2012
Ind. Eng. Chem. Res. 1993,32, 2012-2016
Steady-State-GainIdentification for Multivariable Process Control J o h a n A. Pensar and K u r t V . Waller' Process Control Laboratory, Department of Chemical Engineering, Abo Akademi, SF-20500 Abo, Finland
The steady-state gains of a multivariable process contain much information about the control properties of the process. Identification of the process gains when the process operates in closed loop has advantages to open-loop identification. The process may be more linear in closed loop, the experiments may be designed to give more well-conditioned data, and the ordinary process operation may be less disturbed by the identification experiments. In this paper closed-loop gain identification is illustrated for the distillation process. Open-loop gains for any control structure can then be calculated through control structure transformation rules from closed-loop gains, which are independent of the controlling structure used for the identification. The method is illustrated by results obtained from a nonlinear distillation column model and by experimental data from a pilot-plant column. Introduction Much information about the control properties of a multivariable process can be extracted from the steadystate gains of the process (Grosdidier et al., 1985). A wellknown example is the relative gain array. More recent results concern such important concepts as integral controllability of decentralized control systems (multiloop single input-single output (SISO) systems) (Morari and Zafiriou, 1989). The usual way to identify the gains of a process is to introduce perturbations into the process manipulators and to measure the final resulting changes in the uncontrolled process outputs. In the following this approach is referred to as open-loop identification. An alternative approach has been suggested by Papastathopoulou and Luyben (1990). The identification is made in closed loop running the system under feedback control with integral action in the controllers and making a change in the setpoint of one of the controlled variables. The resulting final changes in the primary manipulators are then measured and then corresponding gains calculated. This approach with the system under feedback control (not necessarily having integral action in the controllers) is labeled closed-loop gain identification in the following. Closed-loop identification seems to be especially advantageous when one wants to obtain process gains for several multiloop SISO control structures that differ as regards the manipulator choice. This is a common problem, e.g., in design of distillation control systems. The gains for any such control structure are obtainable directly from the raw data, as illustrated by Papastathopoulou and Luyben (1990). This fact is a consequence of the fact that the same raw data are obtained independently of which control structure is used in the closed-loop identification. Another advantage of closed-loop identification is the possibility to counteract process nonlinearities. One example from distillation is provided by a change in feed flow rate when the outputs (e.g., compositions) are kept constant. The scaling consistency relationships discussed by Skogestad (1991)state that the intensive variables (such as compositions) remain constant if the extensive variables (e.g., all flows into, in, and from the column) are changed
* Author to whom correspondenceshould be addressed. E-mail address:
[email protected].
Figure 1. The (L, V; D,B ) structure of a distillation column.
in the same ratio. Although the scaling consistency relationships are never exactly true in practice (one reason being that the size of the process equipment is not usually scaled), they can be very useful. In this paper closed-loop gain identification is studied with special reference to the distillation process. By the use of consistency relations and transformation rules valid for continuous distillation, the work of Papastathopoulou and Luyben (1990) is extended to cover identification of disturbance gains as well as gains of possible subsystems of the process (e.g., closed inventory control loops). Openloop gains for any control structure may then be calculated by transformation rules. Closed-Loop Step-Response Identification of Process Gains Consider a continuous distillation column operated with the (L, V; D, B ) structure, i.e., the reflux L controls the top temperature or composition y , the heat input V to the reboiler controls the bottoms temperature or composition x , and distillate flow D and bottoms flow B control the level of the reflux drum and reboiler respectively. (See Figure 1for notation.) If disturbances in feed composition z and feed ,flow rate F a r e considered, a transfer function model of the process can be written:
0888-58~5/93/2632-2012~0~.00/00 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 9, 1993 2013
(la)
(1b) Usually, when a distillation column is modeled by transfer functions, only eq l a is used and not eq lb. However, if one wants to study other control structures (than the (L, V; D , B) structure described in eq 1) by transforming the information of eq 1to the structures in question, knowledge of eq l b is essential (Haggblom and Waller, 1988; 1992). The steady-state part of eq 1can be written compactly: Ay = K,,Au
Av = K,,Au
+ K,Aw + K,Aw
(2a) (2b)
