A novel online affine model identification of multivariable processes using adaptive neuro-fuzzy networks

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

Contents lists available at ScienceDirect

Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

A novel online affine model identification of multivariable processes using adaptive neuro-fuzzy networks Karim Salahshoor ∗ , Morteza Hamzehnejad Department of Automation and Instrumentation, Petroleum University of Technology, Satarkhan st., Tehran, Iran

a b s t r a c t This paper presents a novel systematic identification methodology for online affine modeling of multivariable processes using adaptive neuro-fuzzy networks. The proposed approach introduces an integrated procedure to simultaneously estimate a number of adaptive neuro-fuzzy networks with simple and compact dynamic structures to realize a multivariable affine model identification in real-time. A new fuzzy rule significance concept, based on a generic time-weighted rule activation record (WRAR), together with a measure of time-weighted root mean square (WRMS) error are incorporated to maintain efficient structural and parametric mechanisms for proper adaptation of the resulting neuro-fuzzy networks. An extended Kalman filter (EKF) algorithm is developed to adaptively adjust the neuro-fuzzy free parameters corresponding to the nearest created fuzzy rules. Extensive simulation test studies will be conducted to explore the capabilities of the proposed identification approach to adaptively develop online multivariable affine dynamic models for a highly nonlinear and time-varying continues stirred tank reactor (CSTR) and a highly nonlinear binary distillation column as two challenging benchmark problems. © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Online identification; MIMO affine model; Adaptive neuro-fuzzy model; EKF

1.

Introduction

The need to designing robust and simple control systems has motivated the control community to develop satisfactory model-based control design methodologies for nonlinear processes. Feedback linearization presents a typical control design techniques (Norgaard et al., 2003) which has drawn much attention in recent years. Application of this control design technique, however, is restricted to a certain class of processes whose dynamic mathematical models can be represented by the following “affine structural form”: y(k) = fk (Xk ) + gk (Xk )u(k)

(1)

in which a linear relationship exists between the process output (y) and the process input (u). Where Xk denotes a vector of regressors containing past inputs and outputs, configured at the kth time instant. fk (·) and indicate general nonlinear functions whose models are time-varying as the nonlinear process evolves over time. Despite the model structural limitation, it has been illustrated that this class of dynamic representa-

∗

tion is still very broad for a wide range of physical nonlinear processes (Chen and Khalil, 1995; Narendra, 1992; Henson and Seborg, 1997; Sanner and Slotine, 1995). This puts more emphasizes to develop efficient algorithms to find a solution for exact affine modeling of any nonlinear affine or nonaffine process. Most of industrial processes, however, have inherently non-affine dynamic model structures. Therefore, developing expedient models for the time-varying nonlinear functions fk (·) and gk (·) is not a trivial task. One solution for affine modeling of a non-affine process is based on using the approximate Taylor series expansion of process dynamic model around its input, yielding to NARMR-L2 model proposed by Narendra and Mukhopodhyay (1997). Many industrial processes have multi-input, multi-output (MIMO) dynamic model structures with time-varying characteristics. This makes the development of an exact affine-type model for such practical processes much more complicated and challenging subject. In this work, a new online identification algorithm is proposed to derive an adaptive affine model for MIMO nonlinear and time-varying processes based on a self-organizing neuro-fuzzy approach. The proposed algo-

Corresponding author. Tel.: +98 21 44236062. E-mail address: [email protected] (K. Salahshoor). Received 24 February 2009; Received in revised form 12 July 2009; Accepted 15 July 2009 0263-8762/$ – see front matter © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2009.07.009

156

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

the required model accuracy via appropriate rule addition, modification and pruning mechanisms in a real-time manner. An EKF parameter learning algorithm is developed to adaptively adjust the free network coordinates, corresponding to the already created fuzzy rules which are nearest to the incoming process input pattern. This scheme implements the required rule modification strategy by capturing the most  recent dynamic process data in the identified set of f ik (·) and  gijk (·) functions given by Eq. (2). In case of need for any network structural improvement, two model learning schemes are incorporated to realize appropriate online rule addition or replacement and rule pruning adaptations for multivariable processes. However, to avoid from any inaccurate model structural decision, a complementary measure is utilized in terms of a time-weighted root mean square (WRMS) error measure to assess the model inadequacy at each sampling time instant. The rest of this paper is organized as follows: a new online identification algorithm is presented in Section two to develop an affine-type model for multivariable processes. The EKF parameter learning algorithm is presented in Section three. A set of simulation studies are carried out in Section four to evaluate the performance of the proposed online affine model identification algorithm on a CSTR and a distillation column as two highly nonlinear multivariable benchmark problems. Section five summarizes the resulting concluding remarks.

Nomenclature h (t) C (t) CA qc q CAf Tf Tcf Li and Vi xi and yi

fouling coefficient deactivation coefficient effluent concentration, the controlled variable coolant flow rate, the manipulated variable feed flow rate, disturbance feed concentration feed temperatures coolant inlet temperature liquid and vapor flow from stage i [kmol/min] liquid and vapor composition of light component on stage i Mi liquid holdup on stage i [kmol] D and B distillate (top) and bottoms product flow rate [kmol/min] L = LT and V = VB reflux flow and boilup flow [kmol/min] F and zF Feed rate [kmol/min] and feed composition [mole fraction] qF fraction of liquid in feed ˛ (in distillation column) relative volatility between light and heavy component taul time constant [min] for liquid flow dynamics on each stage  (in distillation column) constant for effect of vapor flow on liquid flow (“K2-effect”)

2. A new online multivariable affine model identification approach based on an adaptive neuro-fuzzy network

rithm presents a novel procedure to integrate simultaneous identification of a number of adaptive neuro-fuzzy models to   estimate the unknown functions f ik (·) and gijk (·), characterizing the affine-type dynamic model structure between the jth input uj ∈ R and the ith output yi ∈ R in a real-time manner at the kth sampling time, leading to 





yi (k) = f ik (Xk ) + gijk (Xk ) uj (k),

i = 1, . . . , b, j = 1, . . . , a.

