A Numerical Approach To Generalized Periodic Sylvester Matrix Equation

Abstract

The paper is dedicated to solving the generalized periodic discrete‐time Sylvester matrix equation, which is frequently encountered in control theory and applied mathematics. An iterative algorithm for this equation is presented. The proposed method is developed from a point of least squares method. The rationality of the method is testified by theoretical analysis. The presented approach is numerically reliable and requires less computation. Two numerical examples illustrate the effectiveness of the raised results.

Hierarchical multi‐innovation generalised extended stochastic gradient methods for multivariable equation‐error autoregressive moving average systems

“…5. Use (19) to calculate the noise vector ŵ (t) and use (20) to calculate the residual vector v^(t). 6.…”

“…For example, a filtering multi-innovation identification approach was proposed for the multivariable system with moving average noise [18]. Typically the measured data can be run through the recursive or iterative schemes for solving matrix equations [19][20][21][22] and for deriving system identification approaches. Recently, the issue of the model recovery was discussed in terms of the non-linear Hammerstein systems via the technique of the orthogonal matching, which is combined the auxiliary model technique [23] and the hierarchical scheme [24].…”

Moving data window gradient‐based iterative algorithm of combined parameter and state estimation for bilinear systems

“…The problem we face is that the parameter matrix A u (t) and vector b are unknown. Therefore, the state estimation in Equations (28) to (30) cannot be realized. The solution is that using the parameter estimates…”

“…Then, the estimate u,k (t) of A u (t) can be obtained by u,k (t) ∶= k (t) +M k (t)u(t). Replacing A u (t) and b in Equations (28) to (30) with their estimates u,k (t) andb k (t), and replacing the unknown information vector v (t), the parameter vector v in Equations (28) to (30) with their estimateŝv ,k (t) in Equation (17) and̂v ,k (t) in Equation (20) givê…”

“…25,26 The iterative algorithm processes a batch of measurement data and makes efficient use of the measurement data at each iteration. [27][28][29] Hence, some iterative methods can be used for finding the solutions to some matrix equations 30,31 and for deriving new signal modeling 32,33 and parameter identification algorithms. 34,35 On the basis of the iterative schemes, this paper aims to design a dynamical moving data window (MDW) to update the collected data to identify the bilinear system.…”