1 Introduction

Takagi–Sugeno models (Takagi and Sugeno 1985) are a powerful practical engineering tool for modeling and control of complex systems. Due to the fuzzy regions (clusters), the nonlinear system is decomposed into a multi-model structure consisting of linear models (Johanson and Murray-Smith 1997). This enables the T–S fuzzy model to approximate virtually any nonlinear system within a required accuracy, provided that enough regions are given (Hwang and Chang 2007).

There are a number of off-line methods for identification of fuzzy model (Kukolj and Levi 2004; Kim et al. 1997; Shing and Jang 1993; Kasabov 1996). In recent years there has been an increased interest in an on-line nonlinear model identification that employ fuzzy logic and its combination with the neural networks (Angelov and Filev 2004; Kasabov and Song 2002; Kim et al. 2005; Leng et al. 2004; Kasabov 2001, 1998a, b; Qiao and Wang 2008; Xu et al. 1993; Hai-Jun et al. 2006; Juang and Lin 1998; Wang et al. 2008; Lin et al. 2001; Wu and Er 2000; Wu et al. 2001; Lin 1995; Tzafestas and Zikidis 2001; Azeem et al. 2003). The methods employ different clustering algorithms for example Widrow–Hoffs LMS algorithm, evolving clustering method (Kasabov and Song 2002), N-first nearest neighborhood heuristic, error reducing and structure-evolving mechanism, modified mountain clustering (Azeem et al. 1999), Kohen’s feature maps and other algorithms. The algorithms use mostly Gaussian or triangular membership functions.

Probably one of the most promising methods for on-line identification of the Takagi–Sugeno fuzzy model is the evolving Takagi–Sugeno model (eTS) (Angelov and Filev 2004). The method is a further development of the evolving rule-based models (eR) (Angelov 2002). It uses the informative potential of a new data sample to update the rule base. To estimate the clusters the recursive clustering algorithm (Angelov 2004) based on subtractive clustering (Chiu 1994) is used. This is an improved version of mountain clustering (Yager and Filev 1993). The width of the membership functions in the original eTS method was fixed, but the method was improved so that the width is also automatically set [exTS (Angelov and Zhou 2006) and eTS+ (Angelov 2010)]. The improved version can therefore detect different shapes of clusters without inverting the fuzzy covariance matrix. The exTS uses local scatter measurement over the input space that resembles the variance.

Recently an extension of the Gustafson–Kessel clustering algorithm for evolving data stream clustering was published as a chapter in a book (Filev and Georgieva 2010). The algorithm is also based on a GK off-line clustering algorithm as ours, but the adaptation of centers and the calculation of the fuzzy covariance matrix are different. A recursive version of Gath-Geva clustering (Soleimani-B et al. 2010) algorithm was also published recently.

The idea behind the proposed algorithm is similar as in (Angelov and Filev 2004). The difference is that they based the algorithm on an off-line subtractive clustering algorithm and the proposed one is based on a Gustafson–Kessel off-line algorithm. Both algorithm use the Gaussian membership function, but eTS is able to add new cluster centers as apposed to the proposed, where the number of clusters are fixed.

The proposed algorithm is developed for on-line identification and adaptation of the fuzzy model. It is developed to be used in a fuzzy predictive functional controller (Škrjanc and Matko 2000) to control nonlinear dynamic systems, where the nonlinearity is time dependent. It can also be used for a fault detection of a non-linear process, where the dynamic of the system changes, such as waste water treatment process. For the basis we chose the GK clustering, because is well known, widely used for constructing fuzzy models that are used for model predictive control and gives good results. The recursive equations are derived directly from equations of the off-line algorithm.

The paper is organized in the following fashion. First the recursive identification method is developed based on GK and least squares algorithm. The use of the method is then demonstrated on an example of monitoring of waste water treatment process and on the control of the pH process. Then the results for Mackey–Glass (M–G) time series prediction are given. At the end some conclusions are made.

