### Abstract

Recently there has been significant growth in the use of fuzzy logic in industrial and consumer products (J. Yen 1995). However, although fuzzy control has been successfully applied to many industrial plants that are mostly nonlinear systems, many critics of fuzzy logic claim that there is no such thing as a stability proof for fuzzy logic systems in closedloop control (Reznik 1997; Farinwata, Filev et al. 2000). Since fuzzy logic controllers are classified as "non-linear multivariable controllers" (Reznik 1997; Farinwata, Filev et al. 2000), it can be argued that all stability analysis methods applicable to these controller types are applicable to fuzzy logic controllers. Unfortunately, due to the complex non-linearities of

most fuzzy logic systems, an analytical solution is not possible. Furthermore, it is important to realize that real, practical problems have uncertain plants that inevitably cannot be modelled dynamically resulting in substantial uncertainties. In addition the sensors noise and input signal level constraints affect system stability. Therefore a theory that is able to deal with these issues would be useful for practical designs. The most well-known time domain stability analysis methods include Lyapunov’s direct method (Wu & Ch. 2000;

Gruyitch, Richard et al. 2004; Rubio & Yu 2007) which is based on linearization and Lyapunov’s indirect method (Tanaka & Sugeno 1992; Giron-Sierra & Ortega 2002; Lin, Wang et al. 2007; Mannani & Talebi 2007) that uses a Lyapunov function which serves as a generalized energy function. In addition many other methods have been used for testing fuzzy systems stability such as Popov’s stability criterion (Katoh, Yamashita et al. 1995; Wang & Lin 1998), the describing function method (Ying 1999; Aracil & Gordillo 2004), methods of stability indices and systems robustness (Fuh & Tung 1997; Espada & Barreiro 1999; Zuo & Wang 2007), methods based on theory of input/output stability (Kandel, LUO

et al. 1999), conicity criterion (Cuesta & Ollero 2004). Also there are methods based on hyper-stability theory (Piegat 1997) and linguistic stability analysis approach (Gang & Laijiu 1996).

most fuzzy logic systems, an analytical solution is not possible. Furthermore, it is important to realize that real, practical problems have uncertain plants that inevitably cannot be modelled dynamically resulting in substantial uncertainties. In addition the sensors noise and input signal level constraints affect system stability. Therefore a theory that is able to deal with these issues would be useful for practical designs. The most well-known time domain stability analysis methods include Lyapunov’s direct method (Wu & Ch. 2000;

Gruyitch, Richard et al. 2004; Rubio & Yu 2007) which is based on linearization and Lyapunov’s indirect method (Tanaka & Sugeno 1992; Giron-Sierra & Ortega 2002; Lin, Wang et al. 2007; Mannani & Talebi 2007) that uses a Lyapunov function which serves as a generalized energy function. In addition many other methods have been used for testing fuzzy systems stability such as Popov’s stability criterion (Katoh, Yamashita et al. 1995; Wang & Lin 1998), the describing function method (Ying 1999; Aracil & Gordillo 2004), methods of stability indices and systems robustness (Fuh & Tung 1997; Espada & Barreiro 1999; Zuo & Wang 2007), methods based on theory of input/output stability (Kandel, LUO

et al. 1999), conicity criterion (Cuesta & Ollero 2004). Also there are methods based on hyper-stability theory (Piegat 1997) and linguistic stability analysis approach (Gang & Laijiu 1996).

Original language | English |
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Title of host publication | New Approaches in Automation and Robotics |

Number of pages | 20 |

Publication status | Published - 2007 |

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### Keywords

- fuzzy control
- automation
- robotics

### Cite this

Soraghan, J. J. (2007). Fuzzy stabilization of fuzzy control systems. In

*New Approaches in Automation and Robotics*