Major factors determined the controllable abundance of the linear continuous-time systems

In my blog article, “Approximate computation of the infinite-time controllable abundance for linear continuous-time systems” (http://blog.sciencenet.cn/blog-3343777-1066456.html), when the matrix $A$ of the SISO linear continuous-time systems $\Sigma(A,b)$ is an any matrix with the n distinct eigenvalues $\lambda_{i}>0,i=1,2,\cdots,n$ , the infinite-time controllable abundance of the systems can be computed approximately as follows.

$v_{c,\infty}=\lim_{N\rightarrow\infty}V_{n}(C_{n}(G_{N}))=\left|\det(P)\right|\left|\left(\prod_{1\leq j_{1}

where the 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 $A$ .

Based on the above results, we can conclude that the major factors determined the controllable abundance of the SISO linear continuous-time systems with the n distinct eigenvalues are as:

        ⅰ. the magnitude of the eigenvalues.

        ⅱ. the distribution of eigenvalues(the eigenvalues are distinct and their distribution are uniform).

        iii. the product of all elements of the each column of the matrix $B$ in the Jordan canonical form;

        iv. the angles among the columns of the matrix $B$ (the determinant of matrix $B^{T}B$ in multi-input case)

        v. the angles between the left eigenvectors of the system matrix $A$ and the column of the matrix $B$

        vi. the angles between the any two left eigenvectors of the system matrix A.