Approximate computation of the infinite-time controllable abundance for linear continuous-time systems

In my blog article “Approximate computation of the controllable abundance for the linear continuous-time systems”(http://blog.sciencenet.cn/blog-3343777-1068213.html), by discreting the continuous-time models, the approximate computing methods of the finite-time controllable abundance for the linear continuous-time systems are proposed.

When the matrix $A$ and eigenvalues $\eta_{i}(i=1,2,\cdots,n)$ of the SISO linear continuous-time systems $\Sigma(A,B)$ satisfy that

     1) the matrix $A$ is a diagonal matrix, i.e., $A=\mathrm{diag-matrix}\{\eta_{1},\eta_{2},\cdots,\eta_{n}\}$ ,

     2) all eigenvalues are differential,

     3) all eigenvalues are real and positive, i.e., the eigenvalues $\eta_{i}>0(i=1,2,\cdots,n)$

the systems matrix and input matrix of the corresponding discrete systems $\Sigma(G,H)$ are respectively

      $G=\mathrm{diag-matrix}\{e^{-\eta_{1}\Delta},e^{-\eta_{2}\Delta},\cdots,e^{-\eta_{n}\Delta}\}$

      $H=\left[\frac{b_{1}}{\eta_{1}}\left(1-e^{-\eta_{1}\Delta}\right),\frac{b_{2}}{\eta_{2}}\left(1-e^{-\eta_{2}\Delta}\right),\cdots,\frac{b_{n}}{\eta_{n}}\left(1-e^{-\eta_{n}\Delta}\right)\right]$

where $\Delta$ is the sampling step, $[b_{1},b_{2},\cdots,b_{n}]=B^{T}$ . And then, the infinite-time controllable abundance of the continuous-time systems $\Sigma(A,B)$ can be computed approximately as follows

$\lim_{N\rightarrow\infty}\mathrm{Vol}\left(R_{dx}\right)= =\left|\left(\prod_{1\leq j_{1}

     When the systems $A$ is not a diagonal matrix and but its eigenvalues are differential, real, ans positive, the infinite-time controllable abundance can be computed approximately as follows

$\lim_{N\rightarrow\infty}\mathrm{Vol}\left(R_{dx}\right)=\left|P\right|\left|\left(\prod_{1\leq j_{1}

where the transformation matrix $P$ is composed of the all right eigenvectors of the system matrix $A$ , the row vector $q_{i}$ is the left eigenvector corresponding the eigenvalue $\lambda_{i}$ of the matrix $A$ .