Environmental Engineering Reference
In-Depth Information
There are several solutions for the c-suboptimal H ? synthesis problem, but
the most popular nowadays is the formulation as an optimization problem with
linear matrix inequalities (LMIs) constraints [ 11 ]. Considering a controller with a
state-space realization
the controller matrices can be found by solving the following optimization
problem:
minimize c ð R ; S ; B k ; C k ; D k Þ ,
subject to
2
3
AR þ B u C k þð I Þ
I
I
4
5 \0 ;
ð B w þ B y D k D yw Þ T
cI n w
I
C z R þ D zu C k
D zw þ D zu D k D yw
cI n z
2
3
ð SA þ B k C y Þþð I Þ
I
I
4
5 \0 ;
ð SB w þ B k D yw Þ T
cI n w
I
C z þ D zu D k C y
D zw þ D zu D k D yw
cI n z
[ 0 ;
RI
IS
with ''['' and ''\'' denoting positive and negative definite matrices, respectively,
and I represents the matrices needed to obtain a symmetric matrix.
After finding the positive definite matrices R and S and matrices B k , C k and D k ,
the controller matrices can be computed from
A k ¼ð A þ B u D k C y Þ T þ SB w þ B k D yw
C z þ D zu D k C y
"
# 1
"
#
ð D zw þ D zu D k D yw Þ T
ð B w þ B y D k D yw Þ T
C z R þ D zu C k
cI
D zw þ D zu D k D yw
cI
B k ¼ N 1 ð B c SB u D c Þ;
C k ¼ð C k D k C y R Þ M T ;
with MN T ¼ I RS.
The optimization problem involved in the H ? -synthesis can be effectively
solved with available software such as Sedumi [ 12 ] and YALMIP [ 13 ]. It is also
available as a command in the Robust Control Toolbox for Matlab. Therefore, the
design process of an H ? optimal control requires only to put the control specifi-
cations in terms of the minimization of the norm in Eq. 4.9 , i.e., to construct the
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