(2)

This approach has several interesting advantages. It enables a multivariable affine-type model to be identified adaptively for a nonlinear MIMO process without a priori dynamic knowledge requirement. Thus, the dimensions of neuro-fuzzy networks are not determined a priori to bet  ter model the unknown nonlinear functions f ik (·) and gijk (·) in Eq. (2). The neuro-fuzzy networks have adaptive structural and parametric capabilities so as to recursively follow up the process time-varying and nonlinear dynamic characteristics with a small set of time-updated fuzzy rules. This is a major advantage over other conventional neuro-fuzzy modeling approaches in which their networks should be conservatively predetermined based on the highest dimension requirements to cover all the possible process working conditions. Hence, the proposed approach possesses enough flexibility to efficiently deal with nonlinear and time-varying process dynamics in a wider range of operating conditions, leading to a compact network model structure with lower realtime computational burden. The proposed online multivariable identification algorithm makes use of a new fuzzy rule significance concept, expressed in terms of a weighted rule activation record (WRAR) measure, to provide a generic time-based contribution record with forgetting factor feature for online performance evaluation of each created fuzzy rule during its life-time. This incorporates an adaptive rule management strategy to maintain

A wide class of nonlinear dynamic processes can be represented by the following nonlinear auto-regressive with exogenous input (NARX) model description (Ljung and Soderstrom, 1983): Y(k) = N(X(k)),

(3)

where X(k) ∈ Rr indicates the input model regression vector, defined as X(k)=[Y T (k − 1), . . . , Y T (k − n), UT (k − d − 1), . . . , UT (k − d − m)], r = nb + ma,

(4)

where N(·) denotes a nonlinear static transition function vector with a dimension similar to the process output vector Y, given by b × 1. U shows the process inputs, configured into a column vector which has the dimension of a × 1. d is an input time delay while n and m represent the process model orders. The idea of deriving affine-type model has already been introduced by Narendra and Mukhopodhyay (1997). They developed a scalar affine model, called as NARMA-L2, using an approximate Taylor series expansion of a general nonlinear process dynamic model around its scalar input u(k). The present work proposes an online multivariable affine model identification approach which is more general and practically oriented, leading to the following distinctive features: (i) the identification procedure does not need to have an already available nonlinear process dynamic model, (ii) the procedure considers the problem as a black-box identification case with no need to a priori process dynamic knowledge, (iii) the adaptive neuro-fuzzy network structures do not require to be set initially and start their learning from scratch without a priori assumed fuzzy rule, (iv) the neuro-fuzzy networks are recur-

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

sively updated via either structural or parametric adaptation mechanism based on the novelty of the incoming process operational data, realizing a more compact and significant set of fuzzy rules, and (v) the adaptive affine model identification approach has been proposed for practical industrial processes having MIMO dynamic structures with nonlinear and timevarying characteristics.

2.1.

Multivariable affine model structure

Consider a MIMO process with a inputs U ∈ Ra and b outputs denoted by Y ∈ Rb . The general affine model identification for such a multivariable process can be represented by the following discrete-time affine-type expression at the kth time instant: 





Y(k) = Fk (Xk ) + Gk (Xk )U(k),



157



where f ik (·) and gijk (·) indicate nonlinear functions, estimated  by the adaptive neuro-fuzzy networks. Thus, Fk (·) vector is implemented using b estimated neuro-fuzzy networks whereas b × a neuro-fuzzy networks should be identified  to model matrix. This implies that each Gk (·) estimated neuro-fuzzy network models only a dynamic portion of the multivariable process, leading to reduction of the requisite size of the neuro-fuzzy networks. Accordingly, the online learning procedure of the networks becomes easier and, hence, the estimated affine model becomes more robust to errors. This implies that b(a + 1) nonlinear functions should be estimated using the adaptive neuro-fuzzy networks to realize the proposed multivariable affine model structure. For this purpose, each nonlinear function relating to the corresponding affine model structure can be represented by the following NARX equation:

(5) yafj (k) = hj (X(k)),

j = 1, . . . , b(a + 1).

(10)

where 





Y(k) = [y1 (k), . . . , yb (k)]

(estimated output vector),

(6)

T

(actual input vector),

(7)

U(k) = [u1 (k), . . . , ua (k)] 



2.2.

T



T

Fk (Xk ) = [f 1k (Xk ), . . . , f bk (Xk )]b×1 ,

⎡ 

⎢ ⎢ ⎣

g11k (Xk ) .. .

Gk (Xk ) = ⎢ 

gb1k (Xk )



· · · g1ak (Xk ) ..

.. .

.

