1510
Znd. Eng. Chem. Res. 1994,33, 1510-1519
Pattern Recognition Based Adaptive Control of Two-Input/Two-Output Systems Using ART2-A Neural Networks Lawrence Megan+and Douglas J. Cooper' Chemical Engineering Department, University of Connecticut, U-222, Storrs, Connecticut 06269-3222
This paper details an applied investigation of pattern recognition based adaptive control for twoinput/two-output systems. Two ART2-A neural networks perform a concurrent analysis of controller error and manipulated input patterns resulting from a set point change or an unmeasured disturbance to the system. This information is then used to adapt the models that describe each input/output relationship. The adaptive strategy is demonstrated on two challenging processes: a pilot-scale continuous distillation column and a simulation of the Shell fundamental control problem. The distillation column demonstrates the applicability of the adaptive strategy to both set point changes and disturbances in a challenging real-world process, while the Shell problem demonstrates the ability of the strategy to handle irregular disturbance dynamics.
Introduction Artificial neural networks (ANN) have emerged as an important topic for chemical engineering research over the past 5 years in such applications as process modeling, fault diagnosis, and process control. Presented is a neural network approach to pattern recognition based adaptive control. Typically, adaptive controllers are dependent upon maintaining a model descriptive of the local operating regime of the process and then utilizing the model within some model based controller. With this in mind, this paper details an approach where the neural networks translate recent trends in process data to process model parameter adjustments. By focusing on the process model, the ANN based adaptive strategy is applicable to anumber of model based control algorithms, such as dynamic matrix control (DMC) or model algorithmic control (MAC). Initial work centered on the application of ANN technologyto single-inputlsingle-output(SISO)processes. However, a substantial number of chemical engineering systems are multi-inputlmulti-output (MIMO) processes with significant interaction between the controlled output and manipulated input variables. More recent work has therefore focused on extending the ANN techniques for data analysis and model adaptation to two-inputltwooutput (2 X 2) processes. A concurrent analysis of the manipulated input and controller error patterns forms the basis for effective model adaptation. Similar to SISO systems, a highly underdamped response to a set point change or disturbance in a MIMO system can be used to infer that one or more of the model gains is too high. However, unlike SISO systems where the model adaptations can be fairly precise (Megan and Cooper, 19921, pattern recognition based adaptation in MIMO systems is limited to a more qualitative level. In SISO systems, the error patterns are fairly consistent from process to process for a given processlmodel mismatch, whereas in higher order systems the error patterns are a function of not only the process/model gain mismatch but also the degree of interaction in the system. The methodology described here decomposes the 2 X 2 system into the individual SISO systems to allow for a qualitative evaluation of the model mismatch. The model parameters
* Author to whom correspondence should be addressed. E-mail:
[email protected]. 'Present address: Praxair, 175 East Park Drive, Box 44, Tonawanda, NY 14151. 0888-5885/94/2633-1510$04.50/0
are then heuristically adjusted to improve controller performance. While there are a number of potential ANN architectures to perform the pattern analysis, this work uses a type of vector quantizing neural network, the ART2-A network of Carpenter et al. (1991). As opposed to a backpropagation neural network (BPN), which learns a relationship given a small subset of potential patterns, the ART2-A maintains a large library of patterns and matches incoming patterns with the most similar of the network exemplar patterns. While the ART2-A lacks the interpolative abilities of the BPN, it is useful when the range of potential patterns makes BPN training impractical. This paper studies 2 X 2 systems and assumes that each input/output relationship can be described by a first order plus dead time model (FOPDT). The focus is on model gain adaptation and hence there are four process model gains to adapt. Following a perturbation to the system, a batchwise analysis of the patterns in the recent history of the manipulated input for each input/output pair is used to determine whether or not adaptation of the corresponding model gain is potentially beneficial. The decision is based on whether or not the set point change or disturbance has shifted the manipulated input from its previous steady-state value, indicating a change in operating regime and a potential change in process character due to nonlinearity. A single ART2-A network is used to make this determination, and is developed in a general nature such that it does not have to be retrained from process to process. If a given input pattern indicates a shift in steady state, a subsequent analysis of each error pattern is used to identify the degree of mismatch between the process model gain and its actual value for each of the two input/output models corresponding to that input variable. Controller aggressiveness is inversely proportional to process gain, so that if the process model gain is too low, the controller action is typically too aggressive. A second generalized ART2-A network classifies each of the two controlled variable error patterns based on the degree of damping. To maintain the process independence of the strategy, the classifications are general in nature, where the error pattern is categorized into one of five classes. The model gain adaptation for a inputloutput pair is proportional to this damping classification, allowing the given process model gain to be adjusted closer to its true value. The presented adaptive strategy, due to its qualitative nature, is best suited for model adaptation in the wake of 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1511 unmeasured disturbances. Dynamics caused by disturbances are difficult to model with traditional methods, as the data may be a reflection of the disturbance itself and not the true input/output relationship. Since disturbances typically occur with a much higher frequency in chemical systems than do set point changes, it is desired to make use of these dynamic situations for model adaptation. Thus the presented adaptive strategy allows some measure of controller/model performance and model adaptation in situations where standard modeling techniques are unreliable. However, this is not to say the presented adaptive strategy is not applicable to set point change dynamics. While traditional model fitting techniques are more