Consistent Estimates of the Parameters of a Linear System

Positive definite solutions of the nonlinear matrix equation X+AHX¯-1Zhou1, Cai2, Lam3 2013Applied Mathematics and Computation320190View full textAdd to dashboardCite

“…However, X = Y ∞ is also a solution of the nonlinear matrix equation (1). Hence, the nonlinear matrix equation (1) has the maximal solution X + = Y ∞ .…”

“…Hence, by applying Lemma 6 again, X − I n − Y + is also a solution of equation (1). In fact, X − is the minimal positive definite solution of equation (1). Otherwise assume that X * ≤ X − is a positive definite solution of (1).…”

“…In fact, X − is the minimal positive definite solution of equation (1). Otherwise assume that X * ≤ X − is a positive definite solution of (1). Then Y * = I n − X * ≥ I n − X − = Y + , is a positive definite solution of (6).…”

“…Therefore, by adopting the existing results on the nonlinear matrix equation (7), we can get corresponding results on the original nonlinear matrix equation (1).…”

“…Especially, the problem of finding fixed points of the nonlinear matrix equation X + A * X −1 A = Q where A and Q > 0 are given and X is unknown, has been extensively studied in the last two decades. The interest of studying such problem mainly relies on its applications in many fields such as the analysis of ladder networks [2], dynamic programming [33], control theory [22], stochastic filtering [1] and statistics [32] (see [3], [18] and [19] for detailed introduction). Some generalized forms of the nonlinear matrix equation X + A * X −1 A = Q have received much attention in recent years (see, for example, [6], [17], [24], and [30]).…”