Optimal control and the linear quadratic regulator

Abstract

In this chapter, we introduce optimal control theory and the linear quadratic regulator. In the introduction, we briefly discuss and compare classical control, modern control, and optimal control, and why optimal control designs have emerged as a popular design method of control in aerospace problems. We then begin by introducing optimal control problems and the resulting Hamilton–Jacobi–Bellman partial differential equation. Then, for linear systems with a quadratic performance index, we develop the linear quadratic regulator. We will cover both finite-time and infinite-time problems and will explore some very important stability and robustness properties of these systems. Central to the design of optimal control laws is the selection of the penalty matrices in the performance index. In the last section, we discuss some asymptotic properties with regard to the penalty matrices that will set the stage for detailed design of these controllers in later chapters.