Mathematical Problems in Engineering 3
3. Design of Nonfragile State Feedback RMPC
with Time-Delay Compensation
In this section, the focus will be on the design of nonfragile
state feedback RMPC with time-delay compensation includ-
ing dierent gain perturbation cases. To begin with, two
nonfragile state feedback control laws u
𝐴
(+|)and u
𝑀
(+
|)with additive gain perturbation and multiplicative gain
perturbation are respectively described as follows:
u
𝐴
(
+|
)
=K
𝐴
+K
𝐴
(
)
x
(
+|
)
+K
𝐴𝑑
x
(
+−|
)
,
(8)
u
𝑀
(
+|
)
=K
𝑀
+K
𝑀
(
)
x
(
+|
)
+K
𝑀𝑑
x
(
+−|
)
,
(9)
where K
𝐴
(),K
𝐴
(),K
𝐴𝑑
(),K
𝑀
(),K
𝑀
(),K
𝑀𝑑
() are
gain matrixes to be determined by RMPC strategies. Assume
that K
𝐴
()and K
𝑀
()have the following forms as in (10)
and (11), respectively, representing additive gain perturbation
case (referred to as Case I) and multiplicative gain perturba-
tion case (referred to as Case II):
K
𝐴
(
)
=ML
(
)
N,
L
(
)
2
≤1,
(10)
K
𝑀
(
)
=UW
(
)
VK
𝑀
,
W
(
)
2
≤1,
(11)
where M, N, U, V are constant matrixes with appropriate
dimensions and L(), W() can represent any unknown
matrixes satisfying norm-bounded conditions by scaling M,
N, U,andV.
For systems (1) subject to constraints, the following
innite horizon quadratic cost function is rst dened:
J
(
)
=
∞
𝑖=0
x
(
+|
)
Q
1
+
u
(
+|
)
R
,
(12)
where Q
1
and R are given positive denite symmetric weight-
ing matrixes. Consider the following min-max optimization
problem, which minimizes the worst case innite horizon
quadratic objective function.
CaseIisasfollows:
min
u
𝐴
(𝑘+𝑖|𝑘)(0≤𝑘≤∞)
max
ΔA(𝑘),ΔA
𝑑
(𝑘),ΔK
𝐴
(𝑘)
J
(
)
.
(13)
Case II is as follows:
min
u
𝑀
(𝑘+𝑖|𝑘)(0≤𝑘≤∞)
max
ΔA(𝑘),ΔA
𝑑
(𝑘),ΔK
𝑀
(𝑘)
J
(
)
.
(14)
In order to design such controllers as Cases I and II, the
upperboundofthecostin(13)or(14)needstobedetermined.
According to Lyapunov-Krasovskii function, a delay state
dependent quadratic function is chosen as follows:
V
(
x
(
))
=
x
(
)
P
+
𝑑
𝑗=1
x
−
S
,
(15)
where P = P
𝑇
>0and S = S
𝑇
>0are positive denite
symmetric matrixes and satisfy the following condition to
guarantee the cost monotonicity:
V
(
,
)
=V
(
x
(
++1|
))
−V
(
x
(
+|
))
≤−
x
(
+|
)
Q
1
−
u
(
+|
)
R
.
(16)
For the cost in (13) or (14) to be nite, V(x(∞ | )) = 0
is assumed. Summing both sides of (16) from =0to =∞,
the worst value of the cost function J()will be bounded by
the following.
Case I is as follows:
max
ΔA(𝑘),ΔA
𝑑
(𝑘),ΔK
𝐴
(𝑘)
J
(
)
≤V
(
x
(
|
))
=
x
(
|
)
P
+
𝑑
𝑗=1
x −|
S
.
(17)
Case II is as follows:
max
ΔA(𝑘),ΔA
𝑑
(𝑘),ΔK
𝑀
(𝑘)
J
(
)
≤V
(
x
(
|
))
=
x
(
|
)
P
+
𝑑
𝑗=1
x −|
S
.
(18)
erefore, the above optimization can be reformulated as
follows.
Case I is
min
P,S
x
(
|
)
P
+
𝑑
𝑗=1
x −|
S
s.t.
(
1
)
,
(
8
)
,
(
16
)
,
(
17
)
.
(19)
Case II is
min
P,S
x
(
|
)
P
+
𝑑
𝑗=1
x −|
S
s.t.
(
1
)
,
(
9
)
,
(
16
)
,
(
18
)
.
(20)
In order to convert the minimization problem in (19) and
(20) into LMI optimization problems that can be eciently
solved by convex optimization algorithm, the upper bound
of the objective function in (17) and (18) is dened as follows.
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