# A Nonlinear Observer Design for the Discrete-time Systems: Exploiting
Matrix-Multiplier-based LMI Approach

Shivaraj Mohite1 *This work was not supported by any organization1Rhineland-
Palatinate Technical University of Kaiserslautern-Landau, Kaiserslautern,
Germany<EMAIL_ADDRESS>
###### Abstract

This letter focuses on the $\mathcal{H}_{\infty}$ observer design for a class
of nonlinear discrete systems under the presence of measurement noise or
external disturbances. A novel Linear Matrix Inequality (LMI) condition is
developed in this method through the utilisation of the reformulated Lipschitz
property, a new variant of Young inequality and the well-known Linear
Parameter Varying (LPV) approach. One of the key components of the proposed
LMI is the generalised matrix multipliers. The deliberate use of these
multipliers enables us to introduce more numbers of decision variables inside
LMIs than the one illustrated in the literature. It aids in adding some extra
degrees of freedom from a feasibility point of view, thus enhancing the LMI
conditions. Thus, the proposed LMIs are less conservative than existing ones.
Later on, the effectiveness of the developed LMIs and observer is highlighted
through the numerical example and an application of state of charge (SoC)
estimation in the Li-ion battery model.

###### Index Terms:

LMI-based nonlinear observer design, Lipschitz systems, $\mathcal{H}_{\infty}$
criterion.

##  I Introduction

Over the past three decades, the topic of observer design for dynamical
systems has received a significant amount of interest from researchers of
control system engineering. This is because the state variables are essential
in the system analysis and feedback control design. Although many of these
variables can be measured through sensors, some remain inaccessible due to the
unavailability of sensors. Thus, the observers play a pivotal role in modern-
day applications that assist in capturing real-time information of systems,
for example, the state-of-charge estimation of LI-ion battery model [1],
autonomous vehicle tracking [2], and so on.

In the literature, numerous approaches have been proposed for linear observer
design, and all have shown high reliability. However, in contrast to the
linear observer, the development of nonlinear observers is an arduous problem.
There is no systematic technique to construct these types of observers. Hence,
an abundant amount of research has been carried out in this area. The several
popular methods of nonlinear observers are as follows: 1) Transformation-based
observers [3]; 2) High-gain observers [4]; 3) Sliding-mode observers [5]; and
4) Linear Matrix Inequality (LMI)-based observers [2]. All these methodologies
are developed for continuous-time nonlinear systems, while the few
methodologies related to discrete-time systems are showcased in [6, 7, 8] and
[9].

Among these techniques, LMI-based techniques are extensively studied by
researchers and some of them are provided in [10, 7, 11] and [9]. Recently,
the authors of [12] had established a new matrix-multiplier-based LMI approach
for continuous-time nonlinear observer design. The LMI condition presented in
[12] is less conservative than the one shown in [2]. It inspires the authors
to derive a new LMI condition for the development of nonlinear observers for
discrete-time systems through the utilisation of new matrix multipliers,
reformulated Lipschitz property and Young inequalities. Further, the
effectiveness of the developed LMI condition is shown through a numerical
example. The performance of the observer is validated through the application
of State-of-charge (SoC) estimation of Li-ion battery.

The rest of this letter is organised as follows: Some preliminaries related to
the nonlinear observer design are illustrated in Section II. Further, Section
III encompasses the contextualization of the problem statement and the
development of the new LMI condition. Segment III-C contains the design of the
observer for the nonlinear systems with linear outputs. The effectiveness of
the new LMI conditions and the observer performance is displayed in Section
IV. Finally, Section V entails some concluding remarks on the proposed
methodology.

Notation: Throughout the article, the following notations are used: $||e||$
and $||e||_{\mathcal{L}_{2}}$ depict the euclidean norm and the
$\mathcal{L}_{2}$ norm of a vector $e$, respectively. The term $e_{0}$
signifies the initial values of $e(t)$ at $t=0$. A vector of the canonical
basis of $\mathbb{R}^{s}$ is illustrated as:
$e_{s}(i)=(\underbrace{0,\ldots,0,\overbrace{1}^{i^{\text{th}}},0,\ldots,0}_{s\,\,\text{components}})^{\top}\in\mathbb{R}^{s},~{}s\geq
1.$ The identity matrix and the null matrix are denoted by $\mathbb{I}$ and
$\mathbb{O}$, respectively. The transpose of matrix $A$ is symbolised as
$A^{\top}$, while, $A\in\mathbf{S}^{n}$ infers that $A$ is a symmetric matrix
of dimension $n\times n$. The repeated blocks within a symmetric matrix are
described by the symbol $(\star)$. $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$
depict minimum and maximum eigenvalues of $A\in\mathbf{S}^{n}$, respectively.
For the aforementioned matrix $A$, $A>0$ ($A<0$) indicates that it is a
positive definite matrix (a negative definite matrix). Similarly, a positive
semi-definite matrix (a negative semi-definite matrix) is showcased by $A\geq
0$ ($A\leq 0$).

