[1]Rıdvan Keskin

[1]Department of Electrical and Electronics Engineering, Zonguldak Bülent
Ecevit University, Zonguldak, 67100, Türkiye

2]Department of Control and Automation Engineering, Yildiz Technical
University, İstanbul, 34220, Türkiye

# Linear matrix inequality based Type-III compensator synthesis for DC-DC
converters

<EMAIL_ADDRESS>Ibrahim Aliskan<EMAIL_ADDRESS>*  [

###### Abstract

Boost, buck-boost, and fly-back DC-DC converters which are utilized in power
lines of any electric vehicles, solar energy, and power factor correction
applications require control systems to regulate the output voltage under
mismatched disturbances i.e. load current and input voltage. In continuous
current mode operation, the converters, however, are bandwidth-limited control
systems due to their non-minimum phase nature. Disturbance rejection
performance of such bandwidth-limited control system is an open problem
especially where input voltage and load current disturbances cannot be
measured. A third-order integral-lead (Type-III) compensator with a
disturbance observer (DOB) can suppress the disturbances and unmodeled
dynamics of the converters. However, synthesizing such a fixed-order control
system under performance constraints is generally challenging. This paper
proposes a simultaneous design of a Type-III compensator and a fixed order DOB
based on $H_{\infty}$ control approach using convex optimization. The
optimization problem is formulated in a convex-concave procedure by including
the estimated disturbance and sensor noise functions. We proposed a two-stage
iterative algorithm to solve the problem in a convex optimization framework.
Convex programming can therefore be used to synthesize an optimal fixed-order
control system by removing the non-convex constraints on the parameter space.
The approach leads to an easily resolvable control algorithm with linear
matrix inequality constraints over parameterized controller parameters due to
the convexity of the problem. The proposed control system is implemented on a
200 W DC-DC multi-phase interleaved boost converter prototype using a
TMS320F28335 digital signal processor. The performance of the approach is
compared with the well-known K-factor design approach for the Type-III
compensators.

###### keywords:

Disturbance observer, Type-III compensator, fixed-order $H_{\infty}$ control,
convex optimization

##  1 Introduction

Boost, buck-boost, and fly-back converters are utilized in power transmission
lines of any electric vehicle, solar energy systems, and high voltage DC-DC
applications azarastemal2021cascade . The converters are improved to operate
in high-power applications through any kind of parallel, cascade, and
interleaved techniques. The converters regulate the output voltage of the
converter-connected systems in the presence of load current and line
variations. Linearization of the nonlinear converter models around an
operating point produces an unstable zero which causes additional phase lag,
extreme overshoots, and unpleasant responses in a set-point step change in the
voltage mode control loop. Therefore, synthesizing a voltage mode compensator
for the converters is challenging shruti2021analytical . In large part of
control loops in industrial applications, popular fixed-order controllers like
PI, PID, lead, lag, lead-lag, and third-order integral-lead (Type-III)
compensators are preferred to overcome such issues. The Type-III compensator,
a special kind of power electronics compensator, provides solid disturbance
rejection capability due to inherent phase boost to maintain a reasonable
phase margin sarrafan2020novel . Unlike advanced robust PI, PD/PID, and lead-
lag controller synthesize methods commonly exist in the literature
hayes2016design , there exist control design methods that consider only
nominal operating conditions for Type-III compensators.

The K-factor method which depends on the designer’s prior control knowledge is
widely used for designing the compensator chen2020interleaved ;
ghosh2016design ; rana2017development ; anzehaee2018augmenting . However, the
design process generally does not consider unavoidable noise, disturbances,
and uncertainties. The Pole-placement method is proposed for designing the
compensator according to ideal closed-loop models keskin2019design . However,
the method is derived from the designer’s prior control knowledge of engineers
cr2011type . Machine-learning or swarm algorithms, for example, particle swarm
optimization banerjee2016improved , genetic algorithm tran2020dual  and
dragonfly algorithm rana2021performance  are used to optimize the parameters
of the compensator considering only the step response of the control system.
The algorithms generally converge to local solutions since they are non-convex
optimization algorithms. The stopping criteria used for the global solution in
such optimization algorithms are usually random and vary according to the
lower and upper bounds of the controller parameters hindi2004tutorial . A
robust non-convex $H_{\infty}$ method which considers the sensor noise and
parameter uncertainties in the design process is proposed to design a Type-III
compensator keskin2021robust . The parameters of the compensator obtained from
resolving the algorithm are different since the non-convex algorithms
generally converge to different local solutions in each run of execution.
These methods may be insufficient for industrial applications since the
methods do not involve sensor noise, parameter uncertainty, and mismatched
disturbances in the design process sedhom2020robust . Robust $H_{\infty}$
structured control techniques guarantee robust stability and performance
challenges, particularly the approaches based on linear matrix inequalities
(LMIs) bagheri2022robust . A fixed-order controller synthesis is restructured
as a constrained convex optimization problem where the local solution is also
the global solution ahmad2017robust . Youla parameterization
rodriguez2020modeling , convex inner approximation erol2019fixed , augmented
Lagrangian ankelhed2012partially  and convex-concave procedure (CCP)
kammer2017decentralized  approaches are utilized to convert the problem into a
constrained convex form. A convex optimization method which includes
converting bilinear matrix inequality constraints to LMI constraints is
proposed dinh2011combining . It is presented that the convex-concave algorithm
converges to a locally optimal solution. A controller synthesis method is
proposed, where the method covers only fixed-order controllers with parameters
to be optimized only in the numerator matrix, such as PI and PID saeki2010low
. Another PID controller synthesis method is presented to assess mutual
interference via frequency response data shinoda2017multivariable . A PID
controller design method whose time constant is fixed at a pre-defined value
is proposed using convex optimization boyd2016mimo . The parameter uncertainty
of the system is represented in the inner loop as mini circles of the loop
transfer function. The time constant of the PID controller can be optimized
iteratively segovia2013noise . Quantitative feedback theory-based automatic
loop-shaping method is proposed for PID controllers in multi-model uncertain
systems mercader2016robust .

The aforementioned methods can only be used for controller types that have
parameters to be optimized in the numerator polynomial. The Type-III
compensator design problem, however, is inherently non-convex since it has
parameters to be optimized in the denominator polynomial. A PI controller with
input filters is synthesized using a frequency-domain approach for multiple
voltage source inverters kammer2018convex . The approach allows designers to
use $H_{2}$, $H_{\infty}$, and $H_{2}-H_{\infty}$ loop-shaping performance
objectives. A controller synthesis approach is proposed for model-based or
data-driven systems karimi2017data .

