Matrix GPBiCG algorithms for solving the general coupled matrix equations

• Published in: IET control theory & applications Vol. 9; no. 1; pp. 74 - 81
• Main Author: Hajarian, Masoud
• Format: Journal Article
• Language: English
• Published: The Institution of Engineering and Technology 02.01.2015
• Subjects: Algorithms; conjugate gradient methods; Control systems; control theory; coupled Markovian jump Lyapunov matrix equation; general coupled matrix equations; generalised product bi‐conjugate gradient; generalised Sylvester matrix equation; GPBiCG algorithm; Joining; Kronecker product; linear matrix equations; Markov processes; Mathematical analysis; Mathematical models; matrix algebra; Operators; system theory; Systems theory; vectorisation operator; conjugate gradient methods; generalised Sylvester matrix equation; control theory; coupled Markovian jump Lyapunov matrix equation; Kronecker product; generalised product bi-conjugate gradient; general coupled matrix equations; linear matrix equations; vectorisation operator; matrix algebra; GPBiCG algorithm; system theory
• ISSN: 1751-8644; 1751-8652; 1751-8652
• DOI: 10.1049/iet-cta.2014.0669

Linear matrix equations have important applications in control and system theory. In the study, we apply Kronecker product and vectorisation operator to extend the generalised product bi-conjugate gradient (GPBiCG) algorithms for solving the general coupled matrix equations ∑lj=1(A)i,1,jX1Bi,1,j+Ai,2,jX2Bi,2,j+…+Ai,l,jXi,l,j) = Di  for  i = 1,2,…,l (including the (coupled) Sylvester, the second-order Sylvester and coupled Markovian jump Lyapunov matrix equations). We propose four effective matrix algorithms for finding solutions of the matrix equations. Numerical examples and comparison with other well-known algorithms demonstrate the effectiveness of the proposed matrix algorithms.