##  II Preliminaries

This section is focused on the recapturing of the mathematical tools and
background results which will be needed in the development of the main
results.

######  Definition 1 ([13])

If there exist two vectors,

$\displaystyle\Psi=\begin{pmatrix}\psi_{1}&\ldots&\psi_{n}\end{pmatrix}^{\top},~{}\text{and}~{}\Phi=\begin{pmatrix}\phi_{1}&\ldots&\phi_{n}\end{pmatrix}^{\top},$

then for all $i=0,\ldots,n$, an auxiliary vector
$\Psi^{\Phi_{i}}\in\mathbb{R}^{n}$ corresponding to $\Psi$ and $\Phi$ can be
defined as:

$\displaystyle\begin{split}\Psi^{\Phi_{i}}=\begin{cases}\begin{pmatrix}\phi_{1}&\ldots&\phi_{i}&\psi_{i+1}&\ldots&\psi_{n}\end{pmatrix}^{\top},&\text{for}\,\,i=1,\ldots,n\\\
\Phi,&\text{for}\,\,i=0.\\\ \end{cases}\end{split}$ (1)

######  Lemma 1 ([2])

If there exists a nonlinear function $h:\mathbb{R}^{n}\to\mathbb{R}^{n}$, then
the ensuing two conditions are equivalent:

  1. (I)

The function $h(\cdot)$ fulfils:

$||h(\Psi)-h(\Phi)||\leq
h_{l}||\Psi-\Phi||,\quad\forall\Psi,\,\Phi\in\mathbb{R}^{n}$ (2)

i.e., it is globally Lipschitz with respect to its argument.

  2. (II)

For all, $i,j=1,\ldots,n$, there exist functions
$h_{ij}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$, and constants
$h_{{{ij}_{\min}}}$ and $h_{{{ij}_{\max}}}$ such that
$\forall\,\Psi,\,\Phi\in\mathbb{R}^{n}$,

$h(\Psi)-h(\Phi)=\sum^{n}_{i=1}\sum^{n}_{j=1}h_{ij}\mathcal{H}_{ij}(\Psi-\Phi),$
(3)

where $\mathcal{H}_{ij}=e_{n}(i)e^{\top}_{n}(j)$, and $h_{ij}\triangleq
h_{ij}(\Psi^{\Phi_{j-1}},\Psi^{\Phi_{j}})$. The functions $h_{ij}(.)$ are
globally bounded as follows:

$h_{{{ij}_{\min}}}\leq h_{ij}\leq h_{{{ij}_{\max}}}.$ (4)

The statement (I) showcases the standard Lipschitz property of nonlinearities,
whereas the recently proposed reformulated Lipschitz property of a nonlinear
function is shown in the statement (II).

######  Lemma 2 ([2])

Let us consider two vectors $X,Y\in\mathbb{R}^{n}$ and a matrix
$Z>0\in\mathbf{S}^{n}$. Then the following matrix inequality holds:

$X^{\top}Y+Y^{\top}X\leq\frac{1}{2}(X+ZY)^{\top}Z^{-1}(X+ZY).$ (5)

##  III New observer design: an enhanced LMI approach

###  III-A Articulating the Problem Statement

Let us consider an ensuing set of equations which represents the class of
disturbance-affected nonlinear system dynamics with nonlinear output:

$\displaystyle\begin{split}x_{k+1}&=Ax_{k}+Gf(x_{k})+B_{1}u_{k}+E\omega_{k},\\\
y_{k}&=Cx_{k}+Fg(x_{k})+B_{2}u_{k}+D\omega_{k},\end{split}$ (6)

where $x\in\mathbb{R}^{n}$ and $y\in\mathbb{R}^{p}$ denote a state vector of
the system, and its output, respectively. $u\in\mathbb{R}^{s}$ indicates the
input provided to the system, and $\omega\in\mathbb{R}^{q}$ depicts
$\mathcal{L}_{2}$ bounded noise/disturbance vectors affecting the system
dynamics and measurements. The matrices $A\in\mathbb{R}^{n\times n}$,
$G\in\mathbb{R}^{n\times m}$, $B_{1}\in\mathbb{R}^{n\times
s},\,B_{2}\in\mathbb{R}^{p\times s}$, $C\in R^{p\times n}$,
$E\in\mathbb{R}^{n\times q}$, $D\in\mathbb{R}^{p\times q}$ and
$F\in\mathbb{R}^{p\times r}$ are known and constant. The functions
$f(\cdot):\mathbb{R}^{n}\to\mathbb{R}^{m}$ and
$g(\cdot):\mathbb{R}^{n}\to\mathbb{R}^{r}$ illustrate the nonlinearities
present in the dynamics and outputs of the system, respectively. Both
functions are assumed to be globally Lipschitz. Further, the detailed form of
$f(\cdot)$ and $g(\cdot)$ are showcased as follows:

$f(x_{k})=\begin{bmatrix}f_{1}(F_{1}x_{k})\\\ \vdots\\\
f_{i}(\underbrace{F_{i}x_{k}}_{\theta_{i}})\\\ \vdots\\\
f_{m}(F_{m}x_{k})\end{bmatrix},~{}\text{and}~{}g(x_{k})=\begin{bmatrix}g_{1}(G_{1}x_{k})\\\
\vdots\\\ f_{i}(\underbrace{G_{i}x_{k}}_{\nu_{i}})\\\ \vdots\\\
g_{r}(G_{r}x_{k})\end{bmatrix},$ (7)

where $F_{i}\in\mathbb{R}^{\bar{n}\times n}~{}\forall~{}i\in\\{1,\ldots,m\\}$
and $G_{i}\in\mathbb{R}^{\bar{p}\times n}~{}\forall~{}i\in\\{1,\ldots,r\\}$.

For the state estimation purposes of the system (6), the subsequent observer
is employed:

$\begin{split}\hat{x}_{k+1}=A\hat{x}_{k}+Gf(\hat{x}_{k})+Bu_{k}+L\big{(}y_{k}-C\hat{x}_{k}-Fg(\hat{x}_{k})\big{)},\end{split}$
(8)

where $\hat{x}_{k}$ and $L\in\mathbb{R}^{n\times p}$ represent the estimated
states and the observer gain matrix, respectively. The estimation error of the
proposed observer (8) is defined as

$e_{k}={x}_{k}-\hat{x}_{k}.$

Thus, from (6) and (8), the estimation error dynamic is computed and it is
given by,

$\begin{split}e_{k+1}&=(A-LC)e_{k}+G\big{(}f(x_{k})-f(\hat{x}_{k})\big{)}\\\
&-LF\big{(}g(x_{k})-g(\hat{x}_{k})\big{)}+(E-LD)\omega_{k}.\end{split}$ (9)

Since $f(\cdot)$ and $g(\cdot)$ are globally Lipschitz, the deployment of
Lemma 1 on the terms $(f(x_{k})-f(\hat{x}_{k}))$ and
$(g(x_{k})-g(\hat{x}_{k}))$ yields:

  1. 1)

There exist functions
$f_{ij}:\mathbb{R}^{\bar{n}}\times\mathbb{R}^{\bar{n}}\to\mathbb{R}$,
$g_{ij}:\mathbb{R}^{\bar{p}}\times\mathbb{R}^{\bar{p}}\to\mathbb{R}$ such that

$f(x_{k})-f(\hat{x}_{k})=\sum_{i,j=1}^{m,\bar{n}}f_{ij}\mathcal{H}_{ij}F_{i}e_{k},$
(10a)
$g(x_{k})-g(\hat{x}_{k})=\sum_{i,j=1}^{r,\bar{p}}g_{ij}\mathcal{G}_{ij}G_{i}e_{k},$
(10b)

where $f_{ij}\triangleq
f_{ij}(\theta_{i}^{\hat{\theta}_{i,j-1}},\theta_{i}^{\hat{\theta}_{i,j}})$ and
$g_{ij}\triangleq
g_{ij}(\nu_{i}^{\hat{\nu}_{i,j-1}},\nu_{i}^{\hat{\nu}_{i,j}})$.

  2. 2)

The functions $f_{ij}$ and $g_{ij}$ satisfy

$\displaystyle{f}_{a_{ij}}$ $\displaystyle\leq f_{ij}\leq{f}_{b_{ij}},$ (11)
$\displaystyle{g}_{a_{ij}}$ $\displaystyle\leq g_{ij}\leq{g}_{b_{ij}},$ (12)

where ${f}_{a_{ij}}$, ${f}_{b_{ij}}$, ${g}_{a_{ij}}$, and ${g}_{b_{ij}}$ are
known constants.

Without loss of generality, let us presume that ${f}_{a_{ij}}=0$ and
${g}_{a_{ij}}=0$. Thus, the inequalities (11) and (12) are reformulated as:

$\displaystyle 0$ $\displaystyle\leq f_{ij}\leq{f}_{b_{ij}},$ (13)
$\displaystyle 0$ $\displaystyle\leq g_{ij}\leq{g}_{b_{ij}},$ (14)

One can refer to [13] for additional information about this.

By substituting (10) into (9), the error dynamic (9) is rewritten in the
following manner:

$\begin{split}e_{k+1}&=\mathbf{A}e_{k}+\mathbf{E}\omega_{k},\end{split}$ (15)

where

$\displaystyle\begin{split}\mathbf{A}&=(A-LC)+\sum_{i,j=1}^{m,\bar{n}}f_{ij}G\mathcal{H}_{ij}H_{i}-\sum_{i,j=1}^{r,\bar{p}}g_{ij}LF\mathcal{G}_{ij}G_{i},\\\
\mathbf{E}&=E-LD.\end{split}$

The objective of the proposed methodology is to estimate the gain matrix $L$
so that 1. The estimation error dynamic (15) is asymptotically stable in the
absence of the disturbances/noise, i.e., at $\omega=0$. 2. When $\omega\neq
0$, the closed-loop system (15) fulfils the ensuing $\mathcal{H}_{\infty}$
criterion:
$||e||_{\mathcal{L}_{2}}\leq\sqrt{\mu||\omega||^{2}_{\mathcal{L}_{2}}+\nu||e_{0}||^{2}},$
(16) where the positive scalar $\sqrt{\mu}$ is known as the noise attenuation
level.

The aforementioned problem statement has garnered a significant amount of
interest from researchers in the domain of control system engineering,
resulting in the establishment of numerous LMI-based methods, for example,
[7], [9], [14], and so on. The LMI conditions provided by each of these
methods are based on several mathematical tools, such as the Schur lemma and
the Young inequality. Though all of these techniques yield less conservative
LMI conditions, there is a scope for further enhancements. In the sequel,
through the exploration of Lemma 2 and newly defined matrix multipliers, a new
LMI criterion is derived.

###  III-B Synthesis of a Novel LMI Condition

This segment of the article is devoted to the formulation of the
$\mathcal{H}_{\infty}$ criterion-based LMI condition which ensures the
asymptotic stability of the error dynamic (15).

Let us consider the quadratic Lyapunov function

$V(e_{k})=e_{k}^{\top}Pe_{k},~{}\text{where}~{}P>0\in\mathbf{S}^{n}$ (17)

for the stability analysis of the error dynamic (15). Further, one can compute
the term $\Delta V_{k}=V(e_{k+1})-V(e_{k})$ by using (15) and (17), which is
illustrated as:

$\begin{split}\Delta
V_{k}&=e_{k}^{\top}\Big{(}-P+\mathbf{A}^{\top}P\mathbf{A}\Big{)}e_{k}+e_{k}^{\top}\Big{(}\mathbf{A}^{\top}P\mathbf{E}\Big{)}\omega_{k}\\\
&+\omega_{k}^{\top}\Big{(}\mathbf{E}^{\top}P\mathbf{A}\Big{)}e_{k}+\omega_{k}^{\top}\mathbf{E}^{\top}P\mathbf{E}\omega_{k}.\end{split}$
(18)

According to [9], the $\mathcal{H}_{\infty}$ criterion (16) is satisfied if
the ensuing inequality is true:

$\mathcal{W}_{k}\triangleq\Delta V_{k}+||e_{k}||^{2}-\mu||\omega_{k}||^{2}\leq
0.$ (19)

From (18) and (19), we obtain

$\begin{split}\mathcal{W}_{k}&=e_{k}^{\top}\Big{(}\mathbb{I}-P+\mathbf{A}^{\top}P\mathbf{A}\Big{)}e_{k}+e_{k}^{\top}\Big{(}\mathbf{A}^{\top}P\mathbf{E}\Big{)}\omega_{k}\\\
&+\omega_{k}^{\top}\Big{(}\mathbf{E}^{\top}P\mathbf{A}\Big{)}e_{k}+\omega_{k}^{\top}\Big{(}\mathbf{E}^{\top}P\mathbf{E}-\mu\mathbb{I}\Big{)}\omega_{k},\end{split}$
(20)

Further, $\mathcal{W}_{k}\leq 0$ if

$\begin{bmatrix}\mathbb{I}-P+\mathbf{A}^{\top}P\mathbf{A}&\mathbf{A}^{\top}P\mathbf{E}\\\
\mathbf{E}^{\top}P\mathbf{A}&\mathbf{E}^{\top}P\mathbf{E}-\mu\mathbb{I}\end{bmatrix}\leq
0.$ (21)

Further, the inequality (21) is altered as:

$\begin{bmatrix}\mathbb{I}-P&\mathbb{O}\\\
\mathbb{O}&-\mu\mathbb{I}\end{bmatrix}+\begin{bmatrix}\mathbf{A}^{\top}\\\
\mathbf{E}^{\top}\end{bmatrix}P\begin{bmatrix}\mathbf{A}&\mathbf{E}\end{bmatrix}\leq
0.$ (22)

The employment of Schur Lemma on (22) resulted in

$\Sigma_{1}+{\mathbf{NL}}\leq 0,$ (23)

where

$\Sigma_{1}=\begin{bmatrix}\begin{bmatrix}\mathbb{I}-P&\mathbb{O}\\\
\mathbb{O}&-\mu\mathbb{I}\end{bmatrix}&\begin{bmatrix}(A-LC)^{\top}P\\\
(E-LD)^{\top}P\end{bmatrix}\\\ \star&-P\end{bmatrix},$ (24)
$\small\mathbf{NL}=\begin{bmatrix}\mathbb{O}&\mathbb{O}&\bigg{(}\sum_{i,j=1}^{m,\bar{n}}f_{ij}G\mathcal{H}_{ij}H_{i}-\sum_{i,j=1}^{r,\bar{p}}g_{ij}LF\mathcal{G}_{ij}G_{i}\bigg{)}^{\top}P\\\
\star&\mathbb{O}&\mathbb{O}\\\ \star&\star&\mathbb{O}\end{bmatrix}.$ (25)

Further, the term $\mathbf{NL}$ is written under the form of (32).
For enhancing comprehensibility, let us consider the subsequent notations:

$\displaystyle\mathbb{U}$
$\displaystyle=\begin{bmatrix}\mathbb{U}^{\top}_{11}&\ldots&\mathbb{U}^{\top}_{1\bar{n}}&\ldots&\mathbb{U}^{\top}_{m1}&\ldots&\mathbb{U}^{\top}_{m\bar{n}}\end{bmatrix}^{\top},$
(26) $\displaystyle\mathbb{V}$
$\displaystyle=\begin{bmatrix}\mathbb{V}^{\top}_{11}&\ldots&\mathbb{V}^{\top}_{1\bar{n}}&\ldots&\mathbb{V}^{\top}_{m1}&\ldots&\mathbb{V}^{\top}_{m\bar{n}}\end{bmatrix}^{\top},$
(27) $\displaystyle\mathbb{M}$
$\displaystyle=\begin{bmatrix}\mathbb{M}^{\top}_{11}&\ldots&\mathbb{M}^{\top}_{1\bar{p}}&\ldots&\mathbb{M}^{\top}_{r1}&\ldots&\mathbb{M}^{\top}_{r\bar{p}}\end{bmatrix}^{\top},$
(28) $\displaystyle\mathbb{N}$
$\displaystyle=\begin{bmatrix}\mathbb{N}^{\top}_{11}&\ldots&\mathbb{N}^{\top}_{1\bar{p}}&\ldots&\mathbb{N}^{\top}_{r1}&\ldots&\mathbb{N}^{\top}_{r\bar{p}}\end{bmatrix}^{\top},$
(29)

where $\mathbb{U}_{ij}$, $\mathbb{V}_{ij}$, $\mathbb{M}_{ij}$ and
$\mathbb{N}_