Chem. Eng. Comm., 191: 10831119, 2004 Copyright # Taylor & Francis Inc. ISSN: 0098-6445 print/1563-5201online DOI: 10.1080/00986440490276452
MODELING AND PREDICTIVE CONTROL OF MIMO NONLINEAR SYSTEMS USING WIENER-LAGUERRE MODELS
PRABIRKUMAR SAHA S. H. KRISHNAN V. S. R. RAO SACHIN C. PATWARDHAN Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai, India
In this work, a Weiner-type nonlinear black box model was developed for capturing dynamics of open loop stable MIMO nonlinear systems with deterministic inputs. The linear dynamic component of the model was parameterized using orthogonal Laguerre filters while the nonlinear state output map was constructed either using quadratic polynomial functions or artificial neural networks. The properties of the resulting model, such as open loop stability and steady-state behavior, are discussed in detail. The identified Weiner-Laguerre model was further used to formulate a nonlinear model predictive control (NMPC) scheme. The efficacy of the proposed modeling and control scheme was demonstrated using two benchmark control problems: (a) a simulation study involving control of a continuously operated fermenter at its optimum (singular) operating point and (b) experimental verification involving control of pH at the critical point of a neutralization process. It was observed that the proposed Weiner-Laguerre model is able to capture both the dynamic and steady-state characteristics of the continuous fermenter as well as the neutralization process reasonably accurately over wide operating ranges. The proposed NMPC scheme achieved a smooth transition from a suboptimal operating point to the optimum (singular) operating point of the fermenter without causing large variation in manipulated inputs. The proposed NMPC scheme was also found to be robust in the face
Received 30 April 2001; in final form 25 October 2002. Address correspondence to Sachin C. Patwardhan, Department of Chemical Engineering, I.I.T. Bombay, Powai, Mumbai, 400 076, India. E-mail:
[email protected] 1083
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P. SAHA ET AL. of moderate perturbation in the unmeasured disturbances. In the case of experimental verification using the neutralization process, the proposed control scheme was found to achieve much faster transition to a set point close to the critical point when compared to a conventional gain-scheduled PID controller. Keywords: Wiener models; Laguerre filters; Predictive control; pH control; Input multiplicity
INTRODUCTION Linear model predictive control (MPC) schemes are now widely used in the process industry for control of key unit operations. However, the use of linear dynamic models for prediction limits their applicability to a narrow range of operation or to systems that exhibit mildly nonlinear dynamics. Many key unit operations in chemical plants, such as high purity distillation columns, reactors, neutralization processes (control of pH), and biochemical reactors exhibit highly nonlinear dynamic behavior. The need to achieve tighter control of strongly nonlinear processes has led to a more general MPC formulation whereby nonlinear dynamic models are used for prediction (Meadows and Rawlings, 1997). Selection of a suitable form of a nonlinear model to represent system dynamics is a crucial step in the development of a nonlinear MPC (NMPC) scheme. The NMPC schemes proposed in literature use either models developed from first principles (Brengel and Seider, 1989; Patwardhan and Madhavan, 1993) or models identified from input-output data (i.e., nonlinear black box models) (Hernandez and Arkun, 1993; Su and McAvoy, 1997; Patwardhan et al., 1998). The first principle models are valid globally and can predict system dynamics over the entire operating range. However, development of a reliable first principle model is, in general, a difficult and a time-consuming task. The nonlinear black box models, on the other hand, have certain advantages over the mechanistic models in terms of development time and efforts. Thus, from a practical viewpoint, development of a NMPC scheme based on a nonlinear black box model appears to be a more attractive option. In the development of a nonlinear black box model, selection of a suitable model structure that can capture nonlinear dynamics over a wide operating range is not an easy task. Significant among the different black box model structures are nonlinear autoregressive with exogenous inputs (NARX) models (Hernandez and Arkun, 1993), Volterra series expansion models and block oriented models (Hammerstein and Wiener structures) (Patwardhan et al., 1998), and artificial neural networks (ANN) (Su and McAvoy, 1997). The determination of model order and model structure of a general NARX model is a difficult exercise even for a single-input, single-output (SISO) system and the difficulties are further
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compounded for multiple-input, multiple-output (MIMO) systems. Volterra series models can be used to model a wide class of nonlinear systems; however, these models are non-parsimonious in parameters and, in turn, difficult to use for modeling MIMO systems. Su and McAvoy (1997) show that recurrent neural networks (RNN) are better suited for the development of nonlinear MPC schemes. These models have output error structure and generate better long-range predictions than feedforward neural networks (FNN) that have been widely employed in process control applications. Development of RNN models is, however, considerably more difficult than development of FNN models. Thus, it is necessary to evolve a scheme for development of a black box model in which the model structure can be selected relatively easily and the resulting model is valid over a wide operating range. One of the linear black box modeling techniques that has been receiving increasing attention in the past decade is the use of orthonormal filter networks such as Laguerre and Kautz filters (Zervos and Dumont, 1988; Wahlberg, 1991, 1994). The orthonormal filter approximation provides a simple and elegant method of representing stable systems. Advantages of using such orthonormal filters for process modeling can be summarized as follows: A good approximation can be obtained with a small number of the terms due to the orthogonal property. Development of orthonormal filter models does not need any explicit knowledge about system time constant and time delay. The estimates of the model coefficients are unbiased even for a truncated series. The use of such orthonormal filters in combination with a memoryless nonlinear map (referred to hereafter as a Wiener-Laguerre model) for modeling of nonlinear dynamical systems was originally proposed by Wiener (see Billings, 1980). Boyd and Chua (1985) have shown that a nonlinear moving average operator of this type is adequate for capturing dynamics of any time invariant nonlinear systems with fading memory. Doyle et al. (1995) and Maner et al. (1996) have proposed a second-order Volterra series model and used this model in nonlinear MPC formulation. One difficulty with this model is that the number of parameters that need to be estimated can be prohibitively large for a multivariable system. The Wiener-Laguerre model can be looked upon as a representation of the Volterra series model that is parsimonious in parameters. Dumont et al. (1994) have used a model of this type for developing an adaptive predictive control scheme for controlling SISO nonlinear systems. Recently, multivariable extensions of the Weiner-Laguerre model have been proposed by Sentoni et al. (1998) and Saha (1999). In both the cases,
