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A general problem encompassing output regulation and pattern generation can be formulated as the design of controllers to achieve convergence to a persistent trajectory within the zero dynamics on which an output vanishes. We develop an optimal control theory for such design by adding the requirement to minimize the H2 norm of a closed-loop transfer function. Within the framework of eigenstructure assignment, the optimal control is proven identical to the standard H2 control in form. However, the solution to the Riccati equation for the linear quadratic regulator is not stabilizing. Instead it partially stabilizes the closed-loop dynamics excluding the zero dynamics. The optimal control architecture is shown to have the feedback of the deviation from the subspace of the zero dynamics and the feedforward of the control input to remain in the subspace. 

When designing feedback controllers to achieve periodic movements, a reference trajectory generator for oscillations is an important component. Using autonomous oscillators to this effect, rather than directly crafting periodic signals, may allow for systematic coordination in a distributed manner and storage of multiple motion patterns within the nonlinear dynamics, with potential extensions to adaptive mode switching through sensory feedback. This paper proposes a method for designing a distributed network that possesses multiple stable limit cycles from which various output patterns are generated with prescribed frequency, amplitude, temporal shapes, and phase coordination. In particular, we adopt, as the basic dynamical unit, a simple nonlinear oscillator with a scalar complex state variable, and derive conditions for their distributed interconnections to result in a network that embeds desired periodic solutions with orbital stability. We show that the frequencies and phases of target oscillations are encoded into the network connectivity matrix as its eigenvalues and eigenvectors, respectively. Various design examples will illustrate the proposed method, including generation of human gaits for walking and running. 

An important objective in the design of feedback control systems is the pattern generation. The term ‘pattern’ denotes the behavior of the individual plant states relative to one another in steady-state, e.g. periodic with a specified frequency and phase offset. Here we solve the optimal, linear, output feedback problem in which the controller is autonomous, achieves pattern generation, and minimizes the L2 norm of the transient portion of the impulse response. Our result reveals the optimal control architecture comprising a linear quadratic regulator and a Kalman filter, along with additional feedback/feedforward to/from a pattern generator, with gains constrained by the regulator equation and its dual, respectively. In contrast to the standard output regulation, the pattern generator is embedded in the feedback loop, allowing the reference signals to be modified autonomously in response to disturbances. A design example illustrates the controller’s ability to recalculate and track the target trajectory following a disturbance. 

The central pattern generator (CPG) is a group of interconnected neurons, existing in biological systems as a control center for oscillatory behaviors. We propose a new approach based on the multivariable harmonic balance to characterize the relationship between the oscillation profile (frequency, amplitude, phase) and interconnections within the CPG, modeled as weakly coupled oscillators. In particular, taking advantage of the weak coupling, we formulate a low-dimensional matrix whose eigenvalue/eigenvector capture the perturbation in the oscillation profile due to the coupling. Then we develop an algorithm to estimate the perturbed oscillation profile of a given CPG, and suggest an optimization to synthesize the interconnections to produce a given oscillation profile.