3阶有复根的线性离散系统的能控丰富性计算

     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=3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)

      $A=\left[\begin{array}{ccc} \lambda_{1} & 0 & 0\\ 0 & \sigma & \mu\\ 0 & -\mu & \sigma \end{array}\right]=\left[\begin{array}{ccc} \lambda_{1} & 0 & 0\\ 0 & \rho_{A}\cos\theta & \rho_{A}\sin\theta\\ 0 & -\rho_{A}\sin\theta & \rho_{A}\cos\theta \end{array}\right]" original="http://latex.codecogs.com/gif.latex?A=\left[\begin{array}{ccc} \lambda_{1} & 0 & 0\\ 0 & \sigma & \mu\\ 0 & -\mu & \sigma \end{array}\right]=\left[\begin{array}{ccc} \lambda_{1} & 0 & 0\\ 0 & \rho_{A}\cos\theta & \rho_{A}\sin\theta\\ 0 & -\rho_{A}\sin\theta & \rho_{A}\cos\theta \end{array}\right]" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$

      $B=\left[\begin{array}{c} b_{1}\\ b_{2}\\ b_{3} \end{array}\right]=\left[\begin{array}{c} b_{1}\\ \rho_{B}\cos\delta\\ \rho_{B}\sin\delta \end{array}\right]" original="http://latex.codecogs.com/gif.latex?B=\left[\begin{array}{c} b_{1}\\ b_{2}\\ b_{3} \end{array}\right]=\left[\begin{array}{c} b_{1}\\ \rho_{B}\cos\delta\\ \rho_{B}\sin\delta \end{array}\right]" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$

其中

    $\rho_{A}=\left(\sigma^{2}+\mu^{2}\right)^{1/2},\quad\theta=\arctan\frac{\mu}{\sigma} " original="http://latex.codecogs.com/gif.latex?\rho_{A}=\left(\sigma^{2}+\mu^{2}\right)^{1/2},\quad\theta=\arctan\frac{\mu}{\sigma} " style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;display:inline;$

    $\rho_{B}=\left(b_{2}^{2}+b_{3}^{2}\right)^{1/2},\quad\delta=\arctan\frac{b_{3}}{b_{2}}" original="http://latex.codecogs.com/gif.latex?\rho_{B}=\left(b_{2}^{2}+b_{3}^{2}\right)^{1/2},\quad\delta=\arctan\frac{b_{3}}{b_{2}}" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;display:inline;$

则生成能控域的向量组

$\widehat{A}_{N}=\left[A^{-N}B,A^{-N-1}B,\cdots,A^{-1}B\right]$

此时有能控丰富性计算如下

  $V_{3}(C_{3}(\widehat{A}_{N}))=\left|b_{1}\rho_{B}^{2}\right|\lambda_{1}^{-1}\rho_{A}^{-2}\sum_{(k_{1},k_{2},k_{3})\in\Omega_{0}^{3},N-1}\left|\lambda_{1}^{-k_{1}}\rho_{A}^{-k_{2}-k_{3}}\sin[(k_{3}-k_{2})\theta]\right.$

                  $\qquad\left.-\lambda_{1}^{-k_{2}}\rho_{A}^{-k_{1}-k_{3}}\sin[(k_{3}-k_{1})\theta]+\lambda_{1}^{-k_{3}}\rho_{A}^{-k_{1}-k_{2}}\sin[(k_{2}-k_{1})\theta]\right|\noindent$

当 $\lambda_{1}=\rho_{A}$ 时，对上述能控丰富性的上界可估计如下

$V_{3}(C_{3}(\widehat{A}_{N}))\leq\frac{3\sqrt{3}\left|b_{1}\rho_{B}^{2}\right|}{2}\left[\frac{1-\lambda_{1}^{-3N+6}}{\left(\lambda_{1}-1\right)\left(\lambda_{1}^{2}-1\right)\left(\lambda_{1}^{3}-1\right)}+\frac{\lambda_{1}^{-2N+1}-\lambda_{1}^{-3N+3}-\lambda_{1}^{-N}+\lambda_{1}^{-3N+4}}{\left(\lambda_{1}-1\right)^{2}\left(\lambda_{1}^{2}-1\right)}\right]$

当 $\lambda_{1}=\rho_{A}>1$ 时，有

$\lim_{N\rightarrow\infty}V_{3}(C_{3}(\widehat{A}_{N}))\leq\frac{3\sqrt{3}\left|b_{1}\rho_{B}^{2}\right|}{2\left(\lambda_{1}-1\right)\left(\lambda_{1}^{2}-1\right)\left(\lambda_{1}^{3}-1\right)}$