(8)

⎤ ⎥ ⎥ ⎥ ⎦



· · · gbak (Xk )

b×a

,

(9)

Adaptive neuro-fuzzy network

Fig. 1 demonstrates the basic configuration of each adaptive neuro-fuzzy model in which fuzzy if-then rules are represented in a layered network structure, composing of five layers. The first layer transmits the received input variables directly to the next layer whose number of nodes is equal to the input vector X dimension. The second layer performs fuzzification. For this purpose, the following Gaussian functions are employed as the membership functions, because they are differentiable, ensure the greatest possible generalization for each fuzzy rule (Angelov and Buswell, 2002) and yet yield a compact formula for the degree of rule fulfillment

Fig. 1 – Neuro-fuzzy model structure.

158

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

in the third layer:

 Aqpj (xp ) = exp

−

(xp − qpj )

2

,

qj 2

q = 1, . . . , Hj , j = 1, . . . , b(a + 1),

(11)

where xp denotes the pth input to the second layer of all the networks, apj and  qj show the center and width of the qth Gaussian function for the pth input and jth network output, respectively. Hj indicates the generated number of adaptive fuzzy rules for the jth separate network output, available at the kth time instant where the general form of the qth rule is as follows:

1. Compute the overall network estimation for each output using Eq. (14). 2. Find the nearest created fuzzy rule to the new input observation data X(k), defined on the basis of Euclidean distance measure determining by ||X(k) − nr,j || = Mini ||X(k) − ij ||, i = 1, . . ., Hj . 3. Calculate the following measures corresponding to the fuzzy rule addition/replacement criteria as

WRARnr,j (k) =



= 1j

aqj is the constant parameter corresponding to the qth rule consequence, defined for the jth estimated network output. In the third layer, each node corresponds to a rule and hence the node output is equal to the firing strength of each rule. Thus, the antecedent connecting part of the if-then fuzzy rules will be determined as follows on the basis of sum–product composition:

Rqj (X) =

Aqpj (xp ) = exp

p=1

= exp

 −

||X − qj qj 2

||2

−

p=1

,

yaf (k) =

qj 2

q = 1, . . . , Hj , (13)

j

Hj Rq (X)aqj q=1 , j = 1, . . . , b(a + 1). Hj

(14)

R (X) q=1 qj



k−i−1 k−k−1 1j nr,j (i) + 1j nr,j (k)

Therefore, b(a + 1) separate neuro-fuzzy networks are developed simultaneously to make an exact MIMO affine model. Eq. (14) represents all the individual neuro-fuzzy  model outputs at each time instant and yaf shows the jth j

network output. The network output vector (Y) evolves the proposed b(a + 1) estimated nonlinear function outputs (i.e.,      [f 1 , . . . , f b , g11 , . . . , g1b , . . . , gab ]).

2.3. Online identification scheme for adaptive neuro-fuzzy model The identification algorithm starts with no a priori fuzzy rule. Following the reception of a new online observation data (X(k) ∈ Rr , yafj (k) ∈ R, j = 1, . . . , b(a + 1)), the model identification procedure starts autonomously to sequentially adapt the structural and parametric free adjustments of the neuro-fuzzy networks as follow corresponding to the multi-input, multioutput (MIMO) affine model, summarized in Eq. (5):

(15)

where WRARnr,j refers to a new global time-weighted rule activation record of the nearest created fuzzy rule based on the most recent sequence of past and present input–output process data organized in X for the jth network output. nr,j (k) indicates the instantaneous measure for evaluating the contribution of the nearest fuzzy rule for the estimation of the jth network output at the kth time instant:

nr,j (k) =

where r denotes the summation of model orders (i.e., n + m). The normalized contribution of each rule will then be computed in the normalization nodes of the fourth layer. The fifth layer gives the estimated network model outputs by



i=1

= 1j WRARnr,j (k − 1) + nr,j (k),

j = 1, . . . , b(a + 1),

k−i−1 1j nr,j (i)

−1 = 1j [WRARnr,j (k − 1) + 1j nr,j (k)]

j

r 2 (xp − qpj )

 k−1

(12)

where xi indicates the ith variable of the model regression vector X. Accordingly, Aq1j , . . . , Aqrj show the membership values dealing with the fuzzy sets of the corresponding terms con sidered in X. yaf denotes the jth estimated model output and



k

i=1

j

r

k−i 1j nr,j (i) = 1j

i=1

If x1 is Aq1j and x2 is Aq2j , . . . , and xr is Aqrj Then yaf is aqj , q = 1, . . . , Hj , j = 1, . . . , b(a + 1),

k

R

Hnr,j j

(X)

R (X) q=1 qj

(16)

.

Thus, evaluating the influence of the nearest rule on the basis of this instantaneous measure can make the neurofuzzy model severely sensitive to measurement noises, leading to unacceptable perturbation in the resulting estimated model. To overcome this problem, WRAR offers an efficient new global measure which incorporates the past fuzzy rule contributions in a time interval whose width can be determined by assigning a forgetting factor indicated by 0 <  1j ≤ 1 in Eq. (15). Another measure which can assess the adequacy of the estimated neuro-fuzzy structure corresponds to the following time-weighted root-mean square (WRMS) error:

   k ||ej (i)||2 EWRMS,j (k) =  2j i−k , j = 1, . . . , b(a + 1), Nj

i=k−(Nj −1)

(17)



where ej (i) = yafj (i) − yaf (i) indicates the jth network outj

put estimation error, manifesting the instantaneous quality of the estimated network. Nj shows the predefined sliding data window for the jth network output and  2j denotes the corresponding time-weighted forgetting factor. This measure can, hence, provide a complementary tradeoff measure to maintain the balance between local and global model estimation requirements. 4. Check the rule addition/replacement criteria:

