Optimal filter reduction via a double projection. (Q2734314)

It is known that in many applications there arises the need of reducing the filter order. Solutions have been given for single or multiple variables cases by the use of orthogonal projection. The digital system under attention (multivariable case) is of order \(n\) (the order of the matrix \(A)\) NEWLINE\[NEWLINEx(k+1)= A\cdot x (k)+B\cdot x(k),\quad y(k)=C\cdot x(k),NEWLINE\]NEWLINE where \(x\) is the state vector and \(A,B,C\) are constant matrices of adequate sizes. It is known that the operational transfer function associated to the considered digital system is NEWLINE\[NEWLINEG(z)= C\cdot (zI-A)^{-1}\cdot B,NEWLINE\]NEWLINE where \(z\) is the complex operational variable and \(I\) the unit matrix of size \(n\). It is asked to find a digital system of rank \(r<n\) (the reduced order system) with the transfer function \(G_r\) so that the norm (a \(H_2\) one) of the difference NEWLINE\[NEWLINEG(z)-G_r(z)NEWLINE\]NEWLINE be minimum. The feature of the present paper lies in choosing \(G_r(z)\) in the following form NEWLINE\[NEWLINEG_r(z)=C\cdot U\cot (z\cdot I-V^T\cdot A\cdot U )^{-1}\cdot V^T\cdot B,NEWLINE\]NEWLINE where \(U\) and \(V\) (whence the term of double projection) belong to the Stiefel manifold, i.e. NEWLINE\[NEWLINEU,V\in\{X\in \mathbb{R}^{n \times r} \mid X^TX=I\}.NEWLINE\]NEWLINE Two solutions to find \(U\) and \(V\) insuring the minimization of the norm difference are proposed: a) deriving a pair of ordinary differentions giving the two matrices, and b) an algorithm (proved to be convergent). Some experiments for \(n=22\), \(r=2\) are presented which validate the consistency of the double projection method. We think the paper represent an interesting contribution in the area of reduced order systems.