where K denotes the gains of the transfer functions G . A traditional way to identify the model above is to change the input variables L and V and measure changes in the output variables in the model, i.e., y, x , D, and B. This method is usually referred to as open-loop identification because the composition (or temperature) control loops are kept open. Perturbations are introduced into the main manipulator signals, i.e., in the (L,V; D, B) structure into L and V , but also into the disturbances z and F. The resulting steady state is measured, and the gains from the perturbations in L, V, z , and F to the process outputs y and x and to the secondary manipulators D and B are calculated. Papastathopoulou and Luyben (1990) have suggested another method to identify open-loop steady-state gains for multivariable processes. Their method is in short to run the system under feedback control with integral action in the controllers and change the setpoint of one of the controlled variables, y or x . The resulting final changes in the primary manipulators are then measured and the corresponding gains calculated. As illustrated by Papastathopoulou and Luyben, this can easily be done directly from the raw data for any structure that differs regarding the manipulator choice, for example, the (LID,VIB;D , B ) structure. An alternative way to calculate the gains is described below. It uses transformations and matrix calculations. In that way, data from as many experiments as desired can be treated simultaneously and integral action is not necessary in any controllers: it may be present in some loops while other loops do not contain integral action. For the calculation of the open-loop process gains from the closed-loop experiments, an inverse model structure that goes from primary outputs and disturbances to manipulators in closed loop is defined:
Av = K:,Ay
+ K&Aw
(3b)
The gain matrix K,, of eq 2a is then obtained from the KGYmatrix simply by inversion (superscript c indicates closed-loop gain matrix), i.e.,
K,, = (K:,)-' By writing the experimental data in the form
(4)
where the superscript indicates different experiments, the extension is straightforward to identification with more than one perturbation at a time, or with more than n experiments for an n X n system (McIntosh and Mahalec, 1991). The symbol t denotes the pseudoinverse given by Xt = X*(XXT)-l, which for the case of a square matrix is reduced to X-l. This gives the least-squares estimate of the gains. The closed-loop identification method by Papastathopoulou and Luyben (1990) treats only the calculation of the K,,.matrix. This is, however, not the only identifiable matrix in closed loop. By writing the system model as in eq 3, all gains are identifiable by methods similar to openloop identification. The closed-loop gains can be transformed into openloop gains by use of methods similar to those used for structure transformations between different open-loop structures (Haggblom and Waller, 1988,1992). The KG, and K:, matrices are obtained from eq 2a, Au = (Kyu)-'Ay - (K,,)-'K,Aw
(6)
and the K:, and K& matrices from eqs 2b and 6: AV = Kvu(Kyu)-'A~ + [K, - K,,(K,,)-'K,l~~ (7) By comparing eqs 3 with eqs 6 and 7, we obtain the following seta of equations for transformation from openloop gains to closed-loop gains:
Analogously, the following sets of equations are valid for transformation from closed-loop gains into open-loop gains:
We note that the transformations are valid for any control structure. The closed-loop identification, and the transformation from closed loop to open loop, illustrated above for a 2 X 2 system, are generally applicable to n X n systems. When calculating the gains with a least-squares method, as eq 5, the process is described by deviation variables that are differences from a chosen reference state, which normally is the nominal steady state. One use of the reference state is to center the data so as to be able to omit the offset term for a linear model. It can be shown (Draper and Smith, 1981) that the offset term for a linear model can be omitted if the data are centered, i.e., if the reference state is chosen as the arithmetic mean of the observations.
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However, the arithmetic mean of the observations may not physically correspond to the steady state defined by the arithmetic mean of the input variables. Fortunately, it can be shown (Hliggblom, 1988)that only the reference state of the input variables needs to be centered. To get a good gain estimate at the nominal steady state for a nonlinear process, the experiments thus need to be symmetrically centered around the nominal steady state of the input variables. This is usually no problem in openloop identification if the identification is performed in the structure in question. In such a case it is usually easy to change the chosen manipulators symmetrically around the nominal steady state by introducing both positive and negative perturbations of equal size. If, however, the experimental data also are used to calculate gains for other structures, e.g., structures using nonlinear combinations of flows, such as flow ratios as manipulators, the experimental data are generally no longer centered around the nominal steady state of the new inputs. This means that the calculation of the openloop gains for other structures than the one used for the identification might be ill-conditioned or inaccurate. If the experiments are performed in closed loop, the input variables will be the same for all structures. If these input variables are centered, i.e., the setpoints are symmetrically changed in both positive and negative directions, the data obtained are thus suitable for calculating the gains of any structure that uses combinations of physical process manipulators. During the closed-loop identification, the process can be kept controlled at all times. This means that all product specifications can be met even during the experiments, provided that the needed setpoint changes are acceptable. This may be the case, e.g., when only one of the products from a distillation column has purity specifications and the other product is controlled only with the purpose to reduce loss of primary product. There are also situations where the needed setpoint changes can be advantageously made so that the average value of the process products meets the specifications. How the gains can be obtained from normal production data is illustrated by McIntosh and Mahalec (1991). Structure Transformations Using Closed-Loop Data Transformation of the closed-loop gains from one structure into another that uses other primary manipulators is very simple, as shown by the following examples. Consider, e.g., a closed-loop model using the (L, V; D, B) structure (here the process model expressed by eqs 3 is written in a more compact form):
A transformation into the (D,V; L, B ) structure now involves only the interchanging of row 1 and row 3
and no manipulations on the single elements have to be performed. If desired, the open-loop gains can then be calculated through eqs 9.