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the MIMO systems were modeled as multiple multiple-input single-output (MISO) models. Sentoni et al. (1998) used ANNs for constructing a nonlinear state output map. Saha (1999) used quadratic polynomials as well as ANN for constructing a nonlinear output map and used these models in nonlinear MPC formulations. The present work aims at developing a scheme for modeling of open loop stable MIMO nonlinear systems with deterministic inputs using a Weiner-type model consisting of orthonormal filters and a memoryless output map. As proposed by Saha (1999), the nonlinear output map is constructed using either quadratic polynomials or ANNs. Note that the latter approach is similar to the approach proposed by Sentoni et al. (1998). The resulting nonlinear Weiner-Laguerre model is then used to synthesize a nonlinear MPC scheme. The efficacy of the proposed modeling and control scheme is demonstrated using the following two benchmark control problems: Simulation study: control of a continuously operated fermenter at the singular optimum operating point Experimental evaluation: control of pH in a neutralization process at the critical point The continuous fermenter belongs to a class of nonlinear systems that exhibit input multiplicity in the desired operating region, i.e., identical outputs are obtained for multiple inputs. The optimum operating point of this system is a singular point where steady-state gain is reduced to zero and the gain changes its sign across the optimum. Consequently, controlling the fermenter at the optimum operating point is known to be challenging (Patrwardhan and Madhavan, 1993; Henson and Seborg, 1992; Biegler and Rawlings, 1991). Similarly, controlling pH of a neutralization process at the point of neutralization is extremely difficult due to large variations in steady-state gain in the neighborhood of this point. Thus, control of pH at the point of neutralization has often been used as a benchmark to test nonlinear control schemes in process control literature. This article is organized in six sections. The development of the proposed MIMO Weiner-Laguerre model and its properties are discussed in the next section. This is followed by the development of a nonlinear MPC scheme based on the proposed models. After this, two sections present results of a simulation case study and experimental evaluation. The main conclusions of this work are presented in the last section. DEVELOPMENT OF WIENER-LAGUERRE MODEL Consider a SISO linear system modeled using a Laguerre filter network and represented as
MIMO NONLINEAR SYSTEMS USING WIENER LAGUERRE MODELS N X
y^ðzÞ ¼
1087
! ci Li ðzÞ uðzÞ
ð1Þ
i¼1
where
Li ðzÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 azÞi1 ð1 a2 ÞT ðz aÞi
denotes the ith order Laguerre filter, N represents number of Laguerre filters used for model development, að1 < a < 1Þ represents the Laguerre filter parameter, T represents sampling interval, y^ represents model output, and u represents manipulated input. Defining a state vector XðkÞ ¼ ½x1 ðkÞ x2 ðkÞ . . . xN ðkÞT where xi ðkÞ represent the output from ith order Laguerre filter at the kth sampling instant, a discrete state space realization of the Laguerre filter network can obtained as Xðk þ 1Þ ¼ FðaÞ XðkÞ þ GðaÞ uðkÞ
ð2Þ
where uðkÞ is the system input, FðaÞ is an N N lower triangular matrix, 2
a 0 0 2 6 ð1 a Þ a 0 6 FðaÞ ¼ 6 að1 a2 Þ ð1 a2 Þ a 6 4 ... ... ... ð1ÞN aN2 ð1 a2 Þ ð1ÞN1 aN3 ð1 a2 Þ . . .
0 0 0 ... ...
3 0 07 7 07 7 05 a
and GðaÞ is an N dimensional vector of the form GðaÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT ð1 a2 ÞT ð1 a2 ÞT a ð1 a2 ÞT . . . . . . ðaÞN1
For the linear model given by Equation (1), the model output can be expressed as weighted sum of the states y^ðkÞ ¼ cT XðkÞ
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where elements of ‘c’ are Laguerre filter coefficients, i.e., c ¼ ½c1 c2 ::: ::: cN . If it is desired to develop a Wiener-Laguerre type SISO nonlinear model, a nonlinear state-output map can be constructed so that the model output is represented as y^ðkÞ ¼ C½xðkÞ
ð3Þ
where Cð:Þ : RN ! R represents a memoryless nonlinear function. In the present work, it is desired to develop a Wiener-Laguerre type model to capture dynamics of MIMO nonlinear systems with deterministic inputs. One possibility is to model an n m MIMO nonlinear system as n MISO nonlinear systems. In order to see how this can be achieved, consider a MISO Wiener-Laguerre model relating the manipulated input vector with the ith output as shown in Figure 1. Let Xij 2 RNij represent the Nij dimensional state vector associated with a realization of a Laguerre filter network consisting of Nij number of filters associated with the jth input. The corresponding state dynamics can be represented as Xij ðk þ 1Þ ¼ Fðaij ÞXij ðkÞ þ Gðaij Þuj ðkÞ
ð4Þ
where aij represents the time scale parameter of the Laguerre filter network correlating ith output with jth input. Defining an augmented state vector Xi ðkÞ and input vector uðkÞ as T Xi ðkÞ ¼ XTi1 ðkÞ XTi2 ðkÞ . . . XTim ðkÞ
uðkÞ ¼ ½u1 ðkÞ
u2 ðkÞ . . . um ðkÞT
Figure 1. State space representation of MISO Wiener-Laguerre model.
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the state dynamics of the MISO model can be represented as Xi ðk þ 1Þ ¼ Fi Xi ðkÞ þ Gi uðkÞ
ð5Þ
where Fi ¼ block diag½Fðai1 Þ
Fðai2 Þ . . . Fðaim ÞNi xNi
Gi ¼ block diag½Gðai1 Þ Gðai2 Þ . . . Gðaim ÞNi xm and Ni ¼
m X
Nij
j¼1
for i ¼ 1; 2; . . . n. The output of the MISO Wiener-Laguerre model can be expressed as ^yi ðkÞ ¼ Ci ½Xi ðkÞ
ð6Þ
where Ci ð:Þ : RNi ! R ðfor i ¼ 1; 2; . . . nÞ represents a memoryless nonlinear state-output map. The above set of n MISO models can be further combined to formulate a MIMO Wiener-Laguerre model as follows: Xðk þ 1Þ ¼ FXðkÞ þ GuðkÞ
ð7Þ
^ yðkÞ ¼ C½XðkÞ
ð8Þ
where XðkÞ ¼ XT1 ðkÞ
XT2 ðkÞ . . . XTn ðkÞ
T
F ¼ block diag½F1 F2 . . . Fm NN T G ¼ GT1 GT2 . . . GTm NN ; and N¼
n X i¼1
Ni
