Abstract
This paper treats the computational method of the optimal H∞ model order reduction (MOR) problem of linear time-invariant (LTI) systems. Optimal solution of MOR problem of LTI systems can be obtained by solving the LMIs feasibility coupling with a rank inequality constraint, which makes the solutions much harder to be obtained. In this paper, we show that the rank inequality constraint can be formulated as a linear rank function equality constraint. Properties of the linear rank function are discussed. We present an iterative algorithm based on augmented Lagrangian method by replacing the rank inequality with the linear rank function. Convergence analysis of the algorithm is given, which is distinct to the now available heuristic methods. Numerical experiments for the MOR problems of continuous LTI system illustrate the practicality of our method.
Highlights
Model order reduction (MOR) problem has received considerable attention since it was put up
In physical or engineering system, the mathematical modeling of the system often results in the high-order controllers, while the simulation and physical implementation of the higher order controllers are more difficult to realize due to the high order
We presented an augmented Lagrangian function (ALF) algorithm for optimal H∞ MOR problem of the linear time-invariant (LTI) system by means of an augmented Lagrangian method
Summary
Model order reduction (MOR) problem has received considerable attention since it was put up. Linear matrix inequality (LMI) is a wellknown convex feasibility problem which can be solved by the semidefinite programming (SDP) interior algorithms [14]. The optimal index γ of H∞ MOR problem is computed by minimizing the model given in [5], and an algorithm to solve the related LMIs feasibility coupling with a rank constraint is proposed in that paper. The contribution of this paper is as follows: an algorithm is presented using the augmented Lagrangian method by means of the equivalent condition to the rank constraints; ALF method has more advantages compared with the PF method. An augmented Lagrangian method solving the NLP problem is proposed, as well as convergence results. Symbols ∇ and ∇2 mean the gradient and Hessian matrix of a function, σ(A) denotes the maximum singular value of a matrix, ‖ ⋅ ‖ denotes the H∞ norm of a rational transfer function, and ⟨A, B⟩ denotes the inner product of two matrices
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