The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise
Given the n-th-order observed system \[ x(k+1)=(A+\sum^{p}_{i=1}v_ i(k)A_ i)x(k)+W_ 1(k),\quad y(k)=(C+\sum^{p}_{i=1}v_ i(k)C_ i)x(k)+W_ 2(k) \] where \(W_ 1(k)\), \(W_ 2(k)\) are n-dimensional and \(\ell\)-dimensional white noise processes, \(v_ i(k)\) is an unit variance white noise, A, \(A_ i\) are \(n\times n\) matrices and C, \(C_ i\) are \(\ell \times n\) matrices, \(i=1,...,p\). The problem is to design an \(n_ e\)-th \((1\leq n_ e\leq n)\) reduced-order state estimator \[ x_ e(k+1)=A_ ex_ e(k)+B_ ey(k),\quad y_ e(k)=C_ ex_ e(k)+D_ ey(k) \] (x\({}_ e\), \(y_ e\) denote \(n_ e\)-dimensional and q- dimensional vectors, \(A_ e\), \(B_ e\), \(C_ e\), \(D_ e\) are \(n_ e\times n_ e\), \(n_ e\times \ell\), \(q\times n_ e\), \(q\times \ell\) matrices) which minimizes the state-estimation error criterion \[ J(A_ e,B_ e,C_ e,D_ e)=\lim_{k\to \infty}E[Lx(k)-y_ e(k)]^ T=R[Lx(k)-y_ e(k)] \] where L and R are some matrices. The optimality conditions for the \(n_ e\)-th reduced-order state estimator for the matrices \(A_ e\), \(B_ e\), \(C_ e\), \(D_ e\) are given. The optimal estimator is characterized by three matrix equations (a modified Riccati equation and two modified Lyapunov equations).
scientific article; zbMATH DE number 4001995
scientific article