S T R U C T U R A L A N A L Y S I S FOR M U L T I V A R I A B L E PROCESS CONTROL G A Y L O R D G. G R E E N F I E L D A N D T H O M A S J . W A R D Chemical Engineering Department. Uarkson Collrge of T r c i i n o l o ~ ~ )Potsdam, .. 2‘. I’.
A method is introduced which retains internal structural information available from process models in feedforward and feedback control analysis of linear constant coefficient systems with the control objective of holding one or more outputs constant. Comparative examples illustrate some of the advantages of this approach over the corresponding terminal or external analysis.
R A ~ S F E Rmatrix models of process systems have been used Tsuccessfully in linear multivariable feedforward control analysis (Bollinger and Lamb, 1962; Haskins and Sliepcevich, 1965; T’erneuil, 1965). Hoivever, such models are inadequate for analytic linear multivariable composite feedforward-feedback control analysis. This paper introduces a structured matrix model for linear process systems that contains more information about the process than the corresponding transfer matrix model. With this additional information. unique optimal solutions of a class of composite feedfonvard-feedback control problems can be obtained. First, the general concept of “structure” in problem formulations is considered. It is then shown that this concept can be applied advantageously to some simple control problems. and finally a general structural matrix method for control analysis is presented. Both feedforward and feedback control have characteristic advantages and disadvantages. Feedforward control has the advantage of rapid response and the disadvantage of sensitivity to model error causing permanent compensation error. Feedback control is characterized by relatively slow response, but also relatively little sensitivity to model error and no steady-state compensation error. Luecke and McGuire (1967) have shown in single-variable control problems that composite feedforwardfeedback control preserves the advantages of both types of control while tending to cancel the disadvantages.
Structural Information in Problem Formulation
To establish the meaning and significance of internal structure in problem formulation, it is helpful to consider the use of available information in problem formulation. Generally, the amount of available information which is contained in a problem formulation is one of the most critical factors in acquiring a problem solution. When the amount of information used is insufficient and provides a nonunique problem statement: problem solutions may be incomplete, nonunique, or impossible. The proper use of the available information is not always a simple proposition. Critical information is sometimes of a subtle nature and easily overlooked. Dynamic programming is an example of the effective use of internal structural information. The information used is the serial or stagewise nature of certain decision and optimization problems. Consider a 10-stage system requiring a one-dimensional optimization of each stage. Analysis by dynamic programming uses a stage-by-stage approach such that 10 one-dimensional optimization problems must be solved. The corresponding solution by conventional calculus lumps the 10-stage problem into a single 10-dimensional problem which is usually impractical to solve. 564
I&EC
FUNDAMENTALS
The conventional calculus formulation can be considered an external or terminal formulation and the dynamic programming formulation an internal or structural formulation. The 10-dimensional problem of the calculus formulation could just as well be five, serial. tivo-dimensional problems, or any combination giving 10 dimensions over-all. Such nonuniqueness is a general characteristic of terminal formulations. Structural Formulation of Multivariable Processes
Internal structural information can also be used in multivariable process control analysis \vith significant advantages. An important characteristic of chemical processes is that theoretical or semiempirical models are usually available (Calvert and Coulman. 1961). This important information is lost if a terminal formulation is used (Kalman. 1963). Although structural considerations are not limited to such problems, this paper considers only linear: lumped-parameter. single-stage multivariable systems. Such systems are described by the set of rn linear differential equations:
Equation 1 represents a system \\ith n inputs or disturbance sources, x J , and rn state variables. j h . They are usually obtained from the nonlinear system model equations by retaining the first-order terms of their Taylor series expansion around the desired steady-state operating point. The Laplace transform of Equation 1 can be expressed as
I’,
=
/k#l
Before proceeding further, it will be worthwhile to consider a simple problem by the structural and the terminal approaches in order to distinguish between the two. Example 1. Let the system be the specific case of Equation 2 with two statevariables, Y1 and Ys:and two manipulable inputs, XI and Xp. For simplicity let a12 = a21 = 0. T h e control objective will be the dynamic uncoupling of Y 1 and ET?, so that variations in Y1do not affect ET2 and vice versa. The system can be represented by a signal flow graph as shown in Figure 1, Q . (Signal flow graphs or node-directed branch diagrams along ivith Mason‘s rule for their direct analysis are discussed in most recent introductory texts on control analysis. Briefly, each node represents a summing junction whose value is equal to the sum of the contribution of all in-
coming branches. T h e contribution of each branch is equal to the product of the branch transfer function and the value of the originating node.) Defining (213
ii,j
=
-,
*
b,,
=
(1 I i
bd ai,
(3% b)
gives the structure of Figure 1, h , where it is apparent that the proportional controllers indicated by dashed flow paths produce dynamic uncoupling-that is, the net forward transfer between the Y 1and Z Zand the Y Zand 21nodes is ^b z1 - 8 2 1 and 3 1 2 - 8 1 2 , respectively, or identically zero. I n terminal formulation. the system can be represented by the signal flow graph of Figure 2, (I, or by the matrix equation
(a )
where
p22
n22(.5 -
-
(b) Figure 2. Terminal flow graph of system
bll)
A
A = ( S -- bii)(s - b z ) - b i h i (4f) T h e approach to dynamic uncoupling with the terminal formulation is to diagonalize the controlled plant transfer maare arbitrary: trix, p’. \yhere PIll and t”??
signal
a. Without control
b. With control
YZXl = P2X2XZXl
+
P2X2H2X2Y2Xl
(6)
Solving explicitly for Y gives T h e control matrix, H , which will provide the desired plant transfer matrix is determined by equating the controlled plant transfer matrix as a function of H to the desired controlled plant transfer matrix, P’ (Kavanagh, 1957). From Figure 2, h, it is seen that
y 2 x 1 = (12x2
-
P2x2Hzxz)-’P2x2Xzx1
(7)
which defines the controlled plant transfer matrix. Equating this to the desired controlled plant transfer matrix provides
plxz
(8)
= (12x2 - P z x z H 2 x 2 ) - ’ P 2 x 2
T h e control matrix found by inverting both sides of Equation 8 is
HZxz
=
PF&
-
Pi,;
(9)
In general the elements of the H matrix will be relatively complex functions determined by the arbitrary choice of PtI1 and P ‘ z ~ .With this simple example, a12 = a21 = 0, which means the system has only feedback intercoupling. In such cases, the proportional controllers obtained directly with the structural analysis can in principle be obtained from a terminal analysis by trial and error. T h e procedure is to find the control objective P’ and the corresponding control matrix H from an infinite set such that the sum of the orders of the polynomials of all elements of the two matrices is minimal. This criterion is satisfied uniquely by Equations loa, b where the indicated sum is two.
s-b, (b)
Figure 1. tem
p i x 2
=
Internal signal flow graph of sysa. Without control
b. Normalized with control
VOL.
6
NO.
4
NOVEMBER
1967
565
In more general cases where there is also feedforward intercoupling ( a 1 2 # 0, a21 # 0 ) the structural analysis would provide feedforward controllers to eliminate feedforward intercoupling and feedback controllers to eliminate feedback intercoupling. Since the feedforward and feedback intercoupling mechanisms cannot be separated with the terminal formulation, it is generally impossible io duplicate terminally the structurally determined controllers even by trial and error.
