Graphical Analysis of Control Systems

The purpose of this paper is to demonstrate some graphical methods for finding the transient response of a control system. A simple position follow-up system is considered for convenience although the method is applicable in the same form for higher order systems or those in which only empirical frequency data is known. The basic procedure is to find the roots of the differential equation which correspond to the exponential transient terms which dominate the response. Doctor Profos¹ of Switzerland points out that the plot of the function which describes the system from error to output is a function of a complex variable of which frequency is the imaginary part and damping is the real part. The Nyquist plot is thus one line of a conformal map with the root of the equation being the value of the variable which makes the function equal to -1. Any line of plot can be calculated for systems with known functions with essentially the same ease as the Nyquist plot by use of some graphical tricks. The amplitude of any transient term is determined from the plot once the root is known by use of a theorem of operational calculus. The development possibilities of the subject seem to be very great as suggested by several topics not yet investigated.