Znd. Eng. Chem. Res. 1992,31, 2532-2538
2632
Strong, B.; Henry, R.P. The Purification of Cobalt Advance Electrolyte wing Ion Exchange. Hydrometallurgy 1976, I, 311-317. Szabadka, 0.Studies on Chelating b i n s . 11. Determination of the Protonation Constanta of a Chelating Resin Containing Iminodiacetic Acid Groups. Talanta 1982,29, 183-181. Tare, V.;Karra, S. B.; Haas, C. N. Kinetics of Metal Removal by Chelating Resin from a Complex Synthetic Waste Water. Water,
Air Soil Pollut. 1984, 24 (4), 429-439. Yoshida, H.; Kataoka, T.; Fujikawa, S. Kinetics in a Chelating Ion Exchanger. Chem. Eng. Sci. 1986,41 (lo),2511-2530. Received for reuiew September 26,1991 Revised manuscript receiued February 19, 1992 Accepted July 11,1992
PROCESS ENGINEERING AND DESIGN Evaluation of Operating Procedures Based on Stationary-State Stability Vital Adion,+Jayant Kalagnanam,t and Gary J. Powers*?+ Department of Chemical Engineering and Department of Engineering and Public Policy, Carnegie Mellon Uniuersity, Pittsburgh, Pennsylvania 15213
The safe and reliable operation of chemical planta depends on high-integrity operating procedures. Such procedures often involve process transition through stationary states. The stability of these states is an important part of the integrity evaluation. The methodology presented in this paper qualitatively identifies families of inherently stable chemical operations and establishes design rules for the synthesis of stable stationary states. These rules have been developed using the RouthHuiwitz conditions for three qualitative levels of dynamic system models. These levels represent the gains of the dynamic system by the signs of the gains (+, 0, -), the signs and their equality (+, 0,-, =), and finally the gains represented by their order of magnitude values (+, 0, -, =, ). Example design rules have been developed during the synthesis of operating procedures for a series of vessels with composition control and temperature control of an exothermic chemical reaction. The most robust design rules apply to systems that can be proven stable by the sign level of gain representation. Less robust rules with a wider range of applicability to dynamic systems have been developed using the equality and order of magnitude representations. 1. Introduction Operating procedures are instrumental in the safe, reliable, and environmentally sound operation of a chemical plant. The synthesis of high-integrity operating procedures commonly incorporates deairable features, such as verifiability of procedure steps and stability of intermediate procees states. Operating procedures takes a process from an initial to a goal state. This state transformation is achieved by applying permissible process operations, without violating process constraints. These constraints are motivated by concerns for the safe and reliable operation of the plant. Environmental impact, quality control, and economic issues also influence the synthesis of operating procedures. The concerns for safety and reliability favor processes which are inherently stable over large ranges of parameter values and operating conditions. Such processes face a lower risk of undesirable scenarios, like explosions, runaway reactions, and unacceptably high vessel pressure. Assessing stability early in the design aims to produce processes with inherently favorable operating characteristics. The methodology presented in this paper qualitatively identifies families of inherently stable chemical operations. Qualitative analysis is used to evaluate stability at three levels of process information. The results of the
* Author to whom correspondence should be addressed. t Department
of Chemical Engineering.
* Department of Engineering and Public Policy.