I -
\ \ /
\
/
Structural Control Analysis
Equation 2 can be represented by a signal flow graph as shown in Figure 3, a. (For clarity in graphical representation, it is assumed that m = 2 without any loss of generality.) For further analysis! the inputs, X, and state variables, Y, are each categorized into three subdivisions. Inputs are classed as follows : [These input classifications and nomenclature are a modification of those of Bollinger and Lamb (1962).] 1. Xkyi 1 is the vector of U inputs which are not measured and are therefore unknown. 2. X F h 1 is the vector of K inputs which are measured (therefore known) and used to actuate control. 3. X!f:2 1 is the vector of M inputs which are manipulated to achieve the control objective. State variables are classed similarly as follows: 1. Y 1 is the vector of N non-control-actuating state variables which are generally unmeasured. 2. Y j 2 1 is the vector of Z state variables which are measured and used to actuate or initiate control. 3. YLCi1 is the vector of C state variables which are controlled. These state variables, also referred to as outputs, are the variables contained in the definition of the control objective. T h e rationale for the deterministic structural control analysis presented here is a structural statement of the invariance principle (Petrov, 1960). Two nodes coupled by a net forward channel of transfer described by the transfer function A (s) are shown in Figure 4> a. The requirement for invariance of the second node with disturbances in the first node is a parallel control channel with the transfer function - A ( s ) (see Figure 4, 6 ) . First consider the simple class of problems where there is an equal number of controlled outputs and manipulable inputs (C = M ) with all state variables contained in the YdE’l matrix (’I =’Z = 0 and C = M ) and no unknown inputs ( U = 0). This system is illustrated in Figure 3, a, where it can be seen that there are t\vo types of interaction. Feedback interactions are represented by 6 1 2 and 6 2 1 . This mechanism of interaction is the only inherent mechanism of interaction between the controlled state variables. The other interaction mechanism is feedforward interaction represented by a12 and a21. T h e first step in the analysis is the normalization operation defined by Equations 3a, b. The normalized system is expressed by the solid lines of Figure 3: b . T h e dashed lines in Figure 3, b , indicate the controllers obtained by applying the invariance principle. These controllers serve three distinct roles: 1. FEEDFORDWARD UNCOUPLING CONTROL.This mode of control is represented by -212 and -221. Its purpose is to cancel the system feedforward intercoupling loops to give a controlled system with no feedforward intercoupling. 2. FEEDFORWARD CONTROL.This mode of control is represented by - a i j and -anj. Its purpose is to compensate for disturbances in the kno\vn inputs. Since feedforward intercoupling has been eliminated by feedforward uncoupling control, these feedforward controllers are the same as if the two X t - Zi- Y t paths were completely independent systems. 566
l&EC F UNDAMENTALS
(b)
Figure 3. tem
Internal signal flow graph of sysa. Without control
b. Normalized with control
(b) Figure 4. The invariance concept a. Net forward channel o f transfer b. Forward channel o f transfer with parallel invariant control
3. FEEDBACK UNCOUPLING CONTROL.This mode of control is represented by - $21 and - $ 1 2 . Its purpose is to cancel the system feedback intercoupling loops to give a controlIed system with no intercoupling. Since feedforward intercoupling has been eliminated by feedforward uncoupling control, these controllers are the same as those obtained in Example 1 where there was no feedforward intercoupling. Because exact disturbance compensation can never be xhieved, dynamic uncoupling is often desirable even with simple problems in order to prevent disturbances in one output
from affecting others. ;Subsequent feedback stabilization is also made much easier with feedback uncoupling. I n fact, with total feedback uncoupling, only C single-feedback loop stabilization problems need be solved rather than a single C2dimensional problem. T h e rather involved method of Foster and Stevens (1967), which applies only to the class of problem just discussed, uses a n alternative formulation in which feedforward and feedback intercoupling are indistinguishable because they are both expressed as just feedback intercoupling. Such a formulation contains no more information than the terminal formulation where both types of intercoupling are expressed as just feedforward intercoupling. T h e simple problems considered so far have been to illustrate the structural concepts of feedforward and feedback uncoupling control and to show the advantages of a structural approach even in such simple cases. Most practical problems would be cases where there are non-control-actuating and control-actuating internal variables ( I S > 0). I n these more complex cases, the process model of the form of Equation 1 contains more information than is requi.red for the control analysis. In order to apply the previously developed structural design techniques, this superfluous information is systematically eliminated such that the resulting system configuration is structurally the same =: 0. as the case where Z T h e concept and usefulness of structural reduction can be most clearly presented by a specific comparative example, which will then be generalized. The example, which Haskins and Sliepcevich (1965) used in their experimental confirmation of the realizability of invariant control systems, will also point out some of the relative conceptual advantages of a structural viewpoint. Example 2. T h e system is the jacketed, continuous stirred tank reaction vessel (CSTR) without reaction shown in Figure 5. T h e assumptions made are : 1. Perfect mixing in the vessel itself 2. Imperfect mixing i n the jacket 3. Significant capacitance in the heat exchange wall 4. Coolant and vessel inlet temperatures, constant 5, All pertinent physical properties, constant 6. System well insulated lVithout going into the derivation of the differential equations, linearized equations of the following form are obtained i T,: for constant T C and (The primes denote perturbations from the steady state, and the bar superscripts denc’te Laplace transformed variables.)
