Reconstructable abundance of the linear continuous-time systems

     In my paper arXiv1705.08064(On Controllable Abundance Of Saturated-input Linear Discrete Systems), the controllable abundance as a new measure for metering the control ability is defined and discussed in detailed. In fact, the new measure can be generalized to the studying on the observe ability and reconstruction ability for linear dynamic systems, and then the reconstructable abundance, as a new measure for accurately describing the reconstruction ability of the state variables of linear continuous-time systems(LCTS), is defined as follows.

1. Definition of unit reconstructable region $R_{s,T}$ of LCTS

    [Definition 1] The $T$ -time unit reconstructable region $R_{s,T}$ of LCTS is a state region constituted by the all possible $T$ -time state $x_{T}$ that can be determined uniquely by the observe sequence \ ${y_{t},t\in[0,T]\}$ of the output variables under the unite range $\left(\left\Vert y_{t}\right\Vert _{\infty}\leq1,t\in[0,T]\right)$ of the sensors or instruments.

2. Definition of the unit reconstructable abundance of LCTS

     [Definition 2] The $T$ -time unit reconstructable abundance of LCTS is defined as the two-tuples $(r_{s,T},v_{s,T})$ , where $r_{s,T}$ and $v_{s,T}$ are the space dimension and volume of the unit reconstructable region $R_{s,T}$ , respectively.

3. Computing on the reconstructable abundance of LCTS $\varSigma(A,C)$ in $n$ -dimensions state space

3.1 $r_{s,T}=\mathrm{rank\;}P_{s,n}$ ,  where

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

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

$R_{s,T}=\left\{ \left.x_{T}\right|y_{t}=Ce^{A(t-T)}x_{T},\left\Vert y_{t}\right\Vert _{\infty}\leq1,t\in[0,T]\right\}$

   The above definition equation of $R_{s,T}$ can be rewritten as follows

$R_{s,T}=\left\{ \left.x_{T}\right|x_{T}=W{}_{s,T}^{-1}z,\forall z\in\widetilde{R}_{s,T}\right\}$

where $W_{s,T}$ and $\widetilde{R}_{s,T}$ are the following reconstructable Gramm matrix and a geometry in n-dimensions space, respectively.

      $W_{s,T}=\int_{0}^{T}e^{-A^{T}t}C^{T}Ce^{-At}\mathrm{d}t$

      $\widetilde{R}_{s,T}=\left\{ \left.z\right|z=\int_{0}^{T}e^{A^{T}(t-T)}C^{T}y_{t}\mathrm{d}t,\left\Vert y_{t}\right\Vert _{\infty}\leq1,t\in[0,T]\right\}$

In fact, the shape, boundary, and volume computing of the geomrtry $\widetilde{R}_{s,T}$ is coincident with the reachable region that can describe the reachable abundance of LCTS, and the reconstrcutable region $R_{s,T}$ can be obtained by the rotation transformation (linear space transformation) of the geometry $\widetilde{R}_{s,T}$ with the transformation matrix $W_{s,T}^{-1}$ . Therefore, the volumes of $R_{s,T}$ and $\widetilde{R}_{s,T}$ satisfy

$\mathrm{Vol}(R_{s,T})=\left|W_{s,T}^{-1}\right|\mathrm{Vol}(\widetilde{R}_{s,T})$

that is, the reconstructable abundance $v_{s,T}$ can be computed as follows

$v_{s,T}=\mathrm{Vol}(R_{s,T})=\left|W_{s,T}\right|^{-1}\mathrm{Vol}(\widetilde{R}_{s,T})$

       Whether the reconstructable abundance defined as above is with the great meaningful for the optimizing of the system dynamics, the optimizing and designing of the control systems, same as the controllable abundance? Whether the reconstructable abundance will turn into a fundamental measure and concept in the related fields, such as dynamic system analysis, state estimation, filtering of the dynamic systems, network communication, and so on?