Soft Comput DOI 10.1007/s00500-014-1529-9 METHODOLOGIES AND APPLICATION Adaptive fuzzy sliding mode controller for a class of SISO nonlinear time-delay systems Hafedh Abid · Ahmed Toumi © Springer-Verlag Berlin Heidelberg 2014 Abstract In this paper, we propose an adaptive fuzzy con- troller for a class of nonlinear SISO time-delay systems. The plant model structure is represented by a Takagi–Sugeno (T– S) type fuzzy system. The T–S fuzzy model parameters are adjusted online. The proposed algorithm utilizes the sliding surface to adjust online the parameters of T–S fuzzy model. The controller is based on adjustable T–S fuzzy parameters model and sliding mode theory. The stability analysis of the closed-loop system is based on the Lyapunov approach. The plant state follows asymptotically any bounded reference sig- nal. Two examples have been used to check performances of the proposed fuzzy adaptive control scheme. Keywords T–S fuzzy system · Sliding mode control · Delay system · Lyapunov–Krasovskii function · Adaptive control 1 Introduction In the last decades, the field of adaptive control of non- linear systems using universal approximator function has attracted much of attention. However, these methods use neural networks or fuzzy logic systems to parameterize the unknown nonlinearities. In the literature, two mains approaches of adaptive control have been expanded (Naren- dra and Annaswamy 1989): the first is recognized as indi- rect adaptive control (IAC) (Ordóñez et al. 1997; Abid and Communicated by V. Loia. H. Abid (B ) · A. Toumi ENIS Sfax, Laboratory of Sciences and Techniques of Automatic control & computer engineering (Lab-STA), Sfax, Tunisia e-mail: abidhafedh@gmail.com A. Toumi e-mail: Ahmad.tomi@enis.rnu.tn Chtourou 2006), whereas the second is known as direct adap- tive control (DAC) (Ordóñez et al. 1997; Abid et al. 2012). All of the previously mentioned studies do not cover the state time delays, which often exist in various engineering systems such as communication networks, biological reactors, man- ufacturing systems and chemical processes. The time delay can frequently affect the performances of closed-loop sys- tem and may be considered as a source of instability. Con- sequently, over the past years, the stabilization problem of time-delay systems has received more attention. To palliate this difficulty, the Lyapunov–Krasovskii function has been used for stability analysis and synthesis. Also, stability analy- sis and control synthesis of Takagi–Sugeno fuzzy systems with delay have been proposed in many researches (Song et al. 2008; Li and Liu 2009). Particularly, approximation- based adaptive control has been treated for nonlinear time- delay systems. In Zhu et al. (2012), the proposed control scheme combines backstepping approach with fuzzy logic system to develop and adaptive control for time-delay sys- tems. The fuzzy logic systems are employed to estimate the unknown continuous functions. In Yongming et al. (2012), backstepping method has been combined with fuzzy systems to develop an adaptive fuzzy control law for nonlinear sys- tem with time delay. In Ge et al. (2004), the proposed fuzzy adaptive tracking controller is addressed for nonlinear strict- feedback systems with time delays. The control law guaran- tees that all the closed-loop signals remain bounded, for all bounded input signals. In Niu et al. (2005), the adaptive H∞ control has been studied on the basis of backstepping and fuzzy networks technique. In Wang et al. (2008), an adaptive fuzzy control scheme for nonlinear delayed systems with lower triangular form has been proposed. The fuzzy logic systems have been used to approximate the unknown non- linear function. In Ho et al. (2005), the stabilization prob- lem has been discussed for nonlinear strict-feedback sys- 123