suited for such situations, as the data are a true representation of the input/output relationship, the ANN strategy is a simpler alternative to more computationally expensive model regression techniques. Thus for comparative purposes,the pattern recognition strategy willbe compared to a more traditional modeling technique, batch regression, with respect to both set point changes and disturbances. The batch regression works in the following manner. After the onset of a perturbation to the system,the dynamic process input/output data are collected in a batchwise fashion over a period of time determined as a function of the time constants of the process. The model parameters for each single-outputltwo-inputsystem are then regressed through a least squares fit of the data. This should be a more accurate method of model updating than the ANN based strategy, at the cost of greater computational expense. Thus the two methods are compared on the basis of accuracy versus computational load. This paper investigates the ability of the adaptive strategy to make useful model adaptations following both set point changes and unmeasured disturbances to the process. The demonstration processes include a demanding pilot-scale process and a challengingprocess simulation. The pilot-scale process, a continuous distillation column, is an 18-plate column 1f t in diameter and 20 f t high and is presently being operated on the methanol/water system. The control problem is dual temperature control of two control plates, one above and one below the feed plate, using the reflux rate and the steam rate to the reboiler as the manipulated inputs. The process simulation is a subset of the Shell fundamental control problem (Prett and Garcia, 1988). The total Shell problem consists of seven potential controlled variables and five potential manipulated variables. This paper chooses a 2 X 2 subset of this system for the process simulation and maintains use of another manipulated input as the source of unmeasured disturbances. This simulation is used to demonstrate the applicability of the ANN adaptive strategy to irregular disturbances. Background Dynamic Matrix Control. This work demonstrates the adaptive strategy using the least squares formulation of dynamic matrix control (Prett and Garcia, 1988; Cutler and Ramaker, 1979). Open loop step response models are used in which data for each input/output pair are fit with a FOPDT model, and the FOPDT differential equation is then integrated to get the model parameters used in DMC. Following model gain adaptations, the step response models used in DMC are regenerated using the new FOPDT model expression. As DMC is well documented in the literature, this section only discusses the aspects directly related to this work. The least squares formulation of DMC is used here, where a series of unconstrained future manipulated input moves
are calculated to minimize the following objective function at sample t: J = (yBP- y’)TQl(y,p- Y’)
+ (Au)~Q~(Au)
(1)
where y‘ and yspare, respectively, the vectors of predicted outputs and future set points. For a two-output system, these two vectors are of dimension 2P, where P is the prediction horizon. Au is the vector of future manipulated variable moves, and is of dimension 2C for a two-input system, where C is the control horizon. Q1 and Q2 are diagonal weighting function matrices whose dimensions are [2P,2Pl and [2C,2Cl, respectively. In this work, the diagonal elements of Q1 are equal to 1 while the diagonal elements of QZare defined as functions of the diagonal elements of the model gain matrix, where the input/output pairings are chosen according to the relative gain array (Bristol, 1966). Thus the diagonal elements of QZare defined: Q~ = ri*Kpii2 where Kpiiis equal toKpllfor the first C elements andKp22 for the second C elements. The values for I’i become the tuning parameters for the controller, analogous to 7C in the IMC controller. They are chosen to give user-specified desired performance in the initial operating regime. This formulation thus defines the penalties on manipulated input moves as a function of the process model while keeping the two terms of eq 1 consistent in units and proportionate in size. The step response model matrix contains the four step response models for each input/output pair. In this work, the elements of the model matrix are initially developed by first regressing a FOPDT model on the open-loop step response of each input/output pair. The elements the matrix are then generated by differentiating the FOPDT equation: (3)
where Kpij is the model gain for the given input/output pair, 7pij is the model time constant, and t&jis the model dead time. Online, a new solution for eq 3 is found each time Kpij is updated. Note that upon model adaptation, both the dynamic model matrix and Q2 are reset to reflect the updated model parameters. An important feature of the adaptive strategy is the relative process independence of the algorithm. To maintain this independence,the controller parameters are defined as functions of the minimum and maximum time constant of each 2 X 2 system, as opposed to being set to constant values. This is done as follows: At = X 7 r
P = ( YT;-)/A~
(4)
C=ZP where 7;”’ and 7;- are the minimum and maximum of the four model time constants, respectively. In this paper, X = 0.2, Y = 5.0, and Z = 0.25. ART2-A Neural Networks. The ART2-A neural network is well-suited for situations where a large number of potential patterns exist, and the goal is to cluster similar patterns to reduce the number of pattern classes to a workable number. The ART2-A groups similar patterns into exemplar patterns and stores these patterns in its nodes. All the patterns in this work are data histories in
1512 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
either the manipulated input or controller error which contain the first 50 data samples following the onset of a perturbation to the system, where the sample time is defined in the previous section. To train the network, the user initially defines a number of nodes, which represents the maximum number of classifications. All nodes are initially uncommitted, which is to say they represent no patterns. The training patterns, Io, are first normalized: 71" = IO/~~IO/~
(5)
where 1 .1 signifies the norm (Euclidean distance) and the operator q performs Euclidean normalization. The normalized input pattern is then filtered to minimize the effects of noise:
where i represents each element of the input vector and 9 is a threshold value, defined as
,,,*(new)
=
J
+ (1- p)zj*('ld))
if J is an uncommitted node if J is a committed node (12)