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The MISO output maps Ci ð:Þ in Equation (6) can be selected in different ways. Note that, instead of developing n MISO output maps, it is also possible to propose one MIMO output map Cð:Þ as given in the equation. In the present work we specifically consider two possibilities (Saha, 1997). Output Maps Polynomial Output Map. Nonlinear map Ci ð:Þ can be selected as a polynomial function of elements of state vector Xi ðkÞ. The resulting nonlinear model is referred to as a Laguerre-polynomial model hereafter. In the present work, we restrict ourselves to only simple quadratic polynomial maps. Thus, the resulting MISO Wiener-Laguerre model can be represented as Xi ðk þ 1Þ ¼ Fi Xi ðkÞ þ Gi uðkÞ ^ yi ðkÞ ¼ Ci Xi ðkÞ þ XTi ðkÞDi Xi ðkÞ
ð9Þ
ð10Þ
where Ci is an Ni 1 vector and Di is a symmetric Ni Ni coefficient matrix. The above model can be looked upon as a truncated second-order MISO Volterra series model. When compared to a Volterra series model, the proposed Wiener-Laguerre model has considerably fewer parameters: No. of MISO model parameters ¼ fm þ Ni ðNi þ 3Þ=2g The above set of n MISO models can be further combined to formulate a MIMO Wiener-Laguerre model as follows: Xðk þ 1Þ ¼ FXðkÞ þ GuðkÞ
ð11Þ 2
3 XT1 ðkÞD1 X1 ðkÞ 6 XT ðkÞD2 X2 ðkÞ 7 2 7: ^ yðkÞ ¼ C½XðkÞ ¼ CXðkÞ þ 6 4 5 ... T Xn ðkÞDn Xn ðkÞ
ð12Þ
where C ¼ block diag½C1 C2 . . . Cn nN : Artificial Neural Network (ANN) Output Map. Another possibility is to use a multilayer perceptron (MLP) type ANN to map the state vector XðkÞ to the output. A multiLayer perceptron type ANN has an inherent capability of representing any arbitrary function with adequate accuracy by proper choice of its weights and biases (Su and McAvoy, 1997). This
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virtue of the ANN coupled with the basic merits of the Laguerre model can potentially generate nonlinear models with wider ranges of applicability. The resulting nonlinear model is referred to as a Laguerre-ANN model hereafter. Inputs to the ANN consist of outputs of the Laguerre filters, i.e., elements of vector XðkÞ. Considering one hidden layer with Nh hidden nodes and n nodes in the output layer and assuming sigmoid transfer function in all the nodes (both hidden and output), the output map can be expressed as vi ðkÞ ¼
N X
½W1 il hl ½XðkÞl ai
ði ¼ 1; 2; . . . ; Nh Þ
ð13Þ
l¼1
qi ðkÞ ¼
zi ðkÞ ¼
1 1 þ ev;i ðkÞ Nh X
;
ði ¼ 1; 2; . . . ; Nh Þ
½W2 ij qj ðkÞ bi
ð14Þ
ð15Þ
j¼1
1 yi ðkÞ ¼ gi 1 þ ezi ðkÞ
;
ði ¼ 1; 2; . . . ; nÞ
ð16Þ
where ½W1 and ½W2 represent ðNh NÞ and ðn Nh Þ dimensional weighting matrices, respectively, and ai and bi represent the associated ðNh 1Þ and ðn 1Þ bias vectors, respectively. Also, hl ð:Þ and gi ð:Þ represent the functions used for scaling state variables f½XðkÞl : l ¼ 1; 2; . . . Ng and the network output, respectively. These equations can be expressed in vector matrix notation in a more compact matrix form as follows: VðkÞ ¼ W1 H½XðkÞ a QðkÞ ¼ SðVðkÞÞ
ð17Þ
yðkÞ ¼ G½SfW2 QðkÞ bg where VðkÞ ¼ ½v1 ðkÞ v2 ðkÞ . . . vNh ðkÞT
ð18Þ
and Sð:Þ denotes the logistic sigmoidal function applied to each element of VðkÞ while Hð:Þ and Gð:Þ denote input and output scaling functions, respectively. Note that the output maps in the LaguerreANN model can be developed either as n MISO maps or as one MIMO nonlinear map.
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Remark1. If the nonlinear system to be modeled has oscillatory modes, a model developed using Laguerre filters may not produce satisfactory results. In such cases, a Kautz-Wiener model can be developed using Kautz filters, which can capture dynamics of systems with complex poles. A SISO linear Kautz filter network model can be written as (Wahlberg, 1994)
yðzÞ ¼
N X
! ci ji ðzÞ uðzÞ
i¼1
where even and odd Kautz filters can be represented as pffiffiffiffiffiffiffiffiffiffiffiffiffi i1 1 c2 ðz bÞ cz2 þ bðc 1Þz þ 1 j2i1 ðz; b; cÞ ¼ 2 z þ bðc 1Þz c z2 þ bðc 1Þz c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i1 ð1 c2 Þð1 b2 Þ cz2 þ bðc 1Þz þ 1 j2i ðz; b; cÞ ¼ 2 z þ bðc 1Þz c z2 þ bðc 1Þz c Here, 1 < b < 1 and 1 < c < 1. In practice, depending upon how each output responds to each input, one may have to use a suitable combination of Laguerre and Kautz filters to develop a MIMO model of the form given by Equation (7).
Model Parameter Estimation The key step in the development of the Laguerre models is the estimation of time scale parameters aij and selection of number of filters Ni for the Laguerre filters. In order to develop the MISO models, the system should be perturbed by simultaneously introducing a random variation of variable frequency and amplitude in all the inputs. The selection of MISO model structure, i.e., whether a Laguerre filter or a Kautz filter should be used between an input-output pair, can be carried out by observing the step response data for each input-output pair. The step response data can also be used to generate meaningful initial guesses for the individual filter parameters aij . The following procedure is proposed for the selection of the number of filters between each input-output pair and estimation of parameters of the resulting polynomial output map for ith output: Step 1: Select Nij ð j ¼ 1; 2; 3; 4; . . . mÞ individually for each input-output pair.
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Step 2: For the chosen values of Nij in Step 1, estimate optimal set of ai ¼ ½ai1 ai2 . . . ai2 T such that the sum of the square of errors between the measured data and estimated model output is minimized, i.e., ð^ yi ; abi Þ ¼ arg min ai ;yi
Ns X
ðyi ðkÞ b yi ðkÞÞ2
k¼1
where y represents the vector of parameters of the nonlinear state-output map, yi ðkÞ represents the ith measured output, and Ns represents the number of data points used for parameter estimation. Step 3: Estimate the following statistical information criteria (Pearson and Ogunnaike (1997)): Akaike information criteria (AIC): "
# Ns 1 X 2 ðAICÞi ¼ Ns ln ei ðkÞ þ 2j Ns k¼1 Here, j represents the total number of terms included in the model and yi ðkÞ represents the prediction error at time instant k. ei ðkÞ ¼ yi ðkÞ b Percent prediction error (PPE): Ns P
ðPPEÞi ¼
½yi ðkÞ ybi ðkÞ2
k¼1 Ns P
100% ½yi ðkÞ yi 2
k¼1
Here, yi is the mean value of the plant output data. Step 4: Repeat Steps 1 to 3 for different choices of Nij ð j ¼ 1; . . . nÞ. The most suitable combination of Nij can be arrived at by comparing different models using the AIC/PPE values and selecting the model that gives a good fit with smaller number of parameters. Note that in the present work the resulting nonlinear least square parameter estimation problems were solved using MATLAB Optimization Toolbox. Remark 2. Since matrices ðFi ; Gi Þ are functions of time scale parameter vector ai alone, the parameter estimation problem in the case of the MISO Laguerre-polynomial model ((9)(10)) can be stated as
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P. SAHA ET AL.
