特征根为单实根的连续系统的能重构丰富性计算

     在博文“线性连续系统的能重构丰富性”(http://blog.sciencenet.cn/blog-3343777-1068621.html)中,定义了能重构丰富性并给出了其计算式如下

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

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

      $W_{s,T}=\int_{0}^{T}e^{-A^{T}t}C^{T}Ce^{-At}\mathrm{d}t$

          $=-\left[\begin{array}{cccc} c_{1}^{2}\frac{e^{-2\lambda_{1}T}-1}{2\lambda_{1}} & c_{1}c_{2}\frac{e^{-(\lambda_{1}+\lambda_{2})T}-1}{\lambda_{1}+\lambda_{2}} & \cdots & c_{1}c_{n}\frac{e-^{(\lambda_{1}+\lambda_{n})T}-1}{\lambda_{1}+\lambda_{n}}\\ c_{1}c_{2}\frac{e^{-(\lambda_{1}+\lambda_{2})T}-1}{\lambda_{1}+\lambda_{2}} & c_{2}^{2}\frac{e^{-2\lambda_{2}T}-1}{2\lambda_{2}} & \cdots & c_{2}c_{n}\frac{e^{-(\lambda_{2}+\lambda_{n})T}-1}{\lambda_{2}+\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ c_{1}c_{n}\frac{e^{-(\lambda_{1}+\lambda_{n})T}-1}{\lambda_{1}+\lambda_{n}} & c_{2}c_{n}\frac{e^{-(\lambda_{2}+\lambda_{n})T}-1}{\lambda_{2}+\lambda_{n}} & \cdots & c_{n}^{2}\frac{e^{-2\lambda_{n}T}-1}{2\lambda_{n}} \end{array}\right]$

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

$\widehat{W}=\lim_{N\rightarrow\infty}W_{s,N}=\left[\begin{array}{cccc} \frac{c_{1}^{2}}{2\lambda_{1}} & \frac{c_{1}c_{2}}{\lambda_{1}+\lambda_{2}} & \cdots & \frac{c_{1}c_{n}}{\lambda_{1}+\lambda_{n}}\\ \frac{c_{1}c_{2}}{\lambda_{1}+\lambda_{2}} & \frac{c_{2}^{2}}{2\lambda_{2}} & \cdots & \frac{c_{2}c_{n}}{\lambda_{2}+\lambda_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{c_{1}c_{n}}{\lambda_{1}+\lambda_{n}} & \frac{c_{2}c_{n}}{\lambda_{2}+\lambda_{n}} & \cdots & \frac{c_{n}^{2}}{2\lambda_{n}} \end{array}\right]$

且其行列式值为

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

此时线性连续时间系统的无穷时间的能重构丰富性为    

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

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

     对特征根为单根时的系统的无穷时间的能重构丰富性能解析计算，为能重构丰富性的优化设计，提高计算效率具有重要意义。由于对于实际的状态观测和滤波问题，关注当前状态 $x_{T}$ 观测和滤波问题的能重构丰富性应比关注初始状态 $x_{0}$ 观测和滤波问题的能观丰富性更有意义。