proposed evaluation may be used as general design rules for the synthesis of stable atationary states in high-integrity operating procedures. 1.1. Recent Work in Operating Procedure Synthesis. The methodologies for operating procedure synthesis have included a variety of symbolic manipulation techniques to generate desirable plans. Fusillo and Powers (1987, 1988a,b) uaed a modified form of means/ends analysis, where operating goals were identified and procedural actions were sought to satisfy the goals. A constraint-guided strategy searched for sequences of actions that did not violate the operating constraints. The methodology decompoeed the complexity of the system by exploiting the existence of stationary states,or states where the operating goals are partially met and the system can wait. These states were used as planning islands, where the status of the system can be verified before the next planning action is taken. Lakshmanan and Stephanopoulos (1988a,b, 1990) developed hierarchical, object-oriented modeling techniques and applied them with a nonlinear planning method to synthesize operating procedures for chemical plants. Models were implemented as objects, allowing relations and methods to be inherited through the hierarchy of models. Their planning methodology involved identifying stationary states and using means/ends analysis to plan procedures for carrying the process between stationary states. The nonlinear planning techniques were based on the propagation of constraints. They also aeveloped specific methodologies addressing qualitative mixing con-
0888-5885/92/2631-2532$03.00/00 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2533 straints and quantitative constraints. Aelion and Powers (1991)developed a unified strategy for the retrofit synthesis of flowsheet structures for attaining or improving operating procedures. Their approach proposed structural modifications aimed specifically at creatingstationary states. The stationary states introduced by the structural operators increased process flexibility and added modularity and verifiability to the operating procedures. Means/ends analysis was used to backtrack from the goal to the initial state. Parta of their strategy were implemented in a computer program, the Procedural and Structural Planner (PSP). 1%. Operating Procedure Synthesis and Stationary States. Stationary states can play an important role in the effective operation of chemical plants. Flexibility, safety, and reliability concerns favor operating procedures and proceee flowsheeta with intermediate states, where the system can wait until the next action is taken. At such states the status of the system can be verified. The system may also conveniently retreat to an intermediate state to avoid complete shutdowns in emergency or maintenance situations. Strabgic inclusion of stationary states into the process often adds modularity to its operating procedures and commonly makes the system more flexible. Fusillo and Powers (1987) described the characteristics of useful stationary states: 1. The system is at steady state or changing very slowly. 2. The values of (moat of) the variables lie between their initial and goal-state values. 3. Connections between a subsystem and its neighbors are hlatad sufficiently, so the subsystems do not interact. Stationary states form because of the presence of simultaneous inverse operations or large capacitance for a physical quantity, such as thermal energy or mass. An example of a stationary state is a distillation column operating at total reflux. This stationary state is realized because the evaluation at the bottom of the column is counteracted by the condensation at the top. A batch reactor may exhibit a stationary state when filled with one of two reagenta and solvent and heated until it reaches a set-point temperature. The stationary state results from capacitance for thermal energy and mas. Since the second reagent is not present, the reactor contenta can wait near the desired temperature and composition without undergoing chemical reaction. 11. Ebaluation of Operating Procedum. Operating proceduree are often required to eatiefy multiple conflicting objectives. All operations must ensure a basic standard of safety and reliability, while providing for quality control and efficiency. Safe and reliable operation is related to operations stability. In addition, stability at intermediate states provides for a more flexible process, which also may translate into economic benefits. The synthesis of operating procedures requires evaluation of stability. Stability can be characterized by detailed numerical process simulation, which requires accurate values for process parameters and conditions. This information is not normally available at the conceptual design stage. In addition, process simulation predicts the transient system behavior for a single point in the process variable space; therefore a what-if analysis would require numerous simulation runs. A preliminary evaluation method for operating procedure stability can be used for cases when precise estimates of system parameters are not available. In the next section we discuss the general problem of stability evaluation of operating procedures. Stability is attained for a range of values for the parameters of the
system. Based on this property, interval representations for parameters are introduced and stability properties of different operations investigated. If needed, the preliminary stability analysis of the operating procedures may be verified using detailed simulation models and pilot plant experiments. The proposed preliminary evaluation method can provide estimates of process variable ranges for stable operation for use by the process designer. 2. Stability In this section we provide definitions for stability which indicate the properties of stable steady states that can be used to evaluate operating procedure stationary states. The Routh-Hurwitz conditions for linear dynamic system stability are used to generate qualitative features of states with inherently stable behavior. Consider a physical system represented by a set of n differential equations as follows: hi = fibl,x2, ...,x,, dt
al, ..., a,, ul, ..., uJ, i = 1, ..., n (1)
The variables xi are state variables, uj are input variables, and ai are parameters of the system. Let xi(t) denote the solutions of eq 1. We investigate the stability of the solutions xi@)at the steady states. The linear approximation to eq 1in the neighborhood of the equilibrium point is as f0110w8:
or, equivalently, -dx=
Ax
dt The characteristic equation for eq 2 is A"
+ klA"-' + k2An-2 + ... + k,,-lA + k, = 0
(3)
where
ki = (-1)'C(ith order principal minors of A)
If the real parta of the roots of the characteristic eq 3 are negative, then the system is stable. If at least one root is positive then the system is unstable. The RouthHurwitz conditions are necessary and sufficient for stability of a linear system near an equilibrium point. The conditions are (1) k j > 0 for all i = 1 , 2 , ...,n, and (2) all the determinants, bj, shown below are greater than zero, Le.
