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Applied Soft Computing 25 (2014) 174–183

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l ho me page: www.elsev ier .com/ locate /asoc

arameter identification of PWARX models using fuzzy distanceeighted least squares method

nkit K. Shaha, Dipak M. Adhyarub,∗

Department of Instrumentation & Control, SVIT, Vasad, Gujarat, IndiaDepartment of Instrumentation & Control, Nirma University, Ahmedabad, Gujarat, India

r t i c l e i n f o

rticle history:eceived 19 July 2013eceived in revised form 4 July 2014ccepted 9 September 2014vailable online 18 September 2014

eywords:ieceWise AutoRegressive eXogenous

a b s t r a c t

PieceWise AutoRegressive eXogenous (PWARX) models represent one of the broad classes of the hybriddynamical systems (HDS). Among many classes of HDS, PWARX model used as an attractive model-ing structure due to its equivalence to other classes. This paper presents a novel fuzzy distance weightmatrix based parameter identification method for PWARX model. In the first phase of the proposedmethod estimation for the number of affine submodels present in the HDS is proposed using fuzzy clus-tering validation based algorithm. For the given set of input–output data points generated by predefinedPWARX model fuzzy c-means (FCM) clustering procedure is used to classify the data set according to

ybrid dynamical systemuzzy-c-means clusteringuzzy distance weight matrixeighted least squares

its affine submodels. The fuzzy distance weight matrix based weighted least squares (WLS) algorithm isproposed to identify the parameters for each PWARX submodel, which minimizes the effect of noise andclassification error. In the final phase, fuzzy validity function based model selection method is applied tovalidate the identified PWARX model. The effectiveness of the proposed method is demonstrated usingthree benchmark examples. Simulation experiments show validation of the proposed method.

© 2014 Elsevier B.V. All rights reserved.

. Introduction

Hybrid dynamical systems (HDS) are systems that involve non-inear continuous dynamics and discrete events with some logicalules [1,2]. A mathematical model of the HDS must change itsontinuous dynamic as the discrete event occurred in the sys-ems. PieceWise AutoRegressive eXogenous (PWARX) models haveeen widely adopted by researcher to represent various HDS dueo their universal approximation properties of PieceWise AffinePWA) maps and their equivalence to other classes of the HDS [3–5].

The PWARX models comprise a set of classical ARX submodels,ach of which is partitioned with a finite number of the polyhe-ral partitions. The identification of PWARX models involves thestimation for the number of submodels present in the HDS, thearameters of each submodel and the regions of the submodelefining the polyhedral partition. In [6] methods for the identifica-ion of the PWA models of the HDS are classified. The identification

roblem becomes more challenging as it includes the estimation ofhe number of submodels along with data classification, parametersnd method to verify the identified model parameters. According

∗ Corresponding author. Tel.: +91 9016805104.E-mail addresses: shah ankit2@gtu.edu.in (A.K. Shah), hetdip@rediffmail.com

D.M. Adhyaru).

ttp://dx.doi.org/10.1016/j.asoc.2014.09.005568-4946/© 2014 Elsevier B.V. All rights reserved.

to [5], four procedures for the identification of PWARX models arealgebraic procedure [7], clustering-based procedure [8], Bayesianprocedure [9] and bounded-error procedure [10]. The PWA systemidentification using mixed-integer quadratic programming (MIQP)and mixed-integer linear programming (MILP) proposed in [11]which is NP-hard and required more computations. However, theestimation of the number of submodels and the selection of sub-models has not been dealt properly in the reviewed literature.

The aim of the paper is to show how to estimate the number ofsubmodels and identification of the parameters of the submodelsassociated with each discrete event along with the classification ofthe data points with reasonable computational complexity. Clus-tering algorithms have been widely used in data classification,pattern recognition, system modeling, machine learning and imageprocessing [12]. In the past few years, many clustering algorithmshave been developed and used for the classification of the datapoints for the PWARX models. Works based on clustering algo-rithms for identification of the PWARX models have been reportedon k-means [8,13,14], support vector classifier [5] and fuzzy clus-tering [15–18]. The clustering based procedure, proposed by [8] and[13] consists of the k-means clustering approach with the assump-

tion that the number of discrete modes given a priori. The statisticalclustering based identification approach given in [14], proposed thealgorithm for estimating the number of submodels, but it requiresan exhaustive search between two predefined limits. However, the

