2阶有复根的线性离散系统的能达丰富性计算(整理中)

     本人的文章arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems) 里定义了线性离散系统的controllable abundance(能控丰富性、能控充裕性)及其计算。当系统状态空间维数 $n=2$ 且矩阵 $A$ 的特征根为一对复根,即系统矩阵可表示为(或经变换可表示为)

        $A =\left[\begin{array}{cc} \sigma & \mu\\ \mu & \sigma \end{array}\right]=\lambda\left[\begin{array}{cc} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{array}\right]$

        $B=\left[\begin{array}{c} b_{1}\\ b_{2} \end{array}\right]=\rho\left[\begin{array}{c} \cos\delta\\ \sin\delta \end{array}\right]$

其中

      $\rho=\textrm{sgn}(b_{1})\left(b_{1}^{2}+b_{2}^{2}\right)^{1/2},\quad\delta=\arctan\frac{b_{2}}{b_{1}}\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right)$

      $\lambda=\textrm{sgn}(\sigma)\left(\sigma^{2}+\mu^{2}\right)^{1/2},\quad\theta=\arctan\frac{\mu}{\sigma}\in\left[\frac{-\pi}{2},\frac{\pi}{2}\right)$

则有能达丰富性计算如下

      $\lambda=1:\quad V_{2}(C_{2}(A_{N}))=\rho^{2}\times\sum_{k=1}^{N-1}(N-k)\left|\sin(k\theta)\right|$

      $\lambda\neq1:\quad V_{2}(C_{2}(A_{N}))=\rho^{2}\times\sum_{k=1}^{N-1}\left|\sin(k\theta)\right|\frac{\lambda^{k}-\lambda^{2N-k}}{1-\lambda^{2}}$

当 $\left|\lambda\right|<1$ ,有

$\lim_{N\rightarrow\infty}V_{2}(C_{2}(A_{N}))=\frac{\rho^{2}\theta\left(1+\lambda^{\pi/\theta}\right)}{\left(1-\lambda^{2}\right)\left[\theta^{2}+\ln^{2}\lambda\right]\left(1-\lambda^{\pi/\theta}\right)}$