2534 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
The Routh-Hurwitz conditions are a set of inequality constraints among the parameters aij of the system. Consider a simple system with the characteristic equation
t
I
,L, f
f w r
The Routh-Hurwitz conditions are k l = -(a11 + (222) > 0
=2ah-( - U12Q21 > 0 If we plot these relations in the kl-k2 parameter space, the region of stability is in the first quadrant. In general, the Routh-Hurwitz conditions define a domain of stability (Gantmacher, 1964). This indicates that stability is not a stringent criterion which restricts the parameters to single values. Rather it constrains the parameters to lie in a domain in the parameter space. This property forms the basis for our qualitative evaluation of the stability of stationary states used in planning operating procedures. 2.1. Qualitative Analysis of Stability. The question of stability may be investigated a t various levels of representation. Consider three levels of qualitative representation for the system gains, ki. Gains are described by (1)(+, 0,-1, (2) (+, 0,-, =I, and (3) (+, 0, -, =, ).
The goal is to investigate if these levels of representation can capture the stability characteristicsof stationary states,
commonly found in chemical operations. Similar approaches to qualitative reasoning have been developed by de Kleer and Brown (1984), Kuipers (1984), Raiman (19861, Mavrovouniotis and Stephanopoulos (1988), Samuelson (19831, Quirk (19681, Jeffries (1977), and others. Recent interest in this approach among control engineers is suggested by Ishida et al. (1981) and Kuipers (1989). The different qualitative representations differ in the degree of resolution with which variables and parameters of a given system are represented. An alternative approach to qualitative stability analysis has been developed by Kharitonov (1979) and is based on interval polynomials. 2.2. Stability Analysis Using the (+, 0, -) Representation. Stability analysis using the necessary and sufficient conditions for stability at this level of representation was presented by Jeffries (1977). A system that is stable based only on sign considerations is called sign stable. The (+, 0, -) level of representation gives necessary and sufficient conditions for sign stability only for systems with feedback loops of length 2. The length of a feedback loop, n, is determined by the presence of the following nonzero elements in a matrix: aij, ai+lj+l,..., (I.~+~,~. Systems which are proven unstable only on sign considerations are called sign unstable. If a system is neither sign stable nor sign unstable, then it is potentially sign stable. Such a system can be stabilized by further restricting the values of the gains. In this case, the (+, 0, -1 representation is not adequate to determine the stability of the given system and greater detail is required. Nonlinearity is an important issue in the design of chemical systems. The presented stability test is capable of providing regions of the phase space of nonlinear systems for which the system is locally stable. Every element in the linearized matrix is a function of the state variables. Hence, by imposing a set of conditions for stability on the qualitative values of these elements, we define in effect a region of stability in the nonlinear system phase space. 2.3. Stability Analysis Using the (+, 0, -, =) Representation. This representation adds equality relations
Figure 1. A reactor subsystem.
