Balanced truncation of perturbative representations of nonlinear systems

The paper presents a novel approach for a balanced truncation style of model reduction of a perturbative representation of a nonlinear system. Empirical controllability and observability gramians for nonlinear systems are employed to define a projection matrix. However, the projection matrix is applied to the perturbative representation of the system rather than directly to the exact nonlinear system. This is to achieve the required increase in efficiency desired of a reduced-order model. Application of the new method is illustrated through a sample test-system. The technique will be compared to the standard approach for reducing a perturbative representation of a nonlinear system

Fast nonlinear model order reduction via associated transforms of high-order volterra transfer functions

We present a new and fast way of computing the projection matrices serving high-order Volterra transfer functions in the context of (weakly and strongly) nonlinear model order reduction. The novelty is to perform, for the first time, the association of multivariate (Laplace) variables in high-order multiple-input multiple-output (MIMO) transfer functions to generate the standard single-s transfer functions. The consequence is obvious: instead of finding projection subspaces about every s i, only that about a single s is required. This translates into drastic saving in computation and memory, and much more compact reduced-order nonlinear models, without compromising any accuracy. © 2012 ACM.published_or_final_versio

Krylov subspaces from bilinear representations of nonlinear systems

Purpose – The paper is aimed at the development of novel model reduction techniques for nonlinear systems. Design/methodology/approach – The analysis is based on the bilinear and polynomial representation of nonlinear systems and the exact solution of the bilinear system in terms of Volterra series. Two sets of Krylov subspaces are identified which capture the most essential part of the input-output behaviour of the system. Findings – The paper proposes two novel model-reduction strategies for nonlinear systems. The first involves the development, in a novel manner compared with previous approaches, of a reduced-order model from a bilinear representation of the system, while the second involves reducing a polynomial approximation using Krylov subspaces derived from a related bilinear representation. Both techniques are shown to be effective through the evidence of a standard test example. Research limitations/implications – The proposed methodology is applicable to so-called weakly nonlinear systems, where both the bilinear and polynomial representations are valid. Practical implications – The suggested methods lead to an improvement in the accuracy of nonlinear model reduction, which is of paramount importance for the efficient simulation of state-of-the-art dynamical systems arising in all aspects of engineering. Originality/value – The proposed novel approaches for model reduction are particularly beneficial for the design of controllers for nonlinear systems and for the design and analysis of radio-frequency integrated circuits

Model order reduction of fully parameterized systems by recursive least square optimization

This paper presents an approach for the model order reduction of fully parameterized linear dynamic systems. In a fully parameterized system, not only the state matrices, but also can the input/output matrices be parameterized. The algorithm presented in this paper is based on neither conventional moment-matching nor balanced-truncation ideas. Instead, it uses “optimal (block) vectors” to construct the projection matrix, such that the system errors in the whole parameter space are minimized. This minimization problem is formulated as a recursive least square (RLS) optimization and then solved at a low cost. Our algorithm is tested by a set of multi-port multi-parameter cases with both intermediate and large parameter variations. The numerical results show that high accuracy is guaranteed, and that very compact models can be obtained for multi-parameter models due to the fact that the ROM size is independent of the number of parameters in our approach

Model reduction of controlled Fokker--Planck and Liouville-von Neumann equations

Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples