The controllable abundance of linear discrete-time systems

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

Definition 1. The uint contrllable region $R_{c,N}" original="http://latex.codecogs.com/gif.latex?R_{c,T}}" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;line-height:25.2px;$ is constituted by all initial state $x_0" original="http://latex.codecogs.com/gif.latex?x_0" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$ that can be controlled to the origin of the state space of the linear discre-time systems with the unit input energy $\left ( \left \| u_k \right \|_\infty \leq \leq 1 \right )" original="http://latex.codecogs.com/gif.latex?\left ( \left \| u(t) \right \|_\infty \leq 1 \right )" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$ in the finite sampling steps N.

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

Definition 2. The controllable abundance of linear discrete-time systems is defined as the two-tuples $\left ( r_N,v_{c,N} \right )" original="http://latex.codecogs.com/gif.latex?(r_T,v_{c,T})" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$ , where $r_N" original="http://latex.codecogs.com/gif.latex?r_T" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$ and $v_{c,N}" original="http://latex.codecogs.com/gif.latex?v_{c,T}}" style="margin:0px;padding:0px;word-wrap:break-word;max-width:620px;$ are the space dimension and the volume of the unit controllable region $R_{c,N}" original="http://latex.codecogs.com/gif.latex?R_{c,T}}" style="font-family:'times new roman';font-size:14px;line-height:25.2px;margin:0px;padding:0px;word-wrap:break-word;max-width:620px;background-color:#ffffff;$ ,  respectively.

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

3.1   $r_n=\textup{rank} P_{c,N}$

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

3.2   $v_{c,N}=\textup{Volume}(R_{c,N})$

where $R_{c,N}=\left \{ x_0 | x_0=-P_{c,N}u_{0,N-1},\left \| u_{0,N-1} \right \|_\infty \leq 1 \right \}$

          $u_{0,N-1}=\left [ u^T_{N-1}, u^T_{N-2}, ..., u^T_0 \right ]^T$

4. The Computation of The Volume of The polyhedron $R_{c,N}" original="http://latex.codecogs.com/gif.latex?R_{c,T}}" style="font-family:'times new roman';font-size:14px;line-height:25.2px;margin:0px;padding:0px;word-wrap:break-word;max-width:620px;background-color:#ffffff;$

$\textup{Volume}(R_{c.N})=2^{r_n}V_{r_N}\left ( C_{r_N}(P_{c,N}) \right )$

where the definitions and computations of  the volume function $V_n(\cdot )$ and the polyhedron $C_n(\cdot )$ 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)