2 Fuzzy c-means and Gustafson–Kessel clustering

Fuzzy c-means (FCM) is probably one of the most known off-line clustering algorithms (Bezdek 1981). It is based on a minimization of the fuzzy c-means objective function. From the minimization problem the equations for the membership degrees (μ i ) and cluster centers (v i ) are obtained:

$$ \mu_{i}(k)=\left(d_{ik}^{2} \sum_{j=1}^{c} \left({\frac{1} {d_{jk}^2}}\right)^{{\frac{1}{\eta-1}}}\right)^{-1} $$
(1)
$$ v_i={\frac{\sum_{k=1}^{n} \mu_{i}^{\eta}(k) x(k)}{\sum_{k=1}^{n} \mu_{i}^{\eta}(k)}}, $$
(2)

where d ik is the Euclidian distance between the observation x(k) (data vector \(x(k)=[x_1(k),\ldots,x_m(k)]\)) and the cluster centroid v i . The FCM algorithm is based on assumption that the clusters are spherical shaped. In number of real problems and dynamical systems the clusters are of different shapes and with different orientations in the data space. To detect different geometrical shapes in data sets Gustafson and Kessel (1979) extended the FCM algorithm by employing an adaptive distance norm for each cluster. Each ith cluster has its own norm-inducing matrix A i which affects the distance norm. Euclidan norm in the FCM is replaced with Mahalanobis distance norm given as follows:

$$ D^2_{ik,A_i}=(x(k)-v_i)^T A_i (x(k)-v_i), $$
(3)

where A i is obtained from fuzzy covariance matrix F i defined as:

$$ F_{i}={\frac{\sum_{k=1}^n \mu_{i}^{\eta}(k)\left(x(k)-v_i \right) \left(x(k)-v_i \right)^T}{\sum_{k=1}^n \mu_{i}^{\eta}(k)}}, $$
(4)
$$ A_i=[\rho_i \ det(F_i)]^{1/p} \ F_i^{-1}, $$
(5)

where p is the number of measured variables and ρ i is a cluster volume, which is usually set to 1 for each cluster. This allows the algorithm to find clusters of approximately equal volumes.

3 Recursive GK clustering

When the behavior of the process that generates the observed data changes during the time, the clustering should be done recursively to obtain the clusters that describe the current behavior. The recursive method can also be used for constructing the fuzzy model on-line.

3.1 The recursive center calculation

To develop the recursive fuzzy clustering algorithm we will first define the cluster centroid vector \(v_i^T=\left[ v_{i1},\ldots, v_{i_m}\right]\) according to the current observation, i.e., the weighted mean of the data according to the current membership degrees. This introduces the notation v i (r), which means the cluster centroid at the time instant r that is obtained by weighting with the current membership degrees. The cluster centroid in the next observation is denoted as

$$ v_{i}(r+1)={\frac{\sum_{k=1}^{r+1} \mu_{i}^{\eta}(k) x(k)} {\sum_{k=1}^{r+1} \mu_{i}^{\eta}(k)}}= {\frac{\sum_{k=1}^{r} \mu_{i}^{\eta}(k) x(k)+ \mu_{i}^{\eta}(r+1) x(r+1)} {\sum_{k=1}^{r} \mu_{i}^{\eta}(k)+\mu_{i}^{\eta}(r+1)}} $$
(6)

where \(\mu_{i}(k), \; k=1,\ldots,r+1\) denotes the membership degree of the observation vector \(x(k)^T=\left[x_{1}(k),\ldots,x_{m}(k) \right],\) k = 1,..., r + 1 to the cluster i at the time instant k. Introducing the relation between the old cluster centroid and a new one as follows:

$$ v_{i}(r+1)=v_{i}(r)+\Updelta v_{i}(r+1) $$
(7)

and taking into account Eq. 6 the Eq. 11 is obtained:

$$ v_{i}(r+1)={\frac{\sum_{k=1}^{r} \mu_{i}^{\eta}(k) x(k)/\sum_{k=1}^{r} \mu_{i}^{\eta}(k)}{\sum_{k=1}^{r+1} \mu_{i}^{\eta}(k)/\sum_{k=1}^{r} \mu_{i}^{\eta}(k)}} +{\frac{\mu_{i}^{\eta}(r+1) x(r+1)}{\sum_{k=1}^{r} \mu_{i}^{\eta}(k)+\mu_{i}^{\eta}(r+1)}} $$
(8)
$$ v_{i}(r+1)={\frac{v_{i}(k) \sum_{k=1}^{r} \mu_{i}^{\eta}(k)} {\sum_{k=1}^{r+1} \mu_{i}^{\eta}(k)}} +{\frac{\mu_{i}^{\eta}(r+1) x(r+1)}{\sum_{k=1}^{r} \mu_{i}^{\eta}(k)+\mu_{i}^{\eta}(r+1)}} $$
(9)
$$ v_{i}(r+1)=v_{i}(k)-{\frac{v_{i}(k) \mu_{i}^{\eta}(r+1)} {\sum_{k=1}^{r+1} \mu_{i}^{\eta}(k)}} +{\frac{\mu_{i}^{\eta}(r+1) x(r+1)}{\sum_{k=1}^{r} \mu_{i}^{\eta}(k)+\mu_{i}^{\eta}(r+1)}} $$
(10)
$$ \Updelta v_{i}(r+1)={\frac{\mu_{i}^{\eta}(r+1) \left(x(r+1)-v_{i}(r)\right)}{\sum_{k=1}^r \mu_{i}^{\eta}(k)+ \mu_{i}^{\eta}(r+1)}} $$
(11)

The term in the denominator of equation can be denoted as \({s_{i}(k+1) \in {\mathbb{R}}^{c}}\) and calculated as:

$$ s_{i}(r+1)=s_{i}(r) + \mu_{i}^{\eta}(r+1) $$
(12)

where s i (r) is defined as follows

$$ s_{i}(r)=\sum_{k=1}^r \mu_{i}^{\eta}(k) $$
(13)

Introducing the forgetting factor, Eq. 12 can be rewritten as:

$$ s_{i}(r+1)= \gamma_v \; s_{i}(r) + \mu_{i}^{\eta}(r+1). $$
(14)

The parameter γ v , (\(0\leq \gamma_v \leq 1 \)) denotes the forgetting factor of a past observation, i.e., the forgetting factor of the past membership degrees. The \(\Updelta v_{i}(r+1)\) can now be written as:

$$ \Updelta v_{i}(r+1)={\frac{\mu_{i}^{\eta}(r+1) \left(x(r+1)-v_{i}(r)\right)}{s_i(r+1)}} $$
(15)

The current membership degree μ i (r + 1) is next defined as follows:

$$ \mu_{i}(r+1)=\left(D_{i,r+1,A_i}^{2} \sum_{j=1}^{c} \left({\frac{1} {D_{j,r+1,A_j}^2}}\right)^{{\frac{1}{\eta-1}}}\right)^{-1} $$
(16)

where \(D_{i,r+1,A_i}^2\) defines the quadratic distance from the cluster centroid as follows:

$$ D_{i,r+1,A_i}^2=\left(x(r+1)-v_i(r) \right)^T A_i \left(x(r+1)-v_i(r) \right), \; 1\le i \le c. $$
(17)

The matrix A i is calculated from fuzzy covariance matrix (Eq.  4) at each step. The equation for the fuzzy covariance matrix must be rewritten in a form suitable for a recursive calculation. The equation for the fuzzy covariance matrix can be written in a following form:

$$ F_{i}(r)={\frac{\sum_{k=1}^r \mu_{i}^{\eta}(k)\left(x(k)-v_i^r \right) \left(x(k)-v_i^r \right)^T}{\sum_{k=1}^r \mu_{i}^{\eta}(k)}} $$
(18)

where v r i stands for the centroid vector of the ith cluster calculated for the set of r samples. The fuzzy covariance matrix at the next sample (r + 1) can be expressed as follows

