Applications of almost periodic functions and equivalent nonlinearities to identification and control of nonlinear systems

Abstract

Correlation techniques for the identification of nonlinear systems are discussed in Chapter 1. The Volterra series expansion
of the response of a nonlinear system is described, together with its counterpart in the frequency domain. Qosscorrelation methods for identifying the kernel functions which occur in this expansion are reviewed, with particular emphasis on techniques or obtaining the linear approximant to a nonlinear system. A crosscorrelation method is also discussed which appears to be unrelated to the Volterra approach. This technique uses a 4-level test signal and is the subject of detailed analysis in later chapters.
The 4-level test signal is discussed in detail in Chapter 2 and it is shown that the linear channel of a nonlinear system may
be identified. The concept of the almost periodic function is presented and it is concluded that by means of almost periodic
fi.ncti&ns it is possible to identify the linear portion of the nonlinear channel in the system. The technique is essentially
based on destroying the ynchronisation of the two 2-level signals originally used to produce the 4-level signal. A generalised
technique is developed to calculate the parameters of the 4-level aperiodic signal in order to app],y the method to any single-valued nonlinearity, assuming that 'a priori' knowledge of the nonlinearity is available.
The technique is extended to permit identification of the impulse response of the linear elements when the nonlinearity
is single valued but contains only even components. This is achieved by modifying the nonlinear characteristic to provide an
odd component.
A further extension of the technique shows that identification of certain nonlinear channels containing elements with memory is possible. Several situations are analysed in detail. An extension is also considered where the nonlinear channel is composed of two linear transfer functions separated by a nonlinearity. Experimental results are included to show the accuracy of the techniques developed.
In Chapter 4 the application of high frequency signals (dither) as a technique to be used in the identification of certain open and pl,osed loop nonlinear systems is discussed. It is shown that certin types of dither permit linearization or elimination of nonlinear channels in certain open and closed systems. It is therefore possible, under these conditions, to identify the linear portions by impulse response or frequency testing techniques, and experimental results show the accuracy achieved. The concept of the equivalent nonlinearity provides a simple interpretation of the action of the dither.
Dither is next considered in its role as a means of stabilizing nonlinear control systems, and the particular case of a third order system containing a hysteresis type relay characteristic is analysed in detail. The concepts of equivalent nonlinearity and describing function are used to derive an expression for the minimum amplitude of dither required to quench limit cycle oscillations in the system.
The system is shown to hAve an effective gain margin once oscillations have been quenched. A typical point in this region is investigated and the three dimensional domain of stability for the system is presented,together with typical trajectories. The system is also shown to exhibit subharmonic resonance apci jump phenomena.
In chapter 6 the use of dither in adaptive control systems is discussed, and the specific case of dither adaptive control system proposed as the solution to a qonstant fuel-rate problem is analysed in detail. The dynamic analysis presented divides the system into two loops via an extension of the equivalent nonlinearity concept.
The final chapter reviews the work described in the body of the thesis, and explores several avenues for future research.