between variables. The symbol u=n means that two variables have the same magnitude. Sufficient conditions for stability of a single feedback loop of length n is given by the following structure: -a
b
0 '.. 4 0
0 a "'
0
1
This is a block diagonal matrix with a corner element. The proof for this result can be found in Kalagnanam (1991). The necessary and sufficient conditions for sign stability and instability at this level are not known. 2.4. Stability Analysis Using the (+, 0, -, =, ) Representation. This level adds order of magnitude relations among the gains in matrix A. The relation aij > (> ((0 a k l represents that aii is much larger (smaller) than akl. The semantics for this representation is based on Mavrovouniotis and Stephanopoulos (1988). The stability analysis at this level of representation consists of investigating the Routh-Hurwitz inequality constraints with the order of magnitude estimates of the gains. For example, the condition kl = -(all + 022) > 0 is proven using order of magnitude arithmetics. 2.5. Stability Analysis Based on Interval PoZynomiah. Another approach to analyzing the stability of dynamic systems is based on the Kharitonov theorem (Kharitonov, 1979). The stability of the interval polynomial can be established by testing the Routh-Hurwitz property of four extreme polynomials. However, the Kharitonov theorem only applies to polynomials with coefficients which are independent of each other. In general, dynamic formulations describingphysical systems rarely satisfy this restriction. Uncertainty in the parameters of linear systems, as in the examples of section 3, leads to dependent perturbations in the coefficients of the characteristic polynomial. Although tests for stability of such systems are known (Zadeh and Desoer, 1963), they are in general computationally intractable. More recent research, for example, by Barmish (19891, attempts to identify classes of problems for which the tests are computationally tractable. Our approach in this paper has been to a priori identify qualitatively stable operations. 3. Examples
In this section the stability conditions are investigated for a series of examples which involve stationary states. These states arise as intermediate states in the synthesis of operating procedures for these processes. The objective is to develop design rules for the synthesis of stable stationary states. The analysis is performed with the system gains represented by (+, 0, -), (+, 0, -, =), and (+, 0, -,
).
3.1. A Series of Process Vessels. Consider a chemical process subsystem which consists of a precooler, a reactor, and a splitter, as shown in Figure 1. Species A and B react
Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 2536 at high temperature to form C. At intermediate temperature the reaction leads to an unwanted product, U. A and B enter the precooler at constant flow rates fA and fe mol/s, reepectively. The flow rates between units and the outputs from the system are also constant. This system has been developed by the application of an operating procedure planning methodology,such as the one presented in Aelion and Powers (1991). The reactor contents are to be kept at a low temperature to avoid all reactions during startup. The precooler is included because the reactor configuration does not provide adequate cooling inside the unit. The side stream out of the reactor, fodp supports purging and sampling. Part of the material in the splitter is recycled back to the precooler. A part of an operating procedure for starting up thisunit requires that the reactor composition be established and verified. The stability with respect to composition is needed for evaluating the candidate stationary state. We assume a well-mixed precooler, reactor, and stream splitter and negligible pipeline volume, as compared to that of the overflow vessels. We also assume that the total input to the system is equal to its output, fo = fout,,+ fout,s. Under these conditions the unsteady-state mole and compositionbalances for each processing unit are provided below. precooler dn, dt = f o + f, - f p dnAp dt
dnfiAp dt
-I-=
drip
XAp
(5)
dXAp
dt + np dt = fdtAo
+ fax& - fpXAp (6)
After substituting eq 5 into 6 and canceling opposite terms, we get
Similarly, for the reactor and splitter we get fP -&Ar- - -xAp dt
n,
fP - -xAr nr
fr -&AS- - -xAr
- -f rx h (9) dt n, n, where nAi= moles of A in unit i, ni = total moles in unit i, xAi = mole fraction of A in unit i, and f = flow rate, mol/h. Combining eqs 7-9 in matrix form we get dx/dt = Ax + B,where
==
[ Jf]* f&Ao -
A =
,
nP
B = 0 0
Defining Cij
Cra
4
s
The C,'s are positive, so
Stability is guaranteed if the Routh-Hurwitz conditions are met. kl = Cop+ Cap+ Cpr+ Cr, > 0
k2 = (Cop + C,p)Cpr + CprCrs + C,(C"p
+ Cap) > 0
The signs of the parameters and the equality relations render the expressions true for all numerical values. The Routh-Hurwitz conditions indicate that the subsystem is stable with respect to composition provided that the inputs remain positive. The signs of the parameters along with the equality relations are sufficient for characterizing stability. Note that the structure of the matrix A in this example is the same as the one presented in section 2.3 except for the extra input term in all. This structure is in general stable. The analysis has shown that the outputs play no role in the subsystem's stability. Therefore it is not necessary to require that the inputs be equal to the outputs. If the total input exceeded the total output for a long enough time to flood the system, the present analysis would not have detected it, because it has been set up to address only the stability with respect to composition. We can generalize the results of this example by noting the following evaluation rule: The stationary state created by a series of nonreactive vessekr with an extensive property (moles)is always stable with respect to the corresponding intensive property (concentration),provided that the input(s) to the system is (are) positive (no backflow). The preceding analysis examined stability with respect to composition. Stability evaluation with respect to mass involves analysis of the mass balances of each individual unit. Note that stability with respect to one variable commonly does not prove stability with respect to any other variable. A full evaluation of this operation would require that models of all important variables be subjected to a similar stability analysis. This approach provides no guidance as to whether all stability questions have been addressed. 3.2. A Series of Vessels with Composition Control. Further analysis by the procedure planner indicates to the process engineer that the feed composition falls outside the acceptable range during startup. The startup method of the previous example will work only if the feed composition matches the reactor composition set-point, xhP. The feed composition does not remain constant, and a procedural planner indicates that composition control could be employed. Figure 2 presents the new configuration. The controller is proportional and modifies the incoming composition by readjusting the flows fA and fB while keeping the total feed flow, f,, constant. We msess the stability of this system, for evaluating the candidate stationary state.