A disturbance observer (DOB) theory deals with online unmodelled system
dynamics estimation and compensation, and it consists of an inverse of the
nominal system model and a typical low-pass filter ($Q$) for the system which
has input disturbances ohnishi1996motion . The filter characterizes the
transient response and uncertainty rejection performances of the DOB. The
various DOB structures and design methods are summarized
sariyildiz2019disturbance . An LMI-based control method for designing
$Q$-filter is proposed assuming the control model is a minimum phase system
wang2004design . A systematic DOB design methodology for non-minimum phase
systems is proposed in wang2020robust . The methodology considers non-convex
constraints for the internal stability of the feedback controller and DOB.
Another non-convex design method of DOB is proposed, where the fixed-order
$Q$-filter consists of a single tuning parameter tena2022performance . A
sequential design method of a DOB and full-order feedback controller is
proposed using the non-convex $H_{\infty}$ mixed-sensitivity approach
huang2022high . A simultaneous design of inverse of the plant and DOB is
proposed for data-driven systems in an $H_{\infty}$ convex optimization
framework, where the feedback controller is pre-designed wang2022frequency .
The mentioned studies are implemented for the systems which have input
disturbances. A genetic algorithm-based DOB design method is proposed for the
systems that have mismatched disturbances, where the inverse of the plant
model is estimated through the concept of equivalent transfer function
gonccalves2022disturbance . We propose an approach in which the fixed-order
feedback controller and the $Q$-filter are synthesized simultaneously in a
convex optimization framework for control systems that have mismatched
disturbances.

In this paper, a simultaneous controller design method for the Type-III
compensator and a fixed-order DOB is proposed to meet model matching and
disturbance rejection performance criteria. The estimated models of the multi-
variable control system are employed to find the parameters of the compensator
and filter. The contributions of this work are summarized as follows:

  * •

An iterative $H_{\infty}$ control algorithm for simultaneous design of the
fully parameterized compensator and DOB is proposed in a convex optimization
framework.

  * •

The infinite-norm constraints of the load current and input voltage
disturbances sensitivity functions are considered in the constrained convex
optimization problem.

The structure of the paper is as follows: Section II presents the control
system of the closed-loop converter. Section III includes the formulation of
the synthesis problem in a convex optimization framework. In Section IV,
tracking performances of the controlled systems are presented under
disturbance events and parameter changes of the converter. Section V presents
the digital signal processor (DSP) based on real-time application results.
Conclusions and discussions of the paper are given in Section VI.

##  2 Low-frequency multi-variable models of the converter

The power network of the IBC circuit is shown in Fig. 1, where $L_{1}$ and
$L_{2}$ are the inductances with parasitic resistances $(r_{l})$. $C_{o}$ is
the filter capacitance with parasitic resistance ($r_{c}$) and $i_{o}(t)$ is
the load current disturbance. $F_{1}$ and $F_{2}$ are the active switches of
the relevant phase leg of the converter with on-state parasitic resistances
($r_{s}$). $E_{1}$ and $E_{2}$ are the passive switches of the relevant phase
leg of the converter. $U$ is the DC duty cycle of the gate signals,
$v_{in}(t)$ is the input voltage, $v_{o}(t)$ is the output voltage, $R_{o}$ is
the load resistance and $i_{out}(t)$ is the load current of the converter.
$V_{g1}(t)$ and $V_{g2}(t)$ are the gate signals of the relevant phase leg of
the converter. The converter consists of four operating modes according to the
on-off states of the active switches. The operating modes are presented in
Fig. 1. The components of the converter are measured using an LCR meter to
find the exact values of the components and parasitic resistances. The nominal
and measured values of the converter are given in Table 1.

Figure 1: The power network of the interleaved boost converter for the duty cycle greater than $0.5$ Table 1: Previous and updated system parameters of the 200 W prototype converter Expr. |  Pre. value |  New value |  Unit
---|---|---|---
$L_{1}$ |  5 |  5.069 |  mH
$L_{2}$ |  5 |  5.085 |  mH
$C_{o}$ |  1 |  0.996 |  mF
$r_{l}$ |  0.5 |  0.585 |  $\mathrm{\SIUnitSymbolOhm}$
$r_{s}$ |  0.036 |  0.036 |  $\mathrm{\SIUnitSymbolOhm}$
$r_{c}$ |  0.05 |  0.01 |  $\mathrm{\SIUnitSymbolOhm}$
$R_{o}$ |  50 |  |  $\mathrm{\SIUnitSymbolOhm}$
$v_{in}(t)$ |  46 |  - |  V
$v_{o}(t)$ |  100 |  - |  V
$f$ |  10 |  - |  kHz

The averaged non-linear models of the converter were derived according to the
operating modes of the converter. The linear transfer functions of the multi-
variable control system are obtained using the state-space averaging approach
keskin2021multi . The low-frequency transfer functions of the converter are
given as

$\displaystyle
G_{m}(s)=\frac{(-V_{o}(C_{o}r_{c}s+1)(R_{o}^{2}(4U-2U^{2}-2)+R_{o}(r_{c}+r_{s})+r_{c}(r_{l}+r_{s})+LsR_{tot})}{(U-1)(C_{o}Ls^{2}(R_{o}^{2}+2R_{o}r_{c}+r_{c}^{2})+C_{o}UR_{o}s(R_{o}(r_{s}-r_{c})-r_{c}^{2}+2r_{s}r_{c}))+T_{c}+T_{s}},$
(1) $\displaystyle
G_{vm}(s)=\frac{-(2R_{o}(R_{tot})(U-1)(C_{o}r_{c}s+1))}{C_{o}Ls^{2}(R_{o}^{2}+2r_{c}+r_{c}^{2})+C_{o}UR_{o}s(R_{o}(r_{s}-r_{c})+r_{c}(2r_{s}-r_{c}))+T_{c}+T_{s}},$
(2) $\displaystyle
G_{im}(s)=\frac{(R_{tot})(C_{o}R_{o}s+C_{o}r_{c}s+1)}{C_{o}Ls^{2}(R_{o}^{2}+2r_{c}+r_{c}^{2})+C_{o}UR_{o}s(R_{o}(r_{s}-r_{c})+r_{c}(2r_{s}-r_{c}))+T_{c}}+T_{s},$
(3) $\displaystyle
T_{c}=s(C_{o}R_{o}(R_{o}(r_{c}+r_{l})+2r_{c}r_{l}+r_{c}^{2})+C_{o}r_{c}^{2}(Ur_{s}+r_{l})+R_{tot}L),$
(4) $\displaystyle
T_{s}=R_{o}^{2}U(2U-4)+UR_{o}(r_{s}-r_{c})+Ur_{s}r_{c}+R_{o}(2R_{o}+r_{c})+R_{tot}r_{c},$
(5) $\displaystyle R_{tot}=R_{o}+r_{c},R_{oc}=R_{o}/R_{tot}.$ (6)

The DC magnitudes and AC deviations are denoted by capital and small letters
with a tilde $(\sim)$, respectively. $\widetilde{u}(s)$ is the AC deviation of
the duty cycle, $\widetilde{i}_{o}(s)$ is the AC deviation of the load current
disturbance, $\widetilde{v}_{o}(s)$ is the AC deviation of the output voltage
and $\widetilde{v}_{in}(s)$ is the input voltage disturbance. The mathematical
transfer function of the plant ($G_{m}(s)$) can not be inverted since it has
the right half plane (RHP) zero. Therefore, we use linear parametric system
identification methods to obtain the minimum phase transfer function of the
plant. Orthogonal pseudo-random binary sequences (PRBS) signals, which have a
length of 1.000.000 samples and two times higher bandwidth than RHP zero, are
applied sequentially to the open-loop converter as input voltage and gate
signals. The PRBS signal applied to the active switches of the converter is
presented in Fig. 2. The autoregressive moving average with exogenous inputs
(ARMAX) approach is preferred to estimate the frequency responses of the
converter. The estimation process is performed using the MATLAB identification
toolbox. The order of the estimated plant model is equal to the mathematical
model of the converter. The transfer functions of the multi-variable control
system are given as

