1171
Znd. Eng. Chem. Res. 1991,30,1171-1181 Cirr= upper bounds for disaggregated capacity expansions, defined in (25) CI(t) = capital investment limitation corresponding to period t Hit = time for which plant i is available for operation during period t NC = number of chemicals in the network NEXP(i) = maximum allowable number of expansions for process i NM = number of markets NP = number of processes in the network NT = number of time periods considered Qio = existing capacity of process i at time t = 0 QEb = lower bounds for the capacity expansions BEit = upper bounds for the capacity expansions
Variables Oijt = production amounts defined in (9)
= capacity expansion of plant i made in period t in order to serve production requirements up to period T(T 2 t ) Nij, = number of batches of product j in plant i during time period t NPV = net present value Pi,, = amount of product j purchased from market 1 at the beginning of period t Qir = total capacity of the plant of process i that is available in period t BEi, = capacity expansion of the plant of process i that is installed in period t cpit,
riit = production rate of
the main product j in process i during period t SJ, = amount of product j sold in market 1 at the end of period '
t
Tij,= time during period t that is allocated in process i to the production scheme characterized by the main product j Wjjt = amount of flow of product j tolfrom process i during time period t y~ = decision variable that is 1whenever there is an expansion for process i at the beginning of time period t and 0 otherwise
Literature Cited Austin, G. T. Shreve's Chemical Process Industries, 5th ed.; McGraw-Hill: New York, 1984. Sahinidis, N. V.; Grosemann, I. E. Reformulation of the Multiperiod MILP Model for Capacity Expansion of Chemical Processes. Accepted for publication in Oper. Res. 1990a. Sahinidis, N. V.; Grossmann, I. E. Reformulation of Multiperiod MILP Models for Planning and Scheduling of Chemical Processes. Comput. Chem. Eng. 1990b,in press. Sahinidis, N. V.; Grossmann, I. E. MINLP Model for Multiproduct Scheduling on Continuous Parallel Lines. Comput. Chem. Eng. 199Oc,in press. Sahinidis, N. V.; Grossmann, I. E.; Fornari, R.;Chathrathi, M. Long Range Planning Model for the Chemical Process Industries. Comput. Chem. Eng. 1989,13(9),1049-1063. Received for review September 27, 1990 Accepted December 12, 1990
Geometric Analysis of the Global Stability of Linear Inverse-Based Controllers for Bivariate Nonlinear Processes Ching-Wei Koung and John F. MacGregor* Department of Chemical Engineering, McMaster University, Hamilton, Ontario, Canada L8S 4L7
Many nonlinear processes have highly structured model mismatches that make a controller, based on an inverse of a linear nominal model, more robust than is predicted from the theories assuming unstructured, norm-bounded model mismatch. This work focuses on the robust stability problem of inverse-based controllers for 2 X 2 processes where process nonlinearities are the dominant source of model mismatch. Some geometric criteria are proposed to examine whether these linear controllers can stabilize a nonlinear process over a wide range of operations. High-purity distillation column control is used as an example of physical processes with severe nonlinearities and highly structured model mismatch.
Introduction All linear model-based control systems can be expressed in the internal model control (IMC)structure (Frank, 1974; Zames, 1981; Garcia and Morari, 1982) shown in Figure 1. In this structure the conventional feedback sontroller is decomposed into a model prediction block (P)and_an approximate model inverse block (Q). The model P is obtained by approximating the nonlinear process P by a linear model about a nominal steady state. Different designs have different Q specifications. Q is always an approximation _of the inverse of the nominal linear model of the process (P), and in particular, the steady-state Q must be the exact inverse of steady-state P (i.e., Q(1) = P%)) in all designs in order to ensure no steady-state offset (i.e., provide integral control action). Hence, Q is referred as the 'inverse-based controller" (Skogestad and Morari, 1987). The closed-loop stability and satisfactory per-
* Address correspondence to this author.
formance can be easily obtained for open-loop stable processes if the model is perfect (Garcia and Morari, 1982). However, no models are perfect in the real world. A filter F is usually introduced to ensure closed-loop stability in the presence of model mismatch. To find the conditions where a model mismatch will cause instability, we must first be able to characterize the model mismatch. The relationship between a single-input-single-output (SISO) process and its model can be solely characterized by a scalar, that is, the magnitude of model mismatch. An application of the relationship between the magnitude and the stability is the gain margin design of SISO controllers in which we choose the controller gain based on an expected bound of increases in process gain to ensure closed-loop stability. On the other hand, a multiple-input-multiple-output(MIMO) process should have its model mismatch represented by a matrix. As the dimension of a process gets larger, too many scalars are required to characterize the mismatch to yield any general and useful criterion. Recent research (Doyle and
0888-588519112630-1171$02.50/0 0 1991 American Chemical Society
1172 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 D
presence of model mismatch, if and only if Re (Xi(GG-’)) > 0 V i
Figure 1. The IMC structure. (a) Using bounded vector norms; (b) using bounded vector lengths and angles.
