Synthesizing reduced-order estimators for discrete random-parameter systems (Q1921523)

The author considers problems of synthesizing reduced-order estimators for a linear discrete-time random parameter system \[ \begin{aligned} x(k+1) &= A(\theta(k)) x(k)+w(k),\\ z(k) &= C(\theta(k)) x(k)+ v(k),\end{aligned} \] where \(x(k)\in \mathbb{R}^n\) is the state vector, \(z(k)\in\mathbb{R}^m\) is the observation vector, \(w(k)\) and \(v(k)\) are discrete white noise sequences, \(\theta(k)\) is a stationary random sequence. \(w(k)\), \(v(k)\) and \(\theta(k)\) are mutually uncorrelated. The matrices \(A\in\mathbb{R}^{n\times n}\) and \(C\in\mathbb{R}^{m\times n}\) depend linearly on \(\theta(k)\). The objective of this paper is to synthesize an unbiased linear estimator \(\widehat y(k)\in\mathbb{R}^r\), \(r<n\), and an operator \(D:\mathbb{R}^r\to\mathbb{R}^n\) such that \[ I=\lim_{k\to\infty} E\{[x(k)- D(y(k))]^T Q[x(k)-D(y(k))]\} \] is minimal, where \(Q\geq 0\) is a weight matrix. The author derived sufficient conditions of mean-square stochastic stability for the linear estimator. The synthesis is given via projection of an optimal linear full-order estimator technique.