TELKOMNIKA ISSN: 1693-6930
Multiobjective H
2
H
∞
Control Design with Regional Pole Constraints Hardiansyah 105
where
, ,
x S
and x
R x
R x
Q x
Q
T T
= =
depend affinely on x , is equivalent to
0,
1
−
− T
x S
x R
x S
x Q
x R
3 In other words, the set of nonlinear inequalities Eq. 3 can be represented as the LMI Eq. 2.
Two standard LMI optimization problems are of interest: 1 LMI feasibility problem. Given an LMI
x F
, the corresponding LMI feasibility problem is to find
feas
x
such that
feas
x F
or determine that the LMI is infeasible. 2 Semi-definite Programming problem SDP. An SDP requires the minimization of a linear
objective subject to LMI constraints: Minimize
x c
T
Subject to x
F 4
where c is a real vector, and
F
is a symmetric matrix that depends affinely on the optimization variable x . This is
a convex optimization problem .
Both these problems can be numerically solved vary efficiently, using currently available software [17,18].
2.2. LMI formulation for multiobjective H
2
H
∞
performance
Consider the linear plant P with input u
, disturbance w
, performance output
∞
z
and
2
z
, the measurement signal
x . The input is generated by state feedback, using the controller
K . The
signal
∞
z
is the performance associated with the H
∞
constraint, the signal
2
z
is the performance associated with the
H
2
criterion. The state space representation of the controlled system can be written as follows:
u D
x C
z u
D x
C z
u B
w B
Ax x
22 2
2
+ =
+ =
+ +
=
∞
12 1
2 1
5 where all the matrices are constant real matrices of appropriate dimension. The illustration of
the controlled system is shown in Figure 1.
Figure 1. Generalized plant After substitution of the state feedback controller
Kx u
=
into Eq. 5, the closed-loop system becomes
x K
D C
z x
K D
C z
w B
x K
B A
x
22 2
2 12
1 1
2
+ =
+ =
+ +
=
∞
6 Let
w z
T
∞
and
w z
T
2
be the closed-loop transfer matrices from the generalized disturbance w to the performance output
∞
z
and
2
z
, respectively:
2
Z
P
K
w u
x
∞
Z
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TELKOMNIKA
Vol. 10, No. 1, March 2012 : 103 – 112 106
=
+ +
=
∞
∞
cl cl
cl w
z
C B
A K
D C
B K
B A
s T
12 1
1 2
7
=
+ +
=
2 22
2 1
2
2
cl cl
cl w
z
C B
A K
D C
B K
B A
s T
8 The goal of multiobjective
H
2
H
∞
control is to find an internally stabilizing controller K
that minimizes the
2
H
performance,
2
2
w z
T
, subject to the
∞
H
performance,
γ
∞
∞
w z
T
and places the closed-loop poles in some LMI stability region
D that will be explained in the next
subsection. In this subsection, pure H
2
and H
∞
synthesis are not given. For proofs and more details, see [20, 21].
We are now ready to give tractable necessary and sufficient conditions for solving the following multiobjective
H
2
H
∞
problem: min
cl2 T
cl
PC C
trace
2
9
s.t.
2
=
− +
+
∞ ∞
T cl
T cl
T cl
cl T
cl cl
P P
I P
C PC
B B
PA P
A γ
10
The optimization problem above is not yet convex because of the products
KP
arising in terms like
P A
cl
. So, defining the variables Y
= Y
T
= P
, L
= K Y
and W
= W
T
and using Schurs complement it is possible to rewrite the problem above as the LMI problem
min W
trace 11
s.t.
2 12
1 12
1
− +
+ I
L D
Y C
D L
YC H
T T
T
γ
12
22 2
22 2
+ +
W L
D Y
C D
L YC
Y
T T
T
13 where
T T
T T
B B
B L
L B
YA AY
H
1 1
2 2
+ +
+ +
=
. 2.3. LMI formulation for regional pole constraints
In the synthesis of control systems, meeting some desired transient performance objectives to ensure fast and well-damped transient response, reasonable feedback gain, etc.
should be considered. Generally, H
2
-norm and H
∞
synthesis design do not directly deal with the transient response of the closed-loop system. In contrast, a satisfactory transient response can
be guaranteed by confining its poles in a prescribe region. For many practical problems, exact pole assignment may not be necessary; it suffices to locate the closed-loop poles in a prescribe
subregion in the complex left half plane.