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A.K. Shah, D.M. Adhyaru / Applie

-means clustering procedures suffer from the drawbacks of poornitialization and influence of outliers. In k-means clustering algo-ithms, data points are partitioned into some clusters such thatach data point is assigned to exactly one cluster. However, in fuzzylustering, a data point may belong to several clusters with degreef membership function. Therefore, the membership function val-es for a data point will represent the degree to which that pointelongs to a particular cluster. The Fuzzy c-means (FCM) clustering

s one of the most popular and efficient unsupervised partitionedlgorithm used in several applications where the clustering is over-apping [19–21]. Moreover, the FCM can handle uncertainty andoise better when there is no information about the structure of theata. The FCM clustering based identification approach proposedy [15] estimates the number of submodels, but the nonlinearegression vector is used to identify the PWARX model. The fuzzylustering based technique for PWA approximation proposed by16,18] assumes that the number of submodels is given for the non-inear systems. In this paper, the FCM clustering algorithm is used

hich enables an estimation of the number of submodels present inhe PWARX models. We show how to estimate the number of sub-

odels using fuzzy cluster validation approach. In this paper, weresent new weighted least squares (WLS) approach for identifica-ion of parameters of the submodels based on fuzzy distance weight

atrix. Once the data points have been classified according to itsiscrete events a linear regression is used to compute the param-ters of the submodels. However, pure linear least squares do notive the optimal value of the parameter due to classification errornd sensitive to outliers [8,22]. In order to solve this shortcoming,uzzy distance weight matrix based weighted least squares (WLS)pproach is proposed which suitably defined confidence measuresnd reducing the influence of outliers on the given data points byeans of membership functions. We also verify the effectiveness

f the proposed methodology on two benchmark PWARX systemsnd a Hammerstein model which is composed of static nonlinearitylong with linear system.

This paper has been organized as follows: Section 2 representshe formulation of the identification problem for PWARX models.ection 3 provides details of the various phases of the proposedethod for the given problem in Section 2. In the same section

wo illustrative benchmark mathematical examples and a Ham-erstein model are used to support the proposed fuzzy weightatrix based parameter identification method. Finally, Section 4

ummarizes the contributions of this paper and possible futureesearch work.

. Problem formulation for PWARX models

Consider the discrete-time HDS with input u(t) ∈ Rnu and output(t) ∈ R at time t having discontinuous dynamics where nu is theumber of inputs. Let y(t − 1) and u(t − 1) be past value of inputsnd outputs, respectively, generated by the system up to time t − 1.WARX models are described by a relationship between past obser-ations and future output y(t) in the form given below:

(t) = f (x(t)) + e(t) (1)

here x(t) ∈ R ⊆ Rp is referred to as the regression vector in termsf past observations of input and output, e(t) ∈ R is a white noisedded to the output, and f is a piecewise affine map function of theorm

(x(t)) = �T�tϕ(x(t)) if x(t) ∈ ��t (2)

here the output f(x(t)) ∈ R is a scalar, �t ∈ {1, . . ., s} is the dis-rete events associated with the HDS, s is the number of submodelsresent in the PWARX model, �T

�t∈ Rn is the parameter vector defin-

ng for each �t in terms of matrixes, and ϕ(x(t)) ∈ Rm is the linear

Computing 25 (2014) 174–183 175

extended regression vector which is a function of x(t) given byϕ(x(t)) = [x(t) 1]T. The regression vector x(t) with the fixed structuredepending only on the past na outputs and nb inputs are defined as

x(t) = [y(t − 1). . .y(t − na)u(t − 1)T . . .u(t − nb)T ]T

(3)

where p = na + nbnu.The polyhedral regions ��t form a complete bounded polyhe-

dron partition of � (i.e. � =⋃s

�t=1�t) and �i ∩ �j = � ; ∀ i /= j. Theeach region ��t is assumed to be convex polyhedron. The switch-ing law between the PWARX models is specified by the rule: ifx(t) ∈ ��t, the �tth submodel dynamic is active.

Assumption 1. The data points z(t) = {y(t), x(t)}Nt=1 are generated

from the PWARX model (1) with given number of discrete regions s,the parameter vectors {��t}s

�t=1 for each submodels, the polyhedralregions ��t and the dimension p for the regression set with themodel orders of the past na outputs and nb inputs.