+
Equations 11 provide the structural form of Figure 6. T h e control objective of the problem is the invariant control of T , with bV as the manipulable input, and Tl’c as a disturbance source. Haskins’ transfer matrix approach provides the following controller “alternatives” which will give invariant control.
+
I
i ’0
i7; 0
Ti ,VV Figure 5.
CSlR
Sc:hernatic of a
These three practical controllers had to be chosen from the infinite family of control functions and control configurations which satisfy the terminal form of the invariance principle. This problem was considered by Haskins to be a multivariable problem. However, from a structural point of view, it would seem unreasonable to call this a multivariable problem when only a single output is being controlled. Since there is only one disturbance source, TVc, there can be only a single over-all channel of information transfer between this input and the output T. T h e structural invariance principle would imply only a single invariant controller to compensate for the single channel of information. If this is true, Equations 12, 13, and 14 must be different forms of the same result. By the definition of the control objective as control of only T , the structural approach defines TI“ and Tco as internal vari-
jacketed
Figure system
6. Basic signal flow graph of CSTR
VOL. 6
NO. 4
NOVEMBER 1 9 6 7
567
’
Figure 7.
s-bii
Modified basic signal flow graph of CSTR system
t’ (b)
(C)
Figure 9.
Structural comparison of controllers
a. Feedforward control b. lnternol variable i’c, actuated control C. lnternol variable f’ri-actuoted control
Figure 8. Structurally reduced signal flow graph of CSTR system a Without control
b. Normalized with control
is seen to be identical to Haskins’ invariant controller of Equation 14. By a similar analysis, the controllers actuated by the internal variables T’c, and F‘tpare
ables. A structure identical to Figure 6 but which would be more appropriate is given in Figure 7 , where T,, and TW are seen in their true role as internal variables. T h e form of Figure 7 can be simplified using Mason’s rule to that of Figure 8, a, where
Ti--’ ~-
T’,
(15a)
Figure 8 also clearly points out the real single variable nature of the control problem. As in the structural analysis of the simpler class of problem the normalization indicated in Figure 8, b, is made where
T h e p’c to Tl-‘ invariant controller,
568
l&EC
FUNDAMENTALS
-b12(s)
- biz
= __
(lic)
all
which correspond to Equations 13 and 12, respectively. The relationship of the controllers of Equations 17a, b, c, to the system is shown in Figures 9, u, 6,and c, respectively. Since these feedforward controllers all originate on the T l ’ c to 2’1 loop, it is misleading to consider them as totally separate possibilities. They might better be thought of as equivalent possibilities to choose between according to convenience. For example, since temperatures are easier to measure than flow rates, feedforward control from one of the internal variables might be preferable in this case. T h e lT’c to 17’ controller involves a simulation of the total TT,’ to Z1 transfer. Feedforward from the internal variables may therefore be generally useful, since the need for simulation of part of the path (TT’, to the variable selected) is eliminated. The less actual simulation that is used the better because there are always simulation errors and analog components or digital computing time can be saved. Haskins referred to the nonzero controllers of Equations 12 and 13 as feedback controllers. However, as he notes, no ordinary feedback controller can produce invariant control without infinite gain requirements. Considering the system structurally, these internal variable actuated controllers simultaneously serve
both feedforward and feedback roles. This is illustrated in Figure 10, where it can be seen that the dashed control loop completes fonvard loops--for example, - 2 3 - F'co Z Z - T'Fv - ii" - Z1 - T'-and feedback loops-for exZ1 - T'. With internalample, T' - Zz - T ' , - 17' variable-actuated-invariant control, however, the feedforward role is maximized. By similar reasoning, the "feedback" uncoupling controllers also serve both feedforward and feedback roles with the feedforward role maximized.
