The simplified computing of the 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 of the linear discrete-time systems is defined as follows

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

where $R_{c,N}$ is the controllable region of the systems, $\mathrm{Vol}(\bullet)$ is the volume computing. It can be proven that the volume computing of the controllable region $R_{c,N}$ is equal to the volume computing of the parallel polyhedron $C_{n}(G{}_{0,N-1}))$ that is produced by the vectors $\{g_{i},i=1,2,\cdots,r\times N\}$ of the matrix $G_{0,N-1}=\{B,AB,\cdots,A^{N-1}B\}(N\geq n+1)$ , where $A$ and $B$ are the $n\times n$ matrix and $n\times r$ matrix, respectively. The corresponding computing equation is as

$V_{n}(C_{n}(G_{0,N-1}))=\sum_{(i_{1},i_{2},\cdots,i_{n})\in\hat{\Omega}_{0,N-1}^{n}}\left|\mathrm{det}([g_{i_{1}},g_{i_{2}},\cdots,g_{i_{n}}])\right|$

where $\hat{\Omega}_{0,N-1}^{n}$ is a set constructed by the the all permutations $(i_{1},i_{2},\cdots,i_{n})$ that pick n different numbers from the set $\{1,2,\cdots,r\times N\}$ and is sorted by there values.

      The above volume computing can be calculated recursively as follows

$V_{n}(C_{n}(G_{0,N-1}))&=\left(1+\left|\mathrm{det}(A)\right|\right)V_{n}(C_{n}(G_{0,N-2}))-\left|\mathrm{det}(A)\right|V_{n}(C_{n}(G_{0,N-3}))" original="http://latex.codecogs.com/gif.latex?V_{n}(C_{n}(G_{0,N-1}))&=\left(1+\left|\mathrm{det}(A)\right|\right)V_{n}(C_{n}(G_{0,N-2}))-\left|\mathrm{det}(A)\right|V_{n}(C_{n}(G_{0,N-3}))" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$

        $+\sum_{j=1}^{r}\sum_{k=1}^{r}\sum_{(i_{1},\cdots,i_{j})\in\hat{\Omega}_{0,0}^{j}}\sum_{(i_{j+1},\cdots,i_{n-k})\in\hat{\Omega}_{1,N-2}^{n-j-k}}\sum_{(i_{n-k+1},\cdots,i_{n})\in\hat{\Omega}_{N-1,N-1}^{k}}\left|\mathrm{det}([g_{i_{1}},g_{i_{2}},\cdots,g_{i_{n}}])\right|" original="http://latex.codecogs.com/gif.latex?+\sum_{j=1}^{r}\sum_{k=1}^{r}\sum_{(i_{1},\cdots,i_{j})\in\hat{\Omega}_{0,0}^{j}}\sum_{(i_{j+1},\cdots,i_{n-k})\in\hat{\Omega}_{1,N-2}^{n-j-k}}\sum_{(i_{n-k+1},\cdots,i_{n})\in\hat{\Omega}_{N-1,N-1}^{k}}\left|\mathrm{det}([g_{i_{1}},g_{i_{2}},\cdots,g_{i_{n}}])\right|" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$

When $r=1$ , we have, $B=[b]$ . And then the above recursive equation can be simplified as follows

  $V_{n}(C_{n}(G_{0,N-1}))&=\left(1+\left|\mathrm{det}(A)\right|\right)V_{n}(C_{n}(G_{0,N-2}))-\left|\mathrm{det}(A)\right|V_{n}(C_{n}(G_{0,N-3}))" original="http://latex.codecogs.com/gif.latex?V_{n}(C_{n}(G_{0,N-1}))&=\left(1+\left|\mathrm{det}(A)\right|\right)V_{n}(C_{n}(G_{0,N-2}))-\left|\mathrm{det}(A)\right|V_{n}(C_{n}(G_{0,N-3}))" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$

                                    $\qquad+\sum_{(i_{2},\cdots,i_{n-1})\in\hat{\Omega}_{1,N-2}^{n-2}}\left|\mathrm{det}([b,A^{i_{2}}b,\cdots,A^{i_{n-1}}b,A^{N-1}b])\right|" original="http://latex.codecogs.com/gif.latex?\qquad+\sum_{(i_{2},\cdots,i_{n-1})\in\hat{\Omega}_{1,N-2}^{n-2}}\left|\mathrm{det}([b,A^{i_{2}}b,\cdots,A^{i_{n-1}}b,A^{N-1}b])\right|" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;display:inline;$

            Based on the above computing equations, the controllable abundance of the linear discrete-time systems can be gotten conveniently and can be used effectively for optimizing the dynamics and kinematics of the open-loop controlled plants and promoting the performances and robustness of the close-loop control systems.