Iterative method to the coupled operator matrix equations with sub-matrix constraint and its application in control

Abstract

A finite iterative algorithm is presented for solving the numerical solutions to the coupled operator matrix equations in Zhang (2017b). In this paper, a new finite iterative algorithm is presented for solving the constraint solutions to the coupled operator matrix equations $[\sum_{i = 1}^{p} A_{1 i} (X_{i}), \sum_{i = 1}^{p} A_{2 i} (X_{i}), \dots, \sum_{i = 1}^{p} A_{pi} (X_{i})] = [M_{1}, M_{2}, \dots, M_{p}]$ , where the constraint solutions include symmetric solutions, bisymmetric solutions and reflexive solutions as special cases. If this system is consistent, for any initial constraint matrices, the exact constraint solutions can be obtained by the introduced algorithm within finite iterative steps in the absence of the roundoff errors. Also, if this system is not consistent, the least-norm constraint solutions can be obtained within the finite iteration steps in the absence of the roundoff errors. Furthermore, if a group of suitable matrices are given, the optimal approximation solutions can be derived. Finally, several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Keywords

Operator matrix equation linear operator least-norm solution optimal approximation solution

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