so thatJlqr= limt z(t)Tz(t)
E
!1
the mean-square deviation of z. Since w is a white noise, we have (see section 5.2.2)
Jlqr = H 22
k
k
the square of the 2 norm of the closed-loop transfer matrix.
H
In our framework, the plant for the LQR regulator problem is given by
AP = A
Bu = B
Bw = I
Cz = 0Q12
Cy = I
Dzw = 0
Dzu = R120
Dyw = 0
Dyu = 0
(the matrices on left-hand side refer to the state-space equations from section 2.5).
This is shown in gure 12.1.
P
w
o
1 2
=
R
z
+
q
r
(sI A) 1 x
B
q
1 2
;
=
;
Q
+
u
y
K
The LQR cost is
22.
Figure
12.1
kH
k
The speci cations that we consider are realizability and the functional inequality
speci cation
H 2
:
(12.1)
k
k
12.1 LINEAR QUADRATIC REGULATOR
277
Standard assumptions are that (Q A) is observable, (A B) is controllable, and
R > 0, in which case the speci cation (12.1) is stronger than (i.e., implies) internal
stability. (Recall our comment in chapter 7 that internal stability is often a redun-
dant addition to a sensible set of speci cations.) With these standard assumptions,
there is actually a controller that achieves the smallest achievable LQR cost, and it
turns out to be a constant state-feedback,
Klqr(s) = Ksfb
;
which can be found as follows.
Let Xlqr denote the unique positive de nite solution of the algebraic Riccati
equationATXlqr+XlqrA XlqrBR 1BTX
;
lqr + Q = 0:
(12.2)
;
One method of nding this Xlqr is to form the associated Hamiltonian matrix
M = A
BR 1BT
;
;
Q
AT
(12.3)
;
;
and then compute any matrix T such that
T 1MT = ~A11 ~A12
;
0 ~A22
where ~A11 is stable. (One good choice is to compute an ordered Schur form of M
see the Notes and References in chapter 5.) We then partition T as
T = T11 T12
T21 T22
and the solution Xlqr is given by
Xlqr = T21T 1
;
11 :
(We encountered a similar ARE in section 5.6.3 this solution method is analogous
to the one described there.)
Once we have found Xlqr, we have
Ksfb = R 1BTX
;
lqr
which achieves LQR cost
Jlqr = Xlqr:
T
r
In particular, the speci cation (12.1) (along with realizability) is achievable if and
only if
p
Xlqr, in which case the LQR-optimal controller Klqr achieves the
T
r
speci cations.
In practice, this analytic solution is not used to solve the feasibility problem for
the one-dimensional family of speci cations indexed by rather it is used to solve
multicriterion optimization problems involving actuator e ort and state excursion,
by solving the LQR problem for various weights R and Q. This is explained further
in section 12.2.1.
278