where if zji* > 9 (13) otherwise This ensures that no new information is learned where there previously was none. The learning parameter,& is set between 0 and 1and determines to what degree each training pattern contributes to the exemplar pattern of the associated winning node. Convergence for the AFtT2-A network is complete once the network exemplar patterns stabilize within a given tolerance and either no new nodes are being committed or the maximum number of nodes has been filled. Once trained and put online, the incoming patterns are normalized by eqs 5-8 and the matching scores are calculated by
Ti = Izj*
M is the length of the input vector and is equal to 50 throughout this work. The filtered input pattern is then renormalized via eq 5 which results in the enhancement of larger values and the attenuation of smaller values. The finalized input pattern has the form:
I = q3qI"
(8)
The next step is to calculate the matching scores, which represent how closely each input pattern corresponds to the exemplar patterns. The matching score for a given node is calculated as
T:=
'
IUCUi)
if j is an uncommitted node
pzj*
if j is a committed node
1
(9)
where
Each vector Zj* is the exemplar pattern for each node j . Committed nodes are those which contain a pattern representation. The closer the value of the product of the input pattern and exemplar pattern is to 1, the greater the correspondence between the two patterns. The "winning" node is
(14)
The winning node is the one which satisfies eq 11. If TJ is greater than the vigilance parameter, p, the classification that corresponds to that winning node is used in the model adaptation strategy. Otherwise, the pattern is ignored and no changes are made to the model based on a lack of sufficient information. The parameter values are set as follows. 9 equals 0.0 as the pattern classification requirements of the network are not precise enough to be affected by small amounts of noise. This work uses very general classification of both input and error patterns. If more precise pattern classification was required, then larger values of 8 would be needed to provide additional noise filtering. For similar reasons, p equals 0.75, which is a low value of the vigilance factor by ART2-A standards. Higher values of p would result in more patterns being rejected online. However, in this work the network exemplars only provide a general pattern classification and do not have to be an exact match to the incoming patterns. ss is set to 0.1, allowing new training patterns to have only a small effect on the established network exemplar pattern. a equals l/(M)O.5, which is the standard specification of Carpenter. Finally, the length of all the input patterns is M = 50, which from eq 4 is equal to 1 0 ~ ~ " 'Higher . values of M would allow the network to see aiarger portion of the dynamic pattern but at the expense of more computationalload and a longer time before pattern classification and subsequent model adaptation. Pattern Recognition Based Model Adaptation
Before the input pattern is committed to the winning node, it must pass a vigilance test to ensure it is a reasonable match to the exemplar pattern. Node J is selected as the winning node if TJ 1 p, where p is a positive constant between 0 and 1. If TJ < p, the input pattern is committed to the next uncommitted node. If all the nodes are committed, then the input pattern is discarded and the algorithm moves on to the next training pattern. Learning in the committed nodes is performed as long as J is committed or there are available uncommitted nodes. The exemplar pattern for node j , zj* , is updated as
For an adaptive controller to find widespread industrial acceptance, it should be applicable to a large class of chemicalprocesses. The authors' previousexperiencewith SISO systems shows pattern recognition techniques meet these criteria (Cooper et al., 1992). In SISO systems, a single idealized process is used as the basis around which the adaptive framework is developed. Unlike such model updating techniques as recursive least squares, pattern recognition does not attempt to explicitly determine model parameters from inputloutput data. Rather, it focuses on determining the multiplicative mismatch between the process and model parameters, which is reflected in the controller error patterns following a perturbation to the system. A set of error patterns developed from the
~ERROR I NETWORK?
.
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1513
-47
Table 1. Error Network Pattern Classifications
NETWORK
MAKER
PROCESS INPUTS
PRoCESS~PROCE88' OYTPYTS
'
description
class
description
1 2
very underdamped moderately underdamped slightly underdamped
4
moderately overdamped very overdamped
3
5
I
i
A
YNYE*S"RED
DISTYRBANCE I
FEEDBACK S 1 0 N A i
Figure 1. ARTB-A network adaptive strategy.
idealized process can then be applied to a number of actual processes. The result is a quantitative assessment of processlmodel parameter mismatch which is then used to make precise adaptations to the model parameters. The challenge in extending this concept to higher order systems is that there is no longer a single set of idealized patterns, as the patterns vary as a function of the degree of interaction in the system. What may be considered a good controller error pattern for a highly interacting system may be considered poor control in a processwith less severe interaction. Consequently, it becomes difficult to make precise quantitative adaptations to such systems, and we are limited to a more qualitative level. For example, a highly underdamped response to a set point change or disturbance within a model based multivariable strategy is still used to infer that one or more of the model gains is too low, though it may not be immediately clear which model parameters are incorrect. If a reliable method is developed for decomposing the problem to find which model gains are mistuned, then at least a qualitative assessment of the model mismatch can be made. This information leads to heuristic adaptations to the model parameters, thereby giving a better representation of the true process. Neural Network Pattern Recognition. The adaptation strategy consists of three units as pictured in Figure 1. Two parallel ART2-A neural networks are used, where one analyzes manipulated input patterns and the second analyzes controller error patterns. A decision maker then determines model gain adaptations based on the network classifications. The overall process model being adapted consists of the two equations that describe the output variables, which in the Laplace domain are of the form K Yi(S)
class
=
,
e-tdils
prl TPilS
+1
ul(s) +
K
e-tdi@
'"+ 1 uz(s); i = 1,2 ,
(15)
?pi$
Thus there are four variables, the four process model gains, that can be potentially adapted following a perturbation to the system. Following such a dynamic event, the algorithm collects the first 50 samples of manipulated input and controller error data and sends these patterns to the ART2-A networks for analysis. The first step in the adaptation process is the analysis of the manipulated input patterns. The input patterns are used to determine whether or not a perturbation to the system has potentially changed the process operating regime and thus the character of the process. In the case of set point changes, this information is used to determine if the set point change is large enough to merit process adaptation. For a very small set point change, for example, the process may not deviate enough from its previous operating regime to significantly change process character. In the case of unmeasured disturbances, the input patterns are used to determine the nature of the disturbance. For example, a step disturbance will shift the
