Linear Optimal Control for Autonomous Pattern Generation

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 $L_{2}$ 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.