corresponding matrix of eigenvectors. Consider the sequence A(")partitioned as (33)
where B,C,and D are the appropriate partitioned submatrices. Using T, a similarity transformation is performed on A(n). The transformation matrix is given by (34)
where ai are positive real-valued numbers used for radii manipulation, and I is the identity matrix of dimension n - m. This operation yields
lmijl IC2*i@*jy C3*ie*jq*li-j14,
i # j , i , j = 1 , 2,..., n (23)
where C1,C,, and C3 are positive constants and a,8, y, e, 7, and 8 are real numbers constrained by a z o , , ! 3 + y + l < a , e + q I c r , and 8 > 1 (24) will be enclosed Then the eigenvalues of M as n in Gershgorin disks centered at mjj, j = 1,2,3, ...,with radii rj = C2*?+7*[{('(t-@)- j-('@)] 2*c3*j'+q*j-(8), J' = 1, 2, 3,... (25)
-
+
where e must be in the interval B+l 1 (A3) For each integer j as n increases to w, the eigenvalues will be enclosed in Gershgorin disks centered at mjj, with radii m
m
r; = CC2*iB*jy + CC,*?*jq*li - j1-B i=l
i=l
(A41
Using a similarity transformation, diagv], the equation becomes m
m
i=l
Ea1
r; = C2*jY+t*Cifl-f+ C3*j?+"*cli- jl"
(A5)
where the summations exist for all i # j. The second summation is broken up as follows: m
;-1
m
C li -jI" = C li -jl" + C li - jl"
i=l
is1
(A6)
i=j+l
c is required to be in the following interval to force the bounds to grow just as fast as the centers move toward left B+lCala-y (A71 This yields, after rearranging the summations,
r; = C2*jY+'*[{(e-8) -j@-'] + C3*jv+"*[{(0)+
;-1
Xi"]
i=l
(A8)
where f represents Riemann's S-function. As j tends to w, the summation in eq A8 converges to an upper bound, S(0). Thus the eigenvalues of M will be enclosed in Gershgorin disks centered at mjj, j = 1, 2,3, ..., with radii r, = c2*>+r*[f(c-@) - j - ( 4 ] + 2*c3+y+~*@), j = 1, 2, 3, ... (A9) Proof of Theorem 4. Let M be an n-dimensional sequence for which the following inequalities hold for each integer n: m,j I-C,*?,
j = 1, 2,
...,n
(A101
lmijl IC2*ifl*j~ + C3*if*jv*li- jl", i # j , i , j = 1 , 2,..., n ( A l l )
2646 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
where C1, C2,and C3 are positive constants and a,@,y, e, 7, and 8 are real numbers constrained by a z o , @ + y + l < a , e + t l I a , and 8 > 1 (A12)
At this stage this proof is identical to that of theorem 3. For each integer j as n increases to a,the eigenvalues will be enclosed in Gershgorin disks centered at mjj,with radii rj
gc2*is*jy + gc3*p*jq*+i- jl-e
i=l
i=1
(A131
Using a similarity transformation, diagO6],the equation becomes 0
(ID
rj = C2*j6+y*CiW + C3*jt+6*Cp-d*li- jl-8 (A14) i=l
i=l
Now the second summation is broken into two sums with the fvst going from i = 1 to j- 1 and the Becond going from j + 1 to =. To proceed further with the construction of the bounds, we shall state the following inequalities that can be easily verified: ie6*li - jl” Ij4*ie6 + je6*li - jl” for i 1 j + 1 (A15) and j-1
Cie6*li- j1-O
i=l
5 C,*u
- l)-fi
for i Ij - 1
(A161
where C6 is a positive constant determined by 6 - e and 8, and p = min(k,8). Now, the existence of a fdte bound in the inequality A16 will be shown. j-1
j-1
i=l
i=l
Ci~d*li- jl-8 ICi-fi*li- j1-b
(A17)
with p defined as above. After multiplying by 0’ - l)”,the right-hand side of eq A17 becomes
E( i-1
As j
j - 1 ) = ( j-l)+( j-1 j-1 2*0’ - 2) j-1 j-1
i*li - j l
-
)+
j-1
=, the summation converges from above to
Note that the bound is independent of j since the s u m converges to a constant. Since the infinite sum converges to a finite value and remains bounded for finite j, there exists a positive constant C6 such that
2*&) Ic,
l (A22)
yields rj = C2*j6+y*[r(&@) - j-(h@)] C3*j~++”*f(6-e)+ C,*Y+q*t(8) + C4*j6+vfi, j = 1, 2, 3, ... (A23)
+
r
where C4 = C6*C3,7 + 6 - 0 Ia,and represents Riemann’s r-function. Thus the eigenvaluesof M will be enclosed in Gerahgorin disks centered at mjj,j = 1 , 2 , 3 , ...,with radii given by eq A23.
Literature Cited Bonvin, D.; Mellichamp, D. A. A Unified Derivation and Critical Review of Modal Approaches to Model Reduction. Znt. J. Control 1982,35,829-848. Bonvin, D.; Rinker, R. G.; Mellichamp, D. A. On Controllii an Autothermal Fixed-bed Reactor at an Unstable State. Chem. Eng. Sci. 1983,38,233-244. Finlayson, B. A. The Method of Weighted Residuals; Academic Press: New York, 1972;p 62. Foss, A. S.; Edmunds,J. M.; Kouvaritakis, B. Multivariable Control System for Two-Bed Reactors by the Characteristic Locus Method. Znd. Eng. Chem. Fundarn. 1980,19,109-117. Franke, D. Eigenvalue Enclosure in Certain Infinite-Dimensional Control Systems Based on Finite-Dimensional Galerkin Models. Proceedings of the Eleventh IMACS World Congress on System Simulation and Scientific Computation, Oslo, Norway; 1985; Vol. 4,pp 37-40. Franke, D. Application of Extended Gershgorin Theorems to Certain Distributed Parameter Control Problems. Proceedings of the Concerence on Decision and Control, Fort Lauderdale, FL; 1986; pp 1151-1155. Klickow, H.-H. Analyse von Regelungen mit verteilen Parametarn durch Einschlieseung der Eigenwerte. Regelungstechnik 1984,32, 165-167. Palazoglu, A.; Owens, S. E. Robustness Analysis of a Fixed-bed Tubular Reactor: Impact of Modeling Decisions. Chem. Eng. Commun. 1987,59,213-227. Ray, W. H. Advanced Process Control: McGraw-Hilk New York. iisi;pp 133-207. Rosenbrock, H. H. State-Space and Multivariable Theory; Wiley Interscience: New York, 1970; p 11. Received for review February 14, 1992 Revised manuscript received August 3, 1992 Accepted August 4, 1992