$\small\begin{split}&G_{n}(s)=\frac{-74.79s+1.811\cdot
10^{7}}{s^{2}+284s+9.129\cdot 10^{4}},\\\
&G_{i}(s)=\frac{-s-200}{0.001s^{2}+0.22s+104.6},\\\
&G_{v}(s)=\frac{8.58s+1.724\cdot 10^{5}}{s^{2}+277.8s+9.126\cdot 10^{4}},\\\
&W_{n}(s)=\frac{s^{2}-5730s+1.177\cdot 10^{5}}{s^{2}+9600s+9.379\cdot
10^{7}},\end{split}$ (7)

where $G_{n}(s)$ is the estimated transfer function of the plant from
$\widetilde{u}(s)$ to $\widetilde{v_{o}}(s)$, $G_{i}(s)$ is the estimated
transfer function of the load current disturbance, i.e from
$\widetilde{i}_{o}(s)$ to $\widetilde{v}_{o}(s)$, $G_{v}(s)$ is the estimated
transfer function of the input voltage disturbance, i.e from
$\widetilde{v}_{in}(s)$ to $\widetilde{v}_{o}(s)$ and $W_{n}(s)$ is the
transfer function of the output voltage sensor noise. The frequency responses
of the mathematical and estimated transfer functions are presented in Fig. 3.

Figure 2: The time domain responses of the output voltage and PRBS signal
Figure 3: The frequency responses of the mathematical and estimated transfer
functions

The conventional 1-DOF control scheme is presented in Fig. 4 (a), where $K(s)$
is the feedback compensator, $G(s)$ is the uncertain plant model, $W_{i}(s)$
is the weighting function of the load current disturbance, $W_{v}(s)$ is the
weighting function of the input voltage disturbance, $n$ is the noise, $r$ is
the set-point, $e$ is the error, $u_{f}$ is the control input of the feedback
loop, $y_{m}$ is the measured output variable, and $y_{n}$ is the noisy
measured output variable. The feedback compensator is designed as a Type-III
compensator which has two real zeros, poles, and pure integral action since it
has inherent phase boost properties chan2015generalized . The bandwidth of the
1-DOF control system is restricted since the plant has the RHP zero.
Therefore, the disturbance rejection performance of the system is limited.

A two-degrees of freedom (2-DOF) control system with a DOB is preferred to
improve the disturbance rejection capability and robustness of the system. The
DOB control scheme is presented in Fig. 4 (b), where $u$ is the equivalent
control input to the plant, $G_{n}^{-1}(s)$ is the inverse of the nominal
plant model, $\hat{d}$ is the estimation of the disturbances and $Q(s)$ is the
low-pass filter of the DOB. The filter is utilized to suppress noise at high
frequencies and disturbances at low frequencies. The uncertain plant model can
be equal to the nominal plant model ($G(s)=G_{n}(s)$) since the plant model is
estimated experimentally. The mentioned two conventional control schemes are
combined in a robust $H_{\infty}$ framework to include the transient response
and disturbance rejection performance conditions. The robust $H_{\infty}$
control design scheme of the proposed approach is presented in Fig. 5, where
$W_{1}(s)$ is the error weighting function, $W_{2}(s)$ is the feedback control
input weighting function and $W_{3}(s)$ is the output variable weighting
function. $z_{1}$ is the error performance output, $z_{2}$ is the performance
output of the control input of the feedback compensator, and $z_{3}$ is the
performance output of the measured variable.

(a) The conventional closed-loop voltage-mode 1-DOF control scheme

(b) The proposed DOB based 2-DOF control scheme

Figure 4: The conventional feedback and DOB control systems for the output
disturbances

Note that the $G(s)$ is used instead of the nominal model transfer function
for simplicity throughout the rest of the paper. The sensitivity functions of
the outer feedback loop are obtained to structure the transient response
ignoring the DOB loop keskin2023robust . The three sensitivity functions are
derived from Fig. 5 as

$\begin{split}&S(s)=\frac{1}{1+G(s)K(s)},\\\
&T(s)=\frac{-W_{n}(s)G(s)K(s)}{1+G(s)K(s)},\\\
&S_{i}(s)=\frac{K(s)}{1+G(s)K(s)},\end{split}$ (8)

where the sensitivity function $S(s)$ is the transfer function from set-point
to error, i.e., $r$ to $e$, $T(s)$ is the complementary sensitivity function,
which represents the transfer function from $n$ to $y_{m}$, and $S_{i}(s)$ is
the input sensitivity function which represents the transfer function from $r$
to $u$. The sensitivity functions of the disturbance loops should be included
in the problem to decrease singular values of the sensitivity function in the
medium frequency area. Therefore, we can mitigate vital resonance modes of the
control system. It enables the reduction of the output voltage oscillations
and overshoots in the presence of uncertainties in the real parameters of the
converter. The function of the output disturbances is derived as

$S_{o}(s)=\frac{W_{v}(s)G_{v}(s)+W_{i}(s)G_{i}(s)}{1+G(s)K(s)},$ (9)

where $S_{o}(s)$ is the sensitivity function from $\widetilde{v}_{in}$ and
$\widetilde{i}_{o}$ to ${y}_{m}$. It emphasizes that the parameter
uncertainties in the plant cause the worst-gain effect at medium frequency
area. Since disturbance transfer functions have resonance gain in the same
frequency area, adding the functions in the problem could represent an
additive uncertainty weighting function of the system. The feedback
compensator is designed to obtain reference tracking and noise suppression
performances. The inner DOB loop is included for further disturbance rejection
and uncertainty suppression. The sensitivity functions of the inner DOB loop
are given as

$\begin{split}&S_{D}(s)=\frac{W_{i}(s)G_{i}(s)+W_{v}(s)G_{v}(s)}{1+Q(s)},\\\
&T_{D}(s)=\frac{G(s)Q(s)W_{n}(s)}{1+Q(s)},\end{split}$ (10)

where $S_{D}(s)$ is the transfer function of the inner loop from
$\widetilde{v}_{in}$ and $\widetilde{i}_{o}$ to $y_{m}$ and $T_{D}(s)$ is the
transfer function of the inner loop from $n$ to $y_{m}$. These functions are
utilized to improve the stability of the filter. The sensitivity functions,
however, may be insufficient to shape the frequency response of the
$Q(s)$-filter. The $Q(s)$ filter is shaped in the frequency domain using
infinite norm constraint of the function $(W_{Q}(s)(1-Q(s))$ where $W_{Q}(s)$
reflects the frequency characteristics of the mismatched disturbances. It is
highlighted that the function is a part of the numerator matrix of the closed-
loop sensitivity function. The frequency characteristics of the function
define the property of disturbance estimation wang2004design . Assuming the
multi-variable system is internally stable, DOB is employed to completely
estimate and reject the disturbance inputs in the case $\lvert
Q(j\omega)\rvert\approx 1$ at low and medium frequencies.

