Nonstatistical formulation and solution of a filtering problem (Q1060183)

The system \(x_ i=Ax_{i-1}+\xi_ i\), \(y_ i=F_ ix_ i+\eta_ i\) is considered. Here the n-vectors \(x_ i\) are unknown, the m-vectors \(y_ i\) are observed. The noises \(\xi_ i\), \(\eta_ i\) satisfy the estimates \((W_ i^{-1}\xi_ i,\xi_ i)\leq w\), \((B_ i^{-1}\eta_ i,\eta_ i)\leq c_ i\) and otherwise are arbitrary. The matrices A, \(F_ i\), \(W_ i=W^*_ i>0\), \(B_ i=B^*_ i>0\) and the values of \(W,c_ i\) are known, the matrix A is stable. It is necessary to obtain an estimate \(\hat x_ i\) of \(x_ i\), i.e. to construct matrices \(H_ i=H^*_ i>0\) and vectors \(\hat x_ i\) such that \((H(x_ i-\hat x_ i),(x_ i-\hat x_ i))\leq 1\). A recursive algorithm for the construction of \(\hat x_ i,..\). \(H_ i\) is given. It is shown that the sequences \(\| \hat x_ i\|,..\). \(\| H_ i\|\) are bounded, if \(\sqrt{1+W}\| A\| <\sqrt{1-n^{-2}}\) and also \(\| \hat x_ i-x_ i\| \to 0\) (for \(i\to \infty)\) if \(W=0\) and the sequences \(\hat x_ i,..\). \(H_ i\) satisfy some restrictions. The results of computer simulation are given. Some generalizations of this problem are formulated.