manipulated input away from its mean value that existed prior to the onset of the disturbance. Thus the process has changed operating regime, potentially changing the process character due to nonlinearity in the process. On the other hand, a pulse or oscillating disturbance will not permanently change the character of the process, so although the error patterns resulting from such a disturbance may look underdamped, no model adaptation is necessary. This oscillatory component will be reflected in the manipulated input patterns, which should not significantly deviate from their local operating regime as the process character is not permanently changed. This information can be used to infer that reliable model adaptation cannot be made based on the given error patterns. Thus the pattern analysis of the input network results in one of two responses for each of the two manipulated input patterns: (1)the input pattern has a step component showing significant deviation from the initial steady state, reflecting a change in process character and the need for analysis of the controller error patterns, or (2) the input pattern is remaining local to the mean value which existed prior to the perturbation and thus no model adaptation is necessary. If the pattern analysis for a given input pattern signifies a potential change in character, the second ART2-A network is activated to analyze the controller error responses. The controller error is defined as
at sample t, where i = 1, 2 in this work. The purpose of the error network is to classify the degree of oscillation in the error patterns and determine how much a given model gain should be adapted based on its associated error pattern. For a binary system, Kpll and KPlzare associated with the error pattern of yl, while Kpzl and KP2z are associated with yz. Thus Kpll and Kplz adaptations are based on the el pattern, while Kpzl and KP22adaptations are based on the e2 pattern. As with previous SISO work, this strategy focuses on mismatch ratios as opposed to actual values of the gain, keeping it independent of the given process. The error network assigns the error pattern exemplars in the network to one of five classifications, based on the degree of damping. The classifications are given in Table 1. The decision maker then calculates the model gain adjustment for a given inputloutput pair, 8 K i j , based on the given class and user-specified selection of the class which represents desired performance. 6,ij is simply a factor by which the present model gain is multiplied by to give the updated model gain. The Decision Maker. The combinatorial analysis of the input and error patterns as implemented in the decision maker works in the following manner: Step 1: The input network analyzes the first input pattern, ul, for a shift in steady state, signifying a potential change in process character. Step 2a: If a shift in input steady state is diagnosed, the error network classifies el and adapts Kpll based on the model gain adjustment parameter &I, since Kpll is the model gain which relates el(y1) to u1. The error
1514 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
a
Table 2. Example Model Gain Adjustment Schedule class
8,ii
class
6.i;
1
2.00 1.50 1.00
4 5
0.67 0.50
2 3
SHIFT IN INPUT STEADY STATE
network then classifies e2 and adapts K,zl based on bK21. The adaptation is done according to the formula
NO SHIFT
P=I
KPij(t)= KPij(t-1)hij Step 2b: If the input network determines u1 has not shifted from steady state,K,11 and Kpzlremain unchanged and no analysis of the error patterns is necessary. Step 3: The input network analyzes the second input pattern, 242, for a shift in input steady state. Then repeat step 2, where, if necessary, Kp12is adapted based on el and K,22 is adapted based on e2. Step 4: Update the controller using the new model parameters. In the case of DMC, update the model parameter matrix as described in the DMC section. To improve the flexibility of the adaptive strategy, the user is able to choose which of the five classes is considered desired performance. The decision maker then uses this selection to calculate the adaptation factor, 6Kij, associated with each class. As can be seen in Table 1, the lower the class number, the more underdamped the error pattern. In the case of an underdamped pattern, it is desired to increase the model gain to decrease controller aggressiveness, while the converse is true for overdamped patterns. Thus, if CDis the desired class, the adaptation for a given class, CI, is calculated:
+ 0.51CD - c,I
c, < c, 6,ij = [le0 + 0.51CD - cI1]-’ for c, > CD 6,ij
= 1.0
for
(18)
For example, if class 3 is chosen as the desired response error pattern, then the adaptation schedule becomes that given in Table 2. Input and Error Network Training. In developing the training patterns it is important to remember that the goal of this work is an adaptation strategy that is process independent. The only limitation is that the process input/ output relationships can be reasonably described with a FOPDT model. As a result, there are an endless number of possible combinations of the four process transfer functions and the training set must take this into account. Also, the training set should be general enough such that the networks only have to be trained once, and not on a process-to-process basis. The development of training patterns for both networks is a continuous process of randomly generating a training pattern and then feeding it to the network for classification, where similar patterns are clustered within the nodes of each network. The first step is to randomly select parameters for four process models, the parameters being Kpij, ~ , i j ,and tdij. These models are then simulated as second order plus dead time processes, which is the lowest order process that can be representative of higher order processes. To give as accurate a representation of the potential patterns as possible, all the training patterns are developed in closed loop with the appropriate controller. Therefore, following the selection of process models, the next step is to generate the DMC model matrix. This is done as described in the DMC section, where the FOPDT differential equation of each transfer function is solved for a step change in input. The DMC parameters are then defined by eq 4.
3
F I
4-5
Figure 2. (a)Subset of input network exemplar patterns. (b) Subset of error network exemplar patterns.