Figure 5: Re-structured control scheme for robust $H_{\infty}$ Type-III
feedback compensator and DOB design problem

The general sensitivity functions of the closed-loop system are given as

$\begin{split}&S_{C}(s)=\frac{(1-Q(s))(W_{i}(s)G_{i}(s)+W_{v}(s)G_{v}(s))}{1+G(s)K(s)},\\\
&T_{C}(s)=\frac{(Q(s)+G(s)K(s))W_{n}(s)}{1+G(s)K(s)},\end{split}$ (11)

where the sensitivity function $S_{C}(s)$ is the transfer function from the
output disturbances to the measured variable, and the complementary
sensitivity function $T_{C}(s)$ is the transfer function from the noise to the
measured variable. For the stability of the inner and outer loop, the infinity
norms of these two functions are also restricted.

##  3 The convex optimization problem structure

This section covers the convex optimization problem which is formulated to the
simultaneous design of the feedback compensator and $Q(s)$ filter in the
fixed-order $H_{\infty}$ framework. The control problem can be structured
using the infinity norm representations of the mentioned sensitivity functions
to satisfy the stability of the control system over defined disturbance
effects. We now omit the complex frequency $(s)$ from this point on unless
otherwise noted. The non-convex optimization problem is formulated as

$\begin{array}[]{clclcl}\displaystyle\min_{\begin{matrix}X,Y,N,M,\gamma\\\
\end{matrix}}&\gamma,\\\\[17.07164pt]
\textrm{s.t.}&\big{\|}W_{1}S\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}W_{3}T\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}W_{2}S_{i}\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}S_{o}\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}S_{D}\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}T_{D}\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}W_{1C}S_{C}\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}W_{3C}T_{C}\big{\|}_{\infty}\leq\gamma,\\\\[5.69054pt]
&\big{\|}(W_{Q}(1-Q))\big{\|}_{\infty}\leq\gamma,\end{array}$ (12)

where $\gamma\in\mathbb{R}^{1\times 1}$ is an auxiliary variable that
represents the upper bounds of the singular values, $X$ is the numerator
vector of the compensator, $Y$ is the denominator vector of the compensator,
$N$ is the numerator vector of the filter, and $M$ is the denominator vector
of the filter. Here, the integral action of the compensator can be achieved in
two different ways. The integral action is added to the denominator of the
compensator. Otherwise, the compensator can be simplified by adding the
integral action to the nominal model. We prefer the second way to reduce the
instability and numerical problems that may occur at certain frequency values
in the compensator synthesis. Therefore, the matrices of the compensator and
filter are expressed as

$\begin{split}&X(s)=[s^{0}\ s^{1}\ ...\ s^{h-1}\ s^{h}]\cdot[X_{0}\ X_{1}\
...\ X_{h-1}\ X_{h}]^{T},\\\ &Y(s)=[s^{0}\ s^{1}\ ...\ s^{o-1}\
s^{o}]\cdot[Y_{0}\ Y_{1}\ ...\ Y_{o-1}\ 1]^{T},\\\ &N(s)=[s^{0}\ s^{1}\ ...\
s^{n-1}\ s^{n}]\cdot[N_{0}\ N_{1}\ ...\ N_{n-1}\ N_{n}]^{T},\\\ &M(s)=[s^{0}\
s^{1}\ ...\ s^{m-1}\ s^{m}]\cdot[M_{0}\ M_{1}\ ...\ M_{m-1}\ M_{m}]^{T},\\\
\end{split}$ (13)

where $X\in\mathbb{R}^{h\times 1}$, $Y\in\mathbb{R}^{o\times 1}$,
$N\in\mathbb{R}^{1\times n}$, and $M\in\mathbb{R}^{1\times m}$ for $h,\ o,\
n,\ m$ are the orders of the related coefficient vectors. The polynomial
representations of the second-order feedback controller and first-order filter
are given as

$\begin{split}&K(s)=\frac{X_{2}s^{2}+X_{1}s+X_{0}}{Y_{2}s^{2}+Y_{1}s+Y_{0}},\\\
&Q(s)=\frac{N_{0}}{M_{1}s+M_{0}}.\end{split}$ (14)

Equation (12) is a non-convex problem since it includes multiplications of
compensator and filter matrices. These performance constraints can be
transformed into iterative LMIs with the transition from the infinity-norm to
second-norm representations. The problem is converted into LMIs in two
separate stages, where the first stage is convex approximations of the
feedback constraints and the second stage is the convex approximation of the
DOB constraints. The sensitivity functions of the general system are used in
both stages.

###  3.1 The formulation of feedback loop constraints

Consider the upper bounds of the sensitivity function with the infinity-norm
that is given by

$\big{\|}W_{1}S\big{\|}_{\infty}\triangleq\max_{\omega}\big{(}W_{1}S(j\omega)\big{)},$
(15)

where $max$ denotes the maximum singular value of the function. Therefore, the
infinity-norm constraint can be written using $S=(1+GK)^{-1}$ as

$\footnotesize\big{\|}W_{1}S\big{\|}\leq\gamma\Leftrightarrow\big{(}W_{1}(1+GK)^{-1}\big{)}^{*}(W_{1}(1+GK)^{-1})\leq\gamma^{2}=\gamma_{1},\\\
$ (16)

where $(\cdot)^{*}$ is the complex conjugate transpose and $\gamma_{1}$ is a
non-negative real number. Equation (12) is rewritten using $K=XY^{-1}$ as

$\begin{split}&\big{(}W_{1}(1+GXY^{-1})^{-1}\big{)}^{*}\big{(}W_{1}(1+GXY^{-1})^{-1}\big{)}\leq\gamma_{1},\\\
&\big{(}YW_{1}\big{)}^{*}\big{(}YW_{1}\big{)}\leq\big{(}Y+GX\big{)}^{*}\gamma_{1}\big{(}Y+GX\big{)},\\\
&(YW_{1})^{*}(W_{1}Y)\underbrace{-(\underbrace{Y+GX}_{\text{J}})^{*}\gamma_{1}(Y+GX)}_{\text{concave
term}}\leq 0.\\\ \end{split}$ (17)

We still have quadratic matrix inequality due to the multiplication of
compensator matrices. Therefore, we define a new variable $J=Y+GX$ as

$\big{(}YW_{1}\big{)}^{*}(W_{1}Y)-J^{*}J\gamma_{1}\leq 0.$ (18)

This inequality is a quadratic convex-concave form. It can be linearized using
first-order Taylor expansion of $J^{*}J$ around a feasible initial point. This
equation corresponds to the factorization of a mathematical quadratic
function. The $J^{*}J$ vector polynomial is equivalent to:

$J^{*}J\approx J^{*}J_{i}+J_{i}^{*}J-J_{i}^{*}J_{i},$ (19)

where the left-hand side polynomial is a quadratic function and the right-hand
side term is a linear function around $J_{i}=Y_{i}+GX_{i}$ initial point.
Multiplying (18) with $-\gamma_{1}^{-1}$ and combining with (19), the LMI
representation of the constraint is given using the Schur complement lemma:

$\displaystyle\left[{\begin{array}[]{cc}J^{*}J_{i}+J_{i}^{*}J-J_{i}^{*}J_{i}&(W_{1}Y)^{*}\\\
W_{1}Y&\gamma_{1}\end{array}}\right]\succeq 0.$ (22)

The other constraints of the feedback loop are convexified using the steps
given between (15-21). Finally, The LMI constraints for the optimization of
feedback controller are given as

$\displaystyle\begin{aligned}
&\left[{\begin{array}[]{cc}J^{*}J_{i}+J_{i}^{*}J-J_{i}^{*}J_{i}&(W_{1}Y)^{*}\\\
W_{1}Y&\gamma_{1}\\\ \end{array}}\right]\succeq 0,\\\
&\left[{\begin{array}[]{cc}J^{*}J_{i}+J_{i}^{*}J-J_{i}^{*}J_{i}&(W_{3}GX)^{*}\\\
W_{3}GX&\gamma_{1}\\\ \end{array}}\right]\succeq 0,\\\
&\left[{\begin{array}[]{cc}J^{*}J_{i}+J_{i}^{*}J-J_{i}^{*}J_{i}&(W_{2}X)^{*}\\\
W_{2}X&\gamma_{1}\\\ \end{array}}\right]\succeq 0,\\\
&\left[{\begin{array}[]{cc}J^{*}J_{i}+J_{i}^{*}J-J_{i}^{*}J_{i}&\Big{(}(W_{v}G_{v}+W_{i}G_{i})Y\Big{)}^{*}\\\
(W_{v}G_{v}+W_{i}G_{i})Y&\gamma_{1}\\\ \end{array}}\right]\succeq
0.\end{aligned}$ (23)

###  3.2 The derivation of DOB constraints

The same analysis can be applied to (12) for the sensitivity and complementary
sensitivity functions of the DOB loop. For the sensitivity function of the
DOB, the infinite norm constraint is given as

$\big{\|}S_{D}\big{\|}_{\infty}\triangleq\max_{\omega}\big{(}S_{D}(j\omega)\big{)}.$
(24)

By employing the same steps given between (15-21), the following equations are
obtained

$\footnotesize\big{\|}S_{D}\big{\|}\leq\gamma_{D}\Leftrightarrow\Bigg{(}\frac{(W_{i}G_{i}+W_{v}G_{v})M}{N+M}\Bigg{)}^{*}\Bigg{(}\frac{(W_{i}G_{i}+W_{v}G_{v})M}{N+M}\Bigg{)}\leq\gamma_{D}^{2}=\gamma_{2},$
(25)
$\footnotesize((W_{i}G_{i}+W_{v}G_{v})M)^{*}((W_{i}G_{i}+W_{v}G_{v})M)-(N+M)^{*}\gamma_{2}(N+M)\leq
0,$ (26)
$\footnotesize\begin{split}&(\underbrace{N+M}_{\text{H}})^{*}\gamma_{2}(N+M)-((W_{i}G_{i}+W_{v}G_{v})M)^{*}((W_{i}G_{i}+W_{v}G_{v})M)\geq
0,\\\ &H^{*}H\approx H^{*}H_{i}+H_{i}^{*}H-H_{i}^{*}H_{i},\\\ \end{split}$
(27)

where $H_{i}=N_{i}+M_{i}$ is a initial point of $H$ variable. Then, using the
Schur complement lemma, the following LMI representation is obtained

$\left[{\begin{array}[]{cc}H^{*}H_{i}+H_{i}^{*}H-H_{i}^{*}H_{i}&((W_{i}G_{i}+W_{v}G_{v})M)^{*}\\\
((W_{i}G_{i}+W_{v}G_{v})M)&\gamma_{2}\\\ \end{array}}\right]\succeq 0.$ (28)

The LMI representation of the constraint of the complementary sensitivity
function, $T_{C}$, can be obtained following the steps given between (22-26).
The LMI representations of the two sensitivity constraints are given as

$\footnotesize\left[{\begin{array}[]{ccc}H^{*}H_{i}+H_{i}^{*}H-H_{i}^{*}H_{i}&((W_{i}G_{i}+W_{v}G_{v})M)^{*}&(GW_{n}N)^{*}\\\
((W_{i}G_{i}+W_{v}G_{v})M)&\gamma_{2}&0\\\ GW_{n}N&0&\gamma_{2}\\\
\end{array}}\right]\succeq 0.$ (29)

The convex constraints are utilized to improve the stability of the DOB loop.
The following constraint is added to the problem to shape the filter. For the
additional constraint of the DOB loop, the constraint is given as

$\big{\|}W_{Q}(1-Q)\big{\|}_{\infty}\triangleq\max_{\omega}\big{(}W_{Q}(j\omega)(1-Q(j\omega))\big{)}.$
(30)

By employing the same steps given between (16-19), the following equations are
obtained as

$\footnotesize\big{\|}W_{Q}(1-Q)\big{\|}\leq\gamma_{D}\Leftrightarrow\big{(}W_{Q}(M-N)\big{)}^{*}W_{Q}(M-N)\leq(M^{-1})^{*}\gamma_{2}M^{-1},$
(31)
$\begin{split}&M^{*}\gamma_{2}M-\big{(}W_{Q}(M-N)\big{)}^{*}W_{Q}(M-N)\geq
0,\\\ &M^{*}M\approx M^{*}M_{i}+M_{i}^{*}M-M_{i}^{*}M_{i},\\\ \end{split}$
(32)

where $M_{i}$ is the initial point of the denominator vector of the filter.
Using the Schur Complement lemma, the LMI representation of the constraint is
obtained as

$\left[{\begin{array}[]{cc}M^{*}M_{i}+M_{i}^{*}M-M_{i}^{*}M_{i}&\Big{(}W_{Q}(M-N)\Big{)}^{*}\\\
W_{Q}(M-N)&\gamma_{2}\\\ \end{array}}\right]\succeq 0.$ (33)

###  3.3 The derivation of general loop constraints

For the general stability of the control system, the infinity norm of
sensitivity functions of the closed-loop system should be minimized. In this
subsection, we convert the non-convex constraints of sensitivity functions
into LMIs assuming that the matrices of the compensator are pre-computed. In
other words, the matrices of the filter are considered fixed when the
compensator matrices are considered variables or vice-versa. For the
sensitivity of the closed loop, the constraint is given as

$\big{\|}S_{C}\big{\|}_{\infty}\triangleq\max_{\omega}\big{(}W_{1C}S_{C}(j\omega)\big{)}.$
(34)

By employing the steps given between (12-15), the following equations are
obtained

