This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of periodically excited dynamical systems, including black box simulation models and physical experiments. The proposed control designs rely on modifications to the classical model reference adaptive control framework and the more recent \({\mathscr {L}}_1\) adaptive control architecture, in which an additional low-pass filter is used to ensure guaranteed transient performance and robustness to time delays in the control input even in the limit of arbitrarily large adaptive gains. In contrast to the proportional control formulations that have been used in the literature on control-based continuation, the proposed control designs achieve stable performance with a minimum of parameter tuning. In the context of a class of linear systems with matched uncertainties, the paper demonstrates the successful integration of adaptive control feedback in control-based continuation. Specifically, the control designs are shown to ensure that the control input stabilizes the sought periodic orbits of the uncontrolled system and vanishes along these orbits, provided that an a priori unknown reference input is chosen appropriately. Numerical results obtained using the coco software package demonstrate how the combination of a nonlinear solver (Newton’s method) with the pseudo-arclength parameter continuation scheme can be used to trace the correct choice for the reference input under variations in an excitation parameter.

本文提出了两种新颖的自适应控制设计，用于基于控制的连续范式中用于跟踪周期性激励动力系统的周期性轨道族的反馈信号，包括黑盒仿真模型和物理实验。拟议的控制设计依赖于对经典模型参考自适应控制框架和最新的\（{\ mathscr {L}} _ 1 \）的修改自适应控制体系结构，其中使用了一个附加的低通滤波器，即使在任意大的自适应增益的限制下，也可确保保证的瞬态性能和对控制输入中的时间延迟的鲁棒性。与文献中基于控制的连续性所使用的比例控制公式相比，所提出的控制设计以最少的参数调整即可实现稳定的性能。在具有匹配不确定性的一类线性系统的背景下，本文证明了自适应控制反馈在基于控制的连续性中的成功集成。具体地，示出控制设计以确保控制输入使不受控制系统的所寻找的周期性轨道稳定并沿着这些轨道消失，前提是先验的未知基准输入适当地选择。使用coco软件包获得的数值结果表明，如何将非线性求解器（牛顿法）与伪弧长参数连续方案的组合用于在激励参数变化的情况下跟踪参考输入的正确选择。