# Evolving fuzzy systems

Plamen Angelov (2008), Scholarpedia, 3(2):6274. | doi:10.4249/scholarpedia.6274 | revision #91244 [link to/cite this article] |

**Evolving fuzzy systems** (EFS) can be defined as self-developing, self-learning fuzzy rule-based or neuro-fuzzy systems that have both their parameters but also (more importantly) their structure self-adapting on-line.

They are usually associated with *streaming* data and on-line (often real-time) modes of operation. In a narrower sense they can be seen as *adaptive* fuzzy systems. The difference is that *evolving* fuzzy systems assume on-line adaptation of system structure in addition to the parameter adaptation which is usually associated with the term *adaptive*. They also allow for adaptation of the learning mechanism. Therefore, *evolving* assumes a higher level of adaptation.

In this definition the English word *evolving* is used with its core meaning as described in the Oxford dictionary (Hornby, 1974; p.294), namely *unfolding; developing; being developed, naturally and gradually*.

Often *evolving* is used in relation to so called evolutionary and genetic algorithms. The meaning of the term *evolutionary* is defined in the Oxford dictionary as *development of more complicated forms of life (plants, animals) from earlier and simpler forms*. EFS consider a *gradual* development of the underlying (fuzzy or neuro-fuzzy) system structure and do not deal with such phenomena specific for the *evolutionary* and *genetic algorithms* as *chromosomes crossover, mutation, selection* and *reproduction*, *parents* and *off-springs*.

## Contents |

## Introduction

The concept of EFS was conceived around the turn of the century (Angelov and Buswell, 2001; Angelov, 2002; Kasabov and Song, 2002) addressing the demands for flexible, adaptive, yet robust and interpretable systems for advanced industry, autonomous systems, intelligent sensors, defence etc.

The (technical) systems that claim to be *intelligent* are very often far from true intelligence. One of the main reasons for this is that the intelligence cannot be fixed or static, it is *evolving*. Learning takes place for the example of human beings, during their whole life from experience and form their own rules, adapt them or ignore and replace with new rules. EFS are the first mathematical constructs that can approximate the human-like reasoning by representing it with dynamically evolving fuzzy rule-based structure and applies a formal (and mathematically sound) learning mechanism to implement it using data streams. The fuzzy rule-based systems of so called Takagi-Sugeno (TS) type (Takagi and Sugeno, 1985) that are considered as a convenient structural framework of EFS are fuzzy systems which can be described as:
\[Rule_i: IF (x_1 \quad is \quad X_{i1}) \quad AND... \quad AND(x_n \quad is \quad X_{in}) \quad THEN \quad (y_i = a_{i0}+a_{i1}x_1+...+a_{in}x_n )\]
where

- \(Rule_i\) is one of several (\(i=1,2,...,R\)) fuzzy rules in the rule base;
- \(x_j; j=1,2,...,n\) are input variables;
- \(y_i\) denotes output of the \(i\)-th fuzzy rule;
- \(X_{ij}\) denotes the \(j\)-th prototype (focal point) of the i-th fuzzy rule;
- \(a_{ij}\)denotes the \(j\)-th parameter of the \(i\)-th fuzzy rule.

In general, EFS can be of different type, e.g. of the so called Zadeh-Mamdani type (Zadeh, 1973; Mamdani and Assilian, 1975).

The original TS type fuzzy system as described above is multi-input-single-output (MISO). EFS can also use multi-input-multi-output (MIMO) TS fuzzy systems which can be described as (Angelov et al., 2004): \[Rule_i: IF (x_1 \quad is \quad X_{i1}) \quad AND... \quad AND (x_1 \quad is \quad X_{in}) \quad THEN \quad (y^1_i = a^1_{i0}+a^1_{i1}x_1+...+a^1_{in}x_n ) \] \(AND... \quad AND(y^m_i = a^m_{i0}+a^m_{i1}x_1+...+a^m_{in}x_n )\)

Note that, in general a MIMO system can be more complicated with outputs being interlinked. TS fuzzy systems can also be represented as a neural networks and are thus neuro-fuzzy systems (Jang, 1993). Therefore, the evolving TS (eTS) can be seen as an evolving neuro-fuzzy system (Angelov and Filev, 2002).