ðb ai ; b yi Þ ¼ arg min ðai ;yi Þ
h
Ns h X
yi ðkÞ jðk; ai ÞT yi
i2
k¼1
jðk; ai Þ ¼ Xi ðkÞT X2i;1 ðkÞ Xi;1 ðkÞXi;2 ðkÞ . . . fX2i;Ni ðkÞ T yi ¼ CTi Di;11 Di;12 . . . Di;Ni Ni
iT
Note that, given a guess of aj , the above parameter estimation problem is linear with respect to yj . Thus, yj can be estimated using the ordinary linear least square method. Taking advantage of this fact, the above optimization problem can be reformulated as " # Ns h iT h i X b b ðb ai Þ ¼ arg min yðkÞ ybðk j yi Þ yðkÞ ybðk j yi Þ ai
k¼1
where h i1 Subject to b yi ¼ E ji ðkÞji ðkÞT Eðji ðkÞyi ðkÞÞ jaik j < 1
for k ¼ 1; 2 . . . Ni
where Eð:Þ represents expected value. This approach can considerably simplify the nonlinear optimization problem in Step 2. Remark 3. The iterative procedure proposed above can be computationally demanding when the output map is a multilayer perceptron. The key step in the development of the nonlinear Laguerre model is selection of time scale parameters ‘a’ and the number of Laguerre filters N. For the case of linear systems Wahlberg (1991) has suggested that the value of time scale parameter ‘a’ should be chosen close to the inverse of the dominant time constant of the system in order to obtain a fast rate of convergence of the series. Thus, instead finding optimum time scale parameter ‘a’, one possibility is that the time scale parameter can be selected based on step response data and the ANN parameters can then be computed using standard training methods such as the back propagation method. PROPERTIES OF WIENER-LAGUERRE MODEL In this section, we discuss some properties of the resulting Weiner-Laguerre model and compare them with a general NARX model of the form yðkÞ ¼ X½yðk 1Þ; yðk 2Þ; . . . ; yðk nÞ; uðk 1Þ; . . . ; uðk mÞ þ eðkÞ ð19Þ
MIMO NONLINEAR SYSTEMS USING WIENER LAGUERRE MODELS
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where Xð:Þ represents a MIMO nonlinear operator and eðkÞ represents the residual vector. Steady-State Characteristics Comparison of steady-state behavior of a nonlinear black box model with that of the process can be looked upon as one test of model validation. For any given steady-state input, say us , the steady-state behavior of deterministic component model (19) can be computed by solving b ys ; u s ys ¼ X½b
ð20Þ
For general nonlinear operator Xð:Þ equations of type (20) the solutions have to be obtained by numerical procedures. Also, the steady-state equation may have multiple solutions, some of which may not be physically meaningful. In contrast to a general NARX model, it is comparatively easy to examine the steady-state behavior of the proposed Wiener-Laguerre model. Since the spectral radius of each matrix Fi is strictly less than one, the corresponding steady-state vector Xs can be computed as Xs ¼ ðI FÞ1 Gus
ð21Þ
for i ¼ 1; 2 . . . n, and the corresponding steady-state output is given by b ys ¼ CðXs Þ
ð22Þ
The above expression also reveals some limitations of the Weiner-Laguerre model when compared to a general NARX model. Clearly, there exists only one steady-state Xs and only one steady-state output vector ys corresponding to each us . Thus, the Weiner-Laguerre model can be used for capturing dynamics of systems with input multiplicity. However, systems with output multiplicity, i.e., having multiple steady states for given us , cannot be modeled using this approach. A general NARX model, on the other hand, can deal with the problem of modeling systems with output multiplicity. Output Error Structure Note that the NARX model estimates current outputs using past measurements. When measurements are corrupted with noise, i.e., yðkÞ ¼ ytrue ðkÞ þ vðkÞ
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where vðkÞ represents measurement noise, the measurement errors are fed back to the model through the nonlinear operator Xð:Þ. Thus, measurement noise affects the output estimation in a complex manner. On the other hand, the Wiener-Laguerre model can be expressed as Xðk þ 1Þ ¼ FXðkÞ þ GuðkÞ yðkÞ ¼ C½XðkÞ þ eðkÞ where yðkÞ represents the measurement of the ith output and feðkÞg represents the residual vector. Clearly, the proposed Wiener-Laguerre model has output error structure, and the estimate of current output does not depend on past measurement errors. Thus, the Wiener-Laguerre model is better suited for dealing with additive unmeasured disturbances or errors in measurements. Su and McAvoy (1997) have shown that nonlinear blackbox models with output error structure are better suited for development of nonlinear MPC schemes due to their better long-term prediction capabilities. BIBO Stability If the state-output map for each output, i.e., Ci ð:Þ : RNi ! R for ði ¼ 1; 2; . . . nÞ, is a bounded function (a function Ci ð:Þ is defined as a bounded function if jCi ð:Þj My < 1 on every bounded subset of the RNi ), then it can be argued that the proposed Wiener-Laguerre model is globally bounded-input, bounded-output (BIBO) stable. This follows from the fact that the linear part is always BIBO stable as the spectral radius of F is strictly less than one. Thus, for any bounded input sequence such that kuðkÞk1 Mu < 1, the state sequence kXðkÞk1 Mx < 1 will be bounded. If Ci ð:Þ is a bounded function, then it follows that the corresponding output sequence will be bounded, i.e., jyi ðkÞj My < 1. Specifically in the case of the Laguerre-ANN model, it can be argued that the output of ANN is always bounded, as the output of the sigmoidal function in the output layer is always bounded between 0 and 1. The open loop BIBO stability characteristics are of vital importance as the proposed NMPC scheme requires the internal model to be open loop stable. Note that it is difficult to arrive at a similar conclusion regarding the BIBO stability for a general NARX model (19). DEVELOPMENT OF NONLINEAR MPC BASED ON THE WEINER-LAGUERRE MODEL In a typical MPC formulation, at every sampling instant an explicit dynamic model is used for predicting the future behavior of the plant
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over a finite number of future time steps, say P, which is called the prediction horizon. A set of ‘M ’ future manipulated input moves uðk=kÞ; uðk þ 1=kÞ; . . . ; uðk þ M 1=kÞ (M is the control horizon) are determined by optimization with the objective of minimizing the sum of the squares of the future prediction errors while taking into consideration the operating constraints. Thus, given a sequence of future control moves, i.e., uðk=kÞ; uðk þ 1=kÞ; . . . ; uðk þ M 1=kÞ, the P step ahead open loop output prediction can be written as follows: Xðk þ j þ 1=kÞ ¼ FXðk þ j=kÞ þ Guðk þ j=kÞ
ð23Þ
ðfor j ¼ 1; . . . P 1Þ where uðk þ M=kÞ ¼ . . . ¼ uðk þ P 1=kÞ ¼ uðk þ M 1Þ ybðk þ j=kÞ ¼ C½Xðk þ j=kÞ
ð24Þ
In order to account for plant-model mismatch and unmeasured disturbances, we assume that the discrepancy between the model output and process output is due to additive step disturbances in the output that persist over the prediction horizon. Thus, similar to the linear dynamic matrix control scheme, a mismatch correction term is incorporated in the prediction model as follows: yc ðk þ j=kÞ ¼ ybðk þ j=kÞ þ dðk=kÞ
for j ¼ 1; 2; . . . P
ð25Þ
dðk=kÞ ¼ yðkÞ ybðk=k 1Þ
ð26Þ
where yðkÞ represents the measured plant output at the kth instant and ybðk=k 1Þ represents the model output at the k’th instant using input sequence up to time k 1. Although simplistic, this type of unmeasured disturbance model approximates slowly varying disturbances and provides robustness to modeling error (Meadows and Rawlings, 1997). Now, given the future set point trajectory fysp ðk þ j=kÞ; j ¼ 1; 2; . . . Pg, the controller design problem can be formulated as
min
8 > > > > >
>
> > > :þ
P X j¼1 M X j¼1
T
½Eðk þ j=kÞ WE ½Eðk þ j=kÞ
9 > > > > > =
> > > ½Duðk þ j=kÞT WU ½Duðk þ j=kÞ > > ;
ð27Þ
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where Eðk þ j=kÞ ¼ ysp ðk þ j=kÞ yc ðk þ j=kÞ Duðk þ j=kÞ ¼ uðk þ j=kÞ uðk þ j 1=kÞ
ð28Þ
subject to yL yc ðk þ j=kÞ yU L
u uðk þ j=kÞ u Du
KþM
¼ Du
KþMþ1
U
for j ¼ 1 . . . P (Output limits) for j ¼ 1 . . . M 1 (Input limits)
¼ . . . ¼ Du
KþP1
ð29Þ
¼0
and WE is a positive definite error weighting matrix while WU is an input weighting matrix. The resulting nonlinear programming problem can be solved using any standard optimization technique such as successive quadratic programming (SQP). The controller is implemented in a moving horizon framework, i.e., only uðk=kÞ is implemented at each sampling instant, and the optimization is repeated at each sampling instant based on the updated information from the plant. In the present work, the constrained optimization problem resulting at every sampling instant was solved using function constr in MATLAB Optimization Toolbox in the simulation studies. For experimental evaluation, a modified version of optimization subroutine frprnm from Numerical Recipes in C (Press et al., 1990) has been used.
