Associated Linear Equations for Volterra operators

Non-linear time-invariant systems may be sometimes represented in the time-domain by a Volterra series which is a non-parametric, time-domain representation, or in the frequency-domain by the so-called higher-order frequency response functions (HFRFs). A non-parametric model in the time-domain gives very little insight into the intimate details of the non-linear system. Conversely the HFRFs are complicated in shape and are difficult to analyse. The graphical display of the HFRFs has the disadvantage of been multidimensional. It is known that the nth-order Volterra operator is a multi-linear-function of a combination of input signals. In fact, Hammerstein and Duffing type systems possess a linear relationship between the nth-order Volterra operator response and inputs composed of a combination of lower-order Volterra operators. This characteristic is used here to model the behaviour of the Volterra operators by linear equations that map the nth-order operator from an excitation of the same order produced by a combination of lower-order operators. This group of equations is referred to here as the Associated Linear Equations (ALEs). These ALEs can be used in simulation, control and as an analytical tool for the Volterra class of non-linear systems. The main advantage is that theory that up to today has been limited to linear systems can be used on this kind of non-linear systems allowing simpler manipulations.