Robust Optimal Proportional-Integral Controller for an Uncertain Unstable Delay System: Wind Process Application

In industrial practice, certain processes are unstable, such as different types of reactors, distillation columns, and combustion systems. To ensure greater maneuverability and improve the speed of response command, certain systems in the military and aviation fields are purposely configured to be unstable. These systems are often more difficult to control than stable systems and are of particular interest to designers and control engineers. Despite all advances in process control over the past six decades, the proportional-integral-derivative (PID) controller is still the most common. The main reasons are the simplicity, robustness, and successful applications provided by PID-based control structures. The design of proportional-integral (PI) controller for time-delay (TD) systems is a traditional and classical problem. On one hand, PI controllers are used in more than 95% of industrial processes. On the other hand, there are TD phenomena in almost all practical control systems. In this paper, we study the robustness of a PI controller in stabilizing systems containing uncertain parameters and delays. A robust controller for an unknown unstable second-order with margin TD is constructed using a generalized Kharitonov theorem for quasi-polynomials. A constructive method based on the Hermite-Biehler theorem is used to obtain all PI gains, which stabilize an uncertain and unstable second-order delay system. Genetic algorithms (GAs) and optimization methods are used to obtain optimal system and control parameters for providing the best PI control that makes the system robust and stable under uncertainty. By minimizing performance criteria such as the integral of square error, integral of absolute error, time-weighted integral of absolute error, and integral of time weighted square error, GAs and particle swarm optimization are used to optimize the PI parameters and system parameters that provide the best control under uncertainties.