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

If (||X(k) − nr,j ||> εj (k) and EWRMS,j (k) > Eth,j ) if WRARnr,j (k) > WRARth,j Allocate a new rule (Hj + 1) with the following coordinates: aHj +1,j = ej Hj +1,j = X Hj +1,j = j ||X − nr,j || where j is an overlap factor determining the extent of the rules responses overlap in the input space. Else Replace the nearest rule with the new data as anr,j = ej nr,j = X nr,j = j ||X − nr,j || End if

this time-varying profile. As a consequence, the identification procedure employs a conservative tight condition in the first initial commissioning stage when there is no a priori dynamic knowledge. This causes the neuro-fuzzy model to be adapted smoothly under such a severe uncertainty whereas the timevarying profile lets the resolution threshold to approach a lower value so as to enable the estimated model to be fine tuned as more information is available. The other thresholds, i.e., WRARth,j and Eth,j are assumed constant values, indicating the lowest acceptable rule contribution and the network adequacy measures, respectively, to evaluate the nearest rule activation record and the weighted output error for the jth model output estimation. WRARnr,j denotes the lowest level of weighted rule activation record of the nearest rule to be considered for pruning the closest rule to the most recent input process data.

2.3.1.

Else Adjust the nearest rule free parameters (anr,j , nr,j ,  nr,j ) using an EKF learning algorithm. End if 5. Check the rule pruning criterion as If WRARnr,j (k) < WRARp,j Remove the nearest rule. Reduce the dimensionality of the model parameter vector. End if where εj (k), WRARth,j , and Eth,j represent decision thresholds for the jth MISO estimated network. εj (k) indicates the minimum input data space resolution required for the nearest fuzzy rule to have significant influence on the most recent input data X(k). The following time-varying scheme is employed to realize different resolution in the course of online identification implementation:

 εj (k) = εmax,j − (εmax,j − εmin,j )

 1 − exp

−(k − 2)

j

,

(18)

where εmax,j and εmin,j determines the largest and smallest distance resolution of interest, while j indicates the speed of

159

Online adaptation of fuzzy rule-based structure

Eq. (12) can be viewed in terms of a computational model in a fuzzy inference system (FIS) employing traditional fuzzy if-then rules. When such a fuzzy rule is created, the coordinates of its membership functions are adjusted on the basis of the new online observation data (X), as described in step 4 of the presented identification scheme. This implies that the membership functions exemplify a characteristic input–output behavior of the process to be to be modeled. Although the number of rules (or membership functions) is automatically determined by the proposed online identification algorithm, it should be noted that the user specified parameter j (i.e., the overlap factor characterizing the extent of the rules responses overlap in the input space for the jth separate network output) affects the number of rules that will be generated. This parameter is utilized to initialize the width of the newly added rule and hence can be set based on different objectives. An appropriate option is suggested to be chosen in the range 1 ≤ j ≤ 2. A large j will result in few rules and hence a coarser model, while a small j can produce excessive number of rules and hence a model that does not generalize well. Therefore, j can be regarded as an approximate specification of the desired resolution of the model, which can be adjusted based on a tradeoff between the resultant local and global requirements of the model. In this paper, j = 1.75 has been observed as a proper tuning in all the simulation case studies.

Fig. 2 – Block-diagram representation of the proposed affine model identification approach.

160

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

2.3.2.

Online parameter learning of fuzzy rules

The learning process of adaptive neuro-fuzzy model involves either a simple addition of a new fuzzy rule or replacement of an already created fuzzy rule based on novelty of the received observation input–output data, as specified in step 4 of the identification algorithm. However, when the observation data do not pass the novelty criteria, the neurofuzzy network parameters, which are characterized by the rules premises (i.e., centers and widths of the membership functions) and consequents, can be updated using the EKF parameter learning algorithm. For multiple inputs, however, the input dimensions grow and this may arise numerical difficulties in parameter learning process. That is, when the number of inputs gets large and hence the number of rules grows, the size of the covariance matrix in the EKF algorithm becomes large and this results in a large learning time and storage requirement, causing a computational overload for online applications. To alleviate this problem, the EKF algorithm is utilized to update the parameters of the rule selected by the new input observation data on the basis of the winner rule strategy. That is, as described in step 4 of the identification algorithm, only the neuro-fuzzy network parameters associated with the nearest rule to the most recent observation data is adapted. This enhances the robustness of the learning algorithm and yet decreases the required computational burden, leading to a fast identification procedure.

2.4.

EKF parameter estimation algorithm

The extended Kalman filter (EKF) is an expedient learning algorithm for nonlinear parameter estimation. This algorithm is utilized for the parameter estimation to adjust the free coordinates of the nearest fuzzy rule, i.e., nr,j ,  nr,j and anr,j . For this purpose, the identified nonlinear neuro-fuzzy network, estimating the NARX model represented by Eq. (10), is first linearized around the free adjustable network parameters as follows: 

∂yaf (k) j

a˙ nr,j =



 ˙ nr,j =

∂yaf (k) ∂nr,j

Hj

=

∂anr,j

j

Rnr,j

,

R q=1 qj

⎡



=

(19)

∂yaf (k) ∂Rnr,j j ∂Rnr,j ∂nr,j

⎤

Hj ⎢ q=1 Rqj − Rnr,j ⎥ ∂Rnr,j = anr,j ⎣  2 ⎦ ∂ , Hj nr,j R q=1 qj

(20)

⎡ 

∂yaf (k) j

˙ nr,j =

∂nr,j



=

∂yaf (k) ∂Rnr,j j ∂Rnr,j ∂nr,j

⎢

= anr,j ⎣

⎤

Hj

R q=1 qj



− Rnr,j ⎥ ∂Rnr,j

Hj R q=1 qj

2 ⎦ ∂ , nr,j (21)

where ∂Rnr,j ∂nr,j ∂Rnr,j ∂nr,j

= 2Rnr,j

= 2Rnr,j

X(k) − nr,j nr,j 2

,

||X(k) − nr,j nr,j 3

(22)

||2

.

