A Graphical Approach to Examining Classical Extremum Seeking Using Bifurcation Analysis

The majority of extremum-seeking literature discusses rigorous stability analysis, thereby limiting its audience to mathematicians and control engineers with strong theoretical backgrounds. Here, we complement these studies by proposing the use of harmonically forced bifurcation analysis to evaluate parameter choices in classical extremum-seeking systems. This method generates a graphical map of limit cycle attractors and how they change with respect to a chosen parameter. Our approach retains the full properties of a harmonically forced system, thereby avoiding the requirement to approximate the dynamics as equilibrium solutions as done in previous studies. Common elements of nonlinear dynamical systems, such as loss of local stability and coexistence of multiple solutions via fold bifurcations are observed. Bifurcation analysis therefore provides an intuitive tool for engineers and new adopters to gain further insight into classical extremum-seeking systems. The link between extremum-seeking control and dynamical system theory is also highlighted. We use an example of an auto-trim system in a nonlinear, longitudinal (fourth-order) flight dynamics model to demonstrate the method. The influence of the forcing frequency, modulation phase, and high-pass filter frequency on the stability and performance of the system is examined using both one- and two-parameter continuation.