$\footnotesize\big{\|}W_{1C}S_{C}\big{\|}\leq\gamma_{C}\Leftrightarrow\Big{(}\frac{P\big{(}1-\frac{N}{M}\big{)}}{1+\frac{GX}{Y}}\Big{)}^{*}\Big{(}\frac{P\big{(}1-\frac{N}{M}\big{)}}{1+\frac{GX}{Y}}\Big{)}\leq\gamma_{C}^{2}=\gamma_{3},$
(35) $\big{(}PY(M-N)\big{)}^{*}\big{(}PY(M-N)\big{)}-(MJ)^{*}\gamma_{3}MJ\leq
0,$ (36) $\begin{split}&(MJ)^{*}\gamma_{3}MJ-(PY(M-N))^{*}PY(M-N)\geq 0,\\\
&M^{*}M\approx M^{*}M_{i}+M_{i}^{*}M-M_{i}^{*}M_{i},\\\ \end{split}$ (37)

where $W_{1C}(W_{i}G_{i}+W_{v}G_{v})$ function is denoted by $P$ to simplify
the equations. It is highlighted that only the $M^{*}M$ polynomials are
linearized around $M_{i}$ initial point since assuming the compensator
matrices are fixed. Then, using the Schur Complement lemma, the following LMI
representation is obtained as

$\small\left[{\begin{array}[]{cc}J^{*}J(M^{*}M_{i}+M_{i}^{*}M-M_{i}^{*}M_{i})&(PY(M-N))^{*}\\\
PY(M-N)&\gamma_{3}\\\ \end{array}}\right]\succeq 0.$ (38)

The LMI representation of the constraint of the complementary sensitivity
function, $T_{C}$, can be obtained following the steps (28-32). Finally,
convex approximations of the general closed-loop constraints are given as

$\footnotesize\left[{\begin{array}[]{ccc}J^{*}J(M^{*}M_{i}+M_{i}^{*}M-M_{i}^{*}M_{i})&\big{(}PY(M-N)\big{)}^{*}&\big{(}W_{T}(MGX+NY)\big{)}^{*}\\\
PY(M-N)&\gamma_{3}&0\\\ W_{T}(MGX+NY)&0&\gamma_{3}\\\
\end{array}}\right]\succeq 0,$ (39)

where $W_{T}$ represents $W_{3C}W_{n}$ function. The structured CCP algorithm
for the 2-DOF control system is proposed in Algorithm 1, where $N$ and $M$
matrices are equal to the $N_{i}$ and $M_{i}$ matrices only in the first
iteration of stage 1. $\varepsilon$ is a pre-specified tolerance, $i$ is the
iteration number, and $l$ is the maximum iteration number. We organize the
proposed algorithm as two stages since the matrices $X$ and $M$ or the
matrices $Y$ and $N$ are in multiplication. The $\gamma_{3}$ can be
$\gamma_{1}$ or $\gamma_{2}$ according to the stage of the algorithm in which
the compensator or filter is optimized. This problem is solved by employing a
sequential solution in a generic loop. Finally, the coefficients of the
compensator and filter are simultaneously optimized due to the proposed
algorithm, which is solved in a similar way to the classical $D-K$ iteration
method in robust control.

###  3.4 K-factor method

The K-factor method is preferred to design of a Type-III compensator the
performances of the proposed 1-DOF and 2-DOF control systems. The method is
the only analytical method used to synthesize the compensator. The method is
based on the principle of placing the zero and poles of the compensator
according to the designer’ preference by analyzing the amplitude-frequency
curve of the nominal system. It is determined the required phase margin and
amplitude gain boost at a selected crossover frequency. The phase boost
expression is given as

$A_{p}=2(tan^{-1}(\frac{f_{c}}{f_{z_{1,2}}})-tan^{-1}(\frac{f_{c}}{f_{p_{1,2}}}))-\frac{\pi}{2},$
(40)

where $A_{p}$ is the desired phase boost value in degree, $f_{c}$ is the
crossover frequency, $f_{z_{1,2}}$ are the frequencies of the zeros,
$f_{p1,2}$ are the frequencies of the poles. The $k$ gain of the method is
given as

$k={(tan(\frac{A_{p}}{4}+\frac{\pi}{4}))^{2}}.$ (41)

The coincident zero pair of the compensator is given as

$f_{z_{1,2}}=\frac{f_{c}}{\sqrt{k}}=\frac{f_{c}}{tan(\frac{A_{p}}{4}+\frac{\pi}{4})}.$
(42)

The coincident pole pair of the compensator is given as

$f_{p_{1,2}}=f_{c}\sqrt{k}=tan(\frac{A_{p}}{4}+\frac{\pi}{4})f_{c}.$ (43)

The mid-band gain of the compensator is derived as

$G_{o}=\frac{\sqrt{1+(\frac{f_{c}}{f_{p_{1}}})^{2}}\sqrt{1+(\frac{f_{c}}{f_{p_{2}}})^{2}}}{\sqrt{1+(\frac{f_{z_{1}}}{f_{c}})^{2}}\sqrt{1+(\frac{f_{c}}{f_{z_{2}}})^{2}}}M_{g},$
(44)

where $M_{g}$ is the gain value at the selected crossover frequency. Finally,
the compensator is given as

$K_{k}(s)=G_{o}\frac{(s/2\pi f_{z1}+1)(s/2\pi f_{z2}+1)}{(s/2\pi
f_{z1}+1)(s/2\pi f_{z2}+1)},$ (45)

where $K_{k}(s)$ is the Type-III compensator designed by using K-factor
method. rana2017development .

###  3.5 Implementation stage

The convex optimization problem is semi-infinite programming since the problem
is defined in the infinite frequency range. By searching for a solution in a
frequency range higher than the closed-loop bandwidth, it is transformed into
a semi-definite programming problem, where any LMIs solver could be used for
the solution of the problem. The problem is solved in a reasonable frequency
number $(L)$ using an inner loop for frequency gridding. In other words, the
problem is discretized with a logarithmically spaced frequency domain into a
finite set $w_{j}={w_{1},\ w_{2},\ ...,\ w_{L}}$ where $j$ is the number of
the inner loop cycles. The reasonable gridding number could be found using the
randomized scenario approach to ensure the constraints with sufficient
probability level campi2018introduction ; dacs2021robust . The reasonable
number for scenarios given as

$L\geq\frac{2}{\epsilon_{r}}(d_{p}-1+\ln{\frac{1}{\beta_{c}}}),$ (46)

where $d_{p}$ is the number of optimization variables, $\epsilon_{r}\in(0,1)$
is the risk index, and $\beta_{c}\in(0,1)$ is the confidence index. The
convex-concave procedure presents a robust solution to original non-convex
optimization problems, although it does not guarantee convergence to the
global solution of the non-convex problem. Robust stability and performance
improvements are achieved by selecting the relevant weighting functions. In
convex-concave programming, the design must start from feasible initial
coefficients of compensator and filter which may be a challenge to find while
the basic iteration will work in all of these variations. In addition, it is
recommended that the initial coefficients of the compensator should be close
to zero for a successful optimization process. In the selection of the
coefficients, it should be noted that the zeros and poles of the compensator
should be negative real values. Alternatively, the initial coefficients of the
compensator can be found via the K-factor method which is a classical approach
for the design of the compensator or any $H_{\infty}$ solver such as
$hinfstruct$ karimi2017data ; dacs2021data .