The data points z(t) = {y(t), x(t)}Nt=1 are generated according to

given Assumption 1 and the number of inputs {��t}s�t=1 and �

are known, the identification problem to solve s linear estimationproblem for each partition. The regression vector x(t) can be classi-fied into �t sets of partitions ��t according to rule if x(t) ∈ ��t, datapoints z(t) = {y(t), x(t)}N

t=1 ∈ ��t .

2.1. Problem statement

Given a set of data points z(t) which are generated by a hybridor piecewise linear system of the form (1) and (2) with Assumption1 hold, the objective is to estimate the number of submodels spresent in the systems and the parameter vectors {��t}s

�t=1 for eachsubmodel along with switching condition based on regions ��t.

The identification of a PWARX model of a finite of data pointsz(t) = {y(t), x(t)}N

t=1 is a complex problem which requires tech-niques to estimate the number of discrete modes at the same timeclassification of the data points according to its discrete events. Theconsidered identification problem consists in finding the PWARXmodel that involves the estimation for the number of submodelsassociated with HDS and parameter vector ��t of the each affinesubmodel.

Example 1. Consider the following single-input-single-output(SISO) PWARX model [8]:

y(t) =

⎧⎪⎨⎪⎩

[1 2][u(t − 1)1]T + ∈ (t) if u(t − 1) = x(t) ∈ X1 = [−4, 0]

[−1 0][u(t − 1)1]T + ∈ (t) if u(t − 1) = x(t) ∈ X2 = [−1, 2]

[1 2][u(t − 1)1]T + ∈ (t) if u(t − 1) = x(t) ∈ X3 = [2, 4]

(4)

For the given PWARX system na = 0, nb = 1 are given a prioriand 50 input samples, u(t) ∈ R, are applied randomly according tothe uniform distribution on [−4,4] to generate the output samples.Here the regression vector is x(t) = [u(t − 1)]T and data points z(t) ={y(t), u(t − 1)}50

t=1. The measured output is corrupted by Gaussiannoise with variance 0.05. Fig. 1 shows 50 sample data points of thegiven system along with the region of partitions.

The problem becomes easy if the number of submodels s is fixed,and the switching conditions are known a priori. The procedureproposed in this paper assumed that the number of submodelspresent in the given PWARX models and switching mechanismbetween the submodels are not given a priori. The framework forthe identification of the PWARX models is summarized as follows:

Phase 1: Estimation of the number of submodels.

In this phase, the measured inputs and output data points are

used to estimate the number of submodels present in the HDS byusing proposed fuzzy clustering based index. Initialize this phasewith the assumption of maximum number of submodels and phase

176 A.K. Shah, D.M. Adhyaru / Applied Soft Computing 25 (2014) 174–183

PWAR

3

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Fig. 1. The data points of the

complete with the optimum number of submodels represent by spresents in the HDS.Phase 2: Classification of the measured data points.

In this phase, the FCM clustering algorithm is applied in order toobtain the classification of measured data points into s partitioneddata points for each submodel.Phase 3: Identification of PWARX submodel parameter.

This phase represents a novel fuzzy distance matrix weightbased weighted least squares (WLS) algorithm in order to obtainPWARX submodel parameter vector {��t}s

�t=1.Phase 4: Submodel selection and validation.

This phase shows the effectiveness of the proposed identifi-cation method for parameters of the PWARX model. The fuzzyfunction based selection criteria is used for the measured data toselect the appropriate submodel and predict the new output of thehybrid system.

. Identification algorithm for PWARX models

The identification algorithm for PWARX models for solving theroblem statement mentioned in the last section composed of theour phases explained as below.

.1. Phase 1: Estimation of the number of submodels

The main difficulty in solving the mentioned problem is tostimate the numbers of submodels present in the given PWARXodel. Xie–Beni’s validity index for FCM clustering based approach

s used to estimate the number of submodel present in the openoop data points of the HDS. Since FCM partition clustering algo-ithm is unsupervised, it requires the number of cluster centers orartitions present in the data points. The choice of optimal num-er of cluster centers or submodels for partitioning the given data

oints depend upon clear separation between the cluster centers,he minimum volume of the cluster and maximum number of dataoints concentrated in the vicinity of the cluster center. FCM algo-ithm of partition clustering allows one piece of data point to belong

X system given in Example 1.

to two or more clusters with the degree of belongingness. Thismethod (developed by Dunn in 1973 and improved by Bezdek in1974) is frequently used in pattern recognition [23]. The fuzzy set isdistinct from a crisp set that it allows the elements to have a degreeof membership function for each cluster.