SIMULATION CASE STUDY: CONTROL OF CONTINUOUS FERMENTER Control of a continuously operated fermenter at its optimum operating point is difficult when the fermentation reaction has substrate and product inhibition kinetics (Henson and Seborg, 1991; Patwardhan and Madhavan, 1993). The system exhibits input multiplicity, i.e., identical steady-state output is obtained for multiple sets of input values. Input multiplicities in general occur due to the presence of competing effects in the process. In the case of a fermenter, at a low concentration of input feed substrate, the growth rate of the biomass dominates the reaction rate and increase in feed substrate concentration results in an increase in product concentration. However, beyond a certain value of the feed substrate concentration the product and substrate inhibition prevails, and product concentration shows a negative gradient with respect to input feed substrate concentration. In other words, the steady-state gain of product and biomass concentrations with respect to input (feed substrate concentration) smoothly changes sign in the desired operating region.
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Consequently, there exists an optimum value of process input at which the fermenter product concentration attains a maximum (optimum). Controlling the fermenter at the optimum point appears to be the most desirable option, economically however, this is difficult to achieve. Morari (1983) observed that input multiplicities of nonlinear systems usually cause robustness problems that cannot be eliminated using linear controllers. The phenomenon of change in the sign of the steady-state gain poses a difficulty even for nonlinear control schemes. From a system theoretic viewpoint, the relative degree of the system becomes undefined and the invertibility is lost at the optimum (singular) point where the steady-state gain is reduced to zero. As a consequence, the global inputoutput linearization based approaches cannot be applied (in a straightforward manner) in the regions of state space where the relative degree is not well defined and invertibility is lost (Slotine and Li, 1991). Thus, in the case of almost all the nonlinear control strategies based on exact linearization, an assumption, often not stated explicitly, is that the steady-state gain of the model cannot change sign in the operating region (Henson and Seborg, 1991). Biegler and Rawlings (1991) observe that use of an unconstrained nonlinear controller may not be sufficient in such cases, even when the model is perfect. When a zero process gain situation is encountered, the perfect model cannot be inverted and the model inversion based controller can become ill-conditioned. Thus, controlling the continuous fermenter at the optimum operating point is challenging and is used as a benchmark to assess the performance of the proposed modeling and control scheme. Characteristic Features of Continuous Fermenter Consider a continuously operated fermenter described by the following set of ordinary differential equations (1) (Patwardhan and Madhavan, 1993; Henson and Seborg, 1992): dX ¼ F1 ðX; S; P; D; Sf Þ ¼ DX þ mX dt
ð30Þ
dS 1 ¼ F2 ðX; S; P; D; Sf Þ ¼ DðSf SÞ mX dt YX=S
ð31Þ
dP ¼ F3 ðX; S; P; D; Sf Þ ¼ DP þ ðam þ bÞX dt
ð32Þ
where X represents effluent cell-mass or biomass concentration, S represents substrate concentration, and P denotes product concentration.
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It is assumed that product concentration (S) and the cell-mass concentration (X) are measured process outputs, while dilution rate (D) and the feed substrate concentration ðSf Þ are process inputs that can be manipulated. Model parameter m represents the specific growth rate, YX=S represents the cell-mass yield, and a and b are the yield parameters for the product. The specific growth rate model is allowed to exhibit both substrate and product inhibition:
m¼
mm 1 PPm S 2
Km þ S þ SKi
:
where mm represents maximum specific growth rate, Pm represents the product saturation constant, Km represents the substrate saturation constant, and Ki represents the substrate inhibition constant: The nominal values of model parameters are listed in Table I. The productivity (Q) of the continuous fermenter can be defined as the amount of product produced per unit time, as Q ffi DP The optimum steady-state operating conditions for the fermenter such that productivity is maximized are S ¼
pffiffiffiffiffiffiffiffiffiffiffi Km Ki ¼ 5:138 g=l
P ¼
Pm ¼ 25:0 g=l 2
Table I Continuous Fermenter: Nominal Model Parameters Parameter YX=S a b mm Pm Km Ki
Nominal value 0.4 g/g 2.2 g/g 0.2 h71 0.48 h71 50 g/l 1.2 g/l 22 g/l
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The optimum can be obtained by setting D ¼ D ¼ 0:1636 h1
and
Sf ¼ Sf ¼ 23:39 g=l
Figure 2 presents the steady-state behavior of the system in the neighborhood of the optimum. As can be observed from these figures, both product concentration ðPÞ and biomass concentration ðXÞ exhibit change in the sign of the steady-state gain across the optimum. The behavior of the biomass concentration ðXÞ with respect to the dilution rate is also similar. Another problem associated with the control of a continuous fermenter is the change in process optimum over time due to a variety of disturbances. The growth rate microorganism is affected by some biological factors such as change in metabolic path, enzymatic deactivation,
Figure 2. Continuous fermenter; comparison of the steady-state behavior of process and Wiener-Languerre models. The curves represent: (a): biomass concentration ðXÞ versus dilution rate ðDÞ (at Sf ¼ 23:4); (b): product concentration ðPÞ versus dilution rate ðDÞ (at Sf ¼ 23:4); (c): biomass concentration ðXÞ versus feed substrate concentration ðSf Þ (at D ¼ 0:164); and (d): product concentration ðPÞ versus feed substrate concentration ðSf Þ (at D ¼ 0:164).