Stein, 1981; Grosdidier et al., 1985; Morari and Zafiriou, 1989) simplified the characterizations of MIMO mismatch by a single magnitude only (measured by a convenient matrix norm) and developed some closed-loop stability criteria. These multivariable stability criteria provide bounds on the mismatch magnitude in the same way as in SISO systems. The essential assumption behind these magnitude criteria is that all mismatch matrices with a specified magnitude can exist for the process. This is the so-called “norm-bounded assumption”, or “unstructured uncertainty description”, for model mismatch. However, in most situations processes are only free to change in a few directions and hence can only exhibit a very limited subset of mismatches with a specified magnitude. In this paper, we refer these processes being subjected to “structured model mismatch”. There are many sources of model mismatch: process input/output nonlinearities, input uncertainties, timevarying phenomena, inaccurate model-parameter identification, low-order model approximation, etc. Many of these mismatch sources are stochastic in nature and characterizable only by an unstructured, norm-bounded assumption. However, the process input/output nonlinearities (probably the most prominent source of mismatch in chemical processes) are deterministic in nature and determined by definite physical and chemical factors. As a result they are usually highly structured. Therefore the unstructured, norm-bounded assumption is seldom valid for a process with nonlinearities as its dominant source of model mismatch, and this is the major reason for the contradiction between the prediction from robustness theories and actual practice. 111-conditioned nonlinear processes are predicted by the magnitude criteria to be destabilized by so small a mismatch magnitude that they are virtually uncontrollable with inverse-based controllers. Nevertheless, these controllers are often quite robust in practice. McDonald et al. (1988) have discussed some aspects of structured versus unstructured model mismatch descriptions on the robustness of high-purity distillation control. Although we recognize that all systems contain some elements of unstructured uncertainty, we are interested here in considering only those effects of structured uncertainty arising from nonlinearities. Closed-loop stability not only depends on the model mismatch but also on the controller design and tuning. A comprehensive analysis, without the norm-bounded assumption, would involve too many variables to yield general results. To simplify the problem and gain a general understanding, we shall investigate a situation where the instability of an inverse-based control system has nothing to do with the controller design and tuning and is solely due to the presence of model mismatch. Garcia and Morari (1985) have shown that a diagonal first-order exponential filter OIfiCl,
vi
(1)
can always stabilize an inverse-based controller in the
(2)
where_Xi is the ith eigenvalue of a matrix and G p P(1) and G = P(1) are the steady-state process gain and nominal model gain matrices, respectively. This condition indicates that, as long as the steady-state gains of the true process and its model satisfy (2), a stable closed-loop system with no steady-state offset can be obtained in spite of the model mismatch simply by making fi’s sufficiently large. In this paper, we will refer a control system satisfying this condition as “IMC-filter stabilizable”. If the steady-state condition in (2) is violated, no inverse-based controller can be stabilized using the IMC filter. In contrast to those magnitude criteria that assume unstructured, normbounded model mismatch, (2) is valid for any type of model mismatch. In SISO systems, this condition corresponds to the well-known uncontrollable situation where a process and its model have opposite signs in their steady-state gains. For -MIMO processes, (2) requires that all the eigenvalues of GG-’ lie in the right half-plane of the complex plane. However, it is not a simple matter to see how changes in G and G can lead to violation of (2). The majo: concern of this paper is to develop conditions on G and G that will ensure that (2) is not violated, and thereby provide some understanding of how nonlinearities and other stuctured uncertainties affect control system stability. The methods developed in this paper are intended for analysis rather than synthesis of linear inverse-based control systems. We will focus on the IMC-filter stabilizability of 2 X 2 processes that have their model mismatch dominated by process nonlinearities. Therefore significant model mismatch exists only when a process is not at its nominal steady state (Le., its nominal operating condition), and this type of mismatch will always be structured. When a nonlinear process changes to another operating condition, the IMC-filter stabilizability of a controller based on the original nominal model can be easily checked by use of (2) where the locally linear process is compared with the nominal model. The objective of this paper, however, is to develop some global criteria that are capable of predicting the IMC-filter stabilizability of inverse-based controllers to control a nonlinear process in a wide range of operating conditions. Since only 2 X 2 systems are considered, this objective can be achieved by using geometric interpretations and analysis of IMC-filter stabilizability, which are presented in the following. High-purity distillation will be used as an example to illustrate how the criteria can be used to analyze the stability of a particular nonlinear process. Mathematical Preliminaries
Steady-State Model Mismatch Due to Process Input/Output Nonlinearities. Many n-input-n-output processes can have their steady-state models in the following form: jji =
fi(al,...,Un)
i = 1, n
(3)
The bars over variables indicate that they are the original variables (i.e., they can be used to develop models from first principles). If deviation variables are used, i.e. y 1. 19. 1 - jj9 1
and u. 1 I u1. - 4 -0
(4)
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1173
Figure 2. Model mismatch descriptions.
where superscript 0 denotes nominal values, (3) can be rewritten in the form of a Taylor series:
(high-order terms) (5a)
components of a gain vector are the values of the process output changes due to an unit change of that input alone. Similarly, AG can be decomposed into model mismatch vectors of each gain vector: Y = G*U GiUl + ... + Gnu,, (74
= (GI+ Al)ul n
=
j=l
+ Aij)uj
i = 1, n
(5c)
where Aii =
IF( a(afi/auk)uk) + (high-order terms) 2k-1
or we may put (5c) into a matrix form:
+ ... + (fin+ An)un
According to the norm-bounded assumption, there are no correlations among the elements in each Ai nor among the Ai)s. For 2 X 2 systems, the geometrical interpretations can be easily visualized in a plane as shown in Figure 2a, where the magnitude of model match in each gain vector can be represented by the radii (r, and r2) of the disks in Figure 2a. Even this bounding of the vector norms (IIAill) is a more structural model mismatch description than bounding the matrix norm (Il&ll). Nominal Model Interpretations i n Polar Coordinates. As an alternative to using Cartesian coordinates, where a gain vector is expressed in terms of its effects on the process outputs explicitly, one can use polar coordinates to describe the gain vector by its length and its angle with respect to the output coordinate axes (see Figure 2b). In this polar coordinate 2 X 2 system, the nominal model gain matrix can be expressed as
G is called the nonlinear steady-state gain matrix and it
is a function o f the input vector U. is the nominal linear steady-state gain matrix. Process nonlinearities make the true gain matrix at another operating point (Le., another specified U )deviate from its nominal value, and hence 4, which has elements Aij, is referred as the additive model mismatch matrix due to process nonlinearities. In the field of robust process control, it is very difficult to investigate robustness properties using a AG whose elements are correlated or constrained as U changes. As a result, AG is usually characterized by its magnitude measured by some norm, IIAoll. When Euclidean norm is used, the model mismatch can be geometrically interpreted as a sphere in n2-dimensional space. This unstructured, norm-bounded type of model mismatch description makes further mathematical developments possible, but is often far from reality. Since a Aij involves partial derivative terms of an nonlinear output function with respect to all the inputs, it is often constrained and has strong correlation with other Aij terms when U changes. The normbounded type of model mismatch descriptions do not allow Ai;s to have such restrictions. All model mismatch within the sphere must be included. Geometrically, we can interpret G in n-dimensional space as a combination of the column vectors. The column vector is called the gain vector of a specific input. The
(7b)
(8) where li and _ei are the length and the angle of a nominal gain vector Gi. In the robustness literature several simple measures of the nominal gain matrix are often used to infer its sensitivity to modeling errors. The most popular measures are the relative gain array (RGA) (Bristol, 1966) and the condition number. Since we will use these measures later, it is useful to show their geometric interpretations. RGA. With the use of the polar coordinate representation in (8), it is easily shown that for a 2 X 2 system the 1-1 element of the RGA is as follows (Koung and Harris, 1989): 711
= 1/( 1 -
gg,)
= 1/(1--)
- cos 0, sin 0,
(W sin fi where @ = (0, - 0,) is the angle between two gain vectors and hence is a measure of collinearity. Since the numer-
1174 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
ator has a range of -1 to 1, the yll element is large if and only if the denominator is closed to 0, that is, if and only if the gain vectors are nearly collinear. On the other hand, processes with orthogonal gain vectors have their yll's within a range from 0 to 1. The magnitudes or lengths of the gain vectors have no effect on the RGA elements, which are determined solely by the angles. Condjtion Number. The condition number K of a matrix G can be defined in many ways. It is often defined as the ratio of the largest to smallest singular value of the matrix. Alternatively it can be defined in terms of the Euclidean norm. With the use of this latter definition and the polar representation in (81, it can be shown that the condition number of a 2 X 2 system is
Well-conditioned processes (Le., where KE approaches ita minimum value of 2) arise when the gain vectors are nearly orthogonal and of equal length, while ill-conditioned processes (Le., where K E approaches infinity) arise from gain vectors of very different lengths, or from collinear gain vectors (Le., sin j3 = 0). Note that unlike the RGA elements KE also depends on the lengths of the gain vectors. Model Mismatch Interpretations in Polar Coordinates. In polar coordinates model mismatches can be specified as errors in the vector lengths or angles. For 2 X 2 systems, the true process gain matrix can be expressed as
]
nlh cos (el + ad nzl2 cos (e2 + a21 G = [nlll sin 0 3 1 + all nd2 sin (e2+ a2) (11) and ni > 0, Iail < A for i = 1, 2, where ni is the multiplicative error in the length and ai is the additive error in the angle of the ith gain vector (Figure 2b). In contrast to the Cartesian representation in (6a) where the error terms (Aiis)affect each gain element independently, the ni and ai correlate the gain elements of ith gain vector. Correlations between the two gain vectors are allowed by forcing certain relationships between ni and nj,or between ai and a? It will be shown later that the polar coordinate representation of model mismatch provides not only the basis of the theorems proposed later, but also a more convenient description of the nonlinearities in some physical processes such as high-purity-distillation columns than the Cartesian representation. The magnitude of the model mismatch, Act can be measured in terms of its Euclidean norm:
stabilizability of inverse-based controllers when a process is moved in a wide operating region where process input/output nonlinearity is the dominate source of model mismatch. Even though (2) is readily available to test for a locally linear process at any specific operating point, a direct application of (2) to asses the global IMC-filter stabilizability in a region means exhaustive tests on as many operating points as possible. In this section, some simple criteria are derived from (2) to identify the geometrical differences between the gain vectors of the true process and those of the nominal process which cause IMC-filter unstabilizability. These geometric criteria provide the basis for the global stability criteria proposed later. These geometric criteria are also very useful for identifying situations when inverse-based controllers for ill-conditioned processes are or are not sensitive to model mismatch. Geometric Criteria for IMC-Filter Stabilizability. For 2 X 2 systems, the condition in (2)-is true if and only if both the determinant and trace of GG-' are positive. In the following we will derive some geometric criteria which determine the signs of the determinant and trace. With the uncertainty description in ( l l ) , GG-' in (2) becomes:
G&'
=
nlZl cos (0, + al) n2Z2cos (0, nlZl sin (0, + a l ) %12 sin (e,
1
zl
COS
el
ll sin el
1, COS e2 l2 sin 0,
I-'
=
1.
+ az) + q)
l x sin fi
nl sin (el + all cos O2 - -nl sin (el + al) sin e, + n2 sin (e2 + q )cos el n2 sin (0, + a,) sin el nlcos (el + all cos € -I,-nl cos (el + all sin e2 + n2 cos (0, + a2)cos el n2 cos (e, + q )sin el
(13)
where j3 8, - el. The determinant and trace of the above matrix can be derived to be
We can see from (14) that the eigenvalues are determined by one nominal process characteristic B which is a measure of collinearity of the gain vectors and, of course, all the modeling error terms in (12) (nl, n2, al, and a2). When 2 there is no model mismatch (i.e., ni = 1 and ai = 0 for i IlA& = [niliCOS (ei + ai) - ii COS e , ] ~+ i=l = 1,2), the trace and determinant are 2 and 1, respectively, [nilisin (ei+ ai) - li sin 1 3 ~ ] ~ j ~ / ~which are consistent with-the fact that a perfect model has all the eigenvalues of GG-' equal to 1. =[112(n12- 2n1 cos a1 + 1) + We will interpret the true process gain vectors (Gl, G2) &2(n22- 2R2 COS a2 + 1)]'/' (12) as a result of rotating and lengthening/shortening the nominal vectors (Gl, G2). Consider this rotation/lengthThe norm-bounded assumption, which assumes that all ening in two steps. First, let G be the intermediate gain of the four error terms in (12) (Le., ni's and ai's) can inmatrix in which only one nominal gain vector is changed. dependently have any value as long as the resulting At each step, the fixed gain vector divides the space into magnitude is within a given bound, excludes the following two half-plana, and the nominal half-plane is defined here situations: (1)A model mismatch has correlations among as the half-plane in which the nominal gain vector to be n:s and/or ai's. (2) Any error term in (12) is constrained. changed is located (Figure 3a). We have the following Under either situation, we say that the model mismatch theorem for the first change: is structured. Theorem 1. For 2 X 2 processes, det (G&) > 0 if and Robust Stability of 2 X 2 Systems: A Geometric only if the changing gain vector remains on its nominal View half-plane (Figure 3a). The proof is given in Appendix I. As mentioned in the Introduction, the objective of this Now in the second step, we obtain the other true gain paper is to derive some global criteria for the IMC-filter vector by rotating/lengthening the other nominal gain
{c
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1175 Table I. Example case
G
1
-0
K&G)
711
det (G)
0.91
0.11
8.5
-1.4
0.1
2
150 -40 -0.5 -0.41
456, l.o
3
[ 2:; 2 11
-0.6,1.6
0.64
0.14
0.06,1.0
0.9
0.006
-1v-l
0.9
0.1
9.1
2.8
-0.1
0.56,5.0
0.5
0.5
2.0
1.0
-0.25
4 5 6
,/'
a
[ $? 0.4 [ -0.5 -0.4 ] [ t5 0.5
nominal half plane for G
64
0.64
100
0.6
4.80
-41
0.09
0.4
-0.005
-39
150
,
(a) (b) Figure 3. Nominal half-plane criteria for det
(a) tr > 0
vedor. Algebraically, the combined effect of changing both nominal vectors to the true gain vectors is given by G&' = (GG-l)(G&l)
(15a)
Since (15b) det (G&') = det (GG-l) det (G&l) we can apply Theorem 1twice (i.e., for det (GG-') and det (G&'>) and have a corollary of Theorem 1: Corollary 1. Suppose that the gain vectors of a 2 X 2 process are obtained by changing the nominal vectors in the two steps defined in (15a), then det (G&') > 0 if and only if the process and nominal gain vectors: (1)remain on their nominal half-planes in both steps or (2) are on the other side of their nominal half-planes in both steps (Figure 3b). As for the trace, since the right-hand side of (14b) has additive terms, it is impossible to derive a general and simple geometric criterion that is both necessary and sufficient for the trace to be positive. However, some general sufficient conditions are obvious. We have the following geometric criteria: Theorem 2. Define an original half-plane as the half-plane, divided by one nominal gain vector, in which the other nominal gain vector is located (Figure 4). tr (GO-') is (1)positive if both gain vectors remain on their original half-planes (Figure 4a) or (2) negative if both gain vectors are outside their original half-planes (Figure 4b). The proof is given in Appendix I. There is no general geometric criterion to determine the sign of the trace, if one gain vector is outside while the other is inside ita original half-plane. Corollary 1 and theorem 2 transform the algebraic criteria on the determinant and trace of a two-dimensional G&I for the IMC-filter stabilizability into criteria on the geometric differences between a process and ita nominal model. It is important to note that the geometric criteria
(b) tr e 0
Figure 4. Original half-plane criteria for tr (Gd-').
only involve the changes in the relative positions of gain vectors (i.e., it is the matter of angle mismatch) and have nothing to do with the mismatch in vector lengths. Therefore, we could have an infinite mismatch magnitude in (12) due to the structured model mismatch where only the vector lengths are increased and the control system is always IMC-filter stabilizable. To better understand the geometric criteria, consider the following example. Example. Consider a linear 2 X 2 process that has its nominal gain matrix and a few matrix characteristics as follows: (16a)
= 0.9,
a= 0.1,
= 2.8, det (G)= -0.1 (16b) Suppose that the nominal model is used in an inversebased controller to control the true process with six hypothetical mismatch gain matrices listed in Table 1. The gain vectors of these six cases are plotted in Figure 5. Case 1has only one gain vector, G2,on the other side of its nominal half-plane, and therefore the determinant is negative (i.e., one eigenvalue has negative real part). None of the gain vectors is rotated outside the nominal and original half-planes in case 2, and hence both determinant and trace are positive (i.e.,both eigenvalues have positive real parts). In case 3, G2is first rotated and shortened to become G2without being on the other side of the nominal half-plane. Then G1is rotated to the other side of ita nominal half plane, divided by G2, and consequently the determinant is negative. Cases 4 and 6 are similar to case 2 where both gain vectors remain in their half-planes. In case 5, both gain vectors are on the other side of their original half-planes, resulting in positive determinant and negative trace (i.e. both eigenvalues have KE(G)
= 9.1,
711
1176 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
Case 4
Case 5
Case 6
Figure 5. Gain vectors in Table I.
negative real parts). The eigenvalues of G&' listed in Table I confirm all these arguments. The above hypothetical cases not only illustrate the proposed geometric criteria, but they also show that no single process characteristics such as those listed in Table I can assess IMC-filter stabilizability. It is obvious from (2) that n scalars are needed to infer IMC-filter stabilizability for an n X n system (Le., n eigenvalues must have positive real parts). Consequently, any attempt to characterize a multivariate model mismatch by a single scalar such as the maximum singular value (a), the minimum singular value (a),the condition number, RGA, or the matrix determinant (Morari and Zdiriou, 1989) can never provide a necessary and sufficient condition for this purpose. In other words, we cannot find any correlation between the model mismatch in terms of these characteristics and the stabilizability. Table I clearly shows that as the mismatch increases the controller may be stabilizable, while a true process with virtually no mismatch in some characteristics is not stabilizable. Geometrically, we can explain why none of these characteristics by itself is capable of inferring the stabilizability. 1. The maximum singular value of a process gain matrix is a measure of its magnitude. We can deliberately increase the gain vector length (G, in case 2) to have a dramatically large mismatch in matrix magnitude and still get a controllable inverse-based control system, as long as we detune the controller enough. On the other hand, the maximum singular value is not affected by pure rotations of the gain vectors (no length changes), and yet certain rotations will lead to instability according to corollary 1 (case 1) and theorem 2 (case 5). 2. The minimum singular value provides a measure for how close the matrix is to singularity. A process with the same minimum singular value as its nominal value can be unstabilizable. Geometrically this situation occurs when we project the G2to other side of its original half-plane and keep G 1unchanged (case 1). On the contrary, a process can have significant mismatch in and still be IMC-filter stabilizable. For example, can be close to 0 when the gain vectors are rotated to become collinear (case 4) or equal to a when the vectors are orthogonal (case 6). 3. Using the geometric expression for the condition number is (lo), it is obvious that a true process can have a condition number much larger than that of the nominal process and the system still be stabilizable, or a condition number the same as (case 5) or smaller than (cases 1 and
3) that of the nominal process, and not be stabilizable. All these cases seem contrary to the usual association between condition number and stability. 4. Similarly, from (Sc), the RGA element, yll, for the true process can be equal to 1 (case 6), smaller than 1(case 2), or even negative (case 4), and the control system based on a nominal model with yll larger than unity is stabilizable. Negative RGA implies that it is not controllable if multiloop control is used (i.e., the pairing must be changed). However, the full model inverse control can handle such model mismatch. On the other hand, the same RGA for a process and the nominal model does not guarantee stability (case 5). Finally, since the positive determinant of GG-' is necessary for IMC-filter stabilizability and
the condition that the determinants of a process and its nominal model have the same sign (cases 2,4, and 6) is also only necessary, and by no means sufficient (case 5). The absolute values of the gain matrix determinants have nothing to do with IMC-filter stabilizability. Applications to Ill-Conditioned Systems. It is generally believed that ill-conditioned processes, which have high condition numbers, are very difficult to control due to their high sensitivity to modeling errors (Skogestad et al., 1988; Morari and Zafiriou, 1989). Typically, a very small magnitude of model mismatch will destabilize a controller based on an ill-conditioned nominal model. Under the norm-bounded assumption, Grosdidier et al. (1985) have shown that any control system with integral action is not controllable regardless of the controller designs and tuning, if and only if
where 11-11 is any compatible norm and K(G)is based on the chosen norm. Therefore the inverse of the condition number of a nominal process is the upper limit for the ratio of mismatch magnitude to the nominal gain matrix magnitude. The higher the condition number of a nominal model, the less mismatch magnitude can be tolerated. In addition, processes with large RGA elements have been shown to be virtually uncontrollable with an inverse-based controller (Skogestad and Morari, 1987; Yu and Luyben, 1987). For 2 X 2 systems, the proposed geometric analysis
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1177
"la
(a)
(b)
(4
Figure 6. Ill-conditioned processes.
for IMC-filter stabilizability is perfectly consistent with these previous works. The sin @ is the common denominator of the RGA (9),the condition number based on Euclidean norm (lo), and the determinant of GG-' (14a). Therefore, a process with nearly collinear gain vectors (i.e., @ 0, T) will have large RGA elements and a high condition number, and the determinant of GG-' may easily become negative with a very small mismatch magnitude due to angle changes. The geometric criteria are very useful to identify exactly what kind of model mismatch will make an ill-conditioned control system very sensitive to the mismatch magnitude. Consider again the nominal process which is ill-conditioned because of its nearly collinear gain vectors. For the model mismatch that affects only one gain vector, very small angle mismatch (Le., 0 or T - 0)can rotate that vector outside its nominal half-plane and therefore the control system is not IMC-filter stabilizable (Figure 6a). The higher the condition number due to the increasing collinearity, the smaller mismatch magnitude is required. On the other hand, a nominal process with orthogonal gain vectors has much larger tolerable angle mismatch. However, if the orthogonal process is ill-conditioned because one gain vector has very small length, a very small mismatch magnitude is enough to move the short vector to the other side of its nominal half-plane and then GG-' has negative determinant (Figure 6b). Therefore, if the model mismatch in an ill-conditioned process is unstructured, inverse-based controllers should not be used to control such a process. On the other hand, when the model mismatch in an ill-conditioned process is highly structured, the control system can be much more robust than is predicted to be from the condition number. For example, if a model mismatch only affects the vector lengths without rotating any vector, corollary 1 and theorem 2 guarantee both determinant and trace are positive. As a result, the control system is always IMC-filter stabilizable regardless of the mismatch magnitudes. Another example is where the model mismatch affects the vector angle changes in such a highly correlated fashion that the gain vectors are rotated with the same angle mismatch (for example, in high-purity-distillation columns): a1 = a2 (19) In this situation (Figure 6c), the conditions in corollary 1 are always satisfied and therefore the determinant is positive. It is even easier to see this result directly from (14a)when (19)is imposed: det (G&') = n1n2> 0 (20)
-
As to the trace, we cannot determine whether it is positive or not using the geometric criteria because this situation does not meet either sufficient condition in Theorem 2. However, for a particular model mismatch that has the
Figure 7. Input contour plot and process gain vectors.
