Computing on the reconstructable abundance of the linear continuous-time systems with the different real roots

     In my blog article “Reconstructable abundance of the linear continuous-time systems”(http://blog.sciencenet.cn/blog-3343777-1083864.html), the reconstructable abundance of the linear continuous-time systems(LCTS), as a new measure for metering the state reconstruct ability, is defined and discussed in detailed, and the computing equation on the new measure is as follows

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

      If the LCTS $\varSigma(A,C)$ is in SISO case and the system matrix $A$ is (or can be transformed as) a diagonal matrix that its eigenvalues $\lambda_{i}(i=1,2,\cdots,n)$ are differential and real root, we have,

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

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

where $C=[c_{1},c_{2},\cdots,c_{n}]$ .

     When $\lambda_{i}\in[0,+\infty)(i=1,2,\cdots,n)$ , we have

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

and its determinant is

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

and then, the infinite-time reconstructable abundance of the LCTS

    $\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}

      The above analytic computing on the infinite-time reconstructable abundance can help to optimize and design on the reconstruct ability, and is with the great significance to promote the computing efficiency. For the practical LCTS, the reconstructable abundance on the observation and filtering of the state $x_{T}$ at the current time $T$ is with more meaningful than the observable abundance on that of the state $x_{0}$ at the initial time.