The very origin of fuzzy systems is closely linked to human expert knowledge (Zadeh, 1965; Zadeh, 1973). They can be seen as a tool and technique to mathematically formalise the uncertainty, subjective information, preferences, experience, intuition, which are otherwise difficult or impossible to describe. The real surge in applications of fuzzy systems in practice was prompted by so called fuzzy control that followed Mamdani's and Takagi and Sugeno's seminal works. Elicitation of membership functions and definition of fuzzy rules based entirely on expert knowledge, however, is associated with a range of difficulties and problems and the practical demand of effective systems, especially in industry, led to the appearance and fast development of techniques that were called data-driven (Babuska, 1998; Angelov, 2002). Data-driven techniques assume that the human-intelligible fuzzy models are designed primarily from data (streams). Expert knowledge can also be incorporated, but is not essential. (Evolving) fuzzy systems can be seen as a tool for knowledge extraction from data (streams). It is of great importance that these methods can be completely automatic and the systems that are designed can be constantly monitored and analysed (that is, they are not of a *black-box* type). TS fuzzy systems are particularly suitable for data-driven algorithms due to their dual nature - they have a fuzzy linguistic antecedent part and a functional (usually linear) consequent part.

## Evolving fuzzy clustering

Clustering is a well known technique for unsupervised learning (when correct answers are not provided) aiming at grouping the data based on their similarity and proximity in the data space. Fuzzy clustering (de Oliveira and Pedrycz, 2007; Lima et al., 2006) is a special case of clustering when the membership to a cluster is a matter of a degree (a real value in the range [0;1]) rather than a binary (Yes/No or 0/1). The degree of membership to a cluster can be determined in different ways, but more often it is defined as a Gaussian in terms of the Euclidean, Mahalonobis or cosine norm. There are well established off-line (e.g. kNN (Duda et al, 2001), tree-based C4.5 (Hastie et al., 2001), fuzzy C-means, FCM (Bezdek, 1974), Mountain/Subtractive clustering (Yager and Filev, 1993; Chiu, 1994) and incremental/on-line approaches, e.g. self-organising maps, SOM (Kohonen, 1988), adaptive resonance theory, ART (Carpenter and Grossberg, 2003). Most of the approaches require the number of clusters to be provided (pre-specified), e.g. the number of neighbours in kNN, the number of neurons in SOM, the number C in FCM etc. Alternatively a user- and problem-specific threshold must be provided. Another problem is that the mean may be infeasible data point because it is a virtual (non-existing physically) point in the data space.

The evolving clustering approach, eClustering (Angelov and Filev, 2004) does not need the number of clusters to be pre-specified; its improved version eClustering+ (Angelov, 2008) also does not have any user- or problem-specific thresholds. It is on-line (one-pass, non-iterative, recursive) and prototype-based (does not use mean of the data points but real data points only). The fact that eClustering is prototype-based, however, can lead as well to several difficult problems, specially those related to the impact of the noise and the data quality in general to the learning process. The procedure of eClustering starts ‘from scratch’. It is based on the recursive estimation of data density (Angelov et. al, 2008). The density, D at a data point, z resembles the probability distribution used in so called Parzen windows (Specht, 1988). It can be described by the following Cauchy function (Angelov and Filev, 2004): \[D=\frac{1}{1+\frac{\sum_{i=1}^{k-1}||z_k-z_i*||^2}{k-1}}\] where \(z*\) denotes the prototype. In (Angelov and Filev, 2003) it has been shown that Cauchy function which is an approximation of the Gaussian can be calculated recursively.

The aim of the clustering for data space partitioning which is used as a technique to automatically identify the structure of fuzzy systems (Babuska, 1998; Angelov and Filev, 2004) differs from the conventional clustering (Duda et al., 2001). The latter one aims to separate the data into groups in such a way that the data that are similar to each other are grouped in the same group and data that are dissimilar to each other are in different groups. In this sense, the conventional clustering does not assume and does not tolerate overlap or partial and mutual membership between clusters. The former, however, tolerates and assumes some overlap ( Figure 1), partial and mutual membership to clusters, respectively antecedents of the fuzzy rules, which is the backbone of the fuzzy systems.