$$ \begin{aligned} F_{i}(r+1)&={\frac{\sum_{k=1}^{r} \mu_{i}^{\eta}(k)\left(x(k)-v_i^{r+1} \right) \left(x(k)-v_i^{r+1} \right)^T }{\sum_{k=1}^r \mu_{i}^{\eta}(k) + \mu_{i}^{\eta}(r+1)}}\\ &\quad+{\frac{\mu_{i}^{\eta}(r+1)\left(x(r+1)-v_i^{r+1} \right) \left(x(r+1)-v_i^{r+1} \right)^T }{\sum_{k=1}^r \mu_{i}^{\eta}(k) + \mu_{i}^{\eta}(r+1)}} \end{aligned} $$
(19)

where v r+1 i stands for the centroid vector of the ith cluster calculated for the set of r + 1 samples.

Taking into account Eq. 18, introducing it into Eq. 19, and using Eq. 12 and Eq. 13, the following recursive expression for the fuzzy clustering matrix approximation is obtained

$$ \begin{aligned} F_{i}(r+1)&= \gamma_c \; {\frac{s_{i}(r)}{s_{i}(r+1)}} F_{i}(r)+ {\frac{\mu_{i}^{\eta}(r+1)}{s_{i}(r+1)}}\\ & \quad\cdot \left(x(r+1)-v_{i}(r+1) \right) \left(x(r+1)-v_{i}(r+1) \right)^T,\\ \end{aligned} $$
(20)

where γ c was introduced as forgetting factor of fuzzy covariance matrix. For recursive Gustafson Kessel method we also need an inverse of the fuzzy covariance matrix. The calculation of the inverse is a slow operation. Therefore we used the Woodbury inverse lemma to gain the recursive calculation of the inverse fuzzy matrix:

$$F_i^{-1}(r+1)={\frac{A-B}{C}} $$
(21)
$$ A=\frac{1} {\gamma_c}\frac{s_i(r+1)}{s_i(r)} F_i(r)^{-1} $$
(22)
$$ B=\frac{1}{\gamma_c} \frac{s_i(r+1)}{s_i(r)} F_i^{-1} {\frac{\mu_{i}^{\eta}(r+1)}{s_{i}(r+1)}} \left(x(r+1)-v_{i}(r+1) \right)\cdot \left(x(r+1)-v_{i}(r+1) \right)^T \frac{1}{\gamma_c} \frac{s_i(r+1)} {s_i(r)} F_i^{-1} $$
(23)
$$ C=1+\left(x(r+1)-v_{i}(r+1) \right)^T \frac{1} {\gamma_c} \frac{s_i(r+1)}{s_i(r)}\cdot F_i^{-1} {\frac{\mu_{i}^{\eta}(r+1)}{s_{i}(r+1)}}\left(x(r+1)-v_{i}(r+1) \right))$$
(24)

Now the inversion of the fuzzy covariance matrix is done only at the initialization step.

3.2 Applying the recursive least squares

The centers of the fuzzy clusters and their distribution are used to define the new membership functions and using the recursive least squares method the fuzzy model is obtained. Using the projection of the cluster onto the independent variables, the input membership functions are obtained. Here we are assuming that the first m − 1 measured variables represent the input variables and the last mth variable in the observation vector (x(k)) represents the output.

In our case we used the Gaussian membership functions. The membership function of the ith cluster and the jth component of x(k) is defined as

$$ \mu_i\left(x_j(k)\right)=e^{-{\frac{(x_j(k)-v_{ij}(k)) (x_j(k) -v_{ij}(k))^T}{2\mu_m\sigma_{ij}^2(k)}}}, \quad i=1,\ldots,c, \; j=1,\ldots,m-1 $$
(25)

where v ij is the jth component of ith cluster center, x j is the jth component of the observation vector and σ ij is the variance (the jth diagonal element of matrix F i ). The overlapping factor μ m defines the overlapping of the membership functions. Usually it can be set to one so that the membership functions resemble the Gaussian distribution. The m − 1 input variables define the input hyperspace. The subspaces in this hyperspace are defined as the Cartesian product of the subspaces. These imply the definition of the membership degree in each subspace as the product of the membership degrees as follows:

$$ \beta_i(k)=\prod_{j=1}^{m-1}\mu_i\left(x_j(k)\right) $$
(26)