For the ART2-A adaptive strategy to be effective, a wide variety of patterns are needed to represent varying degrees of process/model gain mismatch. Thus for the selected set of process models, G,, the process simulation gains are then randomly mistuned from the model values and a unique disturbance is randomly generated and applied to the process simulation. The resulting input and error patterns are then fed to the respective networks for training. The entire procedure is then repeated with a new G,. After a number of iterations, which include a wide variety of model combinations and mistunings, the end product is a network containing a wide range of patterns reflecting varying characteristics of process/model mismatch. Following the above procedure, the input network contains 40 unique patterns, each of which is then classified based on whether the pattern signifies a shift in the input’s steady-state value. Figure 2a shows a subset of the input network exemplar patterns. The top row of patterns are examples of those classified as changing operating regime, indicating a potential change in process character. These patterns tell the decision maker to make a subsequent analysis of the error patterns. The bottom row shows patterns classified as not having the potential for changing process character, so no check of the error patterns is necessary. For a pattern to be classified as shifting steady state it must clearly show a deviation from its previous steady state. For example, although the middle pattern in the bottom row of Figure 2a seems to be deviating from its initial steady state, it is not definitely clear that this is the case and so it is classified as a pattern with no shift in steady state. This approach is chosen to ensure that the ANN adaptive strategy only makes decisions with complete confidence. The error network contains 105 patterns, each of which is then classified into one of the five categories given in Table 1. A subset of the error network exemplars and their corresponding classifications is shown in Figure 2b, with the most highly underdamped patterns in the top row and the most overdamped patterns in the bottom row. For the bottom row of patterns in Figure 2b, the first two patterns are class 4 patterns while the third one is a class 5 pattern. The decision maker then uses these classifications to update the model gains via eqs 17 and 18.
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1515 Batchwise Regression. The presented adaptive strategy, due to its qualitative nature, is best suited for model adaptation in the wake of unmeasured disturbances. Dynamics caused by disturbances are difficult to model with traditional methods, as the data may be a reflection of the disturbance itself and not the true input/output relationship. In such cases, regressing model parameters to the data leads to a poor model and potentially unstable control. The pattern recognition techniques, on the other hand, are designed to make conservative model parameter adaptations when a high confidence level exists in the effectiveness of the adaptation. These techniques also contain an internal review system that restores the original model parameters if model adaptation causesa degradation in performance. More traditional model regression techniques are best suited for set point induced dynamics as the data give a true representation of the input/output relationship. The pattern recognition techniques used for disturbance dynamics are still applicable, but will not offer the same degree of accuracydue to the heuristic approach. However, the improved accuracy of model regression is achieved at the cost of greater computational expense. To investigate these claims, the ART2-A adaptive strategy will be compared to a more traditional modeling technique, batchwise regression. Following a set point change or disturbance to the system, the next 50 samples of both input and both output variables are collected. The two models given by eq 15 are then regressed on the data. Each model regression minimizes the objective function: (19) for i = 1 , 2 , where yi'(t) is the predicted output based on the model parameters. The minimization technique is a typical two-step multidimensional minimization procedure. Given an initial parameter vector, xo,and an initial search direction, so, the objective function is minimized along the initial search direction, after which a new search direction is chosen and the procedure repeated until convergence. This work uses the BFGS technique to calculate the new search direction (Edgar and Himmelblau, 1988). This is a secant method which uses first order information to obtain a new approximation of the Hessian matrix at every iteration, which is then used to calculate the new search direction by
sk = -(Hk)-'Vf(xk)
(20)
where V f ( x k is ) the value of the gradient and k represents the present iteration. Upon selection of a new search direction, a one-dimensional golden section is used to find the step length X which gives the new estimate of the parameter vector that minimizes the objective function along the given search direction according to Xk+l
= Xk
+ Ask
(21)
This procedure is done separately for each of the two equations described in eq 15. The minimization routine for a given model is terminated when both the change in parameter values and the change in the objective function fall below a specified tolerance at a given iteration.
Process Demonstrations Pilot-Scale Distillation Column Demonstration. The first process used to demonstrate the ART2-A adaptive strategy is a pilot-scale continuous distillation
COOLING M T E R
FEED I
I
h-
STEAM
&
BOTTOMS
Figure 3. Pilot-scale distillation column.
column located in the UCONN unit operations laboratory and pictured in Figure 3. The column includes 18 plates, a total condenser, and a partial reboiler. The feed is a methanol/water solution which is 15 wt ?4 methanol. Thermocouples are inserted in 15 plates in the column, the condenser inlet and outlet streams, and the reboiler. The system was initially equipped with five Foxboro control valves and the accompanying Foxboro 761 microprocessorcontrollers. Three of the controllers regulate flow rates: the feed rate, the reflux rate, and the bottoms rate. The fourth controls the steam valve position, where the steam line pressure is held constant at 60 psi. The fifth controls the reflux decanter level by manipulating the distillate flow rate. The steam flow rate is not used as a control variable due to inaccuracies in the steam flowmeter. The control challenge is dual temperature control of two control plates, as the equipment is not available for online measurement and control of the top and bottoms composition. The feed plate is plate 11, where plate 1 is at the top of the column. The controlled outputs are the temperatures on plate 6, T6,and plate 15, TI^. These are chosen based on the available thermocouples and the relative sensitivity of these plates to disturbances to the system. A personal computer (PC) based data acquisition system is used to read thermocouple data from the control plates and to implement the appropriate control valve adjustments. The manipulated inputs are the steam valve position, u1,and reflux rate, u2. The PC determines the control adjustments based on the input temperature data and sends the desired values of the manipulated variables to the controllers. To increase the accuracy of the data acquisition system, the signal from the PC is sent to the set point channel in the controllers, and not directly to the valves. The Foxboro controllers are then left in automatic mode, allowing them to act as a positioners within the PC based control loops. Thus while the PC determines the control valve positions, the Foxboro controllers act as intermediates in the signal transmission process. Since the Foxboro controllers are not directly determining the valve positions, they are not pictured in Figure 3. The initial operating conditions and DMC parameters for the column are given in Table 3 while the process model for the initial operating conditions is given in Table 4. The model in Table 3 was determined through a series of open loop step tests in the manipulated variables. The first element of the relative gain array (RGA) for this system, All, equals 3.17, indicating significant interaction
1516 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 Table 3. Pilot-Scale Distillation Column Parameters column parameters trays = 18 feed tray = 11 control tray no. 1 = 6 control tray no. 2 = 15 feed rate = 0.20 gal/min feed composition = 15 wt % methanol DMC parameters At = 23.0 s
- 46
0.0
0.8
1.6
2.4
3.2
2.4
3.2
P = 60 C = 15
rl = 100.0 rz = 10.0
] [:::!
