Perfect observers for singular 2D linear systems (Q5938122)

The concept of perfect observer is proposed for the class of singular 2D linear systems having the state space representation NEWLINE\[NEWLINE\begin{aligned} Ex_{i+1,j} & =A_0x_{ij}+ A_1x_{i,j+1}+ Bu_{ij}\\ y_{ij} & =Cx_{ij},\;i,j\in \mathbb{Z}^+ \tag{1}\end{aligned}NEWLINE\]NEWLINE where \(x_{ij}\in \mathbb{R}^n\), \(u_{ij}\in \mathbb{R}^m\) and \(y_{ij}\in \mathbb{R}^p\) are the state, input and output vectors respectively and \(E,A_0,A_1,B\) and \(C\) are real matrices of appropriate dimensions.NEWLINENEWLINENEWLINEA singular 2D system NEWLINE\[NEWLINE\begin{aligned} \widehat E\widehat x_{i+1,j} & =\widehat A_0\widehat x_{ij}+ \widehat A_1 \widehat x_{i,j+1}+ \widehat Bu_{ij}+ \widehat Dy_{ij}+ \widehat Fy_{i,j+1}\\ w_{ij}& =\widehat C\widehat x_{ij}+\widehat Hu_{ij}+ \widehat Gy_{ij},\;w_{ij},\widehat x_{ij}\in \mathbb{R}^n, \tag{2}\end{aligned}NEWLINE\]NEWLINE is called the perfect observer of the system (1) if \(w_{ij}=x_{ij}\), \(i,j\in \mathbb{Z}^+\), for any boundary conditions for (1) and (2).NEWLINENEWLINENEWLINENecessary conditions are established for the existence of a perfect observer of (1) with \(\widehat E=E\), \(\widehat A_0=A_0+K_1C\), \(\widehat A_1=A_1+K_2C\), \(\widehat B=B\), \(\widehat D=-K_1\), \(\widehat F=-K_2\), \(\widehat C=I_n\), \(\widehat H=\widehat G=0\), where \(K_1,K_2\in \mathbb{R}^{n\times p}\).NEWLINENEWLINENEWLINEA procedure for determining such a perfect observer is derived, and it is illustrated by an example.