H 2 optimal model reduction of positive systems by the augmented Lagrangian method on the oblique manifold

This paper focuses on the H₂ optimal model reduction problem of positive systems. According to the coefficient matrices of the positive system, the nonnegative orthonormal matrix is taken as the projection matrix, and the H₂ optimal model reduction problem is developed. Since the projection matrix is orthonormal and nonnegative, the H₂ optimal model reduction problem is reformulated as a constrained optimization problem defined on the Stiefel manifold, and further regarded as a constrained optimization problem defined on the oblique manifold. By the augmented Lagrangian function, the constrained optimization problem defined on the oblique manifold is tackled by employing the Dai-Yuan-type conjugate gradient method to solve a series of unconstrained optimization subproblems. When the objective function of a subproblem satisfies some conditions, the iterative sequence produced by the conjugate gradient method is convergent. Finally, numerical experiments illustrate the efficiency of the proposed model reduction method.