Certain Types Of Nonlinear Systems Research Articles

Data‐driven model order reduction of quadratic‐bilinear systems

SummaryWe introduce a data‐driven model order reduction approach that represents an extension of the Loewner framework for linear and bilinear systems to the case of quadratic‐bilinear (QB) systems. For certain types of nonlinear systems, one can always find an equivalent QB model without performing any approximation whatsoever. An advantage of the Loewner framework is that information about the redundancy of the given data is explicitly available, by means of the singular values of the Loewner matrices. This feature is also valid for the proposed generalization. As for the linear and bilinear cases, these matrices can be directly computed by solving Sylvester equations. We begin by defining generalized higher‐order transfer functions for QB systems. These multivariate rational functions play an important role in the model order reduction process. We construct reduced‐order systems for which the associated transfer functions match those corresponding to the original system at selected tuples of interpolation points. Another benefit of the proposed approach is that it is data‐driven oriented, in the sense that one would only need computed or measured samples to construct a reduced‐order QB system. We illustrate the practical applicability of the proposed method by means of several numerical experiments.

High-order system identification with an adaptive recursive second-order polynomial filter

In this letter, an adaptive recursive nonlinear filter based on the Volterra series and an infinite impulse response (IIR) structure is considered. For certain types of nonlinear systems where high-order nonlinearities are recursively generated, we show that the adaptive recursive second-order polynomial filter has improved performance over the well-known (nonrecursive) adaptive second-order Volterra filter and a third-order Volterra filter. This filter represents an alternative to using a traditional Volterra filter whose order has been increased to match that of the system being modeled.

A Numerical Technique for the Solution of Some Non-Linear Systems

Exact solutions for certain types of non-linear systems where the non-linearity is symmetrical, that is to say, gravitational effects have been ignored, and damping is absent, are given by Den Hartog and Mikina. The algebra involved in these solutions is somewhat cumbersome and inconvenient to apply. A numerical method is given by Stormer but the computation is tedious without the use of a digital computer. The method employed by Lourie and Tchekmarev is commendable by virtue of its simplicity.

Equilibrium of Certain Types of Non-linear Systems

Behaviour of motion of certain types of non-linear systems of higher order are studied through Lyapunov's technique. In particular, stable and unstable higher-order singular points, and different types of limit sets, which have not been studied before, are analysed.

A study of asynchronously excited oscillations in nonlinear control systems

This study was initiated as an effort to explain certain oscillatory phenomena observed in an aircraft and weapon control system operated in a particular mode. The oscillations in question were of the undamped limit-cycle type and their presence could very clearly be correlated with the degree of noise corruption of the signals. It is demonstrated in this paper that certain types of nonlinear systems although being inherently stable may be driven into oscillatory modes not only by random signals but also by any high frequency periodic or nonperiodic signal possessing a certain energy content. It is of interest to note that the tendency for hunting and also the hunt frequency usually are completely independent of the frequency of the excitation signal, i.e., the phenomenon is of asynchronous nature. A general theory, the validity of which has been tested by both analog and digital means, is presented and utilized to demonstrate how this phenomenon may be predicted from information on circuit data.