Diagonal balanced truncation of discrete delay systems
The author introduces a new model reduction method for discrete delay systems, different from the traditional method, where the reduced system is also a discrete delay system of a smaller size in the dimension of the state. The author considers the following discrete state delay system \[ G:\begin{aligned} & x(k+1)=A_0x(k)+A_1x(k-N)+Bu(k)\\ & y(k)=Cx(k)+Du(k),\end{aligned} \] where \(x\in \mathbb{R}^n\) is a state, \(u\in\mathbb{R}^r\) is an input, \(y\in\mathbb{R}^p\) is an output, and \((A_0,A_1,B,C,D)\) are constant matrices of compatible size. For a realization \((A_0,A_1,B,C,D)\) of \(G\), one defines the notion of diagonal balanced realization with respect to a diagonal matrix \(\Sigma\in\mathbb{R}^{n\times n}>0\). It is proved that if there exist \(K=K'\in\mathbb{R}^{n\times n}>0\) and \(L=L'\in\mathbb{R}^{n\times n}>0\) satisfying \[ \left[\begin{smallmatrix} A_0 & A_1\\I & 0\end{smallmatrix}\right]\left[\begin{smallmatrix} K & \;\\ & L\end{smallmatrix}\right]\left[\begin{smallmatrix} A_0 & A_1\\ I & 0\end{smallmatrix}\right]-\left[\begin{smallmatrix} K & \;\\ & L\end{smallmatrix}\right] < 0, \] then there exists a diagonal balanced realization of \(G\) and, moreover, \(G\) is stable for all \(N\geq 1\). Also, the author presents a diagonal balanced truncation for discrete delay systems and he investigates the properties of the truncated system. This truncation can be computed irrespective of the state delay. Therefore it can be used for delay systems with a large state delay (where traditional truncation is computationally demanding) and with an unknown state delay (where traditional truncation cannot be used).
scientific article; zbMATH DE number 1355389