Objects to be clustered are denoted by z(t) = (x(t)1, . . ., z(t)p) ∈ Rp;t = 1, . . ., N, vectors in p-dimensional Euclidean space. Given a set ofN-observations in p-dimensional Euclidean space, FCM clusteringaims to partition it into k subsets (k < N) of cluster centers C suchthat it minimize the within-cluster Sum of Squares (WCSS). Clus-

ter centers are c(i) = (c(i)1, . . ., c(i)p)T; i = 1, . . ., k, where k is the

number of clusters and abbreviated symbol C = c(1), . . ., c(k) is usedfor complete collection of cluster centers. The FCM minimizes thefollowing objective function:

J(C) =k∑

i=1

N∑t=1

(�it)m||z(t) − c(i)||2 (5)

where, N is the number of data points, k is the number of clusterswith 2 ≤ k < N, z(t) is tth measure data point in p-dimension, �it isthe degree of membership function for data point z(t) in the ithcluster, m ∈ [1, ∞] is a weighting exponent which determines thedegree of fuzziness in the resulting clusters, ||·||2 is any second ordernorm expressing similarity between any measured data and thecluster center. For normalizing membership function linearly suchthat it sum is a 1 select value of m = 2. As one select value of m isclose to 1, then the data point close to the cluster center is givenmuch more weight than the other cluster centers and the algorithmis behave like to hard clustering. The membership matrix U = [�it]and its elements have membership values between 0 and 1.

The proposed algorithm for estimation of the number of sub-models given in Table 1. The algorithm starts with initializingthe value of fuzziness m, termination criteria of the algorithm

ε ∈ [0, 1] and maximum number of clusters in the given hybridsystem by cmax. Next step is to select k points as the initial num-ber of cluster centers (means) denoted by (c(1)j, . . ., c(k)j). Forthe randomly selected cluster centers compute the element of

A.K. Shah, D.M. Adhyaru / Applied Soft

Table 1Algorithm 1: Estimation of the number of submodels.

1 For given data points z(t) initialize value of cmax, m, ε.2 for k = 2 to cmax

3 initialize the random value of cluster centers (c(1)j , . . ., c(k)j)4 for i = 1 to k5 for t = 1 to N

6 Compute �jit

= (1/||z(t)−c(i)j ||2)1/m−1∑k

l=1(1/||z(t)−c(1)j ||2)

1/m−1

7 End8 End9 Repeat10 for i = 1 to k

11 Compute c(i)j+1 =∑N

t=1�jm

itz(t)∑N

t=1�jm

it

12 for t = 1 to N

13 Update �j+1it

= (1/||z(t)−c(i)j+1 ||2)1/m−1∑k

l=1(1/||z(t)−c(1)j+1 ||2)

1/m−1

14 End15 End

16 Until∥∥Uj+1 − Uj

∥∥ ≤ ε

17 Obtain final value of c and U = [�it]

18 Let comp(k, U) = 1N

k∑i=1

N∑t=1

(�it )4m||x(t) − c(i)||2

19 Let sep(k, U) = mini /= j

∥∥c(i) − c(j)∥∥2

20 Compute cent(k, U) = comp(k,U)sep(k,U)

mt

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t

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c

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s

s

21 End22 kopt = arg min(cent(k, U))23 Return s = kopt

embership matrix U for each data point by minimizing the objec-ive function given in (5) and solution is:

jit

= (1/||z(t) − c(i)j||2)1/m−1

∑kl=1(1/||z(t) − c(1)j||2)

1/m−1(6)

Update the vector elements of cluster center C using the mem-ership function elements by minimizing (5) and solution is:

(i)j+1 =∑N

t=1(�jit

)m

z(t)∑Nt=1(�j

it)m (7)

Repeat the solution of (6) and (7) until the algorithm satisfieshe termination criteria ||Uj+1 − Uj|| ≤ ε.

