A revision of root locus method with applications

• Uniform analysis of rational, fractional, delayed, distributed-parameter systems.
• We derive RL sketching rules for processes with arbitrary open-loop TFs.
• We propose an effective numerical procedure for RL construction.
• We demonstrate RL-based dominant pole-placement design for arbitrary processes.
• We draw RL of closed-loop poles with respect to open loop dead-time.

The paper investigates applications of the root-locus (RL) method to analysis and design of closed loop systems with arbitrary loop transfer functions. Novel analytic sketching rules have been derived first. These rules are applicable to a wide range of transfer functions: rational, fractional and non-rational ones in general, those with time-delays incorporated at various locations, as well as transfer functions describing distributed-parameter systems. An original, straightforward numerical procedure for plotting the root locus has been proposed next. By means of the derived techniques, a generalization of the pole-placement method, which is applicable to control design of both rational and non-rational processes has been proposed also. Finally, it has been shown that RL technique can be very effectively used to investigate the influence of open loop dead-time variations to closed loop poles. It is of particular interest to stress that the techniques proposed and analyzed in this work are exact, in the sense that no rational approximations of infinite-dimensional systems have been utilized. All results have been thoroughly illustrated by numerical examples.