Gramian‐based model order reduction of parameterized time‐delay systems

Abstract

Time-delay systems (TDSs) frequently arise in circuit simulation especially in high-frequency applications. Model order reduction (MOR) techniques can be used to facilitate the simulation of TDSs. On the other hand, many kinds of variations, such as temperature and geometric uncertainties, can have significant impact on the transient responses of TDSs. Therefore, it is important to preserve parametric dependence during the MOR procedure. This paper presents a new parameterized MOR scheme for TDSs with parameter variations. We derive parameterized reduced-order models (ROMs) for TDSs using balanced truncation by approximating the Gramians in the multi-dimensional space of parameters. The resulting ROMs can preserve the parametric dependence, making it efficient for repeated simulations under different parameter settings. Numerical examples are presented to verify the accuracy and efficiency of our proposed algorithm.

Matrix-Interpolation-Based Parametric Model Order Reduction for Multiconductor Transmission Lines With Delays

“…It has a complexity of O(4n 2 q) where q is the reduced order estimated for the model. But it can be seen that the proposed technique is much more efficient than the Gramian based PMOR for TDS [10] which has a complexity of O(n 3 ). Then we have the computation of the singular values for the common projection matrix which uses an economy-size SVD to improve the computation.…”

“…The proposed PMOR technique, it is compared with the Gramianbased PMOR [10] which is also based on state-space interpolation.…”

“…A number of PMOR methods have been developed in recent years for TDSs based on input-output interpolation [8]- [10]. In [8], the approach is based on a multiorder Arnoldi algorithm which is used to implicitly calculate the moments with respect to frequency and the design parameters, as well as the cross-moments.…”

“…These PMOR methods use input-output system level interpolation which are proved to be robust and accurate, but the order of the parametric macromodels may suffer from oversize due to the nature of the input-output system level interpolation. A Gramian-based PMOR for TDS is presented in [10] where an affine model is used to represent the parametric behavior, but the technique is computationally expensive as Cholesky factorization is required for the computation of the projection matrix.…”

“…As the approach is based on matrix interpolation it overcomes the oversize problem in input-output system level interpolation and the technique uses higher-order Krylov MOR to compute the reduced order models (ROMs) for TDS [6], [7]. This is more efficient in comparison to the augmented MOR technique proposed for PMOR TDS in [9], as the augmentation generates an equivalent firstorder system which is larger than the size of the original model, and the Gramian based MOR for PMOR of TDS [10], as the computation of Gramians are expensive due to Cholesky factorization. The proposed approach computes a set of reduced system matrices in a common subspace using higher-order Krylov MOR and interpolates these ROM and the delays in order to generate PMOR for TDSs.…”