Abstract
Owing to the superior capability of fractional differential equations in modeling and characterizing accurate dynamical properties of many high technology real world systems, the design and control of fractional-order systems have captured lots of attention in recent decades. In this paper, an adaptive intelligent fuzzy approach to controlling and stabilization of nonlinear non-autonomous fractional-order systems is proposed. Since dynamic equations of applied fractional-order systems usually contain various parameters and nonlinear terms, the Takagi–Sugeno (T–S) fuzzy models with if-then rules are adopted to describe the system dynamics. Also, as the nonlinear system parameters are assumed to be unknown, adaptive laws are derived to estimate such fluctuations. Simple adaptive linear-like control rules are developed based on the T–S fuzzy control theory. The stability of the resulting closed loop system is guaranteed by Lyapunov’s stability theory. Two illustrative numerical examples are presented to emphasize the correct performance and applicability of the proposed adaptive fuzzy control methodology. It is worth to notice that the proposed controller works well for stabilization of a wide class of either autonomous nonlinear uncertain fractionalorder systems or non-autonomous complex systems with unknown parameters.
1 Introduction
Fractional-Order (FO) calculus has a long history and a retrospect as long as three centuries, and it has attracted increasing attentions in physics and engineering in recent years [1–4]. Nowadays, it has been known that the nature of many real phenomena can be perfectly characterized and modeled using fractional differential equations and many dynamical systems in various applied fields, such as biology [5], medicine [6], physics [7, 8], electro-mechanics [9] and social sciences [10, 11].
5 Conclusions
In this paper, the problem of control and stabilization of uncertain non-autonomous fractional-order systems is investigated. First, the intelligent Takagi–Sugeno (T–S) fuzzy models with if-then rules are constructed to represent the system dynamics. Then, an adaptive approach is adopted to estimate the unknown parameters and uncertainties of the system. Subsequently, based on the T–S fuzzy control technique and Lyapunov’s stability theorem, linear-like control rules associated with some gain matrices provided to ensure that the system states will approach to zero as time goes infinite. After that, the developed controller is applied for stabilization of a large class of fractional-order non-autonomous systems. Two numerical examples are also presented to validate the analytical results of the article and to illustrate that the designed adaptive schemes are feasible in real world applications. It is worth to note that the results of this paper can be applied for control of real fractionalorder systems, such as fractional-order electrical circuits and mechatronic devices, in spite of having limited knowledge about the time-variant and time-invariant parameters of the system. Extending the results of this paper for design of T–S fuzzy controllers for fractional-order systems with input saturation remains as the future work of the authors.