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culture aging, and spontaneous mutations. In terms of model parameters, this implies that the cell-mass yield coefficient YX=S and the maximum specific growth rate mm tend to be especially sensitive to changes in the operating conditions. From the control perspective, these changes can be regarded as unmeasured disturbances because these variations may exhibit significant time-varying behavior. Figure 3 presents the steadystate characteristics of the fermenter for perturbations in model parameter mm . From the analysis of the relative locations of the peaks with respect to the nominal optimum point (set point), the following two types of control problems can be identified (Saha, 1999): Suboptimal operation: In this case, the optimum operating point shifts to a higher value than the nominal value. In such a situation, operation at the set point based on the nominal parameter value will become suboptimal. Unattainable set point: In this case, the optimum operating point shifts to a lower value than the nominal value. In such a situation, the set point based on the nominal parameter value will become unattainable.
Figure 3. Continuous fermenter; comparison of steady-state behavior in presence of parameter perturbations.
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Patwardhan and Madhavan (1998) point out that the situation leading to infeasibility of the set point is potentially more dangerous as the final steady-state error cannot be eliminated and does not change its sign. In the presence of such persistent error, a controller with integral action can cause instability in the unconstrained case, and input saturation or large undamped input oscillations in the constrained case.
Development of Wiener-Laguerre Model The data required for parameter estimation were obtained by perturbing the fermenter in open loop by simultaneously introducing square pulse sequences of random magnitude in Sf and D. A sampling time of 1 hour was selected for model development as the fermenter has open loop settling times close to 40 hours. Two MISO Wiener-Laguerre models were developed relating each output with both the inputs following the procedure given above. The initial guesses for Laguerre filter parameters required for the optimization problems were generated based on rough estimates of dominant time constants computed from the step response data between each input-output pair. Models with different numbers of Laguerre filters were developed and compared on the basis of AIC and PPE values (see Table II). The models with four filters between each input-output pair were found to be adequate for capturing the dynamics of the system. The model validation results using an independent data set along with the corresponding PPE values and validation input sequences are shown in Figure 4. Also, the comparison of the steadystate behavior of the identified MISO models with that of the process steady-state behavior is presented in Figure 2. Clearly, the proposed MISO Wiener-Laguerre models are able to capture the dynamic as well as the steady-state behavior of the fermenter system with reasonable accuracy.
Table II Comparison of AIC and PPE Values for Laguerre Models No. of Filters
AIC (for P)
2 3 4 5
73048.9 74278.6 74966.0 75335.0
% PPE (for P) 0.7083 0.2018 0.0981 0.0650
AIC (for X) 75586.3 76690.1 77108.4 77533.4
% PPE (for X) 1.0519 0.3399 0.2162 0.1335
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P. SAHA ET AL. Table III Intitial and Optimum Operating Conditions for Fermenter Initial Conditions X ¼ 4.3866 S ¼ 1.0335 P ¼ 15.499 D ¼ 0.15 Sf ¼ 12:0
Optimum Conditions X ¼ 7.3044 S ¼ 5.138 P ¼ 25 D ¼ 0.1636 Sf ¼ 23:39
Set point X ¼ 7.3044 P ¼ 25
Closed Loop Simulation Studies The MIMO servo control problem is formulated similar to Patwardhan and Madhavan (1993) where it is desired to shift the fermenter operation from a given suboptimal initial steady state (see Table III) to the nominal optimum operating point. Figures 5 and 6 compare the performances of the proposed nonlinear NMPC scheme (with P ¼ 5; P ¼ 20, and M ¼ 1) with that of the first principle model based NMPC scheme (Q-NMPC) developed by Patwardhan and Madhavan (1993). As can be seen from Figure 5, the proposed NMPC scheme produces initial sluggish output
Figure 4. Continuous fermenter; model validation results for MISO Languerre-polynomial models.
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Table IV pH Control Example: Statistical Criteria for LaguerrePolynomial Model (Simulation) No. of Inputs 3 4 5 6
filters filters filters filters
AIC 862.61 725.99 617.25 7252.41
PPE (in %) 5.69 4.96 4.45 1.87
responses with longer settling time when compared to the Q-NMPC controller. However, as evident from the manipulated input profiles (Figure 6), the short rise time in the case of the Q-NMPC is achieved at the cost of sudden and large variations in the manipulated inputs. On the other hand, the proposed NMPC controller achieves the desired transition with smoother variations in manipulated input and without any such unacceptable large input changes. As stated earlier, perturbations in cell-mass yield coefficient YX=S or maximum specific growth rate mm can be regarded as unmeasured disturbances. To investigate the effect of such disturbances, it was assumed that a simultaneous step jump occurs in the parameter when the set point
Figure 5. Continuous fermenter; comparison of output profiles for servo problem.
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Figure 6. Continuous fermenter; comparison of manipulated input profiles for servo problem.
change is introduced. Figure 7 presents the closed loop behavior of the NMPC controller (P ¼ 5) under the following two situations: mm ¼ 0:44, which renders the specified set point unattainable, and mm ¼ 0:52, which renders the specified set point suboptimal The latter case causes no difficulty to the NMPC controller and the specified set point is achieved with smooth input variations. In the former case, the NMPC moves both outputs close to the specified set point without causing any undamped input oscillations or input saturation even in the presence of the resulting persistent offset in cell-mass concentration. Patwardhan and Madhavan (1993, 1998) have shown that, if the internal model cannot predict change in the sign of the steady-state gain in the operating region, an NMPC or NIMC controller based on such a model can lead to large undamped input oscillations when parameter perturbations render set point unattainable. They have also shown that such oscillatory input behavior can be eliminated if quadratic terms are included in the prediction model so that changes in the sign of gain can be predicted. The ability of the proposed NMPC controller to achieve transitions without input oscillations can be attributed to fact that the
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Figure 7. Continuous fermenter; effect of parameter perturbations on servo responses.
Laguerre-polynomial model can predict changes in signs of steady-state gains in the neighborhood of the optimum fairly accurately. Also, the possibility of input saturation due to persistent offset is avoided because the controller is formulated as a minimization problem over the prediction horizon. As a result, the measured outputs approach close to their respective maximum attainable values (under given parameter perturbation) while maintaining inputs well within the bounds. Similar closed loop behavior is observed for the case where perturbations are introduced in parameter YX=S . Thus, the proposed controller is found to be robust to moderate perturbations in the these unmeasured disturbances.
SIMULATION AND EXPERIMENTAL STUDIES ON NEUTRALIZATION PROCESS Control of pH in the neutralization process is a benchmark problem often used to test nonlinear control schemes in process control. The primary reason for this is the severe static nonlinearity, reflected in the titration curves for the neutralization process. The steady-state pH response to reagent addition shows a highly nonlinear S shaped titration curve. The steady-state process gain changes by several orders (up to
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105 ) of magnitude over a small range of pH values. We carried out experimental evaluation of the proposed modeling and control scheme using a benchmark pH control problem that involves neutralization of a weak acid (acetic acid) with a strong base (sodium hydroxide) carried out in a continuous stirred tank reactor (CSTR) (McAvoy, 1972). Experimental Stirred Tank Reactor Setup The experimental setup is shown in Figure 8. The setup has a mixing vessel with a capacity of 5 liters. The tank is provided with two inlet pipes, each of 10 mm diameter, to introduce the inlet stream of acid and base into the tank and one overflow type outlet. The liquid holdup in the tank is well mixed by a stirrer driven by a variable speed motor. The feed streams are pumped into the vessel by two metering pumps of maximum capacities of 35 liters/h and 138 liters/h for acid and base, respectively. The pumps can be actuated either manually or by a current signal input of 420 mA. In order to avoid the disturbances in pH measurement due to turbulence caused by the agitator in the main tank, a measurement chamber of 0.6 liters capacity is placed in series with the main for pH measurement. A Pentium (100 MHz) processor-based personal computer
Figure 8. Neutralization process; schematic diagram for experimental setup.
