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Iterative Methods Of Several Matrix Equations For Systems And Control

Posted on:2016-10-26Degree:DoctorType:Dissertation
Country:ChinaCandidate:H M ZhangFull Text:PDF
GTID:1108330464965550Subject:Control theory and control engineering
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Matrix equations are often encountered in systems and control theory. How to solve these matrix equations has become an important topic in the mathematics and control theory. The direct method uses the Kronecker product, which triggers the high-dimensional matrices and causes the greatest computation. Some nonlinear matrix equations can not be solved directly. The iterative method becomes an important strategy in finding the matrix equation solution. Several di?erent matrix equations of systems and control are studied by using the iterative strategy. The selected topics of the thesis make a great significance in theory. The major work of this thesis includes:1. By using the hierarchical identification principle, the gradient-based iterative algorithm is established for the generalized coupled matrix equations. It is proved that the algorithm is convergent and a su?cient condition is given for the convergence. By constructing an objective function and using the gradient and the gradient search principle, three iterative algorithms of the generalized coupled Sylvester matrix equations are established. It is proved that the proposed iterative algorithms are convergent. By the convergence analysis, the optimal convergence factor of these iterative algorithms is given.2. Inspired by the least squares-based iterative algorithm and to extend the least squares-based algorithm, a class of iterative algorithm is proposed for solving matrix equation. A class of matrix related to the symmetric positive definite matrix is studied.The range of the eigenvalues of these matrices is explored. By using the range of these eigenvalues, the convergence of this iterative algorithm is proved. The optimal convergence factor of the least squares-based iterative algorithm is settled.3. By introducing the convergence factor and the iterative matrix, the hierarchical identification principle is used to solve the nonlinear matrix equation. An iterative algorithm is established for solving a nonlinear matrix equation. The convergence of this algorithm is proved by using the properties of the symmetric positive definite matrix.Several conclusions are obtained by the convergence analysis and quadratic convergence of the algorithm is obtained.4. By using the real inner product of complex matrices, linear operator, conjugate linear operator and the orthogonality of the finite inner product space, a finite iterative algorithm for solving coupled complex matrix equations is established. The analysis shows that the matrix sequence generated by this algorithm is orthogonal. By using this orthogonal sequence, the finite step convergence of this algorithm is proved.In summary, several matrix equations of systems and control are studied and several iterative algorithms are established, respectively. The e?ectiveness of the proposed algorithms are illustrated by using numerical examples.
Keywords/Search Tags:hierarchical identification principle, gradient-based iterative algorithm, matrix equation, convergence factor, eigenvalue, eigenvector, least squares-based iterative algorithm, finite iterative algorithm
PDF Full Text Request
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