2636 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992
I I
I
4
1
Purge
Fbure 3. Introduction of a temperature eontroller.
fws
fW
Figure 2. Introduction of a composition controller.
The system is described by the matrix dx/dt = Ax B, where
ro
-K
0
+
1
L
where K is the proportional control constant and xhP is the sebpoint composition. Using the Cij notation,
ro
-K
0
1
A h
To 0 - 0 1
A=l+
0 + -O 0+ I 10 0 + - J Note again that all C..'s are positive. The Routh-Hur-
witz conditions are as ?allows. k , = Cop + Cap+ C , + C, > 0 hi = (Cop+ C,)C, + CprC, + C,(Cop + Csp) > 0 k , = C,C,(K + C,J > 0 k4 = KC&,$, >0 62
+ CBp + Cpr + c,l[(Cop + C&op + C,(C, + C,)l - (K + C,)C,Cpr > 0 6, = k,k,ks - kl'k, - k,2
= kiki - ks = [Cop
+ C,C,
6, = k4as The nBeBs8aTy conditions, hi > 0, are satisfied purely on sign and equality conditions. The sufficient conditions, 6, > 0, can be satisfied by assuming that all the elements in matrix A are approximately of the same magnitude, C, and K 5 C. For this analysis we assume the upper limit, K = C. Under these assumptions the analysis indicates that the system is stable. Consider 6, = k.k, - k,: k, = 4C. k, = 5C. and k. = 2c9
This analysis suggests that if the proportional control constant, K,is not larger than the size of the rest of the parameters Cij,then the given structure is stable. This is a sufficient condition for stability. These results can be summarized in the following evaluation rule: If the capacitances of the vessels are approximately equal, the described stationary state is stable provided that the proportional controller gain, K,is not hrger than the average inverse capacitance, C. Since Ci;s are measures of inverse capacitance, the smaller the capacitance, the better the stability of the system. This evaluation rule is leaa accurate than that developed in example 3.1, since equality assumptions were required. 3.3. A Series of Vessels with Temperature Control. Another scenario in the startup planning of this problem involves maintaining a low reador temperature. Consider a situation with isomolar feed, jA= fB, where intermediate temperatures promote the undesirable exothermic reaction A +B U, with reaction rate, r, = k[A][B]. During planning for startup, stationary states are generated which reduce the production of the undesirable by-product, U, by maintaining a low temperature. One of the candidate states is shown in Figure 3. Species U precipitates from the solution and is removed from the bottom of the reador. This plan involves adding a proportional controller to the flowsheet, which tries to keep the reador temperature at or below a set-point value, T., To avoid recycling hot material, the downstream valve to the splitter is now closed until the reactor temperature is achieved. It is assumed that the set-point temperature may be reached before exhausting the reador capacity for material. We assume constant heat capacity, cp! for both the reactor and precooler mixtures. The evaluahon of the stability of this stationary state follows. The system is described by these equations: precooler energy balance
-
dTP mpcp t= focpTo- fpcpTp+Q, Qe
= KJT8p - TtJ
(10) (11)
Q,is the rate of heat generation from the controller, Tt. is the temperature measured by the sensor, and K , is the proportional control constant. Combining eqs 10 and 11, we get: dTP mpcp t= f.c,T.