A transformation into a structure where the manipulators are nonlinear combinations of the flows, as the (Ll D, V; D, E)structure, involves operations on the elements in the affected row(@as illustrated below. Define the following disturbed manipulators where a bar above a symbol denotes the nominal steady state. Aw denotes a perturbation in the primary output setpoints or in the external disturbances.
L = L + KLWAw D = D + K,,Aw
(12a) (12b)
By definition the gain is given by
-
LID - LID
(13) Aw for Aw 0. By use of eqs 12 and linearizing around the nominal steady state (Aw = 01, the desired gain is obtained as a suitably weighted mean of two gains: K(L/D)w
=
DK,, - LK,, K(L,D),=
D2
Modifying the first row in the closed-loop gain matrix for the (L, V; D,B ) structure according to eq 14 results in the closed-loop gain matrix for the (LID, V; D,B ) structure.
Remark: If all the raw data are available, equations like 13 can be used directly for calculations of the gains in question (Papastathopoulou and Luyben, 1990). If, however, the raw data have been condensed into process gains for one structure, such as (L, V; D, B ) in eq 10, these gains can be used (together with data describing the steady state) for calculation of the gains in the structure in question, as illustrated by eq 14. Consistency Relations The steady-state gains for continuous distillation have to follow some basic relations determined by the totaland partialmass balances. Hiiggblomand Waller (1988,1992) give several such consistency relations as connect certain gains to each other. The different control structures are also related to each other according to structure transformation rules (Hiigiggblom and Waller, 1988,1992). In Hiiggblom (1988) these transformations are used for reconciliation of gains identified in different structures. The identification of a process in a number of different control structures requires much work, because every structure has to be implemented in the process control system, controller parameters have to be found (at least for the secondary loops), and finally the identification has to be done. The identification would also interrupt the continuous production and off-specificationproduct would probably be produced during the test. When closed-loop identification is used, several alternatives are possible. One is that the process is identified in one structure in both open and closed loop, and the results are reconciled by the total and partial mass balances, and by the transformations between open-loop and closed-loop gains. The closed-loop identification requires the implementation of only one structure, and the identification may be possible even during continuous
Ind. Eng. Chem. Res., Vol. 32, No. 9,1993 2015 production if the needed changes in the product compositions are allowed. The reconciliation might minimize some suitable norm of the corrections of all experimentally obtained gains subject to the mass balances and the relations between the open-loop and closed-loop gains. The consistency relations based on the total mass balances are for the open-loop case (Hliggblomand Waller, 1988, 1992):
0 = KDz + KBz
The consistency relations for the closed-loop gains can easily be derived in analogy with the derivation of the open-loop gains. The consistency relations derived from the total mass balance for a continuously operated distillation column are the following.
1 = KcDF + KcBF
(17d)
If the outputs are product compositions (and not temperatures as in eqs 17a and 17b), some open-loop consistency relations derived from the component balances also apply (Hiiggblom and Waller, 1988; 1992). Taking the (L, V; D, B ) structure as an example, they are the following.
Corresponding closed-loop consistency relations based on the component balances can also be derived. These relations are the following.
If the outputs are two temperatures in the column and not the product compositions, the consistency relations derived from the component balances do not apply, only the relations based on the total mass balance. The consistency relations between the open-loop and the closed-loop gains were derived above and are given in eqs 8 and 9. A study of gain reconciliation for a nonlinear dynamic distillation column model is made in Pensar and Waller (1992). The main problem studied there is process nonlinearity. Process nonlinearity affects gain identifi-
cation differently in open-loop and closed-loop identification as discussed above. If both positive and negative (step) changes from the nominal steady state are made, the difference between the so obtained gains can be used as a measure of the goodness of the identification of that particular gain (so that a smaller difference is taken to correspond to a more accurate identification). The weight to put on the gain in the reconciliation procedure can be chosen so that it increases with the increased goodness of the identified gain. This is the approach used in the study mentioned. Pensar and Waller compared the accuracy obtained by different identification and by different reconciliation methods. Identifying the gains in closed loop and transforming them into open loop according to eqs 9 increased the obtained accuracy by a factor of about 2 compared to open-loop identification in the studied example. Reconciling the open-loop gains with the closed-loop gains (and with the consistency relations in eqs 16 and 17) increased the accuracy of the so obtained gains by a factor of the order of 5 compared to open-loop identification. An explanation why use of both open- and closed-loop gains might improve accuracy can be found in the fact that the weights put on the gains in the reconciliation process increases with increasing linearity of the gains, as explained above. Thus, the “most linear gains” in both open and closed loop are given most weight in the averaging procedure. Although results like these are bound to be problem specific, it is very likely that, in coping with process nonlinearities in gain identification of a process like distillation, the accuracy can be significantly increased by using closed-loop identification and further increased by making use of both open- and closed-loop identification in combination with consistency relations like those in eqs 16, 17, and possibly 18 and 19.
