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In this paper, we present a new method in the reduction of large-scale linear differential-algebraic equation (DAE) systems. The approach is to first change the DAE system into a parametric ordinary differential equation (ODE) system via the e-embedding technique. Next, based on parametric moment matching, we give the parameterized model order reduction (MOR) method to reduce this parametric system, and a new Arnoldi parameterized method is proposed to construct the column-orthonormal matrix. From the reduced-order parametric system, we get the reduced-order DAE system, which can preserve the structure of the original DAE system. Besides, the parametric moment matching for the reduced-order parametric systems is analyzed. Finally, the effectiveness of our method is successfully illustrated via two numerical examples.

This paper investigates the Riemannian trust-region method for H2 model reduction of bilinear systems. The H2 error norm is treated as a cost function on the Stiefel manifold such that the orthogonality constraint for the projection matrix is plainly satisfied. The property related to the Euclidean gradient is studied. Then, the inner product associated with the Riemannian Hessian is derived, which can simplify the expression of the trust-region subproblem. The trust-region method for H2 model reduction is accordingly established and the convergence is further discussed. Finally, two numerical examples are employed to demonstrate the performance of the proposed method.

In this paper, we discuss the Arnoldi-based model order reduction (MOR) methods for linear systems with inhomogeneous initial conditions, and present a time domain MOR method by general orthogonal polynomials. The basic procedure is to use the expansion coefficient matrix in the orthogonal polynomial space satisfying a simple recurrence formula to generate a projection matrix, which is produced by a modified Arnoldi algorithm. Then, the resulting reduced model matches a desired number of the expansion coefficients of the original system. The approximate error estimate of the reduced model is given. Since the initial conditions are well represented by the subspaces constructed by our algorithm, it can well deal with those systems with inhomogeneous initial conditions. Two benchmark examples in real applications are simulated to illustrate the effectiveness of the proposed method.

In this paper, the optimal H2 model order reduction (MOR) problem for general MIMO linear time-invariant (LTI) systems is studied. The cross Gramian can provide the controllability and observability information of LTI systems at the same time and it thus is used to discuss the MOR problem. We first deal with the general MIMO system, obtaining a set of SISO subsystems. Then, the cost function related to each SISO subsystem is expressed via the cross Gramian. The orthogonality constraint of the cost function makes it is posed on the Stiefel manifold. Then, making full use of the geometry properties of this Stiefel manifold, we propose a feasible and effective iterative algorithm to solve the H2 minimization problem. In addition, we show that our algorithm is rigorously convergent. Finally, a couple of examples related to MIMO LTI systems demonstrate the effectiveness of our algorithm.

This paper presents a Laguerre-based model reduction method for time-delay systems, which naturally retains the second-order-structure of original systems. We expand time-delay systems under the Laguerre function basis and show that Laguerre coefficients satisfy a special linear system and therefore can be calculated iteratively. Then reduced models are produced using projection methods, where projection matrices are defined in terms of the derived Laguerre coefficients, while properties on the preservation of Laguerre coefficients are proved elegantly based on the linear system corresponding to Laguerre coefficients. We also discuss the practical implementation of the approach. A high order Arnoldi procedure with memory saving is provided which leads to an efficient execution of our approach in practice. Numerical examples are simulated to illustrate the feasibility and efficiency of our approach.

In this paper, we present a model order reduction (MOR) method for large nonlinear input–output systems based on tensor Kronecker product expansion with Arnoldi process. We first approximate the nonlinear system with a quadratic form at single-point expansion, and then use a tensor Kronecker product analysis to it. Constructing the projection matrix through solving a linear equation, a reduced quadratic system is produced, which can match the first several expansion coefficients of the original output. What is more, it can preserve the stability and passivity under some certain conditions as well. The error estimation is also well discussed. Finally, the robust behavior of our MOR method is successfully illustrated via two numerical examples.

In this paper, we focus on optimal H2 model reduction methods for bilinear systems. The optimal H2 model reduction problem can be viewed as a minimization problem on Grassmann manifold. By utilizing the geometry of Grassmann manifold, a gradient descent model reduction algorithm and a conjugate gradient model reduction algorithm based on Grassmann manifold are given. These two algorithms can simulate well the behavior of the original system and the order-reduced systems generated by our algorithms can satisfy the minimal stability requirement. Numerical examples are given to illustrate the effectiveness of our methods.

In this paper, we present a new model order reduction (MOR) method based on general orthogonal polynomials for coupled systems in the time domain. By constructing proper projection matrices, the reduced system not only can preserve the structure of the original system but also can match the first several coefficients of the original output. We study the error bound and the stability of the reduced system as well. Finally, two numerical examples are shown to illustrate the effectiveness of the method.

For a class of coupled systems which include differential-algebraic equation (DAE) subsystems, we discuss model-order reduction (MOR) for them via e - embedding technique and Krylov subspace method. First we change the coupled system into a closed-loop form. Then, both one-sided and two-sided Arnoldi algorithms are used to reduce the closed-loop system. To preserve the interconnected structure, each subsystems can be reduced by the two same algorithms. Next, we show some facts on the error, stability, and passivity for the resulted systems. Finally, we demonstrate the effectiveness of our approach in two examples.

We present a model reduction method for bilinear systems based on the Laguerre series expansion of the kernels resulting from the Volterra representation theory. By employing a two-sided projection, the reduced order system preserves a desired number of Laguerre coefficients, thereby approximating the original system faithfully. Furthermore, the relationship between the proposed Laguerre-based methods and the moment matching methods is studied, which reveals that these two approaches are equivalent under some specific conditions.