Optimal Estimation of Dynamic Systems, 2nd Edition

548 Optimal Estimation of Dynamic Systems

Equation (8.157) is identical to Equation (8.97) with the exception that the true state

x(t) is replaced with the estimated state

ˆ

x(t)! In order for Equation (8.157) to truly

achieve the minimum of the LQG loss function, the matrix [Γ

T

k

S

k+1

Γ

k

+ R

k

] must be

positive definite. This condition will obviously always be met since S

k+1

is always

positive definite, which is shown by the form in Equation (8.103). For autonomous

systems, the discrete-time Separation Theorem can be proved using an eigenvalue

separation of the controller and estimator (see exercise 8.30), similar to the steps

leading to Equation (8.135).

8.7 ...