The fuzzy recursive least squares algorithm (Goodwin and Sin 1984, Kasabov and Song 2002, Angelov and Filev 2004) is then used to estimate the local-linear sub-models parameters. We can use version for global optimization case:

$$ \psi_i^T(k+1)=\beta_i(k) \left[1, \; x_{1}(k), \; x_{2}(k),\; \ldots, x_{m-1}(k) \right] $$
(27)
$$ y_i(k)=\beta_i(k) \;x_{m}(k) $$
(28)
$$ P_i(k+1)={\frac{1}{\lambda_r}} \left(P(k)- {\frac{P_i(k) \psi_i(k+1)\psi_i^T(k+1) P_i(k)}{\lambda_r+\psi_i^T(k+1) P_i(k) \psi_i(k+1)}}\right) $$
(29)
$$ \theta_i(k+1)=\theta_i(k)+P_i(k)\psi_i(k+1) \left(y_i(k)-\psi_i^T(k+1)\theta_i(k) \right) $$
(30)

or for local optimization case:

$$ \psi_i^T(k+1)=\left[1, \; x_{1}(k), \; x_{2}(k),\; \ldots, \;x_{m-1}(k) \right] $$
(31)
$$ y_i(k)=\;x_{m}(k) $$
(32)
$$ P_i(k+1)={\frac{1}{\lambda_r}} \left(P(k)- {\frac{\beta_i(k) P_i(k) \psi_i(k+1)\psi_i^T(k+1) P_i(k)}{\lambda_r+\beta_i(k) \psi_i^T(k+1) P_i(k) \psi_i(k+1)}}\right) $$
(33)
$$ \theta_i(k+1)=\theta_i(k)+P_i(k)\psi_i(k+1)\beta_i(k) \left(y_i(k)-\psi_i^T(k+1)\theta_i(k) \right) $$
(34)

where λ r stands for the exponential forgetting factor, which should be set between 0.98 and 1 to deal with time-varying processes (Åström and Wittenmark 1995), P i stands for the covariance matrix, which is set to \({P_i(0)=10^2 \sim 10^5 I, \; I \in {\mathbb{R}}^{m \times m},}\) and θ i represents the parameters of the ith local model.

3.3 The steps of the algorithm

The algorithm can be described in ten steps that are made at each time instant k and initialization step. The clustering algorithm is from step one to step nine. Step nine represents the fuzzy recursive least squares.

  1. 1.

    Step 0: Initialization:

    • define the number of clusters (c), the fuzziness and overlapping (η, η m ), the forgetting factors (\(\lambda_{r}, \gamma_{v}, \gamma_{c}\)), number of measured variables (p) and the cluster volumes (ρ i ),

    • determine the initial value of P i for \(i=1,\ldots,c,\)

    • determine the initial F i for \(i=1,\ldots,c,\)

    • determine the initial centers: v i  = x(k), for \(i=k=1,\ldots,c\) and membership degrees μ i (x(k)) = 1 for \(i=k,\, \mu_i(x(k))=0\) for \(i \neq k,\, i,k=1,\ldots,c,\)

    • calculate the initial s i for \(i=1,\ldots,c\) from Eq. 13,

    • calculate the inverse of the fuzzy covariance matrix.

  2. 2.

    Step 1: calculate the matrix A i from Eq. 5,

  3. 3.

    Step 2: calculate the distance D i from Eq. 17,

  4. 4.

    Step 3: calculate the current membership degree from Eq. 16,

  5. 5.

    Step 4: calculate the s i (r + 1) from Eq. 14,

  6. 6.

    Step 5: calculate the change of cluster centers \(\Updelta v_{i}\) from Eq. 11,

  7. 7.

    Step 6: calculate the new centers from Eq. 7,

  8. 8.

    Step 7: calculate the new fuzzy covariance matrix F i (r + 1) from equation Eq. 20 and inverse fuzzy covariance matrix from Eqs. 22 to Eq. 24,

  9. 9.

    Step 8: calculate the membership functions μ i from Eq. 25,

  10. 10.

    Step 9: calculate the membership degrees β i from Eq. 26,

  11. 11.

    Step 10: apply the recursive least squares Eqs. 27–30 or Eqs. 31–34 and return to Step 1.

Step 0 is executed only once in the procedure. The first three tasks under step zero are done by user everything else is initialized from the algorithm itself. The steps from Step 1 to Step 10 are executed for the time instances \(k\geq c+1\) for every new sample.

The parameters that have to be set in advance are: forgetting factors, number of clusters, fuzziness, overlapping factor, cluster volumes and number of measured variables. For time varying processes, the factors must be lower than one to ensure the forgetting. By lowering the forgetting factors the fluctuations of estimates will increase but the adaptation will be faster. The forgetting factors can be chosen with the help of the rule of thumb (Åström and Wittenmark 1995):

$$ \lambda=1-2/N, $$
(35)

where λ is the forgetting factor and N are the data samples, that affect the estimates.

The fuzziness factor and overlapping factor define the smoothness of the nonlinearity approximation. The higher they are the more smooth nonlinearity we get, but to high values cause bigger identification error and prediction error. The fuzziness factor is usually set η = 2 and the overlapping factor η m  = 1. The cluster volumes are usually se to one.

For the fuzzy covariance and least squares covariance matrix the initial values must be given. The matrix are usually initialized with the identity matrix as: F i (0)≈ I and P i (0)≈ 104 I.

4 Fault detection based on a fuzzy model

In this example the proposed method is used to construct the fuzzy model based on a number of starting samples. The fuzzy model is then used to monitor the process and detect the fault. The process used in the example was a waste water treatment process.

Waste-water treatment plants are large nonlinear systems subject to large perturbations in flow and load, together with uncertainties concerning the composition of the incoming waste-water. The simulation benchmark has been developed to provide an unbiased system for comparing various strategies without reference to a particular facility. It consists of five sequentially connected reactors along with a 10-layer secondary settling tank. The plant layout, model equations and control strategy are described in detail on the web page (http://www.ensic.u-nancy.fr/costwwtp). In our example the phase where the waste-water is purified is monitored. After this phase the moving bed bio-film reactor is used. Schematic representation of simulation benchmark is shown in Fig. 1.

Fig. 1
figure 1

Schematic representation of simulation benchmark

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The detection of sensor faults was applied to the simulation model where the following measurements were used to identify the fuzzy model: influent ammonia concentration in the inflow Q in defined as \(C_{{NH4N}_{in}},\) dissolved oxygen concentration in the first aerobic reactor tank \(C^1_{O_2},\) dissolved oxygen concentration in the second aerobic reactor tank \(C^2_{O_2}\) and the ammonia concentration in the second aerobic reactor tank \(C_{{NH4N}_{out}}.\) The fuzzy model was build to model the relation between the ammonia concentration in the second aerobic reactor tank and the other measured variables:

$$ C_{{NH4N}_{\rm out}}(k)= {\cal G} \left( C_{{NH4N}_{\rm in}} (k),C^1_{{\rm O}_2} (k),C^2_{{\rm O}_2} (k) \right) $$
(36)

where \(\mathcal{G}\) stands for nonlinear relation between measured variables. First 15,000 measurements (sampling time T s  = 120 s) were used to find the fuzzy clusters and to estimate the fuzzy model parameters. At the measurement 17,000 the slowly increasing sensor fault occur, which is than at sample 18,000 eliminated. This means that sensor to measure the ammonia concentration in the second aerobic reactor tank \(C_{{NH4N}_{out}}\) is faulty. The signal with exponentially increasing value was added to the nominal signal. The whole set of measurements is shown in Fig. 2 and 3.

Fig. 2
figure 2

The whole set of measurements. The influent ammonia concentration \(C_{{NH4N}_{in}}\) and dissolved oxygen concentration in the first aerobic reactor tank \(C^1_{O_2}\)

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Fig. 3
figure 3

The whole set of measurements. The dissolved oxygen concentration in the second aerobic reactor tank \(C^2_{O_2}\) and the ammonia concentration in the second aerobic reactor tank \(C_{{NH4N}_{out}}\)

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The fuzzy model is obtained on the set of first 15,000 samples using the proposed method. The identification of the model is done on-line. At the sample 15,000 the model output is calculated for the starting period and the tolerance index are calculated. The fuzziness and overlapping were set to four and ten clusters were used. The forgetting factors for the first period were set to one (no forgetting). From the sample 15,000 the forgetting factors were set to 0.9998. The on-line adaptation is stopped when alarm is raised and started again after the end of alarm. The fuzzy model output \(\hat{C}_{{NH4N}_{out}}\) and the process output \(C_{{NH4N}_{out}}\) for the tuning period are shown in Fig. 4.

Fig. 4
figure 4

The verification of fuzzy model where the fuzzy model output and the process output are shown

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The fault detection index is defined as:

$$ f=\left({\frac{C_{{NH4N}_{out}}-\hat{C}_{{NH4N}_{out}}} {\hat{C}_{{NH4N}_{out}}}}\right)^2 $$
(37)

The fault tolerance index is defined as relative degree of maximal value of fault detection index in the identification or learning phase f tol  = γ max f where in our case γ = 1.5.

The algorithm calculated the value of fault detection index f tol  = 0.156. The fault which occur at the sample 17,000 is detected at the sample 17,425. The detection is delayed, but this is usual when the faults are slowly increasing. Figure 5 shows the process output and the model prediction for the whole experiment.

Fig. 5
figure 5

The monitoring. The model output and the process output for the whole measurement set

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Fig. 6
figure 6

The fault detection index, the fault tolerance index f and the actual and detected fault

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5 Fuzzy predictive functional control

This example demonstrates the use of the proposed recursive clustering method in a combination with the fuzzy predictive functional control algorithm. The method is used for adaptation of fuzzy model when the nonlinearity changes. This example the FPFC is used to control the pH process. The process was adopted from (Henson and Seborg 1994). The scheme of the process is shown on Fig. 7. The acid stream (q 1), buffer stream (q 2) and base stream (q 3) are mixed in tank 1. Prior to mixing the acid stream enters tank 2, which introduces an additional dynamics. The control valves control the acid and base flow. The tank level (h 1) and effluent pH (pH) are measured values. In this example the controlled variable is pH, manipulated by base flow rate (q 3), acid and buffer flows are considered to be unmeasured disturbances.

Fig. 7
figure 7

The scheme of a pH process

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Equations that describe the process are following:

$$ A_2 {\frac{dh_2}{dt}}=q_1-q_{1e} $$
(38)
$$ q_{1e}=C_{v1}h_2^{0.5} $$
(39)
$$ x=[W_{a4}\quad W_{b4}\quad h_1]^T $$
(40)
$$ d=q_2 $$
(41)
$$ u_a=q_3 $$
(42)
$$ y=pH $$
(43)
$$ f(x)=\left[{\frac{q_1}{A_1 x_3}}(W_{a1}-x_1)\quad{\frac{q_1}{A_1 x_3}}(W_{b1}-x_2)\quad\times{\frac{1}{A_1}}(q_1-C_{v4}(h_1+z)^n)\right]^T $$
(44)
$$ g(x)=\left[{\frac{1}{A_1 x_3}}(W_{a3}-x_1)\quad{\frac{1}{A_1 x_3}}(W_{b3}-x_2)\quad {\frac{1}{A_1}}\right]^T $$
(45)
$$ p(x)=\left[{\frac{1} {A_1 x_3}}(W_{a2}-x_1)\quad{\frac{1}{A_1 x_3}}(W_{b2}-x_2)\quad{\frac{1}{A_1}}\right]^T $$
(46)
$$ \dot{x}=f(x)+g(x)u_a+p(x)d $$
(47)
$$ pK_{a2}=-log_{10}(K_{a2})\quad pK_{a1}=-log_{10}(K_{a1}) $$
(48)
$$ c(x,y)=x_1+10^{y-14}-10^{-y}+x_2 {\frac{1+2\times10^{y-pK_{a2}}} {1+10^{pK_{a1}-y}+10^{y-pK_{a2}}}} $$
(49)
$$ c(x,y)=0 $$
(50)

