The reachable abundance of linear discrete-time systems

1. Definition of The Unit Reachable Region $$ of Linear Discrete-time Systems

[Definition 1]. The uint contrllable region $R_{r,N}$ is constituted by all the $N$ -th step state $x_{N}$ that can be reached from origin of the state space of the linear discre-time systems with the unit input energy $\left(\left\Vert u_{k}\right\Vert _{\infty}\leq1,k=0,1,\cdots,N-1\right)$ in the finite sampling steps $N$ .

2. Definition of The Reachable Abundance of Linear Discrete-time Systems

[Definition 2]. The reachable abundance of linear discrete-time systems is defined as the two-tuples $(r_{N},v_{r,N})$ , where $r_{N}$ and $v_{r,N}$ are the space dimension and the volume of the unit reachable region $R_{r,N}$ , respectively.  

3. The Computation of The Reachable Abundance of Linear Discrete-time Systems $\varSigma(A,B)$

3.1 $r_{N}=\mathrm{rank\;}P_{r,N}$

where $P_{r,N}=\left[B,AB,\cdots,A^{n*-1}B\right],\quad n*=\min\{n,N\}$

3.2 $v_{r,N}=\mathrm{Vol}(R_{r,N})$

where

      $R_{r,N}=\left\{ \left.x_{N}\right|x_{N}=P_{r,N}u_{0,N-1},\left\Vert u_{0,N-1}\right\Vert _{\infty}\leq1\right\}$

      $u_{0,N-1}=\left[u_{N-1}^{T},u_{N-2}^{T},\cdots,u_{0}^{T}\right]^{T}$

4. The Computation of The Volume of The polyhedron $R_{r,N}$

$\mathrm{Vol}(R_{r,N})=2^{r_{n}}V_{r_{n}}\left(C_{r_{n}}(P_{r,N})\right)$

where the definitions and computations of the volume function $V_{n}(\bullet)$ and the polyhedron $C_{n}(\bullet)$ are in the "The volume computing of a special polyhedron in n-dimensions space"

Some results are in my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems)