ABSTRACT
This chapter discusses the main concepts of variational theory and then optimal control, a technique for synthesizing a linear quadratic regulator (LQR) which is able to determine the closed-loop eigenvalues on the basis of an optimized criteria. Several problems regarding the variational calculus are first presented with emphasis on the formulation of general optimization problems. The second part of the chapter is devoted to optimal control, first presented in the context of variational theory and then studied in relation to the practical aspects related to the definition of suitable performance indices. The optimal gains are defined and a classical procedure to obtain them is presented. The nonlinear matrix algebraic Riccati equation is discussed. Several methods to find its positive definite solution are discussed. The dual problem of the optimal observer is also dealt with in the chapter. The LQR problem in the frequency domain is also discussed. In order to approach the dual problems of the optimal controller and the optimal observer, Hamiltonian matrices are introduced.
