Abstract
In this paper, we discuss the implementation and tuning algorithms of a variable-, fractional-order Proportional–Integral–Derivative (PID) controller based on Grünwald–Letnikov difference definition. All simulations are executed for the third-order plant with a delay. The results of a unit step response for all described implementations are presented in a graphical and tabular form. As the qualitative criteria, we use three different error values, which are the following: a summation of squared error (SSE), a summation of squared time weighted error (SSTE) and a summation of squared time-squared weighted error (SST2E). Besides three types of error values, obtained results are additionally evaluated on the basis of an overshoot and a rise time of the output signals achieved by systems with the designed controllers.
Highlights
Proportional–Integral–Derivative (PID) controllers are undoubtedly used in most automatic process control applications in the industry today
During the first phase of the research, the simulations were performed with the assumption that there are no constraints set to the values of control signals generated by the controllers
To make figures more transparent/readable, presented simulation time was limited to 50 s
Summary
Proportional–Integral–Derivative (PID) controllers are undoubtedly used in most automatic process control applications in the industry today. They are renowned for example as an excellent tool to regulate flow, temperature, pressure, and many other industrial process variables. PID controllers were built in 1939 by the Taylor and Foxboro instrument companies. Despite the various tuning methods most of present-day controllers are based on those original proportional, integral, and derivative modes. One of the first methods of PID tuning was Ziegler–Nichols, described in [1], which was widely extended along last decades.
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