Observable abundance of linear discrete-time systems

    In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems), a new measure on the control ability of the linear discrete-time systems(LDTS), named as the controllable abundance, is defined. The new measure can be generalized to measuring accurately the observe ability and then the observable abundance of LDTS can be defined as follows.

1. Definition of the unit observable region $R_{o,N}$ of LDTS

[Definition 1]. The $N$ -steps unit observable region $R_{o,N}$ of LDTS is the region in the state space, constituted by the all possible initial state $x_{0}$ that can be determined uniquely by the observe sequence $\{y_{k},k=0,1,\cdots,N-1\}$ of the output variables under the unite range $\left(\left\Vert y_{k}\right\Vert _{\infty}\leq1,k=0,1,\cdots,N-1\right)$ of the sensors or instruments.

2. Definition of the observable abundance of LDTS

[Definition 2]. The $N$ -steps observable abundance of LDTS is defined as the two-tuples $(r_{o,N},v_{o,N})$ , where $r_{o,N}$ and $v_{o,N}$ are the space dimension and volume of the unit observable region $R_{o,N}$ , respectively.

3. Computing on the observable abundance of LDTS $\varSigma(A,C)$

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

$P_{o,N}=\left[\begin{array}{c} C\\ CA\\ \vdots\\ CA^{N-1} \end{array}\right]$

3.2 $v_{o,N}=\mathrm{Vol}(R_{o,N})$ , where $\mathrm{Vol}(\bullet)$ is the volume function,

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

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

For these new definitions, what shape is the unit observable region $R_{o,N}$ ? how to compute its volume? which dual relation is existed in the unit observable region $R_{o,N}$ and unit controllable region $R_{c,N}$ ?