## Evolving TS fuzzy systems

The TS fuzzy system described in the introduction has several elements:

- Antecedent, IF part;
- Fuzzy sets that are connected with logical aggregation operators (AND, OR) and may represent linguistic variables, e.g. \(x_{i1}\) is
*High*; - Membership function of these fuzzy sets (they represent the degree of membership of a particular value to the specific fuzzy set, \(\mu_{i1}\));
- Parameters of the membership function, e.g. focal point/centre and spread (if Gaussian) or left, centre, right (if triangular);
- Number of input variables, \(n\)

- Fuzzy sets that are connected with logical aggregation operators (AND, OR) and may represent linguistic variables, e.g. \(x_{i1}\) is
- Consequents, THEN part;
- Parameters of the consequent part (usually linear), e.g. \(a_{ij}\ ;\)
- Number of outputs, \(m\)

- Number of fuzzy rules, \(R\)

The concept of EFS assumes that each of the above listed elements of the fuzzy system can dynamically evolve (gradually change) by learning from experience based on streams of data ( Figure 2). Different techniques to automatically evolve fuzzy systems from data are described in (Angelov and Filev, 2004; Angelov, 2006; Angelov and Zhou, 2006; Angelov, 2008.

Because the evolving TS (eTS) fuzzy system can be represented as a neural network (Angelov and Filev, 2002) as illustrated in Figure 3 eTS can also be considered as a self-developing or evolving neuro-fuzzy system.

It is the structural framework that can be used to solve a range of problems offering flexibility, adaptation, robustness, and improved precision with small computational efforts (due to the recursive algorithms). The main group of problems that can be solved by eTS include, but are not limited to:

- clustering (see previous section); note: clustering uses only the antecedent part of the fuzzy system;
- novelty detection (see the previous section);
- regression models of the type \(y_t=f(x_t)\ ;\)
- time-series prediction or filtering, e.g. \(\overline{y_t}=f(x_{t-1},...,x_{t-N_x},y_{t-1},...,y_{t-N_x})\) which is also known as non-linear auto-regression with exogenous input, NARX or as non-linear output error, NOE when the past
**predictions**are used instead of the**actual**data; - classification (see next section);
- control (see section 6).

## Evolving fuzzy classifiers

Fuzzy rule-based classifiers with rules that are evolved from streaming data are called evolving fuzzy classifiers (EFC). EFC does not need to know beforehand in how many classes the data will be classified – new classes can be introduced during the learning process (Angelov and Zhou, 2008). Generally, there can be three types of EFC:

### Predicting class label

One possible EFC which has class labels as outputs is eClass0 (Angelov and Zhou, 2008) which follow the conventional structure of fuzzy classifiers (Kuncheva, 2000). The main difference of eClass0 from a conventional fuzzy classifier is its evolving structure. It can also be seen as an extension of eClustering+ with the addition of labels as a consequent part of the rules: \[Rule_i: IF (x_1 \quad is \quad X_{i1}) \quad AND...\quad AND(x_n \quad is \quad X_{in}) \quad THEN \quad (y_i \quad is \quad Class_C)\] where \(Class_C\) is the label of the \(C\)-th class;

It is computationally very light and is suitable for robotics or as a first step of a more complex classification scheme. Additional advantage of eClass0 is its high interpretability.

### Predicting possibility of a features vector to be of certain class

One EFS which predicts the possibility of a feature vector to be of certain class is eClass1 which is based on the first order Takagi-Sugeno type EFS, eTS. eClass1 differs from conventional fuzzy classifiers in the fact that it aims to approximate the usually highly non-linear classification hyper-surface in a form of regression over the features (inputs). Combined with the very flexible structure of eTS+ which involves on-line feature (input variables) selection (Angelov, 2006; Angelov, 2008) it provides and effective tool for classification and automatic analysis of complex problems (with hundreds of variables). eClass1 can be of MIMO type and thus classify into one of m possible classes: \[Rule_i: IF (x_1 is X_{i1}) \quad AND... \quad AND (x_1 is X_{in}) \quad THEN \quad (p^1_i = a^1_{i0}+a^1_{i1}x_1+...+a^1_{in}x_n ) AND... \quad AND (p^C_i = a^C_{i0}+a^C_{i1}x_1+...+a^C_{in}x_n )\] where \(p^1_i\) is the un-normalized possibility the i-th set of features to describe Class 1; \(\frac{p^1_i}{\sum_{j=1}^np^j_i}\) is the normalized possibility the \(i\)-th set of features to describe Class 1; \(C\) is the number of classes

### Multi-model EFS

One possible multi-model EFS method is eClassM (Angelov and Zhou, 2008). This scheme combines several single-model eClass1 classifiers (one for each class). It provides a decoupling of the antecedent part of the fuzzy rule-based classifier if compared to eClass1 MIMO and thus can bring better performance (higher classification rates) for problems with more than two classes. In fact, it comprises, of \(m\) separate eClass1 structures with each one having a binary prediction (Yes/No; 1/0) in terms of the respective class: \[Rule^1_i: IF (x_1 is X_{i1}) \quad AND... \quad AND (x_1 \quad is \quad X_{in}) \quad THEN \quad (p^1_i = a^1_{i0}+a^1_{i1}x_1+...+a^1_{in}x_n )\] \[Rule^2_i: IF (x_1 is X_{i1}) \quad AND... \quad AND (x_1 \quad is \quad X_{in}) \quad THEN \quad (p^2_i = a^2_{i0}+a^2_{i1}x_1+...+a^2_{in}x_n )\] \[Rule^C_i: IF (x_1 is X_{i1}) \quad AND... \quad AND (x_1 \quad is \quad X_{in}) \quad THEN \quad (p^C_i = a^C_{i0}+a^C_{i1}x_1+...+a^C_{in}x_n )\]

## Evolving fuzzy controllers

Fuzzy control has proven its viability through numerous applications ranging from nuclear power plants safety (DaRuan,2005) to home appliances such as washing machines (Driankov et al., 1996). It is probably the ‘best selling’ offspring of the fuzzy sets theory that led to the growth of interest in this area from engineers and practicioners in the end of 20th century. Adaptive control is another well established and recognised area that is widely used in industry for over three decades (Astrom and Wittenmark,1984). One of the problems of controllers (and in particular fuzzy controllers) is their design and tuning. Because the environment in which industrial plants and controllers operate are dynamically changing off-line design and tuning cannot provide the flexibility that is required. Adaptive control theory (and adaptive systems in general) provides a way out of this problem that is based on constant adaptation of the parameters of the (usually linear) systems (controllers). Real problems, however, are usually highly non-linear, non-stationary and, therefore, adaptive control theory has obvious limitations. Evolving Fuzzy Rule-based Controllers (Angelov, 2002; Angelov, 2004) combine the flexibility and non-linear nature of fuzzy TS systems with the advantages of well established adaptive control schemes such as:

### Indirect learning

Indirect learning was proposed by Psaltis et al. (1998) to be used with an off-line trained neural network serving as a controller. The NN was proposed to be pre-trained off line in such a way that to model the inverse dynamics of the plant. Then this NN is used in an adaptive control scheme to produce control action that should bring the output of the plant to the required reference value (if the inverse model of the dynamic of the plant is perfect). If use an EFS instead of the NN in the same scheme the resulting scheme can be re-trained automatically on-line during the control process. Moreover, not only parameters of the controller can be adapted, but also its structure can evolve.

### Predictive model-based control

An alternative for the use of EFS in control schemes is to model the plant dynamics (Zhou and Angelov, 2008). This scheme combines well known model-predictive control scheme (Babuska, 1998) with eTS+ (Angelov, 2008).

### Stability

An important issue of the controllers design is the stability. The problem of stability of the on line structure and parameters learning algorithms when using an evolving neuro-fuzzy recurrent network has been studied in (Rubio, 2008).