(23)

Then, the gradient of unknown parameter vector, rep˙ nr,j , ˙ nr,j ]T , can be utilized in the resented by ˙ nr,j = [a˙ nr,j ,  following EKF algorithmic steps to estimate the free coordi-

Fig. 3 – Schematic diagram of a non-isothermal CSTR. nates of the nearest fuzzy rule: −1 T Kj (k) = Pnr,j (k − 1) ˙ nr,j (k)[Rj (k) + ˙ nr,j (k)Pnr,j (k − 1) ˙ nr,j (k)] ,

(24)

nr,j (k) = nr,j (k − 1) + Kj (k)ej (k),

(25)

T Pnr,j (k) = [I − Kj (k) ˙ nr,j (k)]Pnr,j (k − 1),

(26)

where Kj indicates the Kalman gain, Rj denotes the modeling error covariance, and Pnr,j defines the parameter error covariance matrix, all referring to the nearest fuzzy rule of the jth estimated MISO model incorporated in Eq. (5).

2.5. A new online multivariable identification algorithm for an adaptive affine neuro-fuzzy model A new online multivariable identification algorithm is presented in this section to develop an adaptive affine-type model  for a MIMO process expressed by Eq. (5). For this purpose, Fk (·) includes b adaptive neuro-fuzzy networks while b × a adaptive  neuro-fuzzy networks are utilized to estimate Gk (·) model. As illustrated in the block-diagram representation shown in Fig. 2, the proposed multivariable affine model identification scheme can be described as follows to simultaneously estimate the   unknown nonlinear function vectors Fk (·) and Gk (·): 1. Initialization:  The algorithm is initialized by setting G0 (X1 ) = ˛Y(0) and  F0 (X1 ) = (1 − ˛)Y(0) in which 0 < ˛ < 1 is selected randomly. 2. Set k = 1 to indicate the first sampling time instant.

Table 1 – Nominal CSTR operating condition. q = 100 l/min CAf = 1 mol/l Tf = 350 K Tcf = 350 K V = 100 l hA = 7 × 105 cal/(min K) k0 = 7.2 × 1010 min−1

E/R = 9.95 × 103 K − H = 2 × 105 cal/mol , C = 1000 g/l Cp , Cpc = 1 cal/(g K) qc = 103.41 l/min T = 440.2 K CA = 8.36 × 10−2 mol/l

161

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

3.

3. Calculate Fk (Xk ) using

Simulation test studies



Fk (Xk ) = Y(k) − Gk−1 (Xk )U(k). 4. Perform the online neuro-fuzzy identification algorithm to  estimate Fk (Xk ) as the set of adaptive neuro-fuzzy networks to model Fk (Xk ). 5. Calculate Gk (Xk ) as



Gki (Xk ) = (Y(k) − Fk−1 (Xk ) −

i−1

Gkp (Xk )up (k)

p=1

−

a Gkp (Xk )up (k) p=i+1

ui (k)

,

i = 1, . . . , a.

Assuming ui (k) = / 0 which is a valid proposition at normal process operating condition. 6. Perform the online neuro-fuzzy identification algorithm to  estimate Gk (Xk ) as the set of required neuro-fuzzy networks to model Gk (Xk ). 7. Calculate the process output estimate as 





A set of simulation test studies are carried out in this section to explore the performance evaluations of the proposed online multivariable affine model identification approach. For this purpose, two challenging multivariable benchmark processes are employed.

3.1.

Simulation case study I

A non-isothermal CSTR process, possessing highly nonlinear and time-varying dynamic characteristics with a non-affine dynamic model, will be investigated as a benchmark problem in this case study.

3.1.1.

CSTR process description

Fig. 3 demonstrates the non-isothermal CSTR process in which an irreversible reaction (A → B) takes place. The dynamic behavior of the process is described by the following nonlinear equations in a non-affine model structure (Nikravesh et al., 2000):

Y(k) = Fk (Xk ) + Gk (Xk )U(k). 8. Set k = k + 1 to denote for the next sampling time instant. 9. Go to step 3.



dCA E q = (CAf − CA ) − k0 CA exp − dt v RT

 ϕC (t),

(27)

Fig. 4 – open-loop CSTR responses to +5% step change deviations: (a) and (b) depict the output responses to qc ; (c) and (d) show the output response to q.

162

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169



(− H)k0 CA E q dT exp − = (Tf − T) + dt v Cp RT





× 1 − exp −

hA qC CPC







ϕC (t) +

ϕh (t) (TCF − T).

