The systematic design of PID controllers using Nyquist stabiliy

This paper introduces new and fast ways of systematically calculating the limiting values of the parameters of PID-action controllers for systems that are open-loop stable, or unstable, minimum phase, or non-minimum phase, and can contain explicit time-delay terms. The technique presented, which is based on the axis-crossing form of Nyquist´s stability theorem, can be readily used for the design of PID controllers to meet certain closed-loop stability and performance requirements. A summary of the background development is given and it is then shown how, by using a simple feature available in most symbolic algebra environments, a much faster version of the proposed method, which avoids the need to calculate any Nyquist diagrams, can be obtained for systems with no explicit time-delay term. A recent complete analytical solution to the PID compensator that results in a very significant reduction in the computing time, yielding almost real-time results, is also introduced. The methods proposed can be extended to develop robust controllers for systems with uncertainties, and present a visual characterization of all PID controllers meeting the desired performance criteria for such systems. Other time-domain performance requirements such as the Integral of Time and Absolute Error criterion can also be employed to select an optimal operating point from the allowable set of controller parameters. An application of these tools to the fan-speed control of the Pegasus aircraft gas-turbine engine, used in the Harrier jump-jet, will be presented.