-
A General Matrix Method
I n the previous examples and discussion, the usefulness of structural analysis has been established. The main advantage is that the feedforward and feedback intercoupling mechanisms are distinguishable in the formulation and can be eliminated by separate feedforu ard and feedback uncoupling controllers. Internal variable actuated control has also been found useful and is included in the general analysis. T o simplify this analysis, the following assumptions are made : 1. There are an equal number of manipulable inputs and controlled outputs (.M == C). Otherwise the problem must be solved stochastically, which is beyond the scope of this paper. 2. There are no critical manipulable input constraintsthat is. while there will be physical constraints on the inputs, the required control effort will usually be such that they are not exceeded. This is also to avoid a stochastic analysis. 3. Any time delays arc: negligible compared with the system time constants. This is to ensure realizable controllers without a stochastic analysis. Systems of the type dexribed by Equation 1 can be placed directly in the matrix form:
T h e superfluous information in Equation 18 can now be eliminated much as in Example 2 by using only the information required to make the system structurally similar to the case with no internal variables shown in Figure 3, a. From Equation 18 it is found that
S
which can be represented more concisely as
Equation 20 contains the non-control-actuating internal variables, Yiy21, explicitly. These can be removed using the relationship (also from Equation 18) T h e significance of the partitioning of the matrices and the somewhat involved nome.nclature is cIarified by the matrix signal flow graph of Figure 11.
Figure 10. Illustration of feedforward and feedback roles of a controller VOL. 6
NO. 4
NOVEMBER 1 9 6 7
569
Let BD(s)&9 be the diagonal matrix composed of the principal diagonal elements of B(s)LcA9 and let
Bt(s)Lcx9 = B ( s ) & x c~ ’ ~BD(s)&c;cJ ’
(27)
A structural operation analogous to that between Equations 1 and 2 can be carried out using Equation 27 in Equation 22 to give
YL%
=
(C,C)
[SIcxc - BYs)cxcl
x( CC/+KK +1fi f ) x i + (
-1
(C CIKIW [A(s)cx(c+KT.lf)
x
(c’)y (1) + Bt(s)(C,C)Y(c) ] B(s)cAr 1x1 c x c cx1
(28)
Since for the deterministic problem M = C, A(s)Lc${] is a square matrix. Letting AD(s)LcgY]be the diagonal matrix com( C 11) posed of the principal diagonal elements of A(s)CgA1f provides the normalized result of
,.
(C L / K / . l f )
Yc‘cx‘l = G (s)c x c [ N S ) c x ( C + K + B(s)
g$]Y&
(C/K/\f) .lil)X(C+K+ .U)x 1
+
+ B(s) c xc)Y c ex1
(29)
which is the form of Figure 8, b, where I = 0. In Equation 29,
G (s)c x c
=
ic
[SIc x c - B (S) &‘kc& ]-‘AD (S ) x .If (30)
(C L /K/W ( C If) -1 (C l - / K / . l f ) A(s)Cx(L.+K+.lf) C X C L +ic+.lf) = [ A D ( S ) C X . l f 1
B (s)kc$]
=
s(s)Lc&2=
[AD(s),$‘k:{)]-lB(s)
LcX I
(31) (32)
[AD(~)ic;(’{]]-lBt(s)&c’o xc
(33)
The general deterministic control function obtained directly from Equation 29 by application of the invariance principle is
x$i)l
(C K ) X ( K ) = -A(S)CkK KX1
x.{;)l
-
[A(S)(c,-lf) ex .If -
I-lf X M 1 x
~ (ccc,r)y (c oy ( e ) x r~ r(x1I i) -fi( s)ckc cxi
1.74)
Equation 34 includes invariant feedforward control from the control-activating internal variables and all possible inputs, as well as feedback dynamic uncoupling control. Since the elements of the principal diagonal of B ( J ) are ~ zero, ~ ~ it~does ~ not include feedback stabilization control. However, the dynamic uncoupling control does facilitate stabilization control analysis. As mentioned previously, stabilization will require only the solution of C completely independent single-variable problems which can be solved by familiar classical techniques. An additional incentive for both dynamic uncoupling and internal variable-actuated control is that a t least part of the disturbances due to the unmeasured inputs, X$.k)1;usually affect the outputs by intercoupling with other state variables. Propagation of these disturbances can be eliminated most effectively by such control. Another important point which has not been discussed is the initial organization of the system-that is? given a definite order of the elements of the Yd$)l vector, what is the corresponding order of the X.