Table 4. Continuous Distillation Column Transfer Functions 200.0s
+1
288.0s
+1
-20.42e-'05.08 115.0s
+1
7
0.0
+1
in the system. Contributing to the control challenge are widely varying time constants and large dead times. As a result of these factors and the natural variability of a real process, the desired response pattern is conservatively chosen as a class 4 pattern from Table 1. Thus any error pattern classification by the error network from class 1to class 3 will result in an increase in the given model gain while a class 5 pattern will result in a decrease in the model gain. As mentioned earlier, while the best application of the ART2-A adaptive strategy is with regard to disturbance dynamics, it is applicable to set point changes. In Figures 4-6, two set point changes are made in TI5 after initializing the model gains to half of the values actually exhibited by the process. Thus the initial K, vector is i0.031, -10.21, 0.17, -79.951. Figure 4 shows the response of the fixed model case where this poor model initialization results in underdamped responses to the two set point changes. The nonlinear nature of the process is reflected in the highly underdamped response to the second set point change. Figure 5 shows the results of ART2-A adaptation to the same set point trajectory, where again the desired response pattern is a class 4 pattern. In this figure and the next, adaptation is performed following each set point change. In Figure 5, the first set point change causes a shift in both input patterns, leading to a subsequent analysis of both error patterns. In this case, TSis a class 4 pattern while T15 is a class 3 pattern. The result is both K,ZI and KP22 are increased by a factor of 1.5. An identical classification is made at the second set point change, and thus the final K, vector is [0.031, -10.21, 0.37, -179.891. While this is not entirely accurate, it offers improved performance over the fixed case shown in Figure 4. Figure 6 shows the same set point trajectory where the batchwise regression technique is used following both set point changes. As can be seen, the regression based adaptation gives slightly better performance than the ART2-A strategy and a more accurate model. Following the second adaptation, the final K, vector is [0.053, -18.90, 0.26, -129.951. As expected, the batchwise regression technique is able to give a reasonably accurate description of the set point induced dynamics, at the cost of greater computational expense. The next series of experiments compares the neural network adaptive strategy to batchwise regression in the wake of an unmeasured disturbance. The disturbance in question is a 50% decrease in the feed rate, from 0.20 to 0.10 gal/min. Figure 7 shows the response of both controlled variables when the model is fixed throughout
6
1.6
1
SAMPLE x 1 0-2, t
-1F~9.9e-'~'.~ 237.0s
0.8
Figure 4. Fixed model control of (a) T6 and (b) Ti5 in response to set point changes in Tl5.
1
641 0.0
0.8
1.6
2.4
3.2
0.0
0.8
1.6
2.4
3.2
SAMPLE x lo-*, t
Figure 5. ART2-A network based adaptive control of (a) T8 and (b) 2'15 in response to set pointchanges in T15.
64 0.0
76
0.0
0.8
1.6
2.4
3.2
0.8
1.6
2.4
3.2
SAMPLE x 10-2, t
Figure 6. Batchwise regression based adaptive control of (a) T6 and (b) T15 in response to set point changes in T15.
the trajectory. The initial set point change in Tl5 shows good controller performance preceding the disturbance, which occurs at sample 200. The change in feed forces the system into a region where the model is no longer accurate,
,
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1517
rr
-46
0
0.0
1.0
2.0
3.0
4.0
=-0.5
5.0
j 0
% 76
0.0 1.0
2.0
3.0
4.0
5.0
0
1.0
2.0
3.0
4.0
2.0
3.0
4.0 5.0
5.0
I
-1 -67
0.0
1.0
,x,
NORMUZED PATTERN NENlORK PAHERN ,
20
30
40
50
___
N O R W Z E D PAllERN N W O R K PATERN
10
20
30
40
50
SAMPLE, t
Figure 7. Fixed model control of (a) T6 and (b) T15 in response to step disturbance in feed rate.
641 0.0
10
'
4-0.51
SAMPLE x 10-2, t
,
SAMPLE x 10-2, t
Figure 8. ARTS-A network based adaptive control of (a) 2'8 and (b) 2'15 in response to step disturbance in feed rate.
resulting in the highly underdamped response for the second set point change in T15. Figure 8 demonstrates the ART2-A adaptive strategy on the same trajectory with the identical feed disturbance. As seen in Figure 8, the system responds well to the initial set point change in T15. To isolate the adaptive strategy with respect to disturbance dynamics, no adaptation is made based on the two set point changes in T15. The only adaptation made is based on the feed rate disturbance, which as in Figure 7 occurs at sample 200. The response to the feed disturbance appears to be more aggressive in Figure 8 than in Figure 7. This is due to the typical nonstationary behavior of the column, as the daily column behavior is affected by many uncontrollable environmental conditions such as ambient temperature. Figure 7 shows the experimental run where the feed disturbance had the least effect on subsequent performance. As can be seen, the adaptive case of Figure 8 still offers significant improvement in performance over Figure 7, and this difference would be magnified had other fixed case experimental runs been shown. This disturbance causes a deviation of both variables from set point, triggering the ART2-A adaptive strategy. In this case, both input variables are classified as shifting from steady state, resulting in the subsequent analysis of both error patterns. The TGerror pattern is classified as
Figure 9. (a) Normalized input pattern for reflux rate and (b) normalized error pattern for 2'15 in response to step disturbance in Figure 8.
