UC Irvine

This work revisits higher order averaging with a focus on the analysis of high-amplitude, high-frequency oscillatory systems; a class of systems that arises in the motion planning and stabilization of control-affine systems via oscillatory inputs, and in extremum seeking control. Traditional extremum seeking control suffers from persistent oscillations in the steady state. As a first contribution of this thesis, we extend recent results in the literature to allow for vanishing oscillations in steady state even when the optimal value of the function is unknown apriori.

Next, we investigate the effect of multiple periodic time scales on the average behavior of highly oscillatory systems through a recursive application of averaging methods. By exploiting the multiple-scale nature of our analysis, it is possible to separate the dither signals used for gradient estimation in extremum seeking on a slower time-scale, thereby relaxing the non-resonance conditions that are otherwise necessary in multi-dimensional extremum seeking.

Then, we consider a singularly perturbed version of highly oscillatory systems. In contrast to the results in the literature, we provide explicit formulas for the reduced order averaged system that accounts for the interaction between the fast periodic time scale and the singularly perturbed part of the system. Moreover, we combine recursive averaging results with singular perturbation on multiple time-scales and provide explicit formulas for the reduced order averaged system.

In addition, we provide two applications of the methods studied in this thesis. The first application is concerned with klinotaxis in microorganisms. Specifically, we show that the chemotactic strategy of sea urchin sperm cells is a natural implementation of an extremum seeking control law under a nonholonomic integrator.

In the light of this novel connection, we propose bio-inspired 3D source seeking algorithms for rigid bodies with collocated and non-collocated sensors. Unlike all the results in the literature, our proposed algorithms do not assume any global attitude information.

Finally, we investigate, through formal calculations and numerical simulations, the effects of time delays on highly oscillatory systems. We provide explicit formulas for second order averaging of retarded differential equations with a constant time delay when the delay is infinitesmal or is finite. We show that the dynamics of the second-order averaged system depends on a twice-delayed state of the original system when the delay is finite. Moreover, we show that even an infinitesmal delay may bifurcate a single periodic orbit of the original system into multiple orbits and affect their stability. Our analysis highlights a fundamental trade-off between robustness to infinitesmal time-delays and the domain and speed of attraction for highly oscillatory systems when the frequency of oscillation is large compared to the natural time-scale of the dynamics.