The data points nearer to the cluster center have highestembership function value and the membership function value

radually decreases as data points move away from the cluster cen-er. The good clustering algorithm should generate clusters withmall intracluster deviations, and large inter-cluster separations24]. The global compactness of FCM partitions of the input dataet can be defined as [25]

omp(k, U) = 1N

k∑i=1

N∑t=1

(�it)m||z(t) − c(i)||2 (8)

The above equation of the compactness matrix is combined withhe fuzzy nature of membership function which give more weighto data points which is nearer to the cluster center and less weighto far away data points. For the better partition between data pointsnd cluster center, value of comp(k,U) given in (8) must be smaller.he separation among the cluster centers for the classification ofhe data points can be calculated by

ep(k, U) = mini /= j

||c(i) − c(j)||2 (9)

For complete separation between the cluster centers value ofep(k,U) must be larger which also indicates the proper partitioning

Computing 25 (2014) 174–183 177

of the given data. The ratio of (8) and (9) defined as Xie–Beni’s indexcluster validity for fuzzy based clustering given in [26,27].

cent(k, U) = comp(k, U)sep(k, U)

(10)

A smaller value of cent(k,U) indicates a compact and clear par-titioning of the data points. The proposed algorithm is ended withthe number of submodels present in the PWARX systems indicatedby s at which the minimum value of cent(k,U).

For the given data points z(t) = {y(t), u(t − 1)}50t=1 of Example 1

above mentioned algorithm is applied with initial value of the max-imum number of cluster centers present in the system as cmax = 5,fuzzy ness function m = 2 and termination criteria ε = 10−5. Theresulting values of cent(k,U) for different values of the cluster centerfrom 2 ≤ k ≤ 5 shown in Fig. 2. The estimated optimal values of thecluster center by applying the proposed algorithm are three, whichis the exact number of submodels present in the PWARX model ofExample 1. This phase gives the number of submodel presents inthe hybrid system which is utilized as the number of cluster centerspresented in open loop data of the system for the classification ofdata points.

3.2. Phase 2: Classification of the measured data points

Once the number of submodels is obtained, then it is requiredto classify the measured data points according to discrete eventspresent in the HDS. In general clustering algorithms can be clas-sified into two categories: Hierarchical algorithms and Partitionalalgorithms. Partitional clustering algorithms identify the partitionthat optimizes (usually locally) clustering criteria. Partitional meth-ods have advantages in applications involving large data sets forwhich the construction of a dendrogram is computationally pro-hibitive [28].

Conventional partitional clustering algorithms use hard parti-tion derived from the classical set theory: contain �it membershipfunction value 0 or 1. These algorithms may trap in local min-ima due to poor initialization. The FCM clustering algorithm usessoft classification using shape membership functions whose val-ues between zero and one. The FCM partitional data classificationmethod separates the data points into clusters based on numberof cluster centers given initially. The FCM clustering algorithmis used to classify the measured data point with derived valueof the number of cluster centers from phase 1, and denoted ass = min {cent(k, U)}. The FCM clustering algorithm requires an ini-tial value of cluster center s, amount of fuzziness and terminationcriteria. The FCM clustering based techniques classify the datapoints based on their membership function, the highest mem-bership function indicates the data point belongs to that cluster.The FCM clustering method classifies the data point into s par-titions and also gives the location of the cluster center for eachpartition. The degree of belongingness of each data point to theirrespective each cluster center is also derived through membershipmatrix.

For the given Example 1 FCM clustering algorithm is appliedto data points denoted as z(t) = {y(t), u(t − 1)}50

t=1 with m = 2 andε = 10−5. The value used for number of cluster centers present inthe given example derived from phase 1 is s = 3. The open loop dataclassification result is shown in Fig. 3 along with location of finalcluster centers denoted as �, ♦ and © symbols.

3.3. Phase 3: Identification of PWARX submodel parameters

Once the data have been classified, linear regression can be usedto compute the final parameters of the submodels. However, pureleast squares are not the optimal choice since they are sensitive tooutliers that may be present because of classification errors. In this

178 A.K. Shah, D.M. Adhyaru / Applied Soft Computing 25 (2014) 174–183

rent n

peptctbc

Fig. 2. Value of s for diffe

hase, WLS given in [8] is used to reduce the effect of classificationrrors and outliers by defining a confidence measure on the dataoints. Here, FCM clustering algorithm having another advantagehat it classified the data according to the submodels and also give

luster centers for each submodels. Now, the cluster centers usedo calculate the fuzzy distance weight matrix for each data pointased on distance between data points and their respective clusterenter. The fuzzy distance based weight matrix denoted by W = [ωit]

Fig. 3. Classification of the da

umber of cluster centers.

and the element of given matrix is calculated using s-shape mem-bership function given by

ω = (1/||z(t) − c(i)||2)8∑ if z(t) ∈ c(i) (11)

it s

l=1(1/||z(t) − c(1)||2)8

where i = 1, . . ., s. Each element of the fuzzy distance weight matrixmust fulfill the following constraints:

ta points for Example 1.

d Soft Computing 25 (2014) 174–183 179

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Fig. 4. Submodel selection algorithm.

