Curriculum Development: An Algebraic Approach to Linear-Quadratic Control

Linear-quadratic control is a foundational technique in modern control technology. For example, the linear-quadratic estimator and more generally the Kalman filter are ubiquitous in the field of estimation. Kalman filters are used in diverse applications, such as predicting space weather and estimating spacecraft attitude. Similarly, the linear-quadratic regulator and linear-quadratic Gaussian controller are fundamental control methods with broad application. Moreover, these techniques have extensive application to NASA technologies, such as spacecraft navigation and control, control of lightweight space structures, and control of spacecraft propulsion systems. Despite the importance of linear-quadratic control, many students find the subject challenging to learn. This challenge is due, in part, to the mathematical tools that are adopted to derive the linear-quadratic estimator, regulator, and controller. More specifically, linear-quadratic results are typically derived by using the principle of optimality and the Hamilton-Jacobi-Bellman equation, or by using Pontryagin’s minimum principle and the Euler-Lagrange equation. These mathematical tools can make linear-quadratic control difficult to understand or in the extreme, inaccessible. The objective of this project is to develop a linear-quadratic-control course that provides a rigorous introduction without resorting to mathematical tools, which can make the material difficult to understand. We seek to develop a course that adopts a primarily algebraic approach to linear-quadratic control. This approach relies on linear algebra and undergraduate calculus (e.g., Lagrange-multiplier techniques) to derive and explain the fundamental results in linear-quadratic control. While the course is intended for graduate students, the course will also be open to advanced undergraduates. Our goal is to offer this course during the 2013-to-2014 academic year. The principal investigator is currently coauthoring a textbook manuscript on linear-quadratic control. This manuscript will be leveraged for the proposed course development.