Fuzzy Clustering for the Identification of Takagi ... - Semantic Scholar

Fuzzy Clustering for the Identification of Takagi-Sugeno Fuzzy Models of MIMO Dynamical Systems Janos Abonyi and Ferenc Szeifert University of Veszprem, Department of Process Engineering P.O. Box 158, H-8201, Hungary e-mail: [email protected]

http://www.fmt.vein.hu/softcomp

Abstract The identification of nonlinear multi-input multi-output (MIMO) processes is important and challenging problem. Fuzzy systems have been effectively used to identify complex nonlinear dynamical systems, but mostly single-input single-output systems are considered. This paper presents a compact Takagi-Sugeno fuzzy model that can be effectively used to represent MIMO dynamical systems. For the identification of this model a fuzzy clustering algorithm is proposed. This new approach is demonstrated by means of the identification of a high-purity distillation column, where the results are compared to results obtained by standard linear and other advanced fuzzy clustering based identification tools.

1

Introduction

Fuzzy model identification is an effective tool for the approximation of nonlinear dynamical systems on the basis of measured data [16]. Among the different fuzzy modeling techniques, the Takagi-Sugeno (TS) model [30] has attracted most attention. This model consists of if–then rules with fuzzy antecedents and mathematical functions in the consequent part. The fuzzy sets partition the input space into a number of fuzzy regions, while the consequent functions describe the system’s behavior in these regions. The construction of a TS model is usually done in two steps. In the first step, the fuzzy sets (membership functions) in the rule antecedents are determined. This can be done manually, using knowledge 1

of the process, or by some data-driven techniques. In the second step, the parameters of the consequent functions are estimated. As these functions are usually chosen to be linear in their parameters, standard linear least-squares methods can be applied. The bottleneck of the construction procedure is the identification of the antecedent membership functions, which is a nonlinear optimization problem. Typically, gradient-decent neuro-fuzzy optimization techniques are used [18], with all the inherent drawbacks of gradient-descent methods. An alternative solution are gradient-free nonlinear optimization algorithms [20, 27]. Unfortunately, the severe computational requirements of these Genetic Algorithm based approaches limit their applicability as a rapid model-development tool. Fuzzy clustering in the Cartesian product-space of the inputs and outputs is another tool that has been quite extensively used to obtain the antecedent membership functions [6, 4]. Attractive features of this approach are the simultaneous identification of the antecedent membership functions along with the consequent local linear models and the implicit regularization [21]. Unfortunately, clustering based methods cannot be applied directly for the identification of multivariable processes. Since, only a set of MISO systems can be identified [5], this approach results in complex rulebase where the interaction between the MISO channels is not given explicitly. To avoid this problem in this paper another approach is followed. The nonlinear multivariable process is represented by a MIMO fuzzy model that consists of local linear MIMO ARX models and a new clustering algorithm is proposed that is able to directly generate such MIMO fuzzy models. Hence, the originality of the paper is the development of a new clustering algorithm that can be directly used for the identification of interpretable MIMO TS fuzzy models. For single-output systems it has been shown that the Gath-Geva (GG) clustering allows the identification of TS fuzzy models with exponential membership functions defined on the linear combination of input variables [23]. However, the use of transformed input variables in most cases destroys the transparency of the obtained model. To avoid this problem, in [3] we proposed a new type of clustering algorithm that can be easily used to obtain interpretable TS fuzzy models. This new approach is extended in this paper to MIMO processes and has been applied to the identification of a nonlinear simulated distillation process, which model is a popular example nonlinear identification [10, 11, 26, 29]. The results are compared with results obtained by standard linear and advanced fuzzy clustering based identification tools. The paper is organized as follows. In Section 2, the applied TS fuzzy model is presented. Section 3 describes how local MIMO ARX models can be identified. In Section 4, the new clustering algorithm is proposed. Section 5 presents the application example. Conclusions are given in Section 6. 2

2 Takagi-Sugeno Fuzzy Model of a MIMO process Multiple-input, multiple-output (MIMO) dynamic processes can be represented by the following nonlinear vector function: y(k + 1) = f (y(k), . . . , y(k − na + 1), u(k − nd ), . . . , u(k − nb − nd + 1))

(1)

 T where, f represents the nonlinear model, y = y1 , . . . , yny is an ny dimensional output vector, u =

[u1 , . . . , unu ]T is an nu dimensional input vector, na and nb are maximum lags considered for the outputs and inputs, respectively, and nd is the minimum discrete dead time. While it may not be possible to find a model that is universally applicable to describe the unknown f (.) system, it would certainly be worthwhile to build local linear models for specific operating points of the process. The modeling framework that is based on combining a number of local models, where each local model has a predefined operating region in which the model is the local model is valid is called operating regime based model [25]. This model is formulated as:   nb na c X X X B ij u(k − j − nd + 1) + ci  Aij y(k − j + 1) + βi (x(k))  y(k + 1) = i=1

(2)

j=1

j=1

where the βi (x(k)) function describes the operating regime of the i = 1, . . . , c-th local linear ARX model, where x = [x1 , . . . , xn ]T is a ”scheduling” vector, which is usually a subset of the previous process inputs and outputs,  x(k) = y1 (k), . . . , y1 (k − na + 1), . . . , yny (k − na + 1), u1 (k − nd ), . . . , unu (k − nb − nd + 1)

(3)

The local models are defined by the θ i = {Aij , B j , ci } parameter set. As na and nb denote the maximum lags considered for the previous outputs and inputs, and n d is the minimum discrete dead time, the lags considered for the separate input-output channels can be handled by zeroing the appropriate elements of the Aij and B ij matrices. If there is no a priori knowledge about the order of the nonlinear system, the model orders and the time delays can be directly estimated from input-output data [9]. The main advantage of this framework is its transparency, because the operating regimes of the local models can be represented by fuzzy sets [7]. This representation is appealing, since many systems change behaviors smoothly as a function of the operating point, and the soft transition between the regimes introduced by the fuzzy set representation captures this feature in an elegant fashion. Hence, the entire global model can be conveniently represented by Takagi-Sugeno fuzzy rules [30]. This MIMO 3

Nonlinear Autoregressive with eXogenous Input (NARX) Takagi-Sugeno fuzzy model is formulated by rules as, Ri :

If x1 is Ai,1 and . . . and xn is Ai,n then nb na X X y i (k + 1) = Aij y(k − j + 1) + B ji1 ,...,in u(k − j − nd + 1) + ci , [wi ] j=1

(4)

j=1

where Ai,j (xj ) is the ith antecedent fuzzy set for the jth input and wi = [0, 1] is the weight of the rule that represents the desired impact of the rule. The value of w i is often chosen by the designer of the fuzzy system based on his or her belief in the goodness and accuracy of the i-th rule. When such knowledge is not available wi is set as wi = 1, ∀ i. The one-step-ahead prediction of the MIMO fuzzy model, y(k + 1), is inferred by computing the weighted average of the output of the consequent multivariable models, c X

βi (x(k))y i (k + 1)

(5)

where c is the number of the rules, and βi is the weight of the ith rule, Q wi nj=1 Aj,i (xj ) βi (x(k)) = Pc Qn i wi j=1 Aj,i (xj )

(6)

y(k + 1) =

i=1

To represent the Ai,j (xj ) fuzzy set, in this paper Gaussian membership function is used ! 1 (xj − vi,j )2 Ai,j (xj ) = exp − 2 2 σi,j

(7)

2 the variance of the Gaussian function. where vi,j represents the center and σi,j

The proposed fuzzy model can be seen as a multivariable linear parameter varying system model (LPV), where at the x operating point, the fuzzy model represents the following LTI model y(k + 1) =

na X

Aj (x(k))y(k − j + 1) +

nb X

B j (x(k))u(k − j − nd + 1) + c(x(k))

(8)

j=1

j=1

with Aj (x(k)) =

c X

βi (x(k))Aij ,

i=1

B j (x(k)) =

c X

βi (x(k))B ij ,

i=1

c(x(k)) =

c X

βi (x(k))ci

(9)

i=1

The aim of the remaining part of the paper is to propose a new clustering-based technique for the identification of the model presented above. 4

3 Identification of the Local MIMO ARX Models The Takagi-Sugeno fuzzy model interpolates between local linear models. The model obtained by interpolating the local models can be interpreted as the local dynamic behavior of the entire fuzzy model. This locally interpreted model is not identical to the model obtained by Taylor-series linearization of the fuzzy model at the considered equilibrium. In [1] the origin of this difference with regard to the applied identification method have been analyzed. The results show that local (weighted least squares for each rule) identification of the rule consequent parameters forces the local linear models to fit the system separately and locally, resulting rule consequences that are local linearizations of the nonlinear system [4]. Hence, fuzzy models obtained by local identification (weighted least squares for each rule) typically yields a poor steady-state representation and the model can only be locally interpreted, the model can only be used as a look-up table of linearizations [24]. On the contrary, a fuzzy model obtained by global identification (one least-square solution for the entire model) can result in a qualitatively bad local interpretation of the gain even though approximates the real process well. Therefore, this model can only be used for prediction or local linearization through Taylor expansion. Since in this paper the fuzzy model is interpreted as an LPV system (8), the parameters of the rule consequences are estimated separately by dividing the identification task into c weighted least-squares problems. The fuzzy model can be formulated in the following compact form: y(k + 1)T =

c X i=1

where φ(k) is the regressor vector,

  βi (x(k)) φ(k) I 1×ny θ Ti + e(k),

φ(k) = [y(k)T , . . . , y(k − ny + 1)T , u(k − nTd ), . . . , u(k − nu − nd + 1)T ]

(10)

(11)

θ i is the parameter matrix of the i-th local model (rule), θ i = [Ai1 , . . . , Ainy , B i1 , . . . , B inu , ci ] and e(k) is a zero mean white noise sequence. The output of this model is linear in the elements of the Aij , B ij consequent matrices and the ci offset vector. Therefore, these parameters can be estimated from input-output process data by linear least-squares techniques. The N identification data pairs and the truth values of the fuzzy rules are arranged in the following matrices.  T Φ = φT (1)|φT (2)| · · · |φN (N ) 5

(12)

Y = [y(2)|y(3)| · · · |y(N + 1)]T  

 βi (1) 0 ··· 0      0 βi (2) · · · 0  Bi =    . .. .. ..  .. . . .     0 0 · · · βi (N )

By using this notation, the weighted least squares solution of θ i is:

              

 −1 T θ i = ΦT Bi Φ Φ Bi Y .

(13)

(14)

(15)

As this method forces the local linear models to fit the data locally, it does not give an optimal fuzzy model in terms of a minimal global prediction error, but it ensures that the fuzzy model is interpretable as a Linear Parameter Varying (LPV) system.

4 Clustering for the Identification of MIMO Processes 4.1

Problem Formulation

The previous section has showed how the consequent part of the TS model can be identified by weighed least squares method when the antecedent membership functions (rule-weights) are given. The bottleneck of the identification of TS models is the data-driven identification of the antecedent part of the TS model that requires nonlinear optimization. Hence, for this purpose often heuristic approaches, like fuzzy clustering methods are applied. The objective of clustering is to partition the identification data Z into c clusters, where the available identification data, Z = [Φ, Y] formed from a regression data matrix Φ and a regression vector Y . This means, each observation consists of (n o × na + nu × nb ) + no measured variables, grouped into a row vector zk = [φ(k) y(k + 1)T ], where the k subscript denotes the kth row of the Z matrix. For sake of simplicity and space, in the following the z k = [φk yk ] notation is used according to the partition of zk to regressors and regressand. Furthermore, the transpose of the x(k) vector containing the scheduling variables (that is the subset of φ k ) will be denoted by xk . The clustering obtains the fuzzy partition of the Z data. The fuzzy partition is represented by the U = [µi,k ]c×N matrix, where the µi,k element of the matrix represents the degree of membership, how the zk observation is in the cluster ı = 1, . . . , c. 6

Different cluster shapes can be obtained with with different kinds of clustering algorithm based on different prototype, e.g., point or linear varieties (FCV) or with different distance measure. Mostly, the Gustafson-Kessel clustering algorithm is applied to identify TS models [14]. A drawback of the this algorithm is that only clusters with equal volumes can be found and the resulted clusters cannot be directly used to form membership functions. The Gath and Geva clustering (GG) algorithm does not suffer from these problems. Gath and Geva interpret the data as normally distributed random variables and assume that the normal (Gaussian) distribution with expected value v i and covariance matrix Fi is chosen for generating the datum with a priori probability p(η i ) [12]. In [23, 3] it has been shown how antecedent fuzzy sets and the corresponding consequent parameters of the TS model can be derived form clusters obtained by the GG algorithm. To preserve the partitioning of the antecedent space, linearly transformed input variables can be used in the model. This may, however, complicate the interpretation of the rules. To form an easily interpretable model that does not use the transformed input variables, a new clustering algorithm is proposed, based on the Expectation Maximization (EM) identification of Gaussian mixture models [3]. In this paper this new technique is extended for the identification of MIMO fuzzy models, where each cluster contains an input distribution, a local model and an output distribution. p(φ, y) =

c X

p(φ, y, ηi ) =

=

(16)

p(φ, y|ηi )p(ηi )

i=1

i=1

c X

c X

p(y|φ, ηi )p(x|ηi )p(ηi )

i=1

The input distribution is parameterized as an unconditioned Gaussian [13] and defines the domain of influence of a cluster similarly to multivariate membership functions −1 x exp − 12 (x − vix )T (Fxx i ) (x − vi ) p p(x|ηi ) = n (2π) 2 |Fxx i |




(17)

while the output distribution is taken to be

∗ T −1 exp −(y − φ∗ θ Ti )T (Fyy i ) (y − φ θ i ) p(y|x, ηi ) = no p (2π) 2 |Fyy i |

where the Fyy and Fxx covariance matrices are calculated as N P (xk − vix ) (xk − vix )T p(ηi |zk ) k=1 Fxx i = N P p(ηi |z k )




(18)

(19)

k=1

Fyy i

=

N P

k=1

yk − φk θ Ti




N P

yk − φk θ Ti p(ηi |z k )

k=1

7


T

p(ηi |z k ) (20)

where φ∗ = [φ, I 1×no ] and φ∗k = [φk , I 1×no ]. When the simplicity and the interpretability of the model is important, the F xx cluster weighted covariance matrix is reduced to its diagonal elements similarly to the simplified axis-parallel version of the Gath-Geva clustering algorithm [17] p(x|ηi ) =

n Y

j=1

1 (xj − vi,j )2 exp − 2 2 σi,j

1 q 2 2πσi,j

!

(21)

yy The identification of the model means the determination of the η i = {p(ηi ), vix , Fxx i , θ i , Fi } pa-

rameters of the clusters. Bellow, the EM identification of the model is presented that is re-formulated in the form of Gath-Geva fuzzy clustering.

4.2

Clustering Algorithm

The clustering is based on the minimization of the sum of weighted squared distances between the data points,zk and the cluster prototypes, ηi . J(Z, U, η) =

c X N X

2 (µi,k ) Di,k (zk , ηi )

(22)

i=1 k=1

where the distance measure is consists of two terms and inversely proportional to the probability of the data. The first term is based on the geometrical distance between the cluster centers and the scheduling vector, x, while the second is based on the performance of the local linear models: 1 2 (z , η ) Di,k k i

(23)

= p(ηi )p(x|ηi )p(y|φ, ηi ) = wi

n Y

j=1

1 (xj,k − vi,j )2 exp − 2 2 σi,j

!

∗ T −1 exp −(y − φ∗k θ Ti )T (Fyy i ) (y − φk θ i ) p · no (2π) 2 |Fyy i |




and the µi,k = p(ηi |φ) weight denotes the membership that the zk input-output data is generated by the ith cluster p(ηi |φ) =

c X p(φ|ηi )p(ηi ) i=1

(24)

p(φ)

To get a fuzzy partitioning space, the membership values have to satisfy the following conditions: U ∈R

c×N

|µi,k ∈ [, ], ∀i, k;

c X

µi,k = , ∀k; 
0. Initialize the partition matrix (randomly), U = [µi,k ]c×N . Repeat for l = 1, 2, . . . Step 1 Calculate the parameters of the clusters • Centers of the membership functions:

x (l)

vi

=

N P

(l−1)

µi,k xk

k=1 N P

k=1

(26)

,1≤i≤c (l−1) µi,k

• Standard deviation of the Gaussian membership function:

2 (l) σi,j

=

N P

(l−1)

k=1

µi,k (xj,k − vi,j )2 N P

k=1

,1≤i≤c

(27)

(l−1) µi,k

• Parameters of the local models (see (15), where the weights in the B i matrix are βi (k) = (l−1)

µi,k ) • Covariance of the modeling errors of the local models (20). • A priori probability of the cluster p(ηi ) =

N 1 X µi,k N

(28)

k=1

• Weight (impact) of the rules: wi = p(ηi )

n Y

1 q 2 2πσi,j j=1

(29)

2 (z , η ) by (23). Step 2 Compute the distance measure Di,k k i

Step 3 Update the partition matrix 1

(l)

µi,k = Pc

j=1 (Di,k (zk , ηi )/Dj,k (zk , ηj ))

2/(m−1)

,

1 ≤ i ≤ c, 1 ≤ k ≤ N .

until ||U(l) − U(l−1) || < . 9

(30)

The remainder of this section is concerned with the theoretical convergence properties of the proposed algorithm. Since, this is algorithm is the member of the family of algorithms discussed in [15], the the following discussion is based on the results of Hathaway and Bezdek [15]. Using LaGrange multiplier theory, it is easily shown that for Di,k (zk , ηi ) ≥ 0, (30) defines U(l+) to be a global minimizer of the restricted cost function (22). From this it follows that the proposed iterative algorithm is a special case of grouped coordinate minimization, and the general convergence theory from [] can be applied for reasonable choices of Di,k (zk , ηi ) to shown that any limit point of an iteration sequence will be a minimizer, or at worst a saddle point of the cost function J. The local convergence result in [19] states that if the distance measures Di,k (zk , ηi ) are sufficiently smooth and a standard convexity holds at a minimizer (U∗ , η ∗ ) of J, then any iteration sequence started with U() sufficiently close to U∗ will converge to (U∗ , η ∗ ). Furthermore, the rate of convergence of the sequence will be w-linear. This means that there is a norm k ∗ kand constants 0 < γ < 1 and l0 > 0, such that for all l ≥ l0 , the sequence of errors {el } = {k(Ul , η l ) − (U∗ , η ∗ )k} satisfies the inequality el+1 < γel .

5 Example: Identification of a Distillation Column The examined process is a first-principle model of a binary distillation column (see Figure 1). The column has 39 trays, a reboyler and a condenser. The modeling assumptions are equilibrium on all trays, total condenser, constant molar flows, no vapor holdup, linearised liquid dynamic. The simulated system covers the most important effects for the dynamic of a real distillation column. The studied column operates in LV configuration [28] with two manipulated variables (reflux and boilup rate, u 1 and u2 ) and two controlled variables (top and bottom impurities, y1 y2 ). Further details of the simulation model are described in [28]. A closed-loop identification experiment has been performed by two simple PI controllers. The sampling time is 2 minutes and 10000 samples are collected for the experiment. The identification data is divided into four sections of 2500 samples such that each section corresponds to data gathered around one operating point (see Figure 2). The first 2000 samples of each section is used form model estimation while the last 500 samples are used for validation. The order of the local models is chosen to be na = nb = 2. The process was identified without time delay, nd = 0. Four different models were identified: (1) Linear: MIMO ARX model, (2) FMID: combination of two MISO TS models obtained by Fuzzy Model Identification Toolbox [4] (two times four rules), (3)