Gain Calculation for Unequal Perturbations When experimentally identifying process gains, it is common practice to make both positive and negative step changes from the considered steady state. Because all real processes are more or less nonlinear, the two obtained gains are generally different. To get the gain, defined as the derivative of the output with respect to the input, at the steady state in question, common practice is to take the arithmetic average of the two experimental gains. One case where it may not be easy to make equal steps in both directions is when disturbances are being introduced. A disturbance in, e.g., feed composition z is often fed to the process through some upstream equipment, and it is often not physically possible to get a disturbance of desired size. It is also possible that the disturbance is not easily measured and controlled; e.g., the exact value of the disturbance is available only after some laboratory tests. In these cases it is necessary to be able to compensate for possible nonequalities in the steps by mathematicalmeans. If no further knowledge about the nature of the nonlinearity exists, it would be natural to approximate the input-output relationship with a polynomial, in this case of second order. The gain at any point would then be given by the derivative of the polynomial. The derivative can be expressed as a weighted mean of the gains calculated from the nominal steady state for perturbations upward and downward: K,,Adown + KdownAup K= (20) AUD + Adown Equation 20 is valid for both unevenly and evenly spaced perturbations, where the input-output relationship is approximated by a second-order polynomial.
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The result expressed by eq 20 is quite natural: a smaller step change generally makes the corresponding gain less affected by the nonlinearities. The gain should therefore be given a larger weight in the averaging procedure. (Another story is that smaller perturbations may be more affected by process and measurement noise.)
Experimental Results To test the method on a real process, experiments were performed on the pilot plant distillation column at Abo Akademi (Waller et al., 1988; Waller, 1992). As a result, the followingopen-loop gains were calculated by use of eqs 9 from a number of closed-loop-identification experiments: -0.046 0.063 Kyu= -0.22 0.71
[
1
]
K,, =
1'23 0.48 -1.24 The followingcorresponding gains were calculated from a number of open-loop step-response experiments.
-0.048 0.062 Kyu= -0.24 0.65
[
1
]
K,, = [-0'42 1'27 0.49 -0.94 (The dimensions for the gains in Ky, are "C/(kg/h). The gains in K,, are dimensionless.) Both methods give results that reasonably well resemble each other. In the experiments the outputs y are the temperatures on plates 4 and 14. Since v = [D,BIT,the total material balance results in the consistency relations that the sum of the elements in the columns of the K,, matrix is zero as expressed by eqs 16a and 16b. The experimental results above (which have not been reconciled with respect to these consistency relations) show that in this experiment this result is (nearly) obtained only for changes in V and only in the closed-loop identification. Summary and Discussion Closed-loop identification of process gains has some attractive properties compared to open-loop identification. During a closed-loop step-response identification, the process can be kept controlled at all times. This means that product specifications can be met even during the experiments, provided that the setpoint changes needed are acceptable. It may also be advantageous that the size of the disturbances is defined in reference to process outputs instead of process inputs as in open-loop identification. Calculation of process gains directly from the experimental data obtained in closed loop becomes very simple for structures differing regarding the manipulator choice. A further advantage of closed-loop step-response identification is that data obtained often express more linear relations than data from open-loop step response identification. Used in combination with open-loop identified gains, the accuracy of the identification can be improved by reconciliation. The distinction made above between open- and closedloop identification makes the grasping and the use of the different methods easy. However, fixing a number of variables corresponding to the degree of freedom of the process determines the steady state (neglecting multiple steady states). Then it is actually only a semantic question
which variables are labeled inputs and which are labeled outputs. In closed-loopidentification controlled variables are specified, whereas in open-loop identification independent manipulated variables are so. This also means that the identification could be made in a "mixed mode" by fixing a mixture of controlled and manipulated variables, the sum of their number being equal to the degree of freedom of the process. This fact is also expressed through the transformations and consistency relations expressing the relations between the variables in the process. The choice of which variables to vary in the identification procedure then boils down to finding a suitable combination of variables that makes the identification as well conditioned as possible with respect to noise, uncertainties, nonlinearities, and unmeasured disturbances.
Acknowledgment The results reported have been obtained during a longrange project on multivariable process control supported by the Neste Foundation, the Academy of Finland, Nordisk Industrifond, and Tekes. This support is gratefully acknowledged. The valuable comments of Dr. Kurt-Erik Hegblom during several discussions are also appreciated. We also wish to thank an anonymous reviewer whose constructive comments we have tried to utilize to improve the paper.
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Received for review October 6, 1992 Revised manuscript received May 17,1993 Accepted May 26, 1993