X1 =

X2 =

X3 =

A.K. Shah, D.M. Adhyaru / Applie

ωit ∈ [0, 1], 1 ≤ i ≤ k, 1 ≤ t ≤ N∑ki=1ωit = 1, 1 ≤ t ≤ N

0 <∑N

i=1ωit < N, 1 ≤ i ≤ k

As the data points at vicinity of the cluster center value ωit haveigher value nearer to one. The �tth submodel parameter is esti-ated based on the data point collected in the region ��t. The

arameter vector ��,t is computed for each submodel by minimizingf the objective function

(��t, �it) =∑

z(t) ∈��t

(W)||y(t) − �T�tϕ(x(t))||

2(12)

To minimize the objective function (12) optimally, take theerivative with respect to the parameter ��t

min��t

J = (W)(y − �T�tϕ)

T(y − �T

�tϕ)

∂J

∂��t

= 2(W)(y − �T�tϕ)

T(−ϕ) = 0

(−ϕ)T(y − �T�t

ϕ)(W)T = 0

T�t = (ϕTWϕ)

−1ϕTWy (13)

The above Eq. (13) gives the optimal value of the parameter vec-or for each submodel based on the weight, assign to the data pointsased on distance from the cluster center. To show the effectivenessf the above WLS based method for identification of PWARX modelarameters is applied on Example 1. For each data point calculateshe weight matrix and applied Eq. (13), then obtained parameterectors are

1[1.0257 1.9150], �2 = [−0.9630 − 0.0673], �3

= [0.9770 1.8756]

Above identified parameters providing a good approximation ofhe given PWARX system as in (4). Results are also comparable with8]. However, in [8] researcher has used double clustering algorithmnd region estimation for identified model. In present algorithm weroposed new submodel selection method given in next phase, inhich no need to estimate the boundary of hyperplanes still we

an obtained the result similar to [8].

.4. Phase 4: Submodel selection and validity computation

The purpose of this phase is to select the final model for the esti-ation of one step ahead prediction output and test the efficiency

nd precision of the identified PWARX model parameter. The onetep ahead prediction for output of HDS based on final submodel

y(t) =

⎧⎪⎪⎨⎪⎪⎩

[−0.4 1 1.5][y(t − 1)u(t − 1)1]T + ∈ (t), if x(t) ∈

[0.5 − 1 − 0.5][y(t − 1)u(t − 1)1]T + ∈ (t), if x(t) ∈

[−0.3 0.5 − 1.7][y(t − 1)u(t − 1)1]T + ∈ (t), if x(t) ∈

election procedure proposed in [24]. In this procedure predictedutput is calculated for all identified submodels from the measurednput and output data, but only one output is selected based onalidity function vj. The location of the cluster center resulting from

phase 2 used to calculate the validity function for the measured datapoints. The equation to find validity function is given by

vj = (1/||x(t) − c(j)||2)∑sl=1(1/||x(t) − c(1)j||2)

(14)

where 1 ≤ j ≤ s. The validity matrix is denoted as V = [vj]. The finalone step ahead predicted output out of all submodels output isselected based on highest validity matrix element given by

y(t) = yj(t) if j = index(max(v)) (15)

The complete algorithm to select the submodel for estimationof the final output is shown in Fig. 4. To measure the accuracy ofestimated output y(t) FIT criteria is given by [22]

FIT =(

1 − ||y(t) − y(t)||2||y(t) − mean(y(t))||2

)× 100% (16)

where y(t) is the actual output of the hybrid system and y(t) isthe estimated output.

For Example 1 proposed validation algorithm is applied to anew set of validation data generated with Gaussian measurementnoise variance of 0.06. Fig. 5 shows the result with actual outputas blue lines and the estimated output as red lines. The accuracyof the estimated output measured based on (16), which gives theFIT value 91.8205%. This indicates that the given validation algo-rithm performs very well despite the amount of noise present in theHDS.

