Alternative Derivation of Linear Quadratic Regulator

The purpose of this short note is to provide an alternative derivation of the optimal controller for the linear quadratic optimization problem in Section 2. Consider the linear time-varying system dynamics ˙ x(t) = A(t)x(t) + B(t)u(t), x(t 0) = x 0 , (1) where x(t) ∈ R n , u(t) ∈ R n i , ∀t ∈ [t 0 ,t 1 ]. Define the quadratic cost function J(u) = 1 2 t 1 t 0 u(t) 2 2 + C(t)x(t) 2 2 dt + x(t 1) T Sx(t 1) , (2) where S is a symmetric, positive semi-definite matrix. We are interested in the linear quadratic optimization problem J * := min u∈PC J(u), subject to (1), (3) where we denote by PC the set of piecewise continuous functions on [t 0 ,t 1 ]. The more familiar form of this problem is the linear quadratic regulation (LQR) problem, with both penalties on the state and the control. Specifically, consider a modified cost function ˜ J(u) = 1 2 t 1 t 0 x(t) T Q(t)x(t) + u(t) T R(t)u(t)dt + x(t 1) T Sx(t 1) , (4) where Q(t), R(t) are symmetric matrices, with Q(t) positive semi-definite and R(t) positive definite. The LQR problem is then given by ˜ J * := min u∈PC˜J(u), subject to (1). With appropriate choices of weighting matrices, problems (3) and (5) are in fact equivalent. In particular, let R 1/2 , Q 1/2 be the square root of the matrices R, Q. Consider the pseudo-input˜u = R 1/2 u. Then the system dynamics (1) can be rewritten as ˙ x(t) = A(t)x(t) + B(t)R −1/2 (t) ˜ u(t), x(t 0) = x 0 ,