The observable abundance of the linear continuous-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 the linear continuous-time systems(LCTS) can be defined as follows.

1. Definition of the unit observable region $R_{o,T}$ of LCTS

     [Definition 1] The $T$ -time unit observable region $R_{o,T}$ of LCTS 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 $\{y_{t},t\in[0,T]\}$ 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 observable abundance of LCTS

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

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

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

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

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

$R_{o,T}=\left\{ \left.x_{0}\right|y_{t}=Ce^{At}x_{0},\left\Vert y_{t}\right\Vert _{\infty}\leq1,t\in[0,T]\right\}$   (1)

From the definition equation (1), we have

$R_{o,T} =\left\{ \left.x_{0}\right|x_{0}=W{}_{o,T}^{-1}z,\forall z\in\widetilde{R}_{o,T}\right\}$

where $W_{o,T}$ and $\widetilde{R}_{o,T}$ are the observable Gramm matrix of the LCTS and the continuous geometry in $n$ -dimension as follows,.

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

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

    Therefore, the $T$ -time unit observable region $R_{o,T}$ is a geometry that is obtained by the rotation transformation (linear space transformation) of the geometry $\widetilde{R}_{o,T}$ with the transformation matrix $W_{o,T}^{-1}$ . The shape, boundary and volume of the gerometry $\widetilde{R}_{o,T}$ is same as the reachable region $R_{r,T}$ of LCTS. The volumes of $R_{o,T}$ and $\widetilde{R}_{o,T}$ satisfy

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

that is, the $T$ -time observable abundance $v_{o,T}$ of LCTS can be computed as follows

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

     Whether the observable 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 observable 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?