C CPC qc CP V (28)

The CSTR nominal operating condition has been summarized in Table 1. Fouling and catalyst deactivation phenomena present two main sources to produce challenging time-varying characteristics in the CSTR process dynamic model during its normal operation. These time-varying dynamic characteristics are represented by the following equations (Nikravesh et al., 2000): ϕh (t) = 1 − 0.01t,



ϕc (t) = exp −0.00067

(29)



ER t , T

(30)

where the fouling effect is caused by deposing material on the heat transfer surface whereas catalyst is deactivated due to poisoning.

3.1.2. Online identification of a multivariable affine model under the influence of noise and time-varying characteristics The performance of the proposed online multivariable identification approach will be assessed by considering the non-affine MIMO CSTR dynamic. For this purpose, the CSTR input and output vectors are defined as U = [qc , q]T and Y = [CA , T]T , respectively. Two sequences of white noises with variance of 0.01 are added as the process noises to the CSTR process dynamics, given in Eqs. (27) and (28), to exercise the identification simulation tests under real challenging circumstances. Measurement noises present other harsh environmental condition which can influence the performance of the proposed identification approach. Thus, two white noise sequences with variance of 0.02 will be added to the effluent concentration (CA ) and temperature (T) to mimic as measurement noises. Furthermore, the fouling and catalyst deactivation phenomena will be included to carry out the simulation studies under more severe practical conditions. Fig. 4 demonstrates the CSTR open-loop responses to +5% step change deviation in the individual coolant flow rate (qc ) and feed flow rate (q), respectively. The obtained responses illustrate typical effluent concentration (CA ) and temperature (T) behaviors under noise-free conditions for the observation purposes. A set of multi-level variations in the coolant flow rate (qc ) and a one step change variation in the feed flow rate (q) being superimposed by respective white noise sequences of variances 0.01 will be introduced, as shown in Fig. 5(a) and (b), respectively, to excite the CSTR benchmark process over a wide range of operating conditions starting from its initial normal operating point. Consequent output responses of the highly nonlinear CSTR are depicted in Fig. 6(a) and (b). To estimate an adaptive multivariable affine-type model, the following simple input regressor vector is adopted: X = [CA (k − 1), T(k − 1), qc (k), q(k)].

(31)

Therefore, the following two multi-input, single-output (MISO) affine-type models will be estimated using adaptive

Fig. 5 – Input signal sequences exciting the CSTR process: (a) coolant flow rate (qc ); (b) feed flow rate (q).

neuro-fuzzy networks in a real-time manner: 







CA (k) = f Ca k (X) + gCa qc k (X)qC (k) + gCa qk (X)q(k), 







T(k) = f Tk (X) + gTqc k (X)qC (k) + gTqk (X)q(k).

(32) (33)

Examining theses equations implies that six adaptive neuro-fuzzy networks are required to simultaneously be esti  mated so as to model for f ik (X) and gijk (X) where i and j indicate the input and output variables, respectively. To implement the online multivariable identification algorithm, the free parameters are set as follows during the whole excitation experiment   to adaptively estimate f ik (X) and gijk (X) models: εmin,j = 0.27, εmax,j = 0.98, j = 4000, WRARth,j = 0.32, WRARP,j = 0.15, j = 1.75 and Eth,j = 0.07, j = 1, . . . , 6. The identified results corresponding to the estimated multivariable affine-type model have been illustrated in Figs. 7–9. The comparative results, illustrated in Fig. 7, clearly demonstrate the high accuracy of the identified multivariable affine model to efficiently approximate the actual non-affine CSTR dynamic model at different operating points. This simulation observation can statistically be verified based on the overall mean square error (MSE) measure. The calculated

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

163

Fig. 6 – CSTR output responses to input excitation signals: (a) effluent concentration (CA ); (b) temperature (T).

Fig. 7 – Estimated CSTR multivariable affine model outputs:   (a) effluent concentration (CA ); (b) temperature (T).

MSE measures for the estimated effluent concentration and temperature behaviors are equal to 0.0005 and 0.0870, respectively. This indicates the high quality of the identified multivariable affine model to reproduce the actual CSTR process output responses in the face of process and measurement noises which impose more challenging conditions. The time history evaluations of the created fuzzy rules in Fig. 8(a)–(f) correspond to the identified neuro-fuzzy networks     represented by the two sets of f Ca k , gCa qc k , and gCa qk and f Tk ,   gTqc k , and gTqk model networks, given in Eqs. (32) and (33). The results indicate that the identified adaptive neuro-fuzzy models have compact structures whose most complex one is recursively ended up finally to three adaptive fuzzy rules as illustrated in Fig. 8(f). Fig. 9(a) and (b) illustrate the time updating evolutions of the estimated network parameters,   corresponding to the adaptive gCa qc k and gTqc k neuro-fuzzy models, respectively, for typical observation purposes.

major control objective. The column shows significant nonlinear behavior over its operating region (Sriniwas et al., 1995). Thus, this highly nonlinear process whose model has not been formulated in an affine-type structure will present a proper benchmark case study to explore the performance evaluation of the proposed identification approach.

3.2.

Simulation case study II

Distillation columns are industrially important process operations whose dynamics tend to become more nonlinear as the product purities increase. Skogestad (1997, website) presented a well-developed nonlinear model for a high-purity binary distillation column as a real benchmark problem. This distillation column, operated in the LV-configuration, poses a

3.2.1.