## Application case studies

### Evolving intelligent sensors

One very important area of application of EFS is the development of evolving intelligent sensors. eSensor (Angelov et al., 2008) is an example of such application to real chemical process modelling (courtesy of Dr. A. Kordon, The Dow Chemicals, TX, USA). Another real application of eSensor is for monitoring quality of products in an oil refinery (courtesy of Dr. J. J. Macias Hernandez, Oil Refinery CEPSA, Santa Cruz, Tenerife, Spain), (J.J. Macias Hernandez et al., 2007). eSensor copes with such phenomena as the *drift* and *shift* in data pattern, re-calibration, automatic extraction of interpretable model structure, fast learning, simple structure, no requirement for training the personnel to use it.

### Robotics

Another big area of application is robotics and autonomous systems. Autonomous systems by the virtue of their definition should be able to act without human intervention in unknown possibly harsh environments. It is therefore essential that they are able to learn and to acquire new knowledge on-line. EFS are well suited to address this task. Some examples of application of EFS include:

- Autonomous landmark recognition (Zhou and Angelov, 2006);
- Self-localisation and mapping (Zhou and Angelov, 2007);
- Object detection and tracking (Angelov et al., 2008b);
- Object following (Liu and Meng, 2004; Zhou et al., 2008);
- Collision avoidance (Angelov et al., 2008c).

### WWW mining

Another emerging area of application of EFS is the Internet-based information mining. Limitations of contemporary search engines are well known – the result is usually an unstructured list of items that often does not include the item that is most needed by the user. eClustering+ and eClass can be used to extract and categorise the information provided by the crawler (Evans et al., 2006).

### Machine health monitoring

An area of constant interest for the industry is the fault detection and prognostics. EFS provide a powerful tool for on-line monitoring analysis and prediction of the health of machines and systems in general (Filev and Tseng, 2006).

## References

- Angelov P and Buswell R (2001) Evolving Rule-based Models: A Tool for Intelligent Adaptation, In proc. 9th IFSA World Congress, Vancouver, BC, Canada, pp.1062-1067.

- Angelov P (2002) Evolving Rule-based Models: A Tool for Design of Flexible Adaptive Systems, Springer-Verlag, Heidelberg, New York, ISBN 3-7908-1457-1.

- Angelov P and Filev D (2002) Flexible Models with Evolving Structure , In Proc. IEEE Symposium on Intelligent Systems, Varna, Bulgaria, v.2, pp.28-33.

- Angelov P and D Filev (2003) On-line Design of Takagi-Sugeno Models, In: Lecture Notes in Computer Science 2715 Fuzzy Sets and Systems IFSA2003 (T. Bilgiç, B. De Baets, O. Kaynak Eds.), 2003, pp. 576-584, ISBN 3-540-40383-3.

- Angelov P and D Filev (2004) An Approach to On-line Identification of Takagi-Sugeno Fuzzy Models, IEEE Transactions on System, Man, and Cybernetics, part B - Cybernetics, vol.34, No1, pp.484-498. ISSN 1094-6977.

- Angelov P, Xydeas C, and Filev D (2004) On-line Identification of MIMO Evolving Takagi-Sugeno Fuzzy Models, In Proc. of the Intern. Joint Conf. on Neural Networks and Intern. Conf. on Fuzzy Systems, IJCNN-FUZZ-IEEE, Budapest, Hungary, ISBN 0-7803-8354-0, pp. 55 - 60.

- Angelov P P (2004) A Fuzzy Controller with Evolving Structure, Information Sciences, ISSN 0020-0255, vol.161, pp.21-35.

- Angelov P (2006) Machine Learning (Collaborative Systems) UK patent application, GB-0621734.3, 1 November 2006.

- Angelov P and X Zhou (2006) Evolving Fuzzy Systems from Data Streams in Real-Time, In Proc. 2006 International Symposium on Evolving Fuzzy Systems, Ambelside, Lake District, UK, IEEE Press, pp.29-35, ISBN 0-7803-9719-3.

- Angelov P, A Kordon, X Zhou (2008a) Evolving Fuzzy Inferential Sensors for Process Industry, In Proc. 2008 3rd International Workshop on Genetic and Evolving Fuzzy Systems, Witten-Bomerholz, Germany, pp. 41-46, ISBN 978-1-4244-1613-4.

- Angelov P and X Zhou (2008) Evolving Fuzzy Rule-based Classifiers from Data Streams, IEEE Transactions on Fuzzy Systems, special issue on Evolving Fuzzy Systems, to appear.

- Angelov P, R Ramezani,X Zhou (2008b), Autonomous Novelty Detection and Object Tracking in Video Streams using Evolving Clustering and Takagi-Sugeno type Neuro-Fuzzy System, In Proc. 2008 IEEE International Conference on Fuzzy Systems within the IEEE World Congress on Computational Intelligence, Hong Kong, June 1-6, 2008, to appear.

- Angelov P (2008) Evolving Takagi-Sugeno Fuzzy Systems from Streaming Data, eTS+, In Evolving Intelligent Systems: Methodology and Applications (Angelov P, D Filev, N Kasabov Eds.), John Willey and Sons, 2008, to appear.

- Angelov P, C D Bocaniala, C Xydeas, C Pattchet, D Ansell, M Everett, G Leng (2008c) A Passive Approach to Autonomous Collision Detection and Avoidance in Uninhabited Aerial Systems 2008 EUROSIM: UK Conference on Computer Modelling and Simulation, 1-3 April 2008, Emanuel College, Cambridge, UK, to appear.

- Astrom K and B Wittenmark (1984) Computer Controlled Systems: Theory and Design, Prentice Hall: NJ USA, 1984.

- Babuska R (1998) Fuzzy Modelling for Control, Kluwer Publishers, Dordrecht, The Netherlands.

- de Barros J C and A L Dexter (2006) An Evolving Fuzzy Model for Embedded Applications, In Proc. 2006 International Symposium on Evolving Fuzzy Systems, IEEE press, ISBN 0-7803-9718-5, pp. 49-54.

- Bezdek J (1974) Cluster Validity with Fuzzy Sets, Journal of Cybernetics, vol.3, No 3, pp. 58-71.

- Carpenter G A and Grossberg S (2003) Adaptive Resonance Theory, In M. A. Arbib (Ed.), The Handbook of Brain Theory and Neural Networks, 2nd Edition (pp. 87-90), MIT Press: Cambridge, MA, USA.

- Chiu S L (1994) Fuzzy Model Identification based on Cluster Estimation, Journal of Intelligent and Fuzzy Systems, v.2, pp.267-278

- Da Ruan (2005) Fuzzy control for nuclear reactor operation -- strengths, weaknesses, opportunities and threats, Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology, vol.16 (4), pp.289-295.

- Driankov D, H Hellendoorn, M Reinfrank (1996) An Introduction to Fuzzy Control, Springer Verlag, Heidelberg, Germany.

- Duda R O, P E Hart, D G Stork (2001) Pattern Classification - Second Edition. Wiley-Interscience, Chichester, West Sussex, England.

- Evans A, P Angelov, X Zhou (2006) On-line Evolving Clustering of Web Documents, 2nd Annual Symposium on Nature Inspired Smart Adaptive Systems, Tenerife, Spain, pp.225-230, ISBN 3-86130-926-2.

- Filev D, F Tseng (2006) Novelty detection-based Machine Health Prognostics, In Proc. 2006 International Symposium on Evolving Fuzzy Systems, IEEE press, ISBN 0-7803-9718-5, pp. 193-199.

- Hastie T, R Tibshirani, and J Friedman (2001) The Elements of Statistical Learning: Data Mining, Inference and Prediction. Heidelberg, Germany: Springer Verlag, 2001

- Hornby A S (1974) Oxford Advance Learner’s Dictionary, Oxford University Press, 1974.

- Jang J S R (1993) ANFIS: Adaptive Network-based FUzzy Inference Systems, IEEE Transactions on Systems, Man, and Cybernetics, part B - Cybernetics, vol. 23, No. 3, 1993, pp.665-685.

- Kasabov N and Song Q (2002) DENFIS: Dynamic Evolving Neural-Fuzzy Inference System and Its Application for Time-Series Prediction, IEEE Trans. on Fuzzy Systems, vol.10, No. 2, 2002, pp. 144 - 154.

- Kohonen T (1984) Self-Organization and Associative memory. New York: Springer Verlag.

- Kuncheva L (2000) Fuzzy Classifiers, Physica-Verlag, ISBN 3-7908-1298-6.