{:i)1 vector which allows control and gives the simplest control functions? If the elements of all coefficient matrices of Equations 20 and 21 were nonzero, there would be no issue. However, with some nonzero elements there is generally a certain order which is preferable. The only absolute rule is that X-{$1 must be ordered such that the principal diagonal of the A(s)(g$!)f matrix contains no zero elements. Unless this condition is met, the control function of Equation 34 will not meet the control objectives. A difficulty which is encountered with any multivariable problem is often referred to a s the ”curse of dimensionality.” 570
l&EC FUNDAMENTALS
The matrix manipulations, involving functions of the Laplace operator, become tedious with higher order systems. The authors have analytically reduced a problem of the type C = 2, I = 0, S = 3, but analysis with lY = 4 would be bothersome, = 5 probably prohibitively complex. Still, this is and with much better than the corresponding terminal analysis. Terminal analysis essentially requires a total structural reduction of order C I f S as compared with the reduction of order ,V required by the structural analysis method. Since the difficulty of reduction increases with the power of the order, more complex problems can be treated analytically by the structural method than the terminal methods. Analytic analysis is not essential to structural analysis. Partial internal frequency response analysis can be conducted either numerically or with an analog computer to give sufficient information to determine the control function of Equation 34 even with unstable systems. For stable systems, the Bode plots obtained from such an analysis can be used to determine approximations to the elements of all matrices of Equation 29. \\‘ith unstable systems, the elements of all matrices except the diagonal matrix &(s)cXccan be determined in this manner. Since & (s)e x does not occur in Equation 34, its determination is not essential to deterministic structural analysis. Nevertheless, it can be determined in an indirect manner. Any stabilizing feedback loops (no more than C will be required) can be artificially introduced and the & ( s ) c X C of the artificial system, G’(s)cxc, can be determined. The G(s)cxc of the real system can then be determined by carrying out the inverse of the artificial stabilizing transformation on &’(s)c x c. An important feature of an internal analysis of this type is that a lower order internal approximation provides a higher order terminal approximation when C > 1 and I > 0. A future article will compare the approach of this paper with that of Bollinger and Lamb (1962) and discuss the implications of structural analysis with stochastic control problems.
+
Nom en c lat u re
ufJ
= input coefficient
u t j ( s ) = input coefficient with reduced structure Ztl = normalized input coefficient
A = input coefficient matrix b,, = state variable coefficient b,,(s) = state variable coefficient with reduced structure normalized state variable coefficient bY state variable coefficient matrix B index referring to the number and identity of conC trolled state variables forward loop transfer matrix in structural analysis G feedback matrix with terminal analysis H identity matrix I index referring to the number and identity of control I actuating internal variables index referring to the number and identity of known K inputs index referring to the number and identity of manipM ulated inputs index referring to the number and identity of nons control-actuating internal variables terminal transfer matrix element pi, terminal transfer matri, P Laplace operator s time t temperature of contents and effluent of CSTR T inlet temperature of cooling medium Tci
= outlet temperature of cooling medium
literature Cited
= temperature of feed to C S T R = temperature of the heat exchange wall
Bollinger, R. E., Lamb, D. E., I N D . ENG.CHEM.FUNDAMENTALS 1,
= index referring to the number of identity of unmeas-
ured inputs
V TV ll, XI
TALS4,241 (1965).
= volume of C S T R = flow rate through CSTR = coolant flow rate = inputs = Laplace transformed of inputs =
245 (1962).
