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

     In my blog article, “Analytic computing of the infinite-time controllable abundance of the linear discrete-time systems” (http://blog.sciencenet.cn/blog-3343777-1065279.html), the infinite-time controllable abundance of the SISO linear discrete-time systems $\Sigma(A,b)$ can be computed analytically as follows.

   (1) When the matrix $A$ is a diagonal matrix with the $n$ distinct eigenvalues $\lambda_{i}\in(0,1),i=1,2,\cdots,n$ , we have

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

where $[b_{1},b_{2},\cdots,b_{n}]^{T}=B$ .

    (2) When the matrix $A$ is an any matrix with the $n$ distinct eigenvalues $\lambda_{i}\in(0,1),i=1,2,\cdots,n$ , we have

$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 eigenvector 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 discrete-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 angles between the left eigenvectors of the system matrix $A$ and the input vector b.

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