1.3. ROBUST CONTROL 5
H
1
techniques are studied in many papers and books. Some of the most well known
are introduced here. Zames [1981] is the pioneering work which introduced the H
1
control.
Kwakernak [1993] is a good tutorial paper on H
1
control with some numeric examples. Zhou
and Doyle [1997] and Green and Limbeer [2012] are general texts on robust control and studied
the H
1
control in detail. Gu et al. [2013] and Chiang et al. [2007] are good references to learn
how to design H
1
controllers using MATLAB
®
.
H
1
techniques are applied to a number of DC-DC converters successfully. Naim et al.
[1997] is the main antecedent in the use of H
1
to DC-DC converters. It designed a H
1
con-
troller for a boost converter. Output impedance, audio susceptibility, phase margin, and band-
width of the control loop are the usual measure of performance in the DC-DC switch mode
Pulse Width Modulator (PWM) converters. Reduction of output impedance in non-minimum
phase converters (such as boost and buck-boost) is achieved at the expense of phase-marging
reduction. However, the H
1
controller designed in Naim et al. [1997] minimizes the output
impedance in a wide frequency range without decreasing the phase margin. is paper neglegted
the system uncertainties. Khayat et al. [2017] studied the robust control of boost converters in
presence of uncertainties. Shaw and Veerachary [2017] designed a H
1
for a High Gain Boost
Converter (HGBC). Vidal-Idiarte et al. [2003] designed a H
1
controller to maximize the band
width of the control loop with a perfect tracking of the desired output voltage for boost and buck-
boost converters. Experimental results are compaerd with those obtained using Sliding Mode
Control (SMC) and current peak control. Hernandez [2008] used the H
1
loopshaping to de-
sign a controller for a buck-boost converter. Designed controller showed better performance
in comparison with PID controller. Gadoura [2001] and Gadoura et al. [2002] designed H
1
controller for paralleled buck converter operating in current-mode control (CMC) and voltage-
mode control (VMC).
1.3.3 SYNTHESIS
e synthesis uses the D-K or -K iteration methods [Gu et al., 2013] to minimize the peak
value of the structured singular value of the closed-loop transfer function matrix over the set of
all stabilizing controllers K. e structured singular value of a closed-loop system transfer matrix
M.s/, with uncertainty and singular values is defined as:
k
M
k
D
1
.
M
/
WD
min
2
f
.
/
W det
.
I M
/
D 0
g
:
Usually the controller designed using the synthesis has a high order which makes the
implemetation difficult. A model order reduction procedure is usually required.
e H
1
control design techniques consider the system uncertinty in the unstructured
form so the controller designed using the H
1
techniques is conservative. e synthesis con-
siders the uncertainty structures so its output is less conservative [Bevrani et al., 1999].
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