Commputing on the observable abundance of the linear discrete-time systems(LDTS) with the different real eigenvalue

      In my blog article “Geometric shape and volume computing of observable region of linear discrete-time systems”(http://blog.sciencenet.cn/blog-3343777-1071333.html), the observable abundance is defined and its computing equation is proposed as follows

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

     When the system is SISO system, the system matrix $A$ is a diagonal matrix with the different real eigenvalue $\lambda_{i}(i=1,2,\cdots,n)$ , and $C=[c_{1},c_{2},\cdots,c_{n}]$ , we have

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

And then, when $\lambda_{i}\in[0,1)(i=1,2,\cdots,n)$ , we have

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

    For the determinant value of the matrix $\widehat{P}$ , we can prove

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

where $G$ is a undetermined function about $\lambda_{i}(i=1,2,\cdots,n)$ .And then, the infinite-time onservable abundance of LDTS can be computed as follows.

      $\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}