Systems In Second-order Form Research Articles

Construction of Nonlinear Interpolants With Second-Order Equation Structure in the Loewner Framework

This paper studies the construction of nonlinear interpolants consisting of second-order equations in the Loewner framework. First, conditions are given for which a system of second-order equations interpolates sets of right and left tangential data mappings, which results in second-order tangential generalized controllability and observability mappings allowing for the direct treatment of systems in second-order form. Following this, a family of interpolants with second-order equation structure is given. The results are demonstrated by constructing a reduced order model of a two link robotic manipulator in second-order equation form.

Laguerre-Gramian-based structure-preserving model order reduction for second-order form systems

A new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order form systems is presented in this article. The main task of the proposed approach is to use the Laguerre polynomial expansion of the matrix exponential function to obtain the approximate low-rank decomposition of the Gramians for the equivalent first-order representation of the original second-order form system. The approximate balanced system is generated by a balancing transformation which is directly computed from the expansion coefficients of impulse responses in the space spanned by Laguerre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states with small approximate Hankel singular values (HSVs). The above method has a disadvantage that it may unexpectedly result in unstable systems although the original one is stable. Therefore, modified reduction procedure combined with the dominant subspace projection method is presented to alleviate the limitation. Finally, two numerical experiments are provided to demonstrate the effectiveness of the algorithms.

Structure preserving balanced proper orthogonal decomposition for second‐order form systems via shifted Legendre polynomials

This study considers structure preserving balanced proper orthogonal decomposition for second-order form systems via shifted Legendre polynomials. The proposed approach is to use time interval empirical Gramians of the first-order representation, which are constructed from impulse responses by solving block tridiagonal linear systems, to generate approximate balanced system for the large-scale second-order form system. The balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted Legendre polynomials, without computing the full Gramians for the first-order representation. Then, the reduced second-order model is constructed by truncating the states corresponding to the small approximate singular values. Furthermore, in combination with the dominant subspace projection method, the authors modify the reduction procedure to alleviate the shortcoming of the above method, which may unexpectedly lead to unstable systems even though the original one is stable. The stability preservation of the reduced model is briefly discussed. Finally, the effectiveness of the proposed methods is demonstrated by two numerical experiments.

Stability of linear systems in second-order form based on structure preserving similarity transformations

This paper deals with two stability aspects of linear systems of the form \({I\ddot{x}+B\dot{x}+Cx=0}\) given by the triple (I, B, C). A general transformation scheme is given for a structure and Jordan form preserving transformation of the triple. We investigate how a system can be transformed by suitable choices of the transformation parameters into a new system (I, B1, C1) with a symmetrizable matrix C1. This procedure facilitates stability investigations. We also consider systems with a Hamiltonian spectrum which discloses marginal stability after a Jordan form preserving transformation.

A novel order reduction method for nonlinear dynamical system under external periodic excitations

The concept of approximate inertial manifold (AIM) is extended to develop a kind of nonlinear order reduction technique for non-autonomous nonlinear systems in second-order form in this paper. Using the modal transformation, a large nonlinear dynamical system is split into a ‘master’ subsystem, a ‘slave’ subsystem, and a ‘negligible’ subsystem. Accordingly, a novel order reduction method (Method I) is developed to construct a low order subsystem by neglecting the ‘negligible’ subsystem and slaving the ‘slave’ subsystem into the ‘master’ subsystem using the extended AIM. As a comparison, Method II accounting for the effects of both ‘slave’ subsystem and the ‘negligible’ subsystem is also applied to obtain the reduced order subsystem. Then, a typical 5-degree-of-freedom nonlinear dynamical system is given to compare the accuracy and efficiency of the traditional Galerkin truncation (ignoring the contributions of the slave and negligible subsystems), Method I and Method II. It is shown that Method I gives a considerable increase in accuracy for little computational cost in comparison with the standard Galerkin method, and produces almost the same accuracy as Method II. Finally, a 3-degree-of-freedom nonlinear dynamical system is analyzed by using the analytic method for showing predominance and convenience of Method I to obtain the analytically reduced order system.

On the computation of frequency response matrices for systems in second-order form

The purpose of this paper is to discuss the computation of the frequency response matrices of the form (P +jwR)(- w2M +jwG + K)-1B which are related to systems given in second-order form. For systems of this type, the above computational problem has not been considered in the literature so far. However, efficient and accurate computation of the frequency response matrices which are related to first-order models were recently presented and available where some of these methods depend mainly on the reduction of the state matrix to Hessenberg form using different means. On the other hand, the matrix form above has no Hessenberg analog and it is not so straightforward computationally. A simple partition to the matrix form is proposed so to allow a direct use to these methods. The use of the algorithms of these methods is introduced where their efficiency are compared using the operations count required. For accuracy comparison, we refer to some numerical examples.