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Shape and volume computing of observable region of LDTS

已有 2079 次阅读 2017-8-16 10:43 |个人分类:能观丰富性|系统分类:科研笔记

The geometric shape and the volume computing of the observable region of the linear discrete-time systems


     In my blog article “Observable abundance of linear discrete-time systems” (http://blog.sciencenet.cn/blog-3343777-1071227.html), the $N$ -steps observable abundance $v_{o,N}$ of the linear discrete-time systems(LDTS) is defined as the volume of the $N$ -steps unit observable region $R_{o,N}$ as follows

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

     The $N$ -steps unit controllable region $R_{c,N}$ of LDTS is a parallel polyhedron in $n$ -dimensions space. Which geometric shape is the $N$ -steps unit observable region $R_{o,N}$ defined as above? How to compute its volume?

     From the definition equation (1), we have

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

where $\widetilde{R}_{o,N}$ is a parallel polyhedron defined as follows.

$\widetilde{R}_{o,N}=\left\{ \left.z\right|z=P_{o,N}^{T}y_{0,N-1},\left\Vert y_{0,N-1}\right\Vert _{\infty}\leq1\right\}$

Therefore, the $N$ -steps unit observable region $R_{o,N}$ is a geometry that is obtained by the rotation transformation (linear space transformation) of the geometry $\widetilde{R}_{o,N}$ with the transformation matrix $(P_{o,N}^{T}P_{o,N})^{-1}$ , and then $R_{o,N}$ is a parallel polyhedron also. The volumes of $R_{o,N}$ and $\widetilde{R}_{o,N}$ satisfy

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

that is, the $N$ -steps observable abundance $v_{o,N}$ of LDTS can be computed as follows

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

        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?




https://blog.sciencenet.cn/blog-3343777-1071333.html

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