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along with a 12 bit AD and DA converter card is used for the real-time control of pH. The acid pump and the base pumps are actuated through the computer through the DA converter and intermediate circuits.
First Principle Model for the Experimental Setup It is assumed that the acid stream is well mixed with a stream of base inside the CSTR. The mechanistic model reported for the acetic acid-sodium hydroxide reaction in a CSTR can be obtained as follows (McAvoy, 1972):
V1
dx1 ¼ F1 C1 ðF1 þ F2 Þx1 dt
ð33Þ
V1
dz1 ¼ F2 C2 ðF1 þ F2 Þz1 dt
ð34Þ
V2
dx2 ¼ ðF1 þ F2 Þðx1 x2 Þ dt
ð35Þ
V2
dz2 ¼ ðF1 þ F2 Þðz1 z2 Þ dt
ð36Þ
½Hþ 3 þ ðKA þ z2 Þ½Hþ 2 þ ðKA ðz2 x2 Þ KW Þ½Hþ KW KA ¼ 0
ð37Þ
pH ¼ log10 ½Hþ
ð38Þ
where V1 ¼ volume of the stirred tank (liters) V2 ¼ volume of the annex tank (liters) F1 ¼ acid flow rate (liters/h) F2 ¼ base flow rate (liters/h) C1 ¼ acid concentration (moles/liter) C2 ¼ base concentration (moles/liter) x1 ; x2 ¼ concentration of HAC and AC z1 ; z2 ¼ concentration of Naþ ½Hþ ¼ hydrogen ion concentration KA ¼ acid equilibrium constant, 1.8 1075 KW ¼ water equilibrium constant, 1.0 10714
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Note that the dynamic model equations (33) to (36) are almost linear firstorder differential equations in state variables x1 ; x2 ; z1 ; and z2 ; while Equation (32) relating x2 and z2 with ½Hþ is a highly nonlinear relation. In addition, the pH is a logarithmic function of ½Hþ . It is these equations (37) and (38) that make the process dynamics highly nonlinear. Thus, the nonlinear behavior arises from a nonlinear output map rather than the state dynamics, and the first principle model is similar to a Weiner-type model. In the present study, the base case steady state values for C1 , C2 , and F2 are fixed at 0.2 mole/liter, 0.1 mole/liter, and 40 liters/h, respectively, and F1 is varied from 0 to 32.5 liters/h. For different values of F1 , the steady-state values of pH are calculated, and the resulting variation of pH with the acid flow rate is shown in Figure 9. By observing the steady-state behavior, we can infer that the system is relatively linear in the high or low pH region; however, in the region where pH changes from 5.5 to 11.5, the system exhibits severe nonlinear behavior. The most sensitive point is detected at the point of inflection where the value of pH is 8.78. This point is referred to as the critical point hereafter. It can also be observed that the steady-state gain of the process changes drastically in this operating region. The experimental process has a large open loop settling time ( 1200) and, in principle, a relatively large sampling time (say 30) should be
Figure 9. Neutralization process; comparison of steady-state characteristics.
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selected to keep the number of prediction horizon small. However, it was observed that the system output changes very rapidly when states are close to the critical point, and the resulting dynamics cannot be adequately captured with slow sampling rates. Thus, a sampling period ðTÞ of 4 was selected for modeling and control studies. Open Loop Studies for Model Development From the steady-state curve it may be observed that the pH of a system varies widely within a very narrow range of input manipulation in the neighborhood of the critical point. The modeling studies were first conducted using the dynamic model given by Equations (33)(38) and later repeated for the experimental setup. The dynamic model was simulated with a random acid flow input sequence of 2% variance as shown in Figure 10. To begin with, a Laguerre-polynomial model was developed using the training data set. Table IV presents the statistical information criteria for different choices of the number of Laguerre filters. A model with six filters was chosen for subsequent studies as the improvement in prediction is marginal with a greater number of filters. Figures 10 and 11 compare the model output predictions with those of the system output
Figure 10. Neutralization process (simulation); comparison of model and process outputs using training data set.
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Figure 11. Neutralization process (simulation); comparison of model and process outputs using validation data set.
behavior for training and validation data sets, respectively. The output of the Laguerre-polynomial model compares quite well with the output in the training data set, however, this model completely fails to predict the plant behavior when the validation data set is used. In fact, as evident from the figure the predicted pH values are well beyond the range of 0 to 14 within the first few samples of the data set. Thus, the nonlinear Laguerre model with polynomial nonlinearity is unable to capture the nonlinear system dynamics properly, even when the model fits the training data set quite well. In order to develop a Laguerre-ANN model, the value of time scale parameter ‘a’ was chosen equal to expðT=tÞ where t was equal to the reactor residence time of the CSTR at the critical point (300 s). The MLPs representing state-output maps were trained using MATLAB Neural Network Toolbox. The number of hidden nodes in the hidden layer and the number of input nodes (number of Laguerre filters) in the input layer were selected with an aim of minimizing the various statistical information criteria (see Table V). Although a combination of an input layer with six filters and a hidden layer with eight nodes was found to be the best, not much improvement was observed over the combination with six inputs and six nodes. Thus, the latter combination was chosen for
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Table V pH Control Example: Statistical Criteria for Laguerre-ANN Model (Simulation) No. of Inputs and Hidden Nodes 4 4 4 5 5 6 6 6
inputs inputs inputs inputs inputs inputs inputs inputs
and and and and and and and and
4 6 8 6 8 6 7 8
nodes nodes nodes nodes nodes nodes nodes nodes
AIC 76026.42 76269.74 76301.95 76585.25 76693.65 76702.45 76725.93 76736.27
PPE (in %) 0.240 0.189 0.182 0.137 0.123 0.120 0.118 0.119
subsequent studies. Comparisons of simulated plant output and model predictions for the training data set and the validation data set are shown in Figures 10 and 11, respectively. As evident from these figures, the Laguerre-ANN model is able to capture the plant dynamics quite well and predicts the plant behavior reasonably accurately the even for the validation data set (PPE value for the validation data set is 7.9053%). The steady-state characteristics of the Laguerre-ANN model, developed from simulated data, are compared with those of the steady-state plant behavior in Figure 9. It may be observed that the Laguerre-ANN model is also able to approximate the nonlinear-steady state behavior in the highly nonlinear region reasonably accurately. A similar modeling exercise was carried out using the data sets obtained from the experimental setup. The process was excited with a random input acid flow sequence of 2%variance and the resulting data was used for development of a Laguerre-ANN model. The evaluation of statistical information criteria for the training data set is shown in Table VI. Although an input layer of six filters and a hidden layer of eight nodes Table VI pH Control Example: Statistical Criteria for Laguerre-ANN Model (Experiment) No. of Inputs and Hidden Nodes 4 4 4 5 5 6 6
inputs inputs inputs inputs inputs inputs inputs
and and and and and and and
4 6 8 6 8 6 8
nodes nodes nodes nodes nodes nodes nodes
AIC
PPE (in %)
77762.48 78439.18 78502.48 78281.22 78509.30 78638.82 78775.16
0.37 0.20 0.18 0.23 0.18 0.16 0.14
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appeared to be the most suitable, in order to maintain a similarity between simulation and the experiment, a combination of six inputs and six hidden nodes was considered for closed loop studies. Comparisons of the measured pH and model predictions for the training and validation data sets are shown in Figures 12 and 13, respectively. The PPE value for validation results is as low as 2.8553%, and the model predicts the process behavior quite accurately. Thus, as was expected from the simulation studies, the Laguerre-ANN model developed from experimental data predicted the system behavior in the highly nonlinear region extremely well. Closed Loop Studies From the control viewpoint, controlling the system under consideration close to the critical point (pH ¼ 8.89) is much more challenging task, due to large variation of steady-state gain in this region, than regulating or controlling pH at some other point (such as pH ¼ 7). Thus, a servo problem was formulated in the operating region where the pH is highly sensitive to small input changes and the system exhibits highly nonlinear behavior. The servo control problem was defined as moving the system from a given initial steady state (pH ¼ 7) to a point (pH ¼ 9) very close to the critical point.
