کنترل یادگیری تکراری با یادگیری حالت اولیه جهت سفارش کسری سیستم های غیر خطی

This paper presents a P-type iterative learning control (ILC) scheme with initial state learning for a class of View the MathML sourceα(0≤α<1) fractional-order nonlinear systems. By introducing the λλ-norm and using a generalized Gronwall inequality, the sufficient condition for the robust convergence of the tracking errors with respect to initial positioning errors under P-type ILC is obtained. Based on this convergence condition, the learning gain of the initial learning and input learning updating law can be determined. Unlike the existing methods, the ILC scheme will not fix the initial value on the expected condition at the beginning of each iteration. Finally, the validity of the methods are verified by a numerical example.

Fractional differential calculus dates back from the 17th century, but only until the recent decade was it applied to physics and engineering [1]. It was found that many systems in interdisciplinary fields could be elegantly described with the help of fractional derivatives and integrals, such as viscoelastic systems, dielectric polarization, electrode–electrolyte polarization and electromagnetic waves [2]. Furthermore, many fractional order controllers have so far been implemented to enhance the robustness and the performance of the control systems [3], [4] and [5]. Iterative learning control (ILC) is one of the most active fields in control theories. The objective of ILC is to determine a control input iteratively, resulting in the plant’s ability to track the given reference signal or the output trajectory over a fixed time interval. Owing to its simplicity and effectiveness, ILC has been found to be a good alternative in many areas and applications, e.g., see recent surveys [6], [7] and [8] for detailed results. In recent years, the application of ILC to the fractional-order system has become a new topic [3], [9], [10], [11] and [12]. The authors in [9] were the first to propose the DαDα-type ILC algorithm in frequency domain. In [10], the asymptotic stability of View the MathML sourcePDα-type ILC for fractional-order linear time invariant(LTI) systems was investigated. The convergence condition of open-loop P-type ILC for fractional-order nonlinear systems was tried in [11]. Moreover, the DαDα-type ILC for fractional-order LTI systems was discussed in [3] and [12]. However, in the above methods, the ILC algorithm must fix the initial value on the expected condition at the beginning of each iteration. Motivated by the above mentioned research to the tracking problem of fractional-order systems, the open-loop and closed-loop P-type ILC updating law with initial state learning are applied to a class of fractional-order nonlinear systems. A sufficient condition for the robust convergence of the tracking errors under the proposed P-type ILC is proved. Based on this convergence condition the learning gains can be determined. The rest of this paper is organized as follows. In Section 2, the problem formulation and some preliminaries are presented. The P-type ILC scheme as well as the convergent condition for fractional-order nonlinear time-delay systems is discussed in Section 3. MATLAB/SIMULINK results are shown in Section 4. Finally, some conclusions are drawn in Section 5.

In this paper, based on the generalized Gronwall inequality and the property of fractional calculus, the robust convergence of the tracking errors with respect to initial positioning errors under P-type ILC scheme for a class of fractional-order nonlinear systems with fractional order View the MathML sourceα(0≤α<1) has been proven theoretically. The ILC scheme would not fix the initial value on the expected condition at the beginning of each iteration. The validity of the method is verified by a numerical example. Our future works include the applications of the proposed P-type iterative learning scheme and the DαDα-type ILC updating law for fractional-order nonlinear systems.