Circulant synthesis of central pattern generators with application to control of rectifier systems

While regulations around an equilibrium point or a. reference trajectory have been the focus of recent feedback control theories, generation of autonomous oscillations with a specific pattern plays a crucial role in important control applications such as robotics. The central pattern generator (CPG) is the fundamental neuronal mechanism underlying rhythmic movements of animals, and may provide a new paradigm for controlled oscillations of engineering systems. This paper gives a novel method for synthesizing artificial CPG circuits by utilizing the properties of circulant matrices. We show how neuron models can be interconnected to yield a periodic orbit with prescribed frequency, amplitudes, and phases. Using the synthetic CPG in a feedback loop, we propose a control design method for a class of nonlinear rectifier systems to achieve the optimal state pattern with respect to a certain measure of efficiency, subject to small ripples in the rectified variable. The effectiveness of the proposed method is illustrated by an application to the gait control of an undulatory snakelike system.