same multiplicative error in vector lengths (i.e., nl = n2 = n), the trace in (14b) becomes tr (GG-') = [n(sin @ cos a1- sin a1 cos 0)+ n(sin p cos a1 + sin a1 cos @)]/(sin0) = 2n cos "1 (21) Therefore an ill-conditioned process with model mismatch structures in (19)and (21)can tolerate very large mismatch magnitudes as long as the rotation does not exceed ~ / 2 . Global IMC-Filter Stabilizability. So far we have used the geometric criteria to assess the stabilizability of any specific pair of a linear process and its nominal model much in the same way as one would use (2). However, when these criteria are applied to a process with its dominant model mismatch arising from process input/output nonlinearities, we can take advantage of the continuity of nonlinear behaviors and use the geometric differences proposed in the early theorems to infer globally the IMC-filter stabilizability for certain nonlinear processes. The nonlinear steady-state relationship between the inputs (ul,u2)and outputs (yl, y2)of a 2 X 2 process is best represented by an input contour plot (Figure 7). The values of the two outputs under constant values of u1 are plotted as the solid contours, while the dashed contours are those under constant values of u2 in this figure. An intersection point of one solid and one dashed contour gives the output values corresponding to a specific pair of input values (Le., an operating point). The gain vector of u1 at any operating point is along the tangent line, drawn from the intersection point, of the u2contour. Similarly, the gain vector of u2 is along the line tangent to the u1 contour at the same point (Figure 7). These contour plots are usually easily obtained from the nonlinear model. For linear processes, the input contour plot has the following characteristics: (1)parallel straight lines for each input; (2)a pair of contours of different inputs that intersect exactly once; (3) equal distances between two parallel contours if the contours are plotted for the same amount of change in that input. Consequently, all operating points have the same gain vectors in a linear process and the stabilizability is not a problem at all. Nonlinear processes are always characterized by curved and/or nonuniform contours. In addition, multiple steady states may exist in a nonlinear process. An intersection of two contours of the same input variable indicates that the same pair of output values (yl, y2) can result from different values of that input and, in general, from different values of the other input as well (Figure 8a). This situation has been referred as input multiplicity (Koppel, 1982). Another input multiplicity situation is when the same value of one output is obtained from different value of one input (i.e., y1 and u2 in Figure ab). This type of input multiplicity is characterized by a sign change in one
1178 Ind. Eng. Chem. Res., Vol. 30, No. 6,1991 Y2
H------
I
H (a)
(b)
,
H (C)
Figure 8. Input/output multiplicities.
element of a gain vector (i.e., the element for y1 in Figure 8b). Finally, the output multiplicity appears when different pairs of output values result from the same pair of input values (Figure 812). Unless the process output variables are implicit functions of the input variables, it is impossible to have output multiplicity (for example, the outputs in (3) are explicit functions of the inputs and there is no output multiplicity). When a nonlinear process has all these multiplicities (i.e., almost arbitrarily angle changes) in an operating region, the model mismatch due to process nonlinearities may well be considered as unstructured. However, a large class of nonlinear processes only has a simple one-to-one mapping for the input/output relationship and hence excludes any kind of multiplicity. For these processes, the model mismatch is highly structured because the angle mismatches are severely constrained to avoid the multiplicities. Theorem 3 is useful to examine the stabilizability problem for some nonlinear processes of this class. Theorem 3. Suppose that model mismatch in a 2 X 2 process is only caused by process input/output nonlinearities. If the nonlinearities define a one-to-one mapping for input/output relationship in an operating region, then the determinant of G&l is always positive in the region. Proof. It is equivalent to prove that if the determinant is negative, then there must exist multiplicities. From corollary 1, we know that the determinant is negative if and only if one gain vector is on the other side of its nominal half-plane while the other gain vector is not. As shown in Appendix 11, it is impossible to connect the process gain vectors with the nominal gain vectors by the contours without including some type of multiplicities. The above sufficient condition for the determinant to be positive is global. T o apply this theorem, it is not necessary to actually obtain the contour plot for a nonlinear process as long as we know analytically that the input/output relationship is one-to-one mapping within a given range of inputs (i.e., when the nonlinear model is simple enough). For SISO processes, theorem 3 is equivalent to the well-known condition that if there is no multiplicity then the sign of process gain is always the same as that of the nominal model (Figure 9). Unlike the SISO systems, this theorem alone is not sufficient for 2 X 2 systems to determine whether the control is IMC-filter stabilizable. Only for those processes that also satisfy the sufficient condition for the positive trace in theorem 2 can we say that the stabilizability is guaranteed if there are no multiplicities in an operating region. To see if the sufficient condition is met, we will have to obtain the contour plot and examine how the gain vectors change along the contours. For some processes such as binary distillation columns shown later, it is rather obvious to see that the sufficient condition is satisfied.
A Physical Example: High-Purity-Distillation Control The dual-composition control of high-purity-distillation columns has been intensively studied in recent years be-
I
I
~
U
U
(a) output mutiplicity
(b) input multiplicrty
U (c) no multiplicrty
Figure 9. Multiplicities in SISO processes.
cause it presents unique challenges from severe process nonlinearities and from ill-conditioned nominal steadystate gain matrices (McAvoy, 1983; Skogestad and Morari, 1988; Skogestad et al. 1988). Two control strategies, LV and DV control, particularly have been the focus of attention. LV control of a column refers to controlling the overhead and bottom compositions with reflux flow (L) and boil-up flow (V), while DV control refers to controlling the dual compositions with overhead product flow (0) and boil-up flow (V). The steady-state relationship between LV and DV control is just a linear transformation due to material balances
[ $1 = [ o1
:1[%]
Let the superscripts E and M denote LV (energy balance) and DV (material balance) control respectively. From (22), the gain vectors of LV and DV control are related as GM= GET
E E = [-G,,GL
+
GFl
(23)
The gain vector of distillate flow is simply the negative gain vector of reflux flow, and the gain vector of boil-up flow in DV control is the summation of the two gain vectors in LV control. Furthermore, the GG-' matrices are the same for LV and DV control because GM(eM)-'= GET(G?)-l = @(@)-I
(24)
Therefore if DV control is IMC-filter stabilizable, so is LV control. The nonlinear steady-state relationship between the inputs (two flows) and the outputs (two compositions) in LV and DV control is approximated in this paper by using a shortcut model for binary distillation columns that consists of the following equations (McAvoy, 1983):
--m
In a
where the feed quality is assumed to be 1 (Le., saturated
Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991 1179 fl)
0.0
G,
XD
0.99
1.0
Figure 10. 10 LV process.
Figure 12. Possible relative positions of the nominal gain vectors. (1)/!I = e; (2) B = T - e; (3) /!I = T + e; (4) /!I = 2r - e.
,'p
Figure 11. DV process.
Figure 13. A systematic way to have det (G6-l)
liquid) in this model. Given the product compositions, we can solve (25a) for R and then solve for the other flow rate from material balance equations
F=D+B FXF
DXD
+ BXB
(264 (26b)
L=DR
(264
V=L+D
(26d)
The input contour plot for the LV process in a region around the nominal operating point is shown in Figure 10. The dashed contours in the LV process are for constant boil-up flows and therefore are along the tangent lines of these contours. The solid contours are for constant reflux flows and have Gf's tangent to them. The LV contour plot has two major characteristics: (1)the process nonlinearities are prominent in the severely curved, and not equally spaced, contours arising from the physical limits on the composition (i.e., 0-1 for mole fraction) and (2) contours of the two inputs are almost overlapped and therefore the gain vectors at any operating point in the region will be nearly collinear (i.e., very ill-conditioned). We can sketch the input contour plot for the DV process from Figure 10, using the relationship in (23), without resolving the model in (25). The contours for constant boil-up flows in DV are the same as those in LV except that, by increasing the other input value on these contours, LV and DV will have the compositions moved in opposite directions along a contour. This is due to the fact that G!# = On the other hand, since the gain vector of boil-up flow in DV at any operating point is the summation of LV's two gain vectors which are nearly collinear and point to opposite directions, the contours for constant distillate flows must be nearly normal to those for boil-up flow. Figure 11 is the input contour plot for the DV process. Assume the nonlinear model represents true behavior of the column. We will investigate whether a multivariable controller based on a linear model, which is obtained by locally linearizing the nonlinear model a t a nominal operating point, is IMC-filter stabilizable for any other operating points in the region. The nonlinear model in (25)
-q.
0.
is simple and therefore, by inspections of the equations, we know that the input/output relationship is one-to-one mapping. According to theorem 3, the determinant of GG-'in the region is always positive. For more complicated models, we should examine the contour plots to see if any multiplicity exists. In DV control, the dashed contours in Figure 11 asymptotically approach the two axes. Consequently the two gain vectors at any point, one tangent to and other normal to a dotted contour, will always remain in their original half-planes. This phenomena satisfies the sufficient condition in theorem 2 for the trace GG-'to be positive. We can conclude that, even though the nominal model in DV (or LV) control is very ill-conditioned, a linear controller is globally IMC-filter stabilizable in a wide region. Consider now some numerical results for the mismatch structure in LV control of a high-purity-distillation column with the following nominal steady-state gain matrix:
I
0.8235 -0.8198 0.1502 -0.1541
(27)
obtained by linearizing the nonlinear model in (25) at a nominal operating point with the specifications shown in Appendix 111. The condition number based on Euclidean norm for this model is 171. According to (18)for unstructured, norm-bounded model mismatch, the tolerable mismatch magnitude in Euclidean norm is less than 0.01. Table I1 contains the input changes, process gain matrices, mismatch magnitudes, model mismatch terms in polar coordinates, and the eigenvalues of G&' corresponding to five other operating points in the region. In all cases, the mismatch magnitude is at least 15 times larger than the tolerable magnitude calculated from (18). However, none of these mimatches will make the linear controller based on (27)iMC-filter unstabilizable because the eigenvalues of GG-' are all positive. This good robust stability is due to the highly structured model mismatch. It is obvious in this table that the process nonlinearities
1180 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
/h
/ ' =
iii)
iii)
(c)
(b)
(a) Figure 14. Possible relationship between G and G . Table 11. Some Operating Points in LV Control case &IF, d V f F G
2
-0.3, -0.3
3
0.16, 0.08
5
[ 8:itz [ 0.8129
ll&llE
-0.2302 -0.74031
0.15
-0.8133
2.6, 2.1
Table 111. Nominal Operating Conditions and Column Specifications methanolf methanol/ components ethanol components ethanol
LIF VIF Q a
2.11 2.62 1 1.65
XD XB XF no. of trays
0.99 0.01 0.5 21
always force the model mismatch to have approximately the following structure:
nl = n2 and a1 = a2
(28)
This structure, as shown in (21), has the positive determinant and trace if the angle mismatch does not exceed a/2.
Conclusion The IMC-filter stabilizability of inverse-based controllers is analyzed geometrically for 2 X 2 systems. Several criteria are proposed for this stability problem on the basis of the geometrical differences between the gain vectors of a true process and ita model. The paper focuses on purely deterministic model mismatch due to process input/output nonlinearities, and in this situation the geometric criteria can be applied to infer the global stability of a nonlinear process controlled by a linear inverse-based controller. Inverse-based control of ill-conditioned processes are well-known to be very sensitive to unstructured uncertainties. However, many nonlinear processes have a highly structured model mismatch for which inverse-based controllers, based on the ill-conditioned nominal models, are IMC-filter stabilizable over a wide operating conditions. LV and DV control of high-purity-distillation columns are used as examples of such ill-conditioned nonlinear processes with highly structured model mismatch.
