Visualisation of system stability using the HGRAM graphical display format

Abstract

The HGRAM is a generic name coined to cover a new family of n-dimensional graph-paper. Two previous papers gave details of the procedures for plotting n-dimensional graphs using the HGRAM format and its applications in the design of computer screen formats in universal stations in distributed control systems. One paper described the application of n-dimensional space walk in the visualisation of contours in hyperspace. Another paper described the application of the HGRAM graphical method in graphical optimisation. This paper will focus on the visualisation of the stability of linear and nonlinear control systems using the HGRAM graphical format. Conventional methods such as Routh's method, Bode's plots, Nicol's charts, Root-Locus plots, Z-domain plots. and phase-portraitures are limited to 2-dimensions. It is shown that the root-locus plots could be extended to higher dimensionalities using the HGRAM graphical method. It is also shown that the zone of stability of nonlinear control systems could be visualised using Lyapunov's second method and this is again not limited to two dimensional problems. The accuracy of graphical interpolation is limited by the pixel resolution of VDU's. However, on CAD-workstation, an accuracy of interpolation of up to 4 decimal places could be achieved. On the average, the HGRAM graphical method deviates from high precision numerical methods by only 0.03% when compared to the latters. Many complicated mathematical procedures could be greatly simplified by using the HGRAM graphical method. This means less skilled workers could be put to work on problems which hitherto are performed by gradates.