Figure 12. Neutralization process (experiment); comparison of model and process outputs using training data set.
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Figure 13. Neutralization process (experiment); comparison of model and process outputs using validation data set.
In the chemical industry, gain-scheduled proportional integralderivative (PID) controllers are widely used for controlling pH. Thus, the performance of the proposed NMPC scheme was compared with that of a PI controller with gain scheduling. For developing a PI controller with gain scheduling, the pH steady-state curve was divided into five regions (see Table VII). Within each region, the steady-state curve was approximated as a linear function with a constant slope. It was also assumed that the dynamic behavior of the system within each region is linear and can be approximated as a first-order transfer function. Thus, average values of steady-state gain and time constant were calculated for each region
Table VII Parameters of Gain Scheduled PI Controller Range of pH values From From From From From
6 to 5 7.5 to 6 9.5 to 7.5 9.5 to 12 13 to 12
Gain
Integral Time (s)
70.347 70.031 70.000685 70.066 70.74
352 438 308 609 436
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and were used to design a PI controller in each region using the method of direct synthesis (pole placement). The resulting PI settings for each region are given in Table VII. The nonlinear MPC algorithms were developed using the LaguerreANN models obtained in the reported modeling exercise. The comparison of performances for the gain-scheduled PI controller and unconstrained NMPC (with P ¼ 300, M ¼ 1, WE ¼ 1, and WU ¼ 0) for simulation and experimental studies is presented in Figure 14. Note that the prediction horizon is chosen unusually large as the process has a large open loop settling time ( 1200), and a relatively small sampling time (4 s) was selected to capture sudden changes in the nonlinear region. The response trajectory of the gain-scheduled PI controller shows high overshoot and relatively large settling time (more than 20 minutes). On the other hand, when the proposed NMPC is used the system reaches its set point in less than 10 minutes and with much less overshoot and oscillation. The manipulated acid flow behavior (Figure 14(b)) is also smoother in the case of the NMPC-controlled process when compared to that obtained using the PI controller. It should be noted that the closed loop performances of both controllers in the experimental studies are qualitatively similar to the results of simulation studies. The NMPC-controlled
Figure 14. Neutralization process (simulation and experiment); comparison of servo responses of NMPC and gain-scheduled PI controller.
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process comes to the desired steady state within 10 minutes, as expected from simulation studies. Thus, in simulation as well as in experimental verification, a substantial improvement of closed loop performance is obtained using the proposed NMPC based on the Laguerre-ANN model when compared to the performance of the gain-scheduled PI controller. CONCLUSIONS In this work, a Weiner-Laguerre model was developed for capturing dynamics of open loop stable MIMO nonlinear systems with deterministic inputs. The nonlinear state-output map was constructed either using quadratic polynomial functions or artificial neural networks. The procedure for model identification and the properties of the resulting model, such as open loop stability and steady-state behavior, have been discussed in detail. The identified Weiner-Laguerre model was further used to formulate a nonlinear model predictive (NMPC) scheme. The efficacy of the proposed modeling and control scheme was demonstrated using two benchmark control problems: (a) a simulation study that involved control of a continuously operated fermenter at its optimum (singular) operating point and (b) experimental verification using control of pH at the critical point of the neutralization process. It was observed that the proposed Weiner-Laguerre model is able to capture both the dynamic and steady-state characteristics of the continuous fermenter and the neutralization process reasonably accurately. The proposed NMPC scheme achieves a smooth transition from a sub optimal operating point to the optimum (singular) operating point of the fermenter without causing large variation in manipulated inputs. The proposed NMPC scheme was also found to be robust in the face of moderate perturbation in the unmeasured disturbances (process parameter changes). In particular, when the parameter perturbations render the specified set point unattainable, the NMPC controller is able to maintain stable closed loop operation without causing input saturation or haunting. In the case of experimental verification using the neutralization process, the proposed control scheme was found to achieve much faster transition to a set point close to the critical point of the system when compared to a gainscheduled PID controller. Simulation as well as experimental studies establish the viability of implementing the proposed modeling and control scheme in practice. However, the main limitation of the proposed approach is the simplistic unmeasured disturbance model used while modeling and in predictive control formulation. The possibility of improving the proposed modeling and control scheme based on stochastic disturbance models is currently being investigated.
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NOMENCLATURE a a Ci Ci d Di D F e E k K m M n N P P q S Sf u v V W x X y^ y z
time scale parameter of Laguerre filter (Equation (1)) vector of time scale parameters vector in MISO state-output map (Equation (10)) concentrations in neutralization process model plant model mismatch (Equation (25)) matrix in MISO state-output map (Equation (10)) dilution rate in fermenter model flow rates in neutralization process model measurement noise future prediction error in MPC formulation sampling instant equilibrium constants in neutralization process model number of manipulated inputs control horizon number of outputs dimension of state vector in MIMO Wiener-Laguerre model prediction horizon product concentration in fermenter model output of hidden layer of ANN substrate concentration in fermenter model feed substrate concentration in fermenter model manipulated input vector output of input layer of ANN tank volume in neutralization process model weighting matrix elements of state vector state vector model output measured output output of output layer of ANN
Greek a a b b e m x z C
letters bias vector in input layer of ANN parameter in fermenter model bias vector in hidden layer of ANN parameter in fermenter model modeling error parameter in fermenter model state variable in neutralization process model state variable in neutralization process model symbolic representation of state-output map
REFERENCES Biegler, L. T. and Rawlings, J. B. (1991). In Chemical Process Control — CPC IV, eds. Y. Arkun, and W. H. Ray, 543571, CACHE, Austin, Tex.; AIChE, New York. Billings, S. A. (1980). IEE Proc., 127, Pt.D. (6), 272285.
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