# ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems
## 2 Preliminaries and Problem Statement
: _When the fractional order \(\alpha=1\), the fractional order distributed parameter System (1) is reduced to an integral order one, which has been widely investigated in [11; 16; 17; 18; 19]._  
The corresponding initial and boundary conditions of System (1) are:  
\[\alpha q_{k}(x,t)+\beta\frac{\partial q_{k}(x,t)}{\partial\
u}=0, \tag{2}\] \[q_{k}(x,0)=q_{k0}(x),\ x\in\Omega. \tag{3}\]  
where \((x,t)\in\partial\Omega\times[0,T]\), \(\frac{\partial}{\partial\
u}\) is the unit outward normal derivative on \(\partial\Omega\) and \(\alpha\) and \(\beta\) are known constant diagonal matrices, satisfying:  
\[{}_{t_{0}}^{C}\mathcal{D}_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{t_{ 0}}^{t}(t-\tau)^{-\alpha+n-1}f^{(n)}(\tau)d\tau,\]0 \[{}_{t_{0}}^{C}\mathcal{D}_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{t_{ 0}}^{t}(t-\tau)^{-\alpha+n-1}f^{(n)}(\tau)d\tau,\]1 (5) \[{}_{t_{0}}^{C}\mathcal{D}_{t}^{\alpha}f(t)=\frac{1}{\Gamma(n-\alpha)}\int_{t_{ 0}}^{t}(t-\tau)^{-\alpha+n-1}f^{(n)}(\tau)d\tau,\]2  
The design objective in this paper is to find an iterative learning control law to generate the control input \(u_{k}(x,t)\) such that the system output \(y_{k}(x,t)\) tracks the desired output trajectory \(y_{d}(x,t)\) as accurately as possible when \(k\) goes to infinity for all \((x,t)\in\Omega\times[0,T]\). To this end, the updating law considered in this manuscript is a second order P-type ILC updating law given by:  
\[\left\{ \begin{array}{l}u_{k+1}(x,t)=\gamma u_{k}(x,t)+(1-\gamma)u_{k-1}(x,t) \\ \phantom{u_{k+1}(x,t)=}+\Lambda e_{k}(x,t)+\Gamma e_{k+1}(x,t),\\ q_{k+1}(x,0)=q_{k}(x,0)+Le_{k+1}(x,0),\end{array} \right. \tag{7}\]  
where \(e_{k}(x,t)=y_{d}(x,t)-y_{k}(x,t)\) denotes the tracking error, \(u_{0}(x,t),\ u_{1}(x,t)\) are two initial inputs, \(0<\gamma<1\) is a constant and \(\Lambda\), \(\Gamma\) and \(L\) are unknown parameters to be determined. **Remark 2**. : _The identical initial condition is commonly used in ILC. As pointed out in [37], it will limit the application of ILC. Many modified strategies have been used to avoid this restriction. Borrowed from [37; 38], we designed a P-type ILC updating law (7) with initial state learning where the strict identical initial condition is not required._  
Throughout, this paper, \(\mathbb{R}^{n}\) denotes an \(n\)-dimensional Euclidean space, and \(I\) means an identity matrix of appropriate order. For the \(n\)-dimensional vector \(W=(w_{1},w_{2},\cdots,w_{n})^{T}\), its two-norm for the \(n\)-dimensional vector \(w=(w_{1},\ w_{2},\ \cdots,\ w_{n})\) is defined as \(\|w\|=\sqrt{\sum_{i=1}^{n}w_{i}^{2}}\), and the spectrum norm of the \(n\times n\) order square matrix \(A\) is \(\|A\|=\sqrt{\lambda_{max}(A^{T}A)}\), where \(\lambda_{max}\) represents the maximum eigenvalue. \(\rho(A)\) denotes the spectral radius of matrix \(A\). Let \(L^{2}(\Omega)\) be the Hilbert space. If \(Q_{i}\in L^{2}(\Omega)(i=1,\ 2,\ \cdots,n)\), we define \(Q=(Q_{1},\ Q_{2},\ \ \cdots,\ Q_{n})\in\mathbf{R}^{n}\bigcap L^{2}(\Omega)\), then \(\|Q\|_{L^{2}}=\{\int_{\Omega}(Q(x)^{T}Q(x))dx\}^{\frac{1}{2}}\). For the function \(f(x,t):\ \Omega\times[0,T]\rightarrow\mathbf{R}^{n},f(x,t)\in\mathbf{R}^{n} \bigcap L^{2}(\Omega),\ t\in[0,T]\), we define the norm of \((L_{2},\ \lambda)\) as:  
\[\|f\|_{(L_{2},\ \lambda)}=\sup_{t\in[0,\ T]}\{e^{-\lambda t}\|f\|_{L^{2}}^{2}\}, \ \lambda>0.\]  
For the function sequence \(f_{k}(x,t):\ \Omega\times[0,T]\rightarrow\mathbf{R}^{n}\), \(k\in\{1,\ 2,\ \cdots\}\), the norm of \((L_{2},\ \lambda,\ \xi)\) is defined as:  
\[\|f_{k}\|_