# ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
## 1 Introduction  
Iterative learning control (ILC) is an effective technique for improving the performance of systems that operate repetitively over a fixed time interval [1; 2; 3; 4]. The fundamental idea in ILC is that the task to be carried out by the system is of a repetitive nature. By using such a repetitive nature, it is possible to adjust the input signal such that the output signal follows the reference signal as closely as possible. Owing to its simplicity and effectiveness, ILC is playing an important role in many areas and applications [5; 6; 7]. Distributed parameter systems are a class of complicated infinite-dimensional systems, whose states depend on both spatial position and time [8; 9; 10; 11]. In recent years, the application of ILC to distributed parameter systems has become a new topic [12; 13; 14; 15]. In [12], an open-loop P-type iterative learning controller was designed for the first order hyperbolic distributed parameter system. The P-type and D-type ILC algorithms were studied in [13] for a class of parabolic distributed parameter systems. For a class of single-input single-output coupling nonlinear distributed parameter systems, a P-type learning controller was designed, and the convergence conditions, speed and robustness of the iterative learning algorithm were also discussed in [14]. For uncertain nonlinear distributed parameter systems and state time-delay distributed parameter systems, the P-type ILC was investigated in [15; 16], respectively. A D-type anticipatory ILC scheme was applied to the boundary control of a class of inhomogeneous heat equations in [17]. In [18], a framework in the frequency domain of ILC was designed for linear inhomogeneous distributed parameter systems. By using the Gronwall-Bellman inequality, a close-loop P-type iterative learning algorithm is proposed for linear parabolic distributed parameter systems in [19]. In addition, based on the eigenspectrum and reduced order model, the ILC was proposed for a class of parabolic distributed parameter systems in [20]. Note that the above existing ILC methods for distributed parameter systems only focus on the lower order ILC. That is, to update the input profile, the learning algorithms use only the information from the last cycle. As we all known, the higher order learning algorithm incorporates the information not only from the last cycle, which can lead to better performance in terms of both robustness andconvergence rate [21; 22; 23]. In [24], for a class of parabolic linear distributed parameter systems with uncertainty coefficients, an ILC algorithm with a forgetting factor was proposed, and the conditions for the convergence of the algorithm were established. On the other hand, in recent years, fractional order control systems have attracted increasing interest. 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 [25; 26; 27]. In [28], a fractional order distributed parameter system was used to describe radial ground water flow. The existence and uniqueness of mild and classical solutions for a class of distributed order fractional differential equations had been studied in [29]. By using the differential inequality method, some sufficient conditions for the oscillation of solutions for partial fractional differential equations were presented in [30]. A fundamental solution of a fractional order distributed parameter system was presented in [31]. The numerical solutions of such systems were proposed in the literature by means of finite difference methods [32], spectral collocation methods and others [33; 34]. However, it is worth pointing out that up to now, there is no concern about ILC for fractional order distributed parameter systems. The purpose of this paper is to present a second order P-type ILC algorithm with initial state learning for fractional order linear distributed parameter systems.