a class 4 pattern while the T15 response is classified as a class 1 pattern. Since the T6 pattern is considered the desired response, no changesare made to the corresponding model gains, K,II and Kp12. By eq 17, the class 1pattern results in both KPzl.and Kp22 being increased by a factor of 2.5. Thus, the initial K, vector of [0.062, -20.42, 0.33, -159.903 becomes 10.062, -20.42, 0.83, -399.751 following the adaptation. The second set point change in T15 shows the resulting performance of the updated model. While the model may not be entirely accurate, it is a significant improvement from the fixed model case shown in Figure 7. Figure 9 gives a clearer demonstration of how network pattern classification works. Figure 9a shows the normalized input pattern of the reflux rate, U2, following the disturbance, along with the closest matching input network exemplar pattern. As can be seen, the pattern shows a continual deviation from ita previoussteady state,resulting in the pattern being classified as having the potential of changing process character. Figure 9b shows the normalized error pattern of T16,E2, along with the winning error network exemplar. The highly underdamped nature of the winning network pattern results in the designation of a class 1 pattern and the aforementioned adaptation of Kp2?and Kp22. Figure 10 shows the results of trying to fit a model to the disturbance dynamics. In Figure 10, as in Figure 8, the only model adaptation occurs in response to the feed rate disturbance at sample 200. In this figure, the batchwise regression technique is used to fit the four model gains based on the change in feed rate. As can be seen by the resultant response, the fit is very poor, especially with respect to Kp21 which changes from 0.33 to -0.022. The change in sign causes the resulting loss of stability shown in Figure 10. This again demonstrates the difficulties in trying to model disturbance dynamics. While the ART2-A adaptive strategy only offers a qualitative assessment of the response, it is able to improve performance where more traditional techniques fail. Shell Fundamental Control Problem Demonstration. This section investigates the ART2-A adaptive strategy on a 2 X 2 subset of the Shell fundamental control problem (Prett and Garcia, 1988). The controlled variables are the top end point ( y l ) and the side end point ( y z ) , while the bottoms reflux duty (u1)and the top draw (u2) are chosen as the manipulated variables. The upper reflux duty ( d ) is used as an unmeasured disturbance to the
1518 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
0.1
1
I 641
0.0
1.0
76 0.0
1 .O
4.0
5.0
2.0 3.0 4.0 SAMPLE Y 10-2, t
5.0
2.0
3.0
-0.1
Table 5. Shell Fundamental Control Problem Parameters steady-state parameters y1 = yz = 0.00 u1= UZ = 0.0 d = 0.0 DMC parameters At = 8 min P=31 C=8 rl = 0.01 r2= 0.01
4.05e-"' 50.0s
+1
40.0s + 1 L
50.0s + 1
10.0
12.5
Figure 11. ART2-A network based adaptive control of (a) y1 and (b) yz in Shell problem for disturbance trajectory in Figure 12. 0.37
SAMPLE x 10-2, t
1[:I
1.44e-27' -
40.0s
+
uz(s)
5.39e-'% -
7.5
m 2.5 5.0 7.5 10.0 12.5
~
+1
5.0
-0.1 0.0
Table 6. Shell Fundamental Control Problem Transfer Functions .~~ 50.0s
2.5
SAMPLE x 10-2, t
Figure 10. Batchwise regression based adaptive control of (a) 2'6 and (b) T15 in response to step disturbance in feed rate.
5.88e-278 -
-
0.0
1.83e-Ib 20.0s
+1
2
system. The process and control parameters are given in Table 5. The transfer function matrix is given in Table 6, where the variables are organized such that the first element of the RGA is positive. The value of A11 for this system is 8.46. Note that all of the variables are constrained between -0.5 and 0.5 as described in Prett and Garcia. The desired pattern class is chosen as a class 3 pattern. The Shell problem is used to demonstrate the ability of the ART2-A adaptive strategy to handle irregular disturbance dynamics. Figure 11 shows how the strategy works on the disturbance trajectory in Figure 12. In Figure 11, the model parameters Kpll and KPlz are initially mistuned to 50% of their true values such that the initial model vector is K, = 12.94, 2.03, 6.90, 5.881. This mistuning causes the initially high overshoot seen in the first two set point changes in Figure 11, at samples 10 and 250. At sample 500, a disturbance in the upper reflux duty as pictured in Figure 12 is injected into the system. As can be seen, this includes a step disturbance from 0.0 to 0.2 followed by a decaying oscillation. Due to the highly interactive nature of the process and the mistuning in the gains, both y1 and y2 respond very aggressively, as shown in Figure 11. Due to the prolonged nature of this disturbance, the ART2-A adaptive strategy actually makes two separate analyses of the disturbance dynamics. The first is at sample 550, following the initial aggressive response to the disturbance. Both u1 and u2 are recognized by the
Figure 12. Disturbance trajectory for Shell problem demonstration.
