Abstract
This paper considers a new method to solve the first-order and second-order nonhomogeneous generalized Sylvester matrix equations AV+BW= EVF+R and MVF2+DV F+KV=BW+R, respectively, where A,E,M,D,K,B, and F are the arbitrary real known matrices and V and W are the matrices to be determined. An explicit solution for these equations is proposed, based on the orthogonal reduction of the matrix F to an upper Hessenberg form H. The technique is very simple and does not require the eigenvalues of matrix F to be known. The proposed method is illustrated by numerical examples.
Highlights
Consider the following two homogeneous generalized Sylvester matrix equations: AV + BW EVF, (1)MVF2 + DVF + KV BW. (2)Matrix equation (1) is called a first-order homogeneous generalized Sylvester matrix equation that is closely related to many problems in linear systems theory, such as eigenstructure assignment [1,2,3,4,5] and control of systems with input constraints [6]
As a generalization of the above matrix equations, we have considered the following nonhomogeneous generalized Sylvester matrix equation: AV + BW EVF + R, (3)
Several authors have studied different methods for matrix equation (3). e secondorder nonhomogeneous Sylvester matrix equation was introduced by Duan [16]
Summary
Consider the following two homogeneous generalized Sylvester matrix equations: AV + BW EVF,. Second-order homogeneous generalized Sylvester matrix equation (2) has found applications in many control problems, for example, pole assignment [7,8,9] and eigenstructure assignment [10, 11]. As a generalization of the above matrix equations, we have considered the following nonhomogeneous generalized Sylvester matrix equation: AV + BW EVF + R,. MVF2 + DVF + KV BW + R. where A, E, M, D, K, B ∈ Rn×n, R ∈ Rn×p , and F ∈ Rp×p are the known matrices, while V ∈ Rn×p and W ∈ Rn×p need to be determined. E main goal of this paper is to present algorithms for solving well-known nonhomogeneous generalized Sylvester matrix equations (3) and (4). Consider the following nonhomogeneous generalized firstorder Sylvester matrix equation:. 0 0 0 hp,p− 1 hpp where l1, l2 , . . . , lp are the columns of L, 0 is the zero vector, g is the unknown vector, and S [s1, s2, . . . , sp] is the known real n × p matrix
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