fuzzy3

and the uniqueness of the solution of this equation
depend on the choice of the weighting matrices and the
attenuation level used in the equation. Furthermore,
the choice of the attenuation level directly affects the
transient time of the control signal.
Proc. Instn Mech. Engrs Vol. 218 Part I: J. Systems and Control Engineering
approximator theorem, Wang [7, 8] has presented an
adaptive fuzzy TS-type controller where the conclusions
are updated online. The process is approximated by
Abstract: This paper presents two approaches for the fuzzy sliding mode control of uncertain and disturbed non-linear multi-input multi-output (MIMO) systems. In the first approach, the control law is deduced from a nominal model of the plant. An adaptive fuzzy system is introduced to substitute the switching control and, hence, to eliminate the chattering phenomenon. In the second approach, two other adaptive fuzzy systems are used to approximate and thereby to overcome the unavailability of the nominal model. The stability and the robustness are proven analytically in both cases, and the adaptation laws of the fuzzy systems are deduced from Lyapunov synthesis. A simulation example of a two-link robot is used for illustration.
system. Thus, this approach combines the intelligence and
the versatility of fuzzy control on the one hand and the
rigorous mathematical background of classical techniques
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A HAMZAOUI, N ESSOUNBOULI AND J ZAYTOON
Sliding mode control based approaches have also been used because of their relative simplicity of implementation and their robustness against both plant uncertainties and external disturbances. The tracking of the desired trajectory is achieved through two phases: an approach phase, where the system is controlled to attain a predefined sliding surface, and a sliding phase along the sliding surface [14, 15]. However, the commutation function, in the established control, requires infinite switching of the control signal to maintain the system dynamics on the sliding surface. In practice, actuator limitations, signal delays and other factors prevent true sliding and lead to the chattering phenomenon. One method to eliminate chattering consists in defining a boundary layer around the sliding surface [15], but this entails a larger response time. To overcome this problem, it is necessary to find a trade-off between the thickness of the boundary layer and the time response.
the tracking performance. Several improvements have
therefore been introduced for single-input single-output
(SISO) systems [9–12] and for multi-input multi-output
adaptive fuzzy systems and the adaptation law acts
directly on the numeric conclusions. Lyapunov theory is
used to demonstrate the global stability of the closed-loop
(MIMO) systems [13]. In these approaches, the robustness
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from the existence
I04603 © IMechE 2004
be compatible with the classic tools of control theory.

Therefore, the analysis of stability and robustness becomes
possible. In this sense, the TS fuzzy models form a
The MS was received on 16 May 2003 and was accepted after revision for publication on 9 February 2004. * Corresponding author: Laboratoire d’Automatique et de Microe´lectronique, IUT de Troyes, 9 rue de Que´bec, BP 396, 10026 Troyes Cedex, France.
of adaptive control on the other. However, the resulting
controller cannot attenuate or eliminate the effect of
both external disturbances and approximation errors on
particular class of non-linear systems.
Because industrial processes are becoming increasingly
complex, uncertain and time-varying, adaptive fuzzy
control can be an alternative. Based on the universal
A second fuzzy controller approach was developed in the 1980s by Takagi and Sugeno (TS) [5, 6 ]. In this approach the conclusions of the rules are numeric and the controllers are formulated in an analytical form to
Keywords: sliding mode control, fuzzy logic systems, robustness, non-linear uncertain systems
1 INTRODUCTION
Control problems today require the development of new approaches and an intelligent combination of existing ones to be able to deal with the increasingly complex and sophisticated processes. One approach consists in exploiting methods used by the human expert that in certain cases have given better results than mathematical techniques for the control of complex industrial processes. The theory of fuzzy sets, introduced by Zadeh [1], can be considered as a means of translating this human expertise to a set of rules from which a fuzzy controller can be deduced. The first approaches relating to fuzzy control [2–4] dealt with the so-called symbolic conclusion or Mamdani controller. These controllers require a large calculation time for rule aggregation and for defuzzification. Therefore, they can only be used in the case of slow systems not subject to severe real-time constraints. Furthermore, these approaches are heuristic in nature and do not provide stability and robustness.
287
Fuzzy sliding mode control with a fuzzy switching function for non-linear uncertain multi-input multi-output systems
A Hamzaoui*, N Essounbouli and J Zaytoon Laboratoire d’Automatique et de Microe´lectronique, Faculte´ des Sciences, Moulin de la Housse, Reims, France