input network as stepping away from their previous steady state values, indicating a potential change in character. The error network classifies the y1 response as a class 2 pattern and the y2 response as a class 3 pattern. Thus Kql1 and Kp12 are adjusted by a factor of 1.5. The new gain vector following this adaptation is [4.41, 3.04, 6.90, 5.881, bringing it closer to the true values for the process gains. As the disturbance persists, the ART2-A adaptive strategy recognizes the continuing dynamic nature of the process and makes a second analysis a t sample 600. The input network, in analyzing the input patterns from sample 550 to sample 600, is now looking at predominantly oscillatory responses. It determines that both u1 and u2 are not deviating from their initial steady state and thus no further adaptation is made. The overall result of the ART2-Aadaptive strategy is demonstrated in the following two set point changes, a t samples 750 and 1000,where the improved model eliminates the overshoot that was present in the initial two set point changes. While the change in response from the first two to the last two set point changes is not tremendous, the underdamped response of y1 to the first two set point changes comes a t the expense of harsh manipulation of the manipulated variables, which is eliminated in the last two set point changes as a result of the model adaptation. Figure 13 demonstrates how the batchwise regression technique performs on the identical trajectory. As expected, the regression technique cannot accurately model the disturbance dynamics, and produces a poor model which leads to instability in the system. This further demonstrates the usefulness of the ART2-A adaptive strategy in giving a t least some measure of performance and useful adaptation in a situation where more traditional methods fail.
Ind. Eng. Chem. Res., Vol. 33, No. 6,1994 1519
-~ 50 75
00
25
0.0
2.5
5.0
1.5
10.0
/5
12.5
SAMPLE x 10-2, t
Figure 13. Batchwise regression based adaptive control of (a) y1 and (b) y2 in Shell problem for disturbance trajectory in Figure 12. Conclusion For any adaptive controller to become widely accepted, it must be applicable to a wide range of processes and process conditions. The pattern recognition basedstrategy presented here is designed to maintain process independence while still allowing reasonable process model adaptation. While the interactive nature of 2 X 2 systems limits the methodology to a more qualitative nature, it still allows for model adaptation in situations where more traditionalmethodsfail. Asdisturbance induddynamics may mask the true process character, making model regression on input/output data difficult, disturbance rejection is the ideal situation for the use of pattern based techniques. However, the ART2-A strategy also offers reasonable adaptation to set point induced dynamics without the computational expense of regression techniques. By focusing on input/output trends in the data and multiplicative model parameter adaptation as opposed to explicit parameter prediction, the strategy is applicable to a wide variety of processes. The ART2-A neural networks ensure the usefulness of the pattern recognition methods by maintaining process independence while providing a means of analyzing a wide variety of patterns in an efficient and reliable matter. This is demonstrated in the application of the strategy to a difficult process simulation and an especially challenging pilot-scale distillation process. Acknowledgment is gratefully made to the National Science Foundation through Grant CTS-9008596, to Connecticut InnovationsInc. throughanElias Howe grant, and to the UConn Precision Manufacturing Center. Nomenclature
C = DMC control horizon e d t ) = controller error for controlled output i at sample t Ei = normalized error for controlled output i
3 = filtering operator used in ART2-A network H = Hessian matrix G, = model transfer function matrix I = filtered and normalized network input pattern vector Io = network input pattern vector K#j = steady-state model gain for output/input pair i j M = dimension of network input pattern vector P = DMC prediction horizon sk = batch regression search vector at iteration k At = sample time t = time expressed as integer number of samples t d i j = model dead time for output/input pair i j Tj = network exemplar to network input pattern matching score for node j 61= DMC weighting matrix for controlled outputs 6 2 = DMC weighting matrix for manipulated inputa Au = vector of manipulated input moves ui(t) = manipulated input i Vi = normalized input for manipulated input i XB = bottoms composition XD = distillate composition xk = batch regression parameter vector at iteration k y ; ( t ) = controlled output variable i yi'W = model estimate of true process output i at sample t y.,i(t) = set point of controlled output i at sample t zj* = network exemplar pattern at node j Greek Symbols 01
= network initialization constant
3 , = network learning parameter
r; = DMC tuning parameter
= model gain adjustment for model ij
&j
X = golden section step length 1) =
normalization operator used in ART2-A network
e = network filter threshold value p = network vigilance parameter r#j
= model time constant for output/input pair i j
-# = adjustednetworkinputpatternfortrainingofcommitted
nodes
Literature Cited Bristol, E. H.On A New Measure of Interaction for Multivariable Process Control. IEEE Trans. Auto Control 1966, AC-11, 133134.
Carpenter, G. A.; Grossberg,S.; &sen, D. B. ART2-A: An Adaptive Resonance Algorithm for FLipid Category Learning and Rerognition. Neural Networks 1991,4, 493-504. Cooper, D. J.; Megan, L.; Hinde, R. F. Comparing Two Neural Networks for Pattern Based Adaptive Process Control.AIChE J. 1992.38, 41-55.
Cutler,C. R.; Ramaker, B. L. Dynamic Matrix Control-AComputer Control Algorithm. Presentedatthe AIChE 86th AnnualMeetine. Houston,TX, 1979.
Edgar, T.;Himmelblau,D. M. Optimization of Chemical Processes; McGraw-Hill: New York. NY. 1988. Megan. L.; Cooper, D. J. Neural Network Based Adaptive Control ViaTemporal Pattern Recognition. Con. J.Chem. EM. 1992,70, 1208-1219.
Prett, D. M.; Garcia. C. G. Fundamento1 Process Control; Butterw o n k . Boston. MA. 1988.
Receiued for reuiew October 15, 1993 Reuised manuscript receiued March 18, 1994 Accepted April I, 1994.
*AbstractpublishedinAduanceACSAbslrac.ts,May 1,1994.
