A generalised phase portrait for piecewise linear system analysis

The development of analysis tools for nonlinear dynamical systems is seen by the authors as the great challange now facing control engineers. Within the area of nonlinear systems a number of different classes of nonlinearities exist. One of the most important of these classes is that of piecewise linear elements. This class includes many commonly occuring nonlinear elements such as saturation, dead-zones, slew rate limits, hysteresis and backlash. In addition to these "hard" nonlinearities, many other nonlinear elements can be approximated by a piecewise linear characteristic. Furthermore, many rule-based and "intelligent controllers", such as those based upon expert systems and programmable logic, use logical rules that can be completely modelled as piecewise linear elements. Thus there exists a range of naturally occuring and wilfully created piecewise linear components. This paper describes a method for analysing the stability of control systems which contain these components. Specifically, the method aims to extend the phase portrait method which has frequently proven useful in the study of the stability analysis of 2nd order nonlinear processes.