10

Figure 1. Schematic diagram of the distillation column. FIX: grid-type MIMO TS model with fixed Gaussian membership functions (four rules), (4) Proposed: MIMO fuzzy model where the Gaussian membership functions and the local models are identified by the proposed method (four rules). As the process gain varies in direct proportion to the concentrations, the fuzzy sets - the operating regions of the local models - are defined on the domain of the product impurities x(k) = [y 1 (k), y2 (k)]. This results in the following MIMO TS fuzzy model structure: Ri

:

y i (k + 1) =

If y1 (k) is Ai and y2 (k) is Ai then 2 X i B Ij u(k − j + 1) + ci Aj y(k − j + 1) +

2 X

(31)

j=1

j=1

Figure 2 shows the simulated output of the proposed MIMO fuzzy model model, whose membership functions are depicted in Figure 3. The results are obtained by the free run simulation of the model where the predicted outputs are fed back to the model as inputs. This free run simulation is a very rigorous test of the predictive power of the model, because in this way small errors can accumulate to major ones. Hence, the presented results are quite promising and indicates the usability of the model in model-based control. Table 1 compares the performance of this model to other models. As this table shows, the proposed model gives superior performance. Only the performance of the model obtained by the FMID toolbox is 11

comparable. Although, this model is built up from two MISO fuzzy models, hence this model has two times four rules and 2 × 4 × 4 membership functions that suggests that this model is unnecessarily more complex than the proposed MIMO TS model. Table 1. Prediction errors obtained by different models. (sum of square error (SSE). Model

SSEy1

SSEy2

Linear

11.4e−3

9.5e−3

FMID

6.3e−3

1.7e−3

Fixed

10.3e−3

9.1e−3

Proposed

3.9e−3

3.3e−3

0.025

Top impurity

0.02 0.015 0.01 0.005 0 −0.005 −0.01

0

0.2

0.4

0.6

0.8

1 Time [min]

1.2

1 Time [min]

1.2

1.4

1.6

1.8

2 4

x 10

−3

20

x 10

Bottom impurity

15 10 5 0 −5

0

0.2

0.4

0.6

0.8

1.4

1.6

1.8

2 4

x 10

Figure 2. Measured (–) and predicted (- -) process outputs. The high-purity binary distillation column is a nonlinear process, because as the demanded product purities increase, the gains of the process are decreasing. As the arrangement of the membership functions depicted in Fig. 3 shows, this phenomena is correctly detected by the clustering algorithm. In this example, the number of rules was determined manually. However, the identification of the number of the clusters (rules) is an important task that is our current research, along with with the selection of relevant input variables [2]. Similarly to the algorithm proposed by Gath and Geva [12] and 12

1

membership

0.8 0.6 0.4 0.2 0

0

0.005

0.01

Top impurity

0.015

0.02

0.025

1

membership

0.8 0.6 0.4 0.2 0

0

0.002

0.004

0.006

0.008 0.01 Bottom impurity

0.012

0.014

0.016

0.018

Figure 3. Operating regimes (fuzzy sets) obtained by the proposed fuzzy clustering. Bezdek [8], the application of cluster validity measures like fuzzy hypervolume and density, or cluster flatness [22] can be applied for this purpose.

6 Conclusions In this paper the identification of nonlinear multiple-input, multiple-output systems is discussed. A fuzzy model structure has been proposed, where the multivariable process is represented by a MIMO fuzzy model that consists of local linear MIMO ARX models. The local models and the antecedent part of the fuzzy model are identified by a new clustering algorithm, which clustering algorithm is an effective approximation of a nonlinear optimization problem related to the identification of fuzzy systems. The approach is demonstrated by means of the identification of a high-purity distillation column. The results show that the proposed clustering based identification obtains compact and accurate models for MIMO processes.

Acknowledgement This work was supported by the Hungarian Ministry of Education (FKFP-0073/2001) and the Hungarian Science Foundation (OTKA TO37600). J. Abonyi is grateful for the Janos Bolyai Fellowship of the Hungarian Academy of Science. 13

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