Definition 1. LMI stability region [20]. A subset
D of the complex plane is called an LMI region if
there exist a symmetric matrix
m m
kl
R
×
∈ =
] [
α α
and a matrix
m m
kl
R
×
∈ =
] [
β β
such that
} :
{ ∈
= z
f C
z D
D
14 where the characteristic function
z f
D
is given by
m l
k kl
kl kl
D
z z
z f
≤ ≤
+ +
=
, 1
] [
β β
α
D
f is
TELKOMNIKA ISSN: 1693-6930
Multiobjective H
2
H
∞
Control Design with Regional Pole Constraints Hardiansyah 107
valued in the space of m
m ×
Hermitian matrices. The location of the closed-loop poles of
2
K B
A +
in Eq. 6 concern with the performance of the closed-loop system, i.e., the stability, the decay rate, the maximum
overshoot, the rise time and settling time. Therefore, it is interesting work for control engineers to design the control gain
K such that the closed-loop poles of
2
K B
A +
lie in a suitable subregion of the left half plane. The interesting region for control purposes is the set
, ,
θ α
r S
of complex number
jy x
+ such that
, ,
r jy
x x
+ −
α
and y
x −
tan
θ
15 as shown in Figure 2. Confining the closed-loop poles to this region ensures a minimum decay
rates
α
, a minimum damping ratio
θ ζ
cos =
, and a maximum undamped natural frequency
θ ω
sin r
d
=
θ
in radian. The LMI formulations for the poles of
2
K B
A +
lie in the region ,
,
θ α
r S
are characterized as the following LMIs [20, 21]: if there exists symmetric
P such that
2 P
2 2
+ +
+ +
P K
B A
P K
B A
T
α
16
2 2
− +
+ −
rP K
B A
P P
K B
A rP
T
17 and
sin cos
cos sin
2
+ +
+ +
− +
+ −
+ +
+ +
2 2
2 2
2 2
2
T T
T T
K B
A P
P K
B A
K B
A P
P K
B A
P K
B A
K B
A P
K B
A P
P K
B A
θ θ
θ θ
18
with P
Y KP
L =
= ;
, the above LMIs are equivalent to 2
2 2
+ +
+ +
Y L
B L
B YA
AY
T T
α
19
2 2
− +
+ −
rY L
B YA
L B
AY rY
T T
20
2 2
2 2
2 2
2 2
+ +
+ −
− +
− −
+ +
+ +
T T
T T
T T
T T
L B
YA L
B AY
L B
YA L
B AY
L B
AY L
B YA
L B
YA L
B AY
sin cos
cos sin
θ θ
θ θ
21
From the analysis above, if there exists Y
and L
for Eqs. 19-21, then the poles of
2
K B
A +
lie in the region ,
,
θ α
r S
. 2.4. Multiobjective control design
The combination objectives of robust multiobjective H
2
H
∞
control with regional pole constraints can be characterized as follows:
ISSN: 1693-6930
TELKOMNIKA
Vol. 10, No. 1, March 2012 : 103 – 112 108
min
} ,
{
W trace
L Y
s.t. Eq. 12, Eq. 13 and Eqs. 19-21 22 From analysis above, the most important task in this paper is to find the variable
Y ,
L ,
γ
and W
can be solved using standard optimization techniques. Once a feasible solution Y, L
satisfying Eq. 22 is found, the required state feedback gain matrix can be computed as
1 −
= LY
K 23
which leads to trace
,
2
2
W T
T
w z
w z
≤ ≤
∞
∞
γ
24 The Lyapunov shaping paradigm for multi-objective design provides a greater flexibility
than single-objective optimal design techniques such as H
∞
synthesis or H
2
-norm technique.
Figure 2. Region ,
,
θ α
r S
Figure 3. A SMIB power system
Figure 4. Static fast exciter model
Figure 5. Block diagram of conventional PSS
3. Results and Analysis 3.1. Dynamic model of the power system