Binary distillation column description

The process represents a first principle model of high-purity binary distillation column, depicted schematically in Fig. 10. It is usually called as “column A” in the literature which consists of 39 trays, a reboiler and a condenser. The model is a 2 × 2“open-loop” column with reflux flow rate, L, and boil-up flow rate, V, as the two manipulated variables whereas the controlled variables are the top and bottom product composition, denoted by XD and XB , respectively. The proposed model has been provided by some assumptions as follows: Binary mixture, constant pressure, constant relative volatility, constant molar flows, no vapor holdup, linear liquid dynamics, equilibrium on all stages, total condenser. Note that we do not assume constant holdup on the stages, that is, we include liquid flow dynamics. This means that it takes some time (about (NT − 2) × taul ) from we change the liquid in the top of the column until the liquid flow into the reboiler changes. The distillation column mathematical model equations have been described as follows:

164

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169











Fig. 8 – Time history evolutions of the created fuzzy rules corresponding to: (a) f CA K ; (b) gCA qc K ; (c) gCA qK ; (d) f TK ; (e) gTqc K ; (f)  gTqK .

- Material balance for light component on each stage i:

- Total material balance on stage i: dMi = Li+1 − Li + Vi−1 − Vi . dt

(34)

d(Mi xi ) = Li+1 xi+1 + Vi−1 yi−1 − Li xi − Vi yi , dt

(35)

165

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

Table 2 – Nominal operating conditions for distillation column. NT NF Feed rate, F Feed composition, zF Feed liquid fraction, qF Reflux flow, LT Boilup, V Liquid holdup on all 41 stages, M0i taul 

D yD=XNT B xB = x1 a

41 stages including reboiler and total condenser Feed at stage 21 counted from the bottom 1 [kmol/min] 0.5 [mole fraction units] 1 (i.e., saturated liquid) 2.706 [kmol/min] 3.206 [kmol/min] 0.5 [kmol]a 0.063 min 0 (assuming the vapor flow doesn’t affect the liquid holdup) 0.5 [kmol/min] 0.99 [mole fraction units] 0.5 [kmol/min] 0.01 [mole fraction units]

Including the reboiler and condenser; for more realistic studies you may use M01 = 10 [kmol] (reboiler) and M0NT = 32.1 [kmol] (condenser).

equilibrium: yi =

˛xi , 1 − (˛ − 1)xi

Fig. 9 – Time updating evolution of network parameters: (a)   gCA qc K ; (b) gTqc K .

Fig. 10 – The complete schematic diagram of distillation column. which gives the following expression for the derivative of the liquid mole fraction: dxi (d(Mi xi )/dt) − (xi (dMi /dt)) . = dt Mi

(36)

- Algebraic equations: The vapor composition yi is related to the liquid composition xi on the same stage through the algebraic vapor–liquid

Fig. 11 – Column input excitation signals in: (a) L; (b) V.

(37)

166

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169

Total condenser: i = NT(MNT = MD , LNT = LT ) dMi = Vi−1 − Li − D, dt

(42)

d(Mi xi ) = Vi−1 yi−1 − Li xi − Dxi . dt

(43)

Reboiler: i = 1 (M1 = MB , V1 = VB = V) dMi = Li+1 − Vi − B, dt

(44)

d(Mi xi ) = Li+1 xi+1 − Vi yi − Bxi . dt

(45)

We have written the model such that the states are xi (i = 1, NT) and Mi (i = 1, NT) – a total of 2 × NT states. Also nominal conditions have been presented in Table 2. Further details of the simulated model are described in Skogestad (1997).

3.2.2.

Online identification of a multivariable affine model

Two normal distributed random sequences of variances 0.001 and 0.007 are superimposed on the column input changes in L and V, respectively, to excite the column benchmark around its normal steady-state operating condition (Skogestad, website). Fig. 11(a) and (b) represent the corresponding input exci-

Fig. 12 – Column output responses: (a) XD ; (b) XB .

where ˛ is the relative volatility. Based on the assumption of constant molar flows and no vapor dynamics, we have the following expression for the vapor flows (except at the feed stage if the feed is partly vaporized, where VNF = VNF−1 + (1 − qF )F): Vi = Vi−1 .

(38)

The liquid flows depend on the liquid holdup on the stage above and the vapor flow as follows (this is a linearized relationship; we may alternatively use Francis’ Weir formula, etc.): Li = L0i +

(Mi − M0i ) + (V − V0)i−1 × , taul

(39)

where L0i [kmol/min] and M0i [kmol] are the nominal values for the liquid flow and holdup on stage i. The vapor flow into the stage may also affect the holdup. For packed columns  is usually close to zero. The above equations apply at all stages except in the top (condenser), feed stage and bottom (reboiler).Feed stage: i = NF (we assume the feed is mixed directly into the liquid at the feed stage): dMi = Li+1 − Li + Vi−1 − Vi + F, dt

(40)

d(Mi xi ) = Li+1 xi+1 + Vi−1 yi−1 − Li xi − Vi yi + FzF . dt

(41)

Fig. 13 – Estimated multivariable affine model for column  outputs: (a) top product composition (X D ); (b) bottom  product composition (X B ).

167

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169











Fig. 14 – Time history evolutions of the created fuzzy rules corresponding to: (a) f XD k ; (b) gXD Lk ; (c) gXD Vk ; (d) f XB k ; (e) gXB Lk ; (f)  gXB Vk .

tation signals, resulting into the output column responses in XD and XB , depicted in Fig. 12(a) and (b), respectively. The following input regressor vector is utilized to implement the proposed online multivariable affine identification approach:

XD (k) = f XD k (X) + gXD Lk (X)L(k) + gXD Vk (X)V(k),

X = [XD (k − 1), XB (k − 1), XD (k − 2), XB (k − 2), L(k), V(k)].

