Environmental Engineering Reference
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4.4 H ' Optimal Control Background
Consider an LTI system with the following state space realization
x ¼ Ax þ B w w þ B u u
z ¼ C z x þ D zw w þ D zu u
y ¼ C y x þ D yw w þ D yu u
ð 4 : 8 Þ
n y n u . The signal u is the control input
and w is the disturbance. The signal y is the controlled variable and z is a fictitious
output signal that serves to state the control objectives. The signal z is commonly
named as the performance output.
It is assumed that system in Eq. 4.8 is stabilizable and detectable. That is, there
exists a control law u ¼ K ð s Þ y that stabilizes the closed loop system
n n , D zw 2
n z n w
where A 2
R
R
and D yu 2
R
T zw ð s Þ¼ G zw ð s Þþ G zu ð s Þ K ð s Þð I þ G yu ð s Þ K ð s ÞÞ 1 G yw ð s Þ
with I the identity matrix and G(s) the transfer function of the system in Eq. 4.8
partitioned as
The c-suboptimal H ? synthesis problem consists in finding an internally sta-
bilizing control law u ¼ K ð s Þ y that guarantees an 1 -norm of the closed-loop
transfer function from the disturbance w to the performance output z lower than c.
Being T zw ð s Þ being the closed-loop transfer function from w to z, the control
objective can be formalized as
k T zw k 1 \c ;
ð 4 : 9 Þ
where kk 1 denotes the infinity norm. For a stable system with transfer function
G(s), the 1 -norm is defined as
k G ð s Þk 1 ¼ max
x
r max ð G ð jx ÞÞ
where r max is the maximum singular value and x is the frequency [ 10 ]. In other
words, the 1 -norm is basically the maximum gain of the frequency response of
the transfer function G(s).
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