Characterisation of nonlinear and linear time-varying systems by Laplace transformation

Abstract

In this article we elaborate on various characterisations of nonlinear systems. Specifically, we focus on the frequency-domain characterisation of the class of single-input single-output (SISO) linear time-varying (LTV) systems. We use the model introduced by Wiener using Volterra functional and later elaborated by others. The application of the multidimensional Laplace transform (MDLT) method as a fundamental tool in analysing variable systems and understanding system dynamic behaviours is emphasised. In particular, applications of double or two-dimensional Laplace transform (2DLT) for the analysis and synthesis of LTV circuits and systems are detailed. Examples worked out that illustrate the method and demonstrate its validity in the frequency-domain.

Keywords:

• system characterisation
• time-invariant nonlinear system
• linear time-varying system
• multi-dimensional transfer function
• nonlinear transformation
• frequency-response characterisation
• double Laplace transform
• two-dimensional Laplace transform (2DLT)
• multidimensional Laplace transform (MDLT)

Notes

Notes

1. At his point, one may argue that a generalised exponential function, such as and any linear combination of it, may be considered as the ultimate monotonically variable function.

2. If the limits of integration are t and τ, the composition is of the first kind and if the limits are constants and , the composition is of the second kind. The two functions x(·, ·) and y(·, ·) are said to be permutable if their composition is commutative; i.e., .

3. The delta function is defined as a distribution or a generalised function. It can rigorously be defined as a measure, which accepts as an argument a subset A of set C and return 1 if and 0 otherwise. More specifically, is equal to 1 if τ is in set C and equal to 0 if τ is not in set C. C could be a subset of the complex line and τ could be a complex number (or variable).

4. Note that represents an norm.