XB (k) = f XB k (X) + gXB Lk (X)L(k) + gXB Vk (X)V(k).

(46)

This leads to an online estimation of the following two MISO affine-type model structures using six adaptive neurofuzzy networks: 















(47) (48)

168

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169













Fig. 15 – Time updating evolutions of network parameters: (a) f XD k ; (b) gXD Lk ; (c) gXD Vk ; (d) f XB k ; (e) gXB Lk ; (f) gXB Vk . For this purpose, the free parameters are set as follows to   estimate f ik (X) and gijk (X) networks: εmin,j = 0.39, εmax,j = 0.89, j = 1200, WRARth,j = 0.22, WRARP,j = 0.01, j = 1.75 and Eth,j = 0.01, j = 1, . . . , 6. The identified multivariable affine model results have been illustrated in Figs. 13–15. Fig. 13 depicts the comparative results in terms of the estimated affine model outputs. The results clearly demonstrate the high accuracy of the identified

multivariable affine model in approximating the highly nonlinear binary distillation benchmark whose dynamic model has been developed in a non-affine structure. The corresponding MSE measures, calculated as 1.617 × 10−5 and 2.767 × 10−5 , for the top and bottom product compositions, respectively, verify the adequacy of the identified neuro-fuzzy networks to adaptively reproduce the actual column outputs. Figs. 14 and 15 show the real-time adjustments of the identified neuro-fuzzy networks to adaptively tune their structures and parameters, respectively. Fig. 14(a)–(f) depict the time history evolutions of the created fuzzy rules to realize the

chemical engineering research and design 8 8 ( 2 0 1 0 ) 155–169



individual structural adaptations corresponding to f ik (X) and  gijk (X) models expressed in Eqs. (47) and (48). The obtained results indicate the simplicity of the estimated neuro-fuzzy model structures in which the identified affine-type model has at most three adaptive fuzzy rules for each structure. Fig. 15(a)–(f) show the real-time updating of the estimated net work parameters, corresponding to the six adaptive f ik (X) and  gijk (X) neuro-fuzzy models.

4.

Conclusions

Development of an adaptive affine-type dynamic model for practical multivariable processes with nonlinear and timevarying characteristics provides a significant contribution, enabling the utilization of a variety of nonlinear model-based control design techniques. A new online multivariable identification scheme has been proposed in this paper to accomplish a practically demanding modeling objective. The proposed scheme presents a new procedure to simultaneously identify a set of complementary adaptive neuro-fuzzy networks to realize an online multivariable affine model structure. The procedure introduces several functional evaluating metrics, such as WRAR, to facilitate the implementation of automatic model development with structural and parametric capabilities. This naturally leads to efficient and compact neuro-fuzzy networks, contributing a tradeoff between local and global model estimation requirements over the whole process operating points spectrum. The extensive simulation test studies carried out on a CSTR and a binary distillation column, as two well-known benchmarks with highly nonlinear and timevarying characteristics, demonstrated the capability of the proposed identification approach to automatically estimate a set of simultaneous adaptive neuro-fuzzy networks with simple and compact dynamic structures so as to develop an accurate affine-type model framework.

169

References Angelov, P. and Buswell, R., 2002, Identification of evolving fuzzy rule-based models. IEEE Trans Fuzzy Syst, 10(5): 667–677. Chen, F.C. and Khalil, H.K., 1995, Adaptive control of a class of nonlinear discrete-time systems using neural networks. IEEE Trans Autom Control, 40(5): 791–801. Henson, M.A. and Seborg, O.E., (1997). Nonlinear Process Control. (Prentice-Hall). Ljung, L. and Soderstrom, T., (1983). Theory and Practice of Recursive Identification. (MIT Press, London). Narendra, K.S., 1992, Adaptive control of dynamical systems using neural networks, in Handbook of Intelligent Control: Neural, Fuzzy and Adaptive Approaches, White, D.A. and Sofge, D.A., Sofge, D.A. (eds) (Van Nostrand-Reinhold, New York), pp. 141–183. Narendra, K.S. and Mukhopodhyay, S., 1997, Adaptive control using neural network and approximation models. IEEE Trans Neural Networks, 8(3): 475–485. Nikravesh, M., Farell, A.E. and stanford, T.G., 2000, Control of nonisothermal CSTR with time varying parameters via dynamic neural network control (DNNC). Chem Eng J, 76: 1–16. Norgaard, M., Ravn, O., Poulsen, N.K. and Hansen, L.K., (2003). Neural Networks for Modeling and Control of Dynamic Systems. (Springer-Verlag, London), pp. 142–147 Sanner, R. and Slotine, J.J., 1995, Stable adaptive control of robot manipulators using neural networks. Neural Comput, 7(4): 753–790. Skogestad, S., 1997, Dynamics and control of distilation columns. Chem Eng Res Des (Trans IChemE), 7(5): 539–562. Skogestad, S., MATLAB Distillation Column Model (“Column A”), available at: http://www.nt.ntnu.no/users/skoge/book/ matlab m/cola/cola.html. Sriniwas, G.R., Arkun, Y., chien, I.L. and Ogunnaike, B.A., 1995, Nonlinear identification and control of high-purity distillation column: a case study. J Process Control, 3: 149–162.