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T race(P ). Consequently, minimization of the trace
of M
1
implies minimization of the trace of P . In
a similar way it can be shown that minimization of
the traces of M
2
and M
3
implies minimization of the
traces of R
1
and R
2
, respectively. Thus, minimiza-
tion of
¯
J in (36) implies minimization of the guaran-
teed cost of the uncertain time-delay system (1)-(5)
with performance index (6). The optimality of the so-
lution of the optimization problem (36) follows from
the convexity of the objective function and of the con-
straints.
4 EXAMPLE
Consider the second order system of the form of
equation (1)with nominal matrices,
A
1
=
·
−2 1
0 1
¸
, A
2
=
·
−0.2 0.1
0 0.1
¸
,
B
1
=
·
0
1
¸
, B
2
=
·
0
0.1
¸
The uncertainty matrices on the state and con-
trol as well as on the delay matrices are according to
(2),
∆A
1
=
·
0 0
0.1 0.1
¸
, ∆A
2
=
·
0 0
0.03 0.03
¸
,
∆B
1
=
·
0
0.1
¸
, ∆B
2
=
·
0
0.03
¸
and hence, the rank-1 decomposition may be
chosen such that,
D
1
=
·
0
0.1
¸
, D
2
=
·
0
0.03
¸
, E
1
= E
2
=
·
1
1
¸
,
F
1
=
·
0
0.1
¸
, F
2
=
·
0
0.03
¸
, G
1
= G
2
= 1
The state and control weighting matrices are,
respectively, Q = I
2
, R = 4. It is β
1
= 0.3,
β
2
= 0.2.
By solving the LMI objective minimization problem
one obtains the guaranteed cost
¯
J = 25.1417.
The optimal rank-1 decomposition is obtained
with s
1
= 0.6664, s
2
= 2.2212, t
1
= 2.5000,
t
2
= 8.3333. Finally, the corresponding control gain
is K = [−0.2470 −6.0400].
5 CONCLUSIONS
In the previous sections, the problem of guaranteed
cost control has been studied for the class of uncer-
tain linear systems with state and input time varying
delays. A constant gain linear state feedback control
law has been obtained by solving an LMI feasibility
problem. The closed-loop system is then quadrati-
cally stable and preserves acceptable performance for
all parameter uncertainties and time varying delays
of a given class. The system performance deviates
from the optimal one, in the sense of LQR design of
the nominal system, due to uncertainties and time de-
lays. However, the performance deterioration is lim-
ited and this is expressed in terms of a performance
upper bound, namely the guaranteed cost. In order to
make the GC as small as possible, one has to solve
an LMI minimization problem. The minimal upper
bound corresponds to the ”optimal” rank-1 decompo-
sition of the uncertainty matrices.
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INPUT DELAYS
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