Extremum Seeking Control : Stability, Accuracy, and Applications

Abstract: In many applications, there is a variable that indicates the overall performance and that must be maximized, such as the output of a hydro power turbine or a mineral processing plant, or that must be minimized, such as CO2 emissions or the consumption of resources. Extremization of this variable (maximization or minimization) through adjusting the influencing manipulated variables is occasionally required without prior knowledge of the optimal values of the manipulated variables or of the optimized variable. Extremum seeking control (ESC), which is an on-line concept for the optimization of dynamic systems, can achieve this task.Many types of ESC have been proposed in the literature, and the majority of these approaches are based on the gradient descent optimization method. The most common type of ESC is the classic ESC, which is based on adding a sinusoidal perturbation to the manipulated variables followed by the use of a band pass filter (BPF) to find an estimate of the gradient of the output with respect to the input. The plant is then extremized by adjusting the manipulated variables to make this estimate approach zero.In this work, an alternative approach called phasor ESC is proposed, which is based on estimating the phasor of the plant output at the perturbation frequency rather than the gradient. Stability analysis of the phasor ESC is presented, including local and semi-global practical asymptotic stability for general non-linear dynamic plants. As an improvement of the existing stability analysis of the classic ESC, a less constrained semi-global practical asymptotic stability condition is also presented. In perturbation-based ESC, the output will continuously oscillate due to the input perturbation. As noted in the literature, the averaged system may not converge to the optimum point, and an average offset from the optimal value will emerge. This offset is negligible in the case of a small perturbation signal but will increase as the perturbation amplitude increases. In this work, a modification for the classic ESC and phasor ESC is proposed that improves the accuracy by reducing this offset. Two applications of ESC are considered: cone crushers and hydro power plants. Cone crushers are used for reducing the size of minerals and are considered to be a key component of many mineral processing plants. On-line optimization of the throughput is proposed based on the concept of ESC. A novel model of cone crushers is presented that can predict the flow and the size distribution of the output as a function of the primary manipulated variables, and simulations of ESC control on this model are presented.For hydro power plants, which are considered to be the most important source of renewable energy, phasor ESC is proposed for constructing and correcting the combinator, which ensures the optimal and efficient operation of the Kaplan turbine. Moreover, phasor ESC is proposed as a tool for maximum power point tracking in micro hydro plants by manipulating the turbine speed, and a test rig is used to conduct an experiment to validate this approach.