A study on ILC for linear discrete systems with single delay

论文题目就是随笔的题目，以后的随笔的命名都是如此，特此说明。
博客园的文章也是我的blog，请勿转载
论文的主要内容是偏理论研究的，引入了离散矩阵延迟指数函数，来处理具有单时滞线性离散系统。对于离散延迟矩阵指数函数其定义为：
emGt:={Θ,if t∈Z−∞−m−1E,if t∈Z−m0E+Gt!1!(t−1)!+G2(t−m)!2!(t−m−2)!+⋯+Gs(t−(s−1)m)!s!(t−(s−1)m−s)!if t∈Z(s−1)(m+1)+1s(m+1),s=1,2,…e_{m}^{G t}:=\left\{\begin{array}{l} \Theta, \quad \text { if } t \in \mathbb{Z}_{-\infty}^{-m-1} \\ E, \quad \text { if } t \in \mathbb{Z}_{-m}^{0} \\ E+G \frac{t !}{1 !(t-1) !}+G^{2} \frac{(t-m) !}{2 !(t-m-2) !}+\cdots+G^{s} \frac{(t-(s-1) m) !}{s !(t-(s-1) m-s) !} \\ \text { if } t \in \mathbb{Z}_{(s-1)(m+1)+1}^{s(m+1)}, \quad s=1,2, \ldots \end{array}\right. emGt:=⎩⎪⎪⎪⎨⎪⎪⎪⎧Θ, if t∈Z−∞−m−1E, if t∈Z−m0E+G1!(t−1)!t!+G22!(t−m−2)!(t−m)!+⋯+Gss!(t−(s−1)m−s)!(t−(s−1)m)! if t∈Z(s−1)(m+1)+1s(m+1),s=1,2,…
现在解释一下单时滞系统，我的理解是在系统的状态方程，状态的x(k)与之前状态有关，百科的解释是信号传递有时间延迟的系统，如下所示：
{xk(t+1)=Axk(t)+A1xk(t−m)+Buk(t),t∈Z0Txk(t)=φ(t),t∈Z−m0yk(t)=Cxk(t)+Duk(t)\left\{\begin{array}{l} x_{k}(t+1)=A x_{k}(t)+A_{1} x_{k}(t-m)+B u_{k}(t), \quad t \in \mathbb{Z}_{0}^{T} \\ x_{k}(t)=\varphi(t), t \in \mathbb{Z}_{-m}^{0} \\ y_{k}(t)=C x_{k}(t)+D u_{k}(t) \end{array}\right. ⎩⎨⎧xk(t+1)=Axk(t)+A1xk(t−m)+Buk(t),t∈Z0Txk(t)=φ(t),t∈Z−m0yk(t)=Cxk(t)+Duk(t)
第一式的第一项与k-m有关。
论文更加偏向理论性的研究和证明，提出的控制律为uk+1(t)=uk(t)+L1⋅ek(t)u_{k+1}(t)=u_{k}(t)+L_{1} \cdot e_{k}(t)uk+1(t)=uk(t)+L1⋅ek(t)，作者主要是利用离散延迟矩阵来对单时滞系统的状态进行转换，转换成只含有控制信号的表达式，如下所示：
xk(t)=AtemB1tA−mφ(−m)+∑j=−m+10A(t−j)emB1(t−m−j)[φ(j)−Aφ(j−1)]+∑j=1tA(t−j)emB1(t−m−j)Buk(j−1)\begin{aligned} x_{k}(t)=& A^{t} e_{m}^{B_{1} t} A^{-m} \varphi(-m)+\sum_{j=-m+1}^{0} A^{(t-j)} e_{m}^{B_{1}(t-m-j)}[\varphi(j)-A \varphi(j-1)] \\ &+\sum_{j=1}^{t} A^{(t-j)} e_{m}^{B_{1}(t-m-j)} B u_{k}(j-1) \end{aligned} xk(t)=AtemB1tA−mφ(−m)+j=−m+1∑0A(t−j)emB1(t−m−j)[φ(j)−Aφ(j−1)]+j=1∑tA(t−j)emB1(t−m−j)Buk(j−1)
文章对离散矩阵延迟指数函数有一个简化的表示，利用G的特征值来表示，如下所示：
emGt=Pdiag⁡(emλ1t,emλ2t,…,emλnt)P−1,t∈Ze_{m}^{G t}=\operatorname{Pdiag}\left(e_{m}^{\lambda_{1} t}, e_{m}^{\lambda_{2} t}, \ldots, e_{m}^{\lambda_{n} t}\right) P^{-1}, \quad t \in \mathbb{Z} emGt=Pdiag(emλ1t,emλ2t,…,emλnt)P−1,t∈Z
代码见Github

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A study on ILC for linear discrete systems with single delay

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