Optimal Estimation of Dynamic Systems, 2nd Edition

Optimal Control and Estimation Theory 533

Table 8.1: Continuous-Time Linear Quadratic Regulator

Model

˙

x(t)=F(t)x(t)+B(t)u(t), x(t

0

)=x

0

Gain L(t)=R

−1

(t)B

T

(t)S(t)

˙

S(t)=−S(t) F(t) −F

T

(t)S(t)

Riccati Equation

+S(t)B(t) R

−1

(t)B

T

(t)S(t) −Q(t), S(t

f

)=S

f

Control Input u(t)=−L(t)x(t)

Note the similarity between the formulation presented here and the continuous-time

Kalman filter in Table 3.4, which leads to the duality results of §5.4.1. A summary

of the continuous-time LQR is shown in Table 8.1. Once the weight matrices R(t)

and Q(t) are chosen, the matrix Riccati solution in Equation (8.60) is integrated

backward in time with boundary conditions given by Equation ...