The parameters are given in Table 1. The change of the dynamics of the process is simulated by changing the buffer input stream q 2 to 0.8 ml/s and the acid stream q 1 to 10 ml/s.

Table 1 Operating conditions for pH process
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To show the cluster center reposition the test in an open-loop was made with the random input signal. The regression vector was chosen as x(i) = [u a (i)  y(i)]T. Figure 8 shows the cluster centers and the input–output data distribution before the change. Figure 9 shows the cluster centers and the input–output data distribution after the change.

Fig. 8
figure 8

The input–output sample distribution before the disturbance. The small dots present the samples, big dots the end cluster position and the rectangles the starting positions

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Fig. 9
figure 9

The input–output sample distribution after the disturbance. The small dotspresent the samples, big dots the end cluster position and the rectangles the starting positions

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For the control the starting fuzzy model was identified on-line in an open loop using random generated input signal. This model was then used to control the process using fuzzy predictive functional controller. The control algorithm is detail described in (Dovžan and Škrjanc 2010) and (Škrjanc and Matko 2000). The forgetting factors were set to 0.999 and the resetting of covariance matrices was used (Åström and Wittenmark 1995). Figure 10 shows the control experiment. In the experiment the adaptive version of the FPFC and nonadaptive version are compared.

Fig. 10
figure 10

Control of the process

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The control of the adaptive version is better than one with the nonadaptive. The disturbance rejection is faster Fig. 11 and has less overshoot.

Also after the model adapts its self to new process dynamics the control is better (Fig. 12).

The most important is that the error between the model and process output is smaller wit the adaptive control (Fig. 13).

The control with the adaptive FPFC is better than with the non-adaptive. There is less overshoot and better regulation after the disturbance. In Table  2 the sum squared errors are given. It can be seen that the error between model prediction and real process output is substantially smaller with the adaptive FPFC.

Table 2 The sum squared error
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Table 3 Comparison of recursive GK method and other fuzzy identification methods on a 85 steps ahead prediction of the M–G time-series
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Fig. 11
figure 11

Disturbance rejection

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Fig. 12
figure 12

Control after the disturbance

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Fig. 13
figure 13

Model matching after the disturbance

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6 Mackey–Glass (M–G) time series prediction

To compare the proposed method to others, the results for the benchmark problem of 85-steps ahead Mackey-Glass time series (Mackey and Glass 1977) prediction are given. The detailed description of the experiment is given in (Kasabov and Song 2002). For the validation of the model we used the non-dimensional error index (NDEI), defined as the ratio of the root mean square error to the standard deviation of the target data. The results from other methods are taken from (Kasabov and Song 2002) and (Angelov 2010) and are shown in Table 3.

Some of the methods are also tested on the 6-step prediction. Therefore an experiment was also made for 6-step prediction. The proposed rGK approximated the 6-step prediction experiment [described in (Paiva and Dourado 2001)] with NDEI 0.0859 (10 clusters).

7 Conclusion

In this paper a method for on-line fuzzy model identification was presented. The method is based on a recursive Gustafson–Kessel algorithm and recursive fuzzy least squares. The rGK algorithm was derived from an off-line GK algorithm. Guidelines for parameters settings are given and examples of its use. The results for M–G time series prediction are given.

The proposed method can be used for adaptive fuzzy control, fault detection, model based design of experiments and other areas, where fuzzy models with time invariant parameters are inappropriate or where recursive identification should be used. The performance of the proposed method is comparable to other established on-line methods.