Analytic computing of the infinite-time controllable abundance of the linear discrete-time systems

     In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems) , the controllable abundance $v_{c,N}$ of the linear discrete-time systems $\varSigma(A,B)$ is defines as follows

$v_{c,N}=\mathrm{Vol}(R_{c,N})$

where $R_{c,N}$ is the controllable region and $\mathrm{Vol}(\bullet)$ is the volume computing. For the SISO linear discrete-time systems, the controllable abundance $v_{c,N}$ , the volume of the controllable region $R_{c,N}$ can be computed as follows

$v_{c,N}=\sum_{(i_{1},i_{2},\cdots,i_{n})\in\Omega_{0,N-1}^{n}}\left|\mathrm{det}([A^{i_{1}}b,A^{i_{2}}b,\cdots,A^{i_{n}}b])\right|$

      If the n eigenvalues $\lambda_{i}(i=1,2,\cdots,n)$ of matrix $A$ are distinct and $\lambda_{i}\in[0,1)$ , the controllable abundance under $N\rightarrow\infty$ can be proven as

$\lim_{N\rightarrow\infty}v_{c,N}=\left|\left(\prod_{1\leq j_{1}

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