Example 2. To verify the proposed identification algorithm weconsider the second single-input-single-output (SISO) PWARX sys-tem with more complexity given in [10,14] as

4y(t − 1) − u(t − 1) + 10 < 0

4y(t − 1) − u(t − 1) + 10≥0 and 5y(t − 1) + u(t − 1) − 6 ≤ 0

5y(t − 1) + u(t − 1) − 6 > 0(17)

Then consider PWARX system is a function of both input andoutput terms in parameters and partitions, which makes the iden-

tification and estimation task more complex. For the given PWARXsystem with na = 1, nb = 1 are given a priori and 200 output sam-ples are generated by applying input samples u(t) ∈ R and thenoise signal ∈(t) randomly according to the uniform distribution

180 A.K. Shah, D.M. Adhyaru / Applied Soft Computing 25 (2014) 174–183

stima

o{xi

ciHvt

Fig. 5. Actual output and e

n [−5,5] and [−0.2, 0.2], respectively. The 200 data points z(t) =y(t), x(t)}200

t=1 be generated for the given system in (17) where(t) = [y(t − 1)u(t − 1)]T. Generated data points are shown in Fig. 6(a)n the regression space R2.

For the given data points the number of discrete modes wasalculated through the algorithm given in phase 1 with assum-

ng that the maximum number of discrete mode presents in theDS are 5, fuzziness m = 2 and termination criteria ε = 10−5. Thealues of s with respect to different values of the cluster cen-er shown in Fig. 7, which give optimal value of the discrete

Fig. 6. Data classification in the regr

ted output for Example 1.

modes present in the system is 3. The estimated number of dis-crete modes matches with the actual number of submodels presentin the system. Once the number of submodels was derived, themeasured data points are then required to classify into differ-ent clusters. FCM clustering based data classification algorithmapplied to classify the data point to a number of cluster center as 3

derived from phase 1. The classification results shown in Fig. 6(b)with location of cluster centers for each cluster. The dash-dottedlines represent the boundaries of hyperplanes regions from left toright.

ession space R2 of Example 2.

A.K. Shah, D.M. Adhyaru / Applied Soft Computing 25 (2014) 174–183 181

ifferen

tugm

Fig. 7. Value of s index for d

Now, for each data point weights are calculated with respect toheir belonging cluster center derive from FCM clustering algorithmsing (11). Now, applying the weighted least square algorithmiven in phase 3 to calculate the parameter vectors for each sub-odels and obtained values of it given as

�1 = [ −0.3902 0.9586 1.6511 ]

�2 = [ 0.3467 −0.8094 −0.2953 ]

�3 = [ −0.2102 0.4425 −2.0834 ]

Fig. 8. Actual output and predic

t number of cluster centers.

Above identified parameters gives the best approximation of thegiven PWARX model in (17). To test the above-derived parametersthe validation algorithm given in phase 4 is applied for differentvalidation data set generated by the uniformly distributed noisesignal of [−0.2, 0.2]. The validation function is calculated from each

measured data points and based on that final estimated outputgiven with red line in Fig. 8. Using Eq. (16), the FIT for the esti-mated output is 74.426%. Results are also comparable with [10,14].However, in [10] and [14] researchers have used estimation of the

ted output for Example 2.

182 A.K. Shah, D.M. Adhyaru / Applied Soft

bno

EHcidi

a

y

n

e

y

if x

if x

if x

at

Fig. 9. Hammerstein process model.

oundary hyperplanes on the regressor space. In present algorithmo need to estimate the boundary of hyperplanes with assumptionf regressor space. Still we obtained the result similar to [10,14].

xample 3. The proposed identification approach is applied toammerstein model shown in Fig. 9, which is configured as a seriesonnection of static nonlinear blocks with a dynamic linear time-nvariant (LTI) system block. We consider SISO Hammerstein modelescribed in [14]. Let us consider static nonlinearity function ℵ(·)

s a static PWA saturation function given as

(t) = ℵ(u(t)) =

⎧⎨⎩

2, if u(t) > 2

u(t), if − 1 ≤ u(t) ≤ 2

−1, if u(t) < −1

The plant G is a LTI system whose dynamic is given by

(t) = −0.5y(t − 1) − 0.1y(t − 2) + a(t − 1)+ ∈ (t)

where ∈(t) is the zero means normally distributed measurementoise with variance of 0.04.

