g~es267cost of the terminal period.

g~es
267
cost of the terminal period.
Q, R = positive symmetric matrices
dot or inner products and ' denoted transposition.
The Hamiltonian is
The co-state vector equation, given by (21) is
At the terminal period k=N, the transversality condition (23)
p(N) = Sx(N)
The optimal control given by (22) is
-1
uk =- R B'pk+l
Substitution into (29) gives
-1
xk+l = Axk- BR B'pk+l, x(o) = x 0
pk = Q.xk + A'pk+l ,p(N) = Sx(N)
From Kalman equation (8.14), we have
Substitution into (34) to eliminate pk and pk+l, gives
-1
xk+l = Axk - BR B'Kk+lxk+l
Solving, as in chapter 8
Kkxk = Q.xk + A'Kk+l LI+BR-1B'Kk+l]Axk
(30)
(31)
(32)
(33)
(34)