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1562 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
Y i2 i < y , i = 1… p,
where, K is the linear state feedback gain; ∑ x is the state
variables covariance; ∑ is the i th variable of output
covariance ∑ y ;
y i
2
y i is constraint of
Lemma2: Schur complements [25]:
For symmetric matrix:
⎡S
S = ⎢
⎣S
11
21
∑ yi .
The following three conditions are equivalent:
1) S < 0
T −1
22 12 11 12 0
2) S 11 < 0 , S − S S S <
−1
T
11 12 22 12 0
3) S 22 < 0 , S − S S S < .
B. Extended MVC 3 Problem and Its LMI Solution
In this work, constraints on manipulative variables
were included in the MVC 3 scheme. The extended MVC 3
can be described as:
S
S
12
22
⎤
⎥
⎦
p
m
min qi ∑ yi + rj ∑u j
∑ x, KY , i, Uj
i= 1 j=
1
∑ ∑ (6)
s.t. ( A+ BK) ∑ ( A+ BK) T + E∑ E
T = ∑ (7)
x w x
T
yi ϕiC
C ϕi
∑ = ∑ x (8)
T
uj ϕ jK
K ϕ j
∑ = ∑ x (9)
2
i
∑ < y , i = 1... p
(10)
yi
2
j
∑ < u , j = 1... m. (11)
uj
The main design target of the extended MVC 3 is
obtaining linear feedback gain K to make object function
(6) minimum, additionally, to make the steady-state
control outputs and the covariance of manipulative
variables satisfy a set of bounds respectively. The above
problem can be converted to a convex form of LMI as
follows.
Theorem 1: If and only if ∃W, X ≥ 0 and Y i ,
U , i = 1... p, j = 1... m s.t.
j
p m
min qY i i+
rU j j
XWYU , , j j i= 1 j=
1
∑ ∑ (12)
⎡
T
X −E∑ E AX BW⎤
w +
s.t. ⎢
⎥>
0
T
⎢⎣( AX + BW ) X ⎥⎦
(13)
⎡ Yi
ϕiCX⎤ ⎢
0
T T ⎥ >
⎣( CX ) ϕi
X ⎦
⎡ U j ϕ jW⎤ ⎢ T T ⎥ > 0
⎢⎣
W ϕ j X ⎥⎦
(14)
(15)
Y i2 i < y , i = 1... p
(16)
U j2 j < u , j = 1... m
(17)
−1
Then, u= Kx( k) = WX x( k)
is the extended MVC 3 linear
feedback controller of system (2), satisfying covariance
constraints.
Proof: If the closed-loop system (2) is stable, the steady
state covariance matrix can be expressed
T
as ∑ x = lim{E[ x( k) x ( k)]}
, and ∑
x
satisfies (7). From
k→∞
the definition of covariance, it is easily to get the
T
T
expression ∑ = ϕC∑ x C ϕ , ∑ = ϕ K∑ x K ϕ ,
where,
yi i i
2
u j
∑
yi
< y i
,
uj j j
2
j
∑ < u .From Lemma 1,
∃∑ x < X , Makes the state covariance constraint (7) is
T
equivalent to LMI (13). Let ϕiCXC ϕ i < Yi, i = 1, … p ,
T
ϕjKXK ϕ j < U j, j = 1, … m , and Y i2 i < y , i = 1... p ,
U j2 j < u , j = 1... m.Then:
p
m
p m
qi ∑ yi + rj ∑u j ≤ qY i i + rU j j
i= 1 j= 1 i= 1 j=
1
∑ ∑ ∑ ∑ .
Then, minimizing the function
ensure the object function
p m
qY i i + rU j j
i= 1 j=
1
p
m
qi yi + rj uj
i= 1 j=
1
∑ ∑ will
∑ ∑ ∑ ∑ be
minimized too. Still use Lemma 1, (14)-(17) can be
derived.
From the definition of extended MVC 3 problem,
controller feedback solution K ∗ can be solved by
∗ ∗ ∗−1
K = W X with LMI. Where, W ∗ and X ∗ denote the
optimal solution matrices of the extended MVC 3 problem.
Conditions (14)-(17) are exactly that required to
determine the feasibility of the extended MVC 3 problem.
If the problem turns out to be infeasible, then the
2 2
bounding region defined by yi
and u j terms should be
enlarged.
C. Performance Evolution Based on Extended MVC 3
For closed-loop system (5) the multi-variable form of
LQG performance benchmarks can be defined as
T
T
J
LQG
= E(
y Qy)
+ λ E( u Ru)
. In order to evaluate
controllers under constrains, the covariance constrains are
included in the LQG performance evaluation benchmarks.
New performance evolution is exactly an advanced
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