##  4 Simulation results of the converter

The two-phase converter circuit is designed for continuous current mode
operation for medium and high power transmission conditions. The values of the
passive components are preferred as high values so that the converter remains
in continuous current mode over the entire load operating range. The
computations of the constrained convex optimization problem are solved by the
convex modeling framework YALMIP using the MOSEK software lofberg2008modeling
. The problem is solved with 200 grinding points in the $10^{2}-10^{5}$ rad/s
frequency range. The initial coefficients of the compensator are chosen as
0.0001 so that the initial compensator is equal to an integrator. The initial
coefficients of the filter are chosen as $N_{i}$=120 and $M_{i}$=[1 120]. The
achieved objectives values are $\gamma_{1}=0.813$ and $\gamma_{2}=1$. In the
K-factor method, high phase contribution is required since the phase margin
and amplitude gain of the system are low based on the amplitude-frequency
response of the open-loop converter which is presented in Fig. 3. The
compensator is designed for a phase margin over 70 degrees and a gain margin
over 12 dB, all of which are regarded as typical requirements for disturbance
rejection performance of a converter system. Therefore, the oscillations of
the output voltage during disturbance events can be mitigated. The compensator
coefficients are obtained by assuming that the required phase boost ($A_{p}$)
is 172 degrees and gain loss ($M_{g}$) would be -40 dB at 2300 Hz crossover
frequency.
The aforementioned weighting functions are selected as follows:

$\begin{split}&W_{1}(s)=\frac{s+395}{1.4s+3.95\cdot 10^{-7}},\ \
W_{2}(s)=\frac{s+1850}{0.04s+18500},\\\\[5.69054pt]
&W_{3}(s)=\frac{s+400}{15s+2000},\ \ W_{v}(s)=\frac{8\cdot
10^{-8}s+3200}{s+2094},\\\\[5.69054pt] &W_{i}(s)=\frac{1\cdot
10^{-8}s+400}{s+2094},\ \ W_{Q}(s)=\frac{s+140}{1.4s+1.4},\\\
&W_{1C}(s)=\frac{s+3700}{1.3s+0.0000271},\ \
W_{3C}(s)=\frac{s+3700}{0.04s+18500}.\end{split}$ (47)

In order to reduce the effect of converter parameter changes on the output
voltage, the amplitude, and frequency values of the input voltage and load
current weighting functions are selected high. Finally, the optimal
coefficients of the compensators and the filter are given as

$\begin{split}&K_{x}(s)=\frac{918.9s^{2}+3.558\cdot 10^{5}s+3.439\cdot
10^{7}}{s^{3}+1.37\cdot 10^{4}s^{2}+4.687\cdot 10^{7}s},\\\\[5.69054pt]
&K_{k}(s)=\frac{5.472\cdot 10^{7}s^{2}+5.282\cdot 10^{10}s+1.275\cdot
10^{13}}{482.7s^{3}+3.822\cdot 10^{8}s^{2}+7.564\cdot 10^{13}s},\\\
&K_{2x}(s)=\frac{1692s^{2}+4.1\cdot 10^{5}s+1.47\cdot 10^{8}}{s^{3}+1.4\cdot
10^{4}s^{2}+10^{8}s},\\\ &Q(s)=\frac{98.77}{s+99.6},\\\ \end{split}$ (48)

where $K_{x}$ is the compensator of the 1-DOF system and $K_{2x}$ is the
compensator of the 2-DOF system which is optimized with the proposed method.

Figure 6: The performance results of the controlled systems in the time
domain: (a) Set-point tracking performances (b) The input voltage changes (c)
The load current changes

The low-high power performances of the controlled closed loop converter are
presented in Fig. 6. The control systems guarantee to track the variable set
point. However, the control systems present different performances in
problematic disturbance events. To examine the performances of the control
systems in detail, Fig. 6 is divided into four different operating points as
in Figs. 7 and 8. The set-point tracking performances of the control systems
at starting of the converter are presented in Fig. 7. The settling time of the
$K_{k}$ controlled system is three times higher than the settling time of the
proposed 1-DOF control system. The set-point of the output voltage is changed
with a range of 10$\%$ at the operation point stepwise from $v_{o}=100$ V to
$v_{o}=90$ V at $t=0.3$ s and from $v_{o}=90$ V to $v_{o}=100$ V at $t=0.7$ s.
The control systems have no overshoots at the starting and the changes of the
set-point (Point 1). The input voltage disturbance rejection performances of
the control systems are presented at high and low power transmissions in Figs.
7 and 8 (Points 2, 4).

Figure 7: The performance results of the control systems under the problematic
events: (a1, b1) Point 1: Set-point tracking at the start-up and around the
nominal operating point ($i_{out}=5$ A), (a2, b2) Point 2: Step changing of
input voltage at high power transmission ($i_{out}=5$ A)

Fig. 8 demonstrates the rejection performances of the load disturbance effects
on the output voltage caused by varying the load current around high and low-
power working conditions. The load resistance is changed stepwise from
$R_{o}=20\ $\mathrm{\SIUnitSymbolOhm}$$ to $R_{o}=80\
$\mathrm{\SIUnitSymbolOhm}$$ at $t=2$ s, from $R_{o}=80\
$\mathrm{\SIUnitSymbolOhm}$$ to $R_{o}=150\ $\mathrm{\SIUnitSymbolOhm}$$ at
$t=2.5$ s, and from $R_{o}=150\ $\mathrm{\SIUnitSymbolOhm}$$ to $R_{o}=240\
$\mathrm{\SIUnitSymbolOhm}$$ at $t=3$ s (events 5, 6 and 9). The proposed
1-DOF and 2-DOF control systems have nearly no oscillation under the external
disturbance events due to boosting the phase margin of the controlled systems
to the requested level. The $K_{x}$ controlled system has nearly 80 ms under
set-point changes and 50 ms settling times under input voltage and load
current changes. The $K_{k}$ controlled system, however, has 230 ms and 140 ms
settling times under set-point input voltage, and load current changes,
respectively. The inductances and related parasitic resistances of the
converter are changed stepwise from the nominal value to half of the nominal
value at 2.65 s in Fig. 8 to show the uncertainty rejection performances of
the controlled systems (event 7). At 2.85 s the components are increased
stepwise to the nominal values (event 8). The effect of the events on the
output voltage is ignored. The quantitative performance metrics, i.e. settling
time and maximum overshoots, are introduced to assess the control
performances. The performance metrics of the simulation results under set-
point tracking and disturbance rejection are given in Table 2. It is seen that
the proposed 1-DOF control system decreases the overshoot 60$\%$ and the
settling time 81$\%$ according to the $K_{k}$ controlled system considering
point 3. The 2-DOF control system presents solid uncertainty rejection
performance in the presence of changes in converter parameters. The system has
minimum settling times at the parameter changes. However, proposed 1-DOF and
2-DOF control systems seem to have similar disturbance rejection performances
at the load current changes.