n1
nz
1.13
1.11
0.93
A2
a1
a2
-0.12
-0.13
1.0, 0.2
0.93
0.11
0.11
1.0, 2.1
0.98
0.97
-0.17
-0.15
0.1, 0.8
0.91
0.95
-0.16
-0.17
0.8, 0.2
0.89
0.90
-0.17
-0.19
0.7, O.OOO1
A19
Nomenclature det = determinant of a matrix D = apparent disturbance vector D' = real disturbance vector D = distillate flow fi = ith filter time constant fi = nonlinear function for output variable i F = feed flow F = filter transfer function g..= ij element in a steady-state gain matrix = the gain vector of distillate flow Gi = true process gain vector of input variable i GL = the gain vector of reflux flow . Gv = the gain vector of boil-up flow Gi = nominal process gain vector of input variable i G = real process gain matrix G = nominal process gain matrix G = intermediate gain matrix where only one gain vector is changed li = length of ith gain vector L = reflux flow ni = multiplicative error in ith gain vector length N = number of trays N , = minimum number of trays = real process transfer function P = nominal process transfer function Q = controller transfer function in IMC structure ri = norm-bounded magnitude of model mismatch of gain vector i R = reflux ratio R, = minimum reflux ratio S = set-point change vector tr = trace of a matrix T = linear transformation between LV and DV control ui = ith input variable of a process U = process manipulated input vector V = boil-up flow X = composition in mole fraction yi = ith output variable of a process
dD
Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991 1181
cl
Y = process controlled output vector
z = z-domain variable
Greek Letters = relative volatility ai = additive model mismatch in the angle of gain vector i j3 = the angle between two gain vectors yi, = the i-j element in RGA Ai = model mismatch in the ith gain vector Aij = model mismatch in the ijth element of a gain matrix A0 = model mismatch in a gain matrix G K = condition number of a matrix Bi = the angle of gain vectorj Xi = the ith eigenvalue of GG-' 8 = the maximum singular value -u = the minimum singular value (Y
Appendix Appendix I: Proof for Theorems 1 and 2. For an arbitrary 2 X 2 process, the model is represented by two nominal gain vectors in which we can always find a e which has a range between 0 and a/2 to replace 0 to characterize the angle between the nominal gain vectors (Figure 12). The determinant can then be rewritten as sin (c + a1- a,) det (G6-l) = n1n2 (A-la) sin e or sin (e - cy1 + a,) det (GG-9 = n1n2 (A-lb) sin c depending on the relative positions of the vectors in Figure 12 (Le., G, may be in one of the four quadrants, (1)to (4), defined by the dashed lines). Therefore the determinant is positive if and only if (note that n's and sin t are positive) -e < (Y1 - (Y2 < * - e (A-2a) or -t
< a, - a1 < a - e
(A-2b)
By setting either
al or a, to be zero (i.e., specifying an unchanged vector) in (A-2), we can see that the changing vector should remain in its nominal half-plane to have a positive determinant. Theorem 1 is thus proven. The trace of GG-' can also be rewritten in term of e: [nlsin (e - al)+ n2 sin (t + a,)] tr (GG-')= (A-3a) sin t or [nlsin (e + al)+ n2 sin (e - a,)] tr (GB-') = (A-3b) sin c depending on the relative positions of the vectors in Figure 12. The trace is positive if both terms in the numerator on the right-hand side of (A-3a) or (A-3b) are positive, and is negative if both terms are negative. Since sin (e f ai) > 0 if 0 < e f ai < a (A-4)
geometrically, (A-4) corresponds to theorem 2. Apendix 11: Connecting a Pair of G and 6 Where det (GG-') < 0 by the Contours. Given an arbitrary G , we can draw the gain vectors o f G at certain point we choose and have negative det (GG-')in a systematic way in Figure 13 (1)passing the chosen point, draw a parallel line to the nominal gain vector that will remain in its
nominal half-plane (i.e., in this figure); (2) at the chosen Eoint, first reproduce the other nominal gain vector (i.e., G,in this figure) and then rotate it anywhere to the other side of this line to obtain G,;(3) at the same point, reproduce G1and rotate it anywhere, without crossing the new line extended from G, to obtain G1. To connect such a pair of G and G (te., four gain vectors) in Figure 13 there are three possibilities: (a) the two gain vectors of the both inputs are connected by one contour (Figure 14a); (b) the two gain vectors of one input are connected by one contour, while the two gain vectors of the other input belong to two separate contours (Figure 14b); (c) the four gain vectors belong to four contours (Le., two for each input) (Figure 14c). Possibility a is the situation of output multiplicity in Figure 8a. In possibility b, the contour passing Gzin Figure 14b has to meet one of the following condition: (i) intersect with the contour passing Gz(i.e., an input multiplicity), (ii) intersect once again with the contour of G1(i.e., an output multiplicity), or (iii) curve in a way that there must exist a horizontal or vertical gain vector on this contour (i.e., an input multiplicity). Possibility c has the same situations as in possibility b and will not be repeated here. Appendix 111: Nominal Operating Conditions and Column Specifications. See Table 111.
Literature Cited Bristol, E. H. On a New Measure of Interactions for Multivariable Process Control. IEEE Trans. Autom. Control 1966,11,133-134. Doyle, J. C.; Stein, G. Multivariable Feedback Design: Concepts for a Classical/Modern Synthesis. IEEE Trans. Autom. Control 1981, AC-26, 4-16. Frank, P. M. Design of Control Circuits with Prescribed Behavior; G. Braun: Karlsruhe, 1974. Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Ind. Eng. Chem. Process Des. Dev. 1982,21, 308-323. Garcia, C. E.; Morari, M. Internal Model Control. 2. Design Procedure for Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1985,24,472-484. Grosdidier, P.; Morari, M.; Holt, B. R. Closed-Loop Properties from Steady-State Gain Information. Ind. Eng. Chem. Fundam. 1985, 24, 221-235.
Koppel, L. B. Input Multiplicities in Nonlinear, Multivariable Control Systems. AIChE J. 1982,28, 935-945. Koung, C. W.; Harris, T. J. Sensitivity of the Relative Gain Array. Presented at the 39th Canadian Chemical Engineering Conference, Hamilton, Ontario, Canada, 1989; paper A2d. McAvoy, T. J. Interaction Analysis; ISA Research Triangle Park, NC, 1983. McDonald, K.; Palazoglu, A.; Bequette, B. W. Impact of Model Uncertainty Descriptions for High-Purity Distillation Control. AIChE J . 1988,34, 1996-2004. Morari, M.; Zafiriou, E. Robust Process Control; Prentice-Hall: Englewood Cliffs, NJ, 1989. Skogestad, S.; Morari, M. Implications of Large RGA Elements On Control Performance. Ind. Eng. Chem. Res. 1987,26,2323-2330. Skogestad, S.; Morari, M. LV Control of a High-Purity Distillation Column. Chem. Eng. Sci. 1988,43, 33-48. Skogestad, S.; Morari, M.; Doyle, J. C. Robust Control of Ill-Conditioned Plant: High-Purity Distillation. IEEE Trans. Autom. Control 1988,33, 1092-1195. Yu, C.-C.; Luyben, W. L. Robustness with Respect to Integral Controllability. Ind. Eng. Chem. Res. 1987, 26, 1043-1045. Zames, G. Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms and Approximate Inverses. IEEE Trans. Autom. Control 1981, 26, 310. Received for review July 5, 1990 Revised manuscript received October 19, 1990 Accepted November 19, 1990
