Machine Learning Control – Taming Nonlinear Dynamics and Turbulence

There are none." John Doyle, IEEE Transactions on Automatic Control, 1978 [86] The most well-developed theory of control generally applies to a linear system or to the linearization of a nonlinear system about a fixed point or a periodic orbit. Linear control theory has many applications in fluid dynamics, such as the stabilization of unstable laminar boundary layers. Although the governing equations may be nonlinear, successful stabilizing controllers will regulate the system to a neighborhood where the linearization is increasingly valid. In this chapter we introduce linear systems (Sec. 3.1) and explore H2 optimal control problems, including the linear quadratic regulator (LQR) in Sec. 3.2 and Kalman filters in Sec. 3.3. These problems are chosen because of their simplicity, ubiquitous application, well-defined quadratic cost-functions, and the existence of known optimal solutions. Next, linear quadratic Gaussian (LQG) control is introduced for sensor-based feedback in Sec. 3.4. Finally, methods of system linear system identification are provided in Sec. 3.5. This chapter is not meant to be an exhaustive primer on linear control theory, although key concepts from optimal control are introduced as needed to build intuition. Note that none of the linear system theory below is required to implement the machine learning control strategies in the remainder of the book, but they are instead included to provide context and demonstrate known optimal solutions to linear control problems. In many situations, H∞ robust control may be more desirable to balance the trade-off between robustness and performance in systems with uncertainty and unmodeled dynamics, and the MLC methods developed here may be generalized to other cost functions. For a more complete discussion of linear control theory, excellent books include [93, 251].