Calvert, S., Coulman, G., Chem. Eng. Progr. 57, 45 (1961). Foster, R. D., Stevens, \V. F., A.I.Ch.E. J . 13, 6 (1967). Haskins, D. E., Sliepcevich: C. kf., IND.E N G . C H E h $ . FUSD.AME~.-
input variable vector transform
= state variables
Laplace transformed state variables state variable vector transform = internal summing node in structural analysis = normalized intrrnal summing node in structural analysis =
=
Kalman, R . E.. J.S.Z.A..Z/I. Control, Ser. A 1, 152 (1963). Kavanagh, R. J., A I E E Trnns 7 6 , Part 11, 95 (1957). Luecke, R. H., McGuire, hi. L., “.Analysis of Optimal Composite Feedforward-Feedback Control,” 6lst X.1.Ch.E. National Meeting, Houston, Tex., 1967. Petrov, B. S . :“The Invariance Principle and the Conditions for Its Application during the Calculation of Linear and Nonlinear Systems,” “Automatic and Remote Control,” Proceedings Congress of the I.F..A.C., Moscow, 1960: p. 117, Butterworth‘s, London, 1961. Verneuil, V. S.: Jr., “Feedforward Control of Multivariable Processes: Continuous and Sampled Data,” Ph.D. thesis in chemical enTineerin5, University of Delaware, 1965. RECEIVED for review September 26, 1966 ACCEPTED June 22. 1967
STRUCTURAL AND TERMINAL ANALYSIS I N MULTIVARIABLE PROCESS CONTROL G A Y L O R D G. G R E E N F I E L D A N D T H O M A S J . W A R D Chemical Engineering Department, Clarkson College fo, Technology, Potsdam, LY. Y .
A method of control analysis for continuous multivariable processes uses a structural description of a process. It makes maximum use of available process information to yield optimum feedforward and feedback control configurations. Comparison is made with the corresponding terminal analysis method, and examples illustrate the significant advantages of the structural approach.
N A
recent article Greenfield and Ward (1967) introduced a
I method of multivariable process control analysis which allows maximum use of av.ailable information on a process system. The method starts with a mathematical description of the process which corresponds to the actual physical structure of the system. Structural information which is superfluous to the control objective is systematically eliminated, leaving only the physical structure essential to the control analysis. Many conceptual errors made in the past due to less specific mathematical description are thus avoided, since the relationship of the control system to the process system is more explicit. Horowitz (1 959) points out that as :many controllers can be formulated as there are independent measures of the system performance. T h e structural method allows for the use of all independent measures of the system performance to provide the maximum number of control degrees of freedom. These additional control degrees of freedom are used to impose additional control constraints or objectives on the system. The resulting control configuration provides feedforward, internal-variable-actuated, and dynamic uncoupling control. Appropriate feedback control degrees of freedom a.re left uncommitted to allow subsequent and independent feedback stabilization control analysis. The control functions associated with this configuration contain the minimum number of state variables, so that they can be implemented with a minimum of approximation. Other advan-
tages include maximum .‘fredforward” compensation of unknoivn input disturbances, and less sensitivity to model parameter errors, since all possible information on the performance of the actual system is used. Previous methods of analysis used a “black box’‘ or terminal mathematical description of the process. This eliminates a portion of the physical structure critical to control analysis. Greenfield and Lb‘ard (1967) discussed some of this previous rvork. This paper provides a direct comparison with the work of Bollinger and Lamb (1962, 1963, 1965). while illustrating the use of the structural method in two applications. Bollinger and Lamb’s work is an extensive development of a general method of multivariable control analysis based on a terminal description of a process. If a process model is available, however, their terminal description eliminates useful process information and leads to unnecessary difficulties and complexities in the control analysis. Structural Formulation
Any linear, lumped parameter single or multistage process can be described by the set of equations
2
dy, = aI,xj dt 3=1
+
VOL. 6
m
biryr
i
=
1, 2 , . . .,m
(1)
k=1
NO. 4 NOVEMBER 1 9 6 7
571