- fpcpTp+KJT, - Td (12)
reactor energy balance mrc,
dT, = fpc,Tp + ,Q
(13)
Q,,
= k[AI[BI(-Wd
(14)
Q, is the heat generation from the reaction. The feed is isomolar, so [A] and [B] remain constant, because they
Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992 2537 are depleted at the same rate. We assume constant AHm and linear k with the reactor temperature, k = XT,. Defining M s X[A][B](-AHH,),a new constant, produces the following reactor energy balance.
dTr m,cp dt = fpcpTp= MT, temperature sensor energy balance dTta mtacp,ta d t - hA(T, - Tb) In eq 16 c ~and , A ~ are the heat capacity and the area of the temperature sensor, respectively, and h is the heabtransfer coefficient between the sensor and the reactor liquid. Combining eqs 12, 15, and 16 in matrix form, we get dT/dt = AT + B,where
= a2(2a63
= k362
>0
>0
These results have shown that the Routh-Hurwitz conditions are satisfied provided that a, is at least 1 order of magnitude smaller than a. Therefore the order of magnitude assumptions are sufficient for guaranteeing stability. The following evaluation rule may be deduced from this example: I f the effect of the exothermic reaction (azz)is much smaller than the effect of the other gains, then the described system is stable. This condition is sufficient for stability. This rule requires assumptions about the order of magnitude of the gains in the system, and therefore requires more detailed information than examples 3.1 and 3.2. 4. Discussion
, B =
A=
[
4 1 1
0
4 1 3
;21
a22
0
a32
4 3 3
[rTBp]
]
Note that one diagonal element in matrix A is positive. The system can, therefore, become unstable even without the interactions across units, represented by the off-diagonal terms. In physical terms the positive element, a22, represents the exothermic reaction which can promote itself through the reaction constant and drive the system unstable unless counteracted by the remaining diagonal elements. The Routh-Hurwitz conditions are kl = all + a33- a22> 0
- a11a22 - a22a33 > 0 k3 = a13a21a32 - a11a22a33 > 0 a2 = (al12aa3+ a11a332+ a11a222 + a33a222)k2
= a11a33
(a1I2a22+ 3alla22433 + a22%j2)> 0 63 = k362 > 0
-
In this example no Routh-Hurwitz conditions are satisfied based on the (+, 0, -, =) representation. An order of magnitude restriction of the parameters, all = 013 azl = a% = aa = a and an 0
k2 = a(a - 2a22)> 0 k3 = a2(a- a22)> 0
High-integrity operating procedures utilize stationary states for the safe and reliable operation of chemical processes. Stability analysis is a critical aspect of evaluating stationary states. At the conceptual design level, this evaluation is confounded by the lack of precise information of the gains of the system. Qualitative representations can provide a basis for investigating inherent stability at the conceptual design level. This paper uses ideas from qualitative control theory to present a method for evaluating the stability of stationary states which arise in the synthesis of operating procedures. The role of this stability test in a complete design system is the identification of families of inherently safe chemical operations, and the establishment of design rules for the synthesis of stable stationary states. Many chemical processing systems are potentially stable. The complexity of the feedback loops in these systems determines the level of detailed information required for assessing stability. Examination of the Routh-Hurwitz conditions needa a combinatorial number of computations. We have not investigated the potential memory problems in symbolic or numerical sign determinations of large expressions. We have considered three qualitative levels of representation for the gains in a lindynamic system. The conditions for system stability have been developed for each level of representation. Design rules that permit the synthesis of stable stationary states in operating procedures were derived from the stability conditions. Systems of feedback length 2 can be classified by using the (+, 0, -) gain representation as sign stable, sign unstable, and potentially sign stable. Potentially sign stable systems of feedback length 2 can be proven stable with additional information, such as the (+, 0, -, =) and the (+, 0, -, =, ) gain representations. Systems of feedback lengths larger than 2 cannot be proven sign stable and always require additional information to be proven stable. Design rules based on these observations allow the process designer to synthesize stable stationary states for use in operating procedure synthesis. The system complexity determines the information required for applying the stability design rules. Design rules that can be based on the sign representation are more robust than rules derived using equality and order of magnitude relations. Acknowledgment
This work was supported by the National Science Foundation, Division of Design and Manufacturing Sys-
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