Figure 8: The performance results of the control systems under the problematic events: Eliminating the effect of parameter uncertainties on the output voltage (a1, b1) Point 3: Step changing of the load resistance and the inductances ($v_{in}=50$ V), (a2, b2) Point 4: The input voltage changes ($i_{out}=0.4$ A) Table 2: Performance indexes of the compensators (event 1-13=e1-13) Point | Max over. ($\%$) | Sett. time (ms)
---|---|---
$K_{x}$ | $K_{k}$ | $2DOF$ | $K_{x}$ | $K_{k}$ | $2DOF$
Starting (e1) | 0 | 0 | 1.5 | 110 | 285 | 35
Point1 (e2) | 0 | 0 | 0.1 | 75 | 180 | 70
Point 2 (e3) | 2.3 | 5.8 | 1.5 | 90 | 162 | 90
Point 2 (e4) | 2.29 | 5.77 | 1.95 | 74 | 154 | 74
Point 3 (e5) | 4 | 12.5 | 5.05 | 60 | 150 | 60
Point 3 (e6) | 0.6 | 1.8 | 1.7 | 15 | 75 | 15
Point 3 (e7) | 0.2 | 0.6 | 0.1 | 25 | 100 | 10
Point 3 (e8) | 0.19 | 0.52 | 0.1 | 23 | 125 | 10
Point 3 (e9) | 0.2 | 0.5 | 0.6 | 24 | 125 | 30
Point 4 (e10) | 3.99 | 8.58 | 3 | 80 | 180 | 80
Point 4 (e11) | 5.5 | 17.3 | 4.5 | 55 | 230 | 55
Point 4 (e12) | 2.1 | 6.1 | 0.85 | 65 | 140 | 50
Point 4 (e13) | 1.73 | 3.75 | 0.9 | 60 | 130 | 50

Table 2 shows that the proposed control systems provide better robustness than
the other compensator in the presence of variations of all parameters. The
proposed control systems mitigate the oscillations under disturbance effects
and provide solid tracking and uncertainty rejection performance under
variations of the circuit parameters in terms of overshoot and settling time
performance metrics. By adding disturbance loops to the synthesis process,
output voltage oscillations are prevented in the presence of parametric
uncertainties. The results demonstrate that, as expected, the proposed convex
method outperforms the K-factor-based control strategy in the case of set-
point tracking, disturbance, and uncertainty rejection performances.

##  5 Experimental results

The proposed control systems are implemented in a TMS320F28335 DSP-based
experimental setup to validate the practical performance of the method. The
general experimental setup and the gate driver circuit are demonstrated in
Figs. 9 and 10. The PWM blocks of the DSP are used to realize 180 degrees of
phase shifting. The MOSFETs of the converter are IRF-1310N with 36
$m$\mathrm{\SIUnitSymbolOhm}$$ conduction resistance; the passive switches are
SB5100 Schottky diodes with negligible resistances. The TLP-350 optocouplers
are used for gate isolation. Fig. 11 demonstrates the start-up of the
converter and the set-point tracking performance with the control input signal
(duty cycle of the PWM signals).

Figure 9: Experimental test bench: (A) host computer, (B) load resistor, (C)
TMS320F28335 DSP, (D) 2-phase interleaved boost converter, (E) switching
connection of the Sorensen programmable power supply, (F) Sorensen XG 300-5
power supply, (G) LCR meter, (H) ADS-3202A oscilloscope to monitor the PWM
signals Figure 10: The electronic application and drive circuits of the
control system

Firstly, the starting process of the 1-DOF control system is shown in Fig. 11
(a), where the input voltage and load resistance are set to 52 V and 87
$\Omega$. The set-point value is changed between 22-26 seconds to analyze the
set-point tracking performance in detail. The controlled system has no
overshoots or undershoots at the starting point. It has a nearly 180 ms
settling time similar to the simulation study presented in Fig. 7 (a1). Set-
point tracking performance is good at rising edge changes of the set-point
signal although an overshoot occurs on the falling edges of the set-point
signal as in the simulation study presented in Fig. 6 (a). There is no steady-
state error in the presence of the sensor noise and the disturbance effects.
Fig. 12 demonstrates output voltage responses under the input voltage and
output resistance changes. To ensure a constant output voltage, the current
and input voltage amplitudes are changed over a certain period in Fig. 12. The
control system mitigates the voltage oscillations on the output voltage of the
converter which is exposed to input voltage changes. The converter maintains
its stability and performance over a wide input voltage range. The circuit is
subjected to a wide range of load changes as shown in Fig. 12 (b). The
tracking performance of the 2-DOF controlled system is demonstrated in Fig.
13. The system has nearly no overshoots at set-point changes. The load current
rejection performance of the 2-DOF system is presented to a wide range of load
changes in Fig. 12 (b). The disturbance rejection performance of the 2-DOF
system is presented under the load current and input voltage changes in Fig.
14. The output regulation is achieved after each load and input voltage
change. In step load changes, the overshoots in the output voltage are quite
low. The control systems provide solid performance above the rated load and in
no-load operations. The experimental performances demonstrate that the
proposed 1-DOF and 2-DOF control systems provide satisfactory dynamic
performance with regard to output voltage tracking and robustness to load and
input voltage changes.

(a) Starting of the converter: The output voltage (a1) and control input/duty
cycle (a2)

(b) Set-point tracking performance: The output voltage (b1) and control input
(b2)

Figure 11: Real-time application results of the proposed compensator in the
case of set-point changing: the output voltage and control input

(a) Input voltage changes: The output voltage (a1) and input voltage (a2)

(b) Load changes: The output voltage (b1) and load current (b2)

Figure 12: Real-time application results of the proposed compensator under
crucial input voltage and load disturbances: the output voltage and control
input (a)  Figure 13: Real-time application results of the proposed 2-DOF
control system in the case of set-point changes: the output voltage and
control input

(a) Step input voltage changes: The output voltage and input voltage

(b) Load changes: The output voltage and load current

Figure 14: Real-time application results of the proposed 2-DOF control systems
under input voltage and load disturbances

The slight differences between simulation and experiment results in Figs. 7, 8
and 11, 12 are due to minor differences in the disturbances. It is important
to note that such a disconformity is due to the difference in sensitivity of
the programmable power supplies in simulation and experiments, and the
inevitable errors implied in practice by the measuring sensors. Also, heating
of the passive components or sensor noise can cause such minor mismatches.

## Discussion and Conclusion

This paper proposes a simultaneous $H_{\infty}$ Type-III compensator and DOB
synthesis approach for the multi-phase DC-DC power converters in a convex
optimization framework. The proposed $H_{\infty}$ control schematic involves
significant sensitivity functions to systematically take into account all
relevant system constraints. It simplifies the design of error, control input,
and noisy output weighting functions. We have proposed an iterative LMI
algorithm to solve the design of the compensator and filter simultaneously.
The proposed optimization problem is solved with several iterations in which
the compensator is chosen as a unit integrator at the initial of the problem.
Although the variation of the input voltage is significant (± 33$\%$) in the
experimental setup, the output voltage of the 1-DOF controlled system has a
small deviation of 200 ms and returns to the nominal set-point value without
oscillations. The load current is changed between +25$\%$ and -75$\%$ of the
nominal value. The output voltage presents small overshoots less than 1.5$\%$
of the nominal value during the load variations. The proposed 2-DOF control
system is capable of rejecting input voltage and load current disturbances.
Consequently, the effectiveness and performance of the proposed approaches
against the parameter uncertainty and low-high load operation are validated in
simulation and real-time applications. It is shown that the simulation and
experimental results are consistent. The proposed 2-DOF control system is
robust to disturbances and sensor noise.

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