The Hammerstein process model shown in Fig. 9 can bexpressed as a PWARX model by

(t) =

⎧⎪⎪⎨⎪⎪⎩

[−0.5 − 0.1 0 2][y(t − 1)y(t − 2)u(t − 1)1]T + ∈ (t),

[−0.5 − 0.1 1 0][y(t − 1)y(t − 2)u(t − 1)1]T + ∈ (t),

[−0.5 − 0.1 0 − 1][y(t − 1)y(t − 2)u(t − 1)1]T + ∈ (t),

For the given Hammerstein model na = 2, nb = 1 are given a priorind 200 output samples are generated by applying normally dis-ributed input samples u(t) ∈ R with zero mean and variance of 4. For

Fig. 10. Actual output and predi

Computing 25 (2014) 174–183

(t) ∈ X1 = u(t − 1) − 2 > 0

(t) ∈ X2 = u(t − 1) − 2 ≤ 0 and u(t − 1) + 1 ≥ 0

(t) ∈ X3 = u(t − 1) + 1 < 0

the given data points z(t) = {y(t), x(t)}200t=1, we estimate the number

of submodels s with cmax = 5, fuzziness m = 2 and termination crite-ria ε = 10−5. The derived values of s with respect to different valuesof the cluster center using a proposed estimation algorithm is 3.For s = 3, the measure data points are classified using a FCM clus-tering algorithm. Using (11), weights are calculated with respect totheir belonging cluster center for each data point. The parametersfor each submodel are obtained using the proposed WLS algorithmgiven in phase 3 as

�1 = [ −0.4512 −0.0459 0.0737 1.6808 ]

�2 = [ −0.4672 −0.0743 0.7215 −0.0964 ]

�3 = [ −0.5113 −0.1073 −0.0006 −0.9736 ]

Above identified parameters gives the best approximation of thegiven PWARX model in (17). To test the above-derived parametersthe validation algorithm given in phase 4 is applied to differentdata set having measurement noise with variance of 0.05. The vali-dation function is calculated from each measured data points andbased on that final estimated data given with dotted lines in Fig. 10.Using (16), the FIT value of the estimated data is 72.826%. Thisexample indicates that the proposed approach is applicable to aHammerstein model.

4. Discussion and conclusions

In this paper, a new fuzzy weight matrix based method is usedfor the identification of hybrid system with unknown number ofdiscrete modes. This method is applicable to complex PWARX

model and noisy systems. In the literature [8,11,13,16], the numberof submodels has to be given a priori. In order to resolve this issue,

cted output for Example 3.

d Soft

Xo

[oaiparomobp

cmpnnof

anpPfip

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[Appl. Soft Comput. 12 (5) (2012) 1555–1565.

A.K. Shah, D.M. Adhyaru / Applie

ie–Benni based approach proposed in this paper which requiresnly maximum number of submodels present in the systems.

In work done by Ferrari-Trecate et al. [8] and Bemporad et al.10], the classification of given input and output data points basedn double clustering process. In order to reduce classification errorsnd the effect of outliers, the FCM clustering algorithm is usedn this paper to classify the data points. Classification of the dataoints using FCM also gives the location of cluster centers, whichre used for the identification and prediction of the output. In [15],egression vector obtained as nonlinear functions of past input andutput. As in [10,14,15] a parameter of PWARX system is esti-ated using pure least squares method which is sensitive from

utliers and misclassification. For this reason, fuzzy weight matrixased weighted least squares parameter identification method isroposed for linear and affine regression vector.

Finally, the proposed method has been designed such that itan validate the identified parameters of PWARX models usingodel selection criteria. The main contribution of the paper is to

ropose methodology for PWARX parameter identification, whereumber of models is not necessary to know, switching surfaceseed not to determine and the system dynamics are discontinu-us. The suggested methodology has been applied and validatedor two different PWARX systems and a Hammerstein model.

An interesting issue to decide the order of the PWARX modelnd implementations of the proposed method to some hybrid andonlinear systems such as in [29] are still open. Moreover, otherossible future topic of research is to design the inputs for theWARX models such that not only all the affine dynamics are suf-ciently excited, but also accurate shaping of the boundaries isossible.

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[2] P.J. Antsaklis, A brief introduction to the theory and applications of hybrid sys-tems, introductory article for the special issue on hybrid systems: theory andapplications, Proc. IEEE 88 (7) (2000) 879–887.

[3] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability ofpiecewise affine and hybrid systems, IEEE Trans. Autom. Control 45 (10) (2000)1864–1876.

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