- Lima E, F Gomide, R Balini (2006) Participatory Evolving Fuzzy Modeling, In Proc. 2006 International Symposium on Evolving Fuzzy Systems, IEEE press, ISBN 0-7803-9718-5, pp. 36-41.

- Liu P X, M Q-H Meng (2004) Online Data-Driven Fuzzy Clustering with Applications to Real-Time Robotic Tracking, IEEE Transactions on Fuzzy Systems, vol.12, No 4, 2004, pp.516-523.

- Macias-Hernandez, J J, P Angelov, X Zhou (2007) Soft Sensor for Predciting Crude Oil Distillation Side Streams using Takagi Sugeno Evolving Fuzzy Models, In Proc. 2007 IEEE International Conference on Systems, Man, and Cybernetics, Montreal, Canada, ISBN 1-4244-0991-8/07, pp.3305-3310.

- Mamdani E H and S Assilian (1975) An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller, International Journal of Man-Machine Studies, vol. 7, 1975, pp.1 - 13.

- Psaltis D, A Sideris, A A Yamamura (1998) A Multilayered Neural Network Controller, IEEE Transactions on Control Systems Management, vol. 8, pp. 17-21.

- Rubio Avila J J (2008) Stability Analisys for an On-Line Evolving Neuro-Fuzzy Recurrent Network, In Evolving Intelligent Systems: Methodology and Applications, John Wiley and Sons - IEEE Press, to appear.

- Specht D (1991) A general regression neural network, IEEE Transactions on Neural Networks, vol. 2 , No 6, pp. 568-576.

- Takagi T and Sugeno M (1985) Fuzzy identification of systems and its application to modeling and control, IEEE Trans. on Syst., Man and Cybernetics, part B – Cybernetics, vol. 15, pp. 116-132.

- de Oliveira J V and W Pedrycz (2007) Advances in Fuzzy Clustering and Applications, John Willey and Sons, Chichester, UK, ISBN 978-0-470-02760-8.

- Yager R R and D P Filev (1993) Learning of Fuzzy Rules by Mountain Clustering, Proc. of SPIE Conf. on Application of Fuzzy Logic Technology, Boston, MA, USA, pp.246-254

- Zadeh L A (1965) Fuzzy sets. Information and Control vol. 8 No3, pp. 338-353.

- Zadeh L A (1973) Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man and Cybernetics SMC, vol.3, pp. 28-44.

- Zhou X-W and P Angelov (2006) Real-Time joint Landmark Recognition and Classifier Generation by an Evolving Fuzzy System, In Proc. 2006 IEEE World Congress on Computational Intelligence, Vancouver, BC, Canada, pp.6314-6321, ISBN 0-7803-9489-5.

- Zhou X and P Angelov (2007) An Approach to Autonomous Self-localization of a Mobile Robot in Completely Unknown Environment using Evolving Fuzzy Rule-based Classifier, In Proc. First 2007 IEEE International Conference on Computational Intelligence Applications for Defense and Security, Honolulu, Hawaii, USA, pp.131-138.

**Internal references**

- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

- Ludmila I. Kuncheva (2008) Fuzzy classifiers. Scholarpedia, 3(1):2925.

- Robert Babuska and Ebrahim Mamdani (2008) Fuzzy control. Scholarpedia, 3(2):2103.

- Milan Mares (2006) Fuzzy sets. Scholarpedia, 1(10):2031.

- Laura Gardini, Gian-Italo Bischi, Christian Mira (2007) Maps with vanishing denominators. Scholarpedia, 2(9):3277.

- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

- James Murdock (2006) Unfoldings. Scholarpedia, 1(12):1904.

## Recommended reading

- Angelov P (2002) Evolving Rule-based Models: A Tool for Design of Flexible Adaptive Systems, Springer-Verlag, Heidelberg, New York, ISBN 3-7908-1457-1.

- Angelov P, D Filev, N Kasabov Eds., Evolving Intelligent Systems: Methodology and Applications, John Willey and Sons, 2008, to appear.

- Kuncheva L (2000) Fuzzy Classifiers, Physica-Verlag, ISBN 3-7908-1298-6.