![](bg3.png)
Transfer matrix of the controller is
G
f
(s) = K
c
c
[sI − A+ BK
c
c
+ K
c
f
C]
−1
K
c
f
(31)
To compute K
c
c
for the LQG/LTR controller the fol-
lowing Riccati equation is to be solved
P
ρ
A+ A
T
P
ρ
+C
T
C−
1
ρ
P
ρ
BB
T
P
ρ
= 0 (32)
for ρ → 0 and then the controller gain K
c
c
is calculated
as
K
c
c,ρ
= −
1
ρ
B
T
P
ρ
. (33)
The following LTR result holds (Athans,
1986): if the plant G(s) is minimum-phase
then lim
ρ→0
G(s)G
f,ρ
(s) = Φ(s), where
Φ(s) = C(sI − A)
−1
K
c
f
and G
f,ρ
(s) is calculated
from (31) for K
c
c,ρ
. The dual LTR result, i.e. when
the weighting matrix Q = Q
0
+ ρM for ρ → ∞ can be
found in (Kulcsar, 2000). It is easy to see from (26)
that asymptotically
lim
T
s
→0
G
f
(γ) = G
f
(s) = K
c
[sI − A+ BK
c
)]
−1
K
f
(34)
and full recovery holds that is G(s)G
f
(s) = Φ(s), so
the δ model approach and continuous-time case are
asymptotically equivalent. Obviously, it holds K
p
=
K
f
for T
s
→ 0.
To compute K
c
f
in (31) the following Riccati equation
is to be solved
AP
µ
+ P
µ
A
T
+ L
T
L−
1
µ
P
µ
C
T
CP
µ
= 0 (35)
and then the filter gain K
c
f
is calculated as
K
c
f,µ
=
1
µ
P
µ
C
T
. (36)
where µI and L
T
L are intensity matrices for measure-
ment and system noise, respectively.
6 ADAPTIVE CONTROL
The SISO ARMAX model is given by
A(q
−1
)y
t
= B(q
−1
)u
t
+C(q
−1
)e
t
(37)
where A(q
−1
),B(q
−1
) and C(q
−1
) are polynomials
in the backward shift operator q
−1
, i.e. A(q
−1
) =
1 + a
1
q
−1
+ ... + a
n
q
−n
,B(q
−1
) = b
1
q
−1
+ ... +
b
n
q
−n
,C(q
−1
) = 1 + c
1
q
−1
+ ... + c
n
q
−n
and y
t
is the
output , u
t
is the control input, and {e
t
} is assumed
to be a sequence of independent variables with zero
mean and variance σ
2
e
. Unknown system parameters
θ = (a
1
,...,a
n
,b
1
,...,b
n
,c
1
,...,c
n
)
T
(or corresponding
parameters of δ model) are estimated on-line to ob-
tain an updated model at time t, i.e.
ˆ
θ
t
(or corre-
sponding δ model) which is in turn used for updat-
ing the lqg adaptive control of the system. The pa-
rameter estimates of δ model can be used for tun-
ing the continuous-time LQG/LTR control assuming
the sampling period is small enough. In this way a
continuous-time system identification problem can be
omitted.
ARMAX model (31) has an equivalent innovation
state space representation
x
t+1
= Fx
t
+ gu
t
+ k
p
e
t
(38)
y
t
= h
T
x
t
+ e
t
(39)
where g = (b
1
,...,b
n
)
T
, k
p
= (c
1
− a
1
,...,c
n
−
a
n
)
T
, h
T
= (1,0, ...,0)
F =
−a
1
1 ... 0
. . ... 0
−a
n−1
. ... 1
−a
n
. ... 0
,
k
p
is the stationary gain vector for the associated
Kalman predictor corresponding to (3)
ˆx
t+1/t
= F ˆx
t/t−1
+ gu
t
+ k
p
˜y
p
t
(40)
where ˜y
p
t
= y
t
− h
T
ˆx
t/t−1
and σ
2
˜y,p
is the variance of
˜y
p
t
for which it holds σ
2
˜y,p
= σ
2
e
.
The actual model used for LQG/LTR control signal
u
t
calculation is obtained for current parameter esti-
mates
ˆ
θ
t
.
The investigated problem is to check out how the
approximated δ model used in adaptive LQG/LTR
control can be used in tuning the continuous-time
LQG/LTR control.
The issue of stability of the proposed adaptive
LQG/LTR control system is of course crucial. This
depends on the asymptotic convergence of parameter
estimates, particularly taking into account that in gen-
eral the parameter estimation in LQG adaptive control
even in the lack of modelling error, does not assure
the convergence to the true parameters. Closed loop
stability and good performance cannot be guaranteed
especially during the transient stage.
7 SIMULATIONS
Consider as an example a third-order minimum-
phase actual system obtained by discretizing the
continuous-time system
G(s) =
s+ 5
(s+ 1)(s+ 2)(s+ 3)
=
−s+ 1
(s+ 1)(s+ 2)
+
1
s+ 3
ADAPTIVE LQG CONTROL WITH LOOP TRANSFER RECOVERY
511