在博文“离散系统能观域几何形状与体积计算”(http://blog.sciencenet.cn/home.php?mod=space&uid=3343777&do=blog&id=1067410)中,定义了能观丰富性并给出了其计算式如下

$v_{o,N}=\mathrm{Vol}(R_{o,N})=\left|(P_{o,N}^{T}P_{o,N})^{-1}\right|\mathrm{Vol}(\widetilde{R}_{o,N})$

当系统为SISO时，系统矩阵为 $A$ 为对角阵且特征值 $\lambda_{i}(i=1,2,\cdots,n)$ 都为单实根, $C=[c_{1},c_{2},\cdots,c_{n}]$ ,则有

$P_{o,N}^{T}P_{o,N}= =\left[\begin{array}{cccc} c_{1}^{2}\frac{1-\lambda_{1}^{2N}}{1-\lambda_{1}^{2}} & c_{1}c_{2}\frac{1-\lambda_{1}^{N}\lambda_{2}^{N}}{1-\lambda_{1}\lambda_{2}} & \cdots & c_{1}c_{n}\frac{1-\lambda_{1}^{N}\lambda_{n}^{N}}{1-\lambda_{1}\lambda_{n}}\\ c_{1}c_{2}\frac{1-\lambda_{1}^{N}\lambda_{2}^{N}}{1-\lambda_{1}\lambda_{2}} & c_{2}^{2}\frac{1-\lambda_{2}^{2N}}{1-\lambda_{2}^{2}} & \cdots & c_{2}c_{n}\frac{1-\lambda_{2}^{N}\lambda_{n}^{N}}{1-\lambda_{2}\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ c_{1}c_{n}\frac{1-\lambda_{1}^{N}\lambda_{n}^{N}}{1-\lambda_{1}\lambda_{n}} & c_{2}c_{n}\frac{1-\lambda_{2}^{N}\lambda_{n}^{N}}{1-\lambda_{2}\lambda_{n}} & \cdots & c_{n}^{2}\frac{1-\lambda_{n}^{2N}}{1-\lambda_{n}^{2}} \end{array}\right]$

      当 $\lambda_{i}\in[0,1)(i=1,2,\cdots,n)$ ，有

$\widehat{P}=\lim_{N\rightarrow\infty}P_{o,N}^{T}P_{o,N}=\left[\begin{array}{cccc} \frac{c_{1}^{2}}{1-\lambda_{1}^{2}} & \frac{c_{1}c_{2}}{1-\lambda_{1}\lambda_{2}} & \cdots & \frac{c_{1}c_{n}}{1-\lambda_{1}\lambda_{n}}\\ \frac{c_{1}c_{2}}{1-\lambda_{1}\lambda_{2}} & \frac{c_{2}^{2}}{1-\lambda_{2}^{2}} & \cdots & \frac{c_{2}c_{n}}{1-\lambda_{2}\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{c_{1}c_{n}}{1-\lambda_{1}\lambda_{n}} & \frac{c_{2}c_{n}}{1-\lambda_{2}\lambda_{n}} & \cdots & \frac{c_{n}^{2}}{1-\lambda_{n}^{2}} \end{array}\right]$

对 $\widehat{P}$ 的行列式值的值，有

$\det\left(\widehat{P}\right)=G\left[\prod_{1\leq j_{1}

此时，离散系统的无限时间能观丰富性为

      $\lim_{N\rightarrow\infty}v_{o,N}=\frac{1}{\det\left(\widehat{P}\right)}\left|\left(\prod_{1\leq j_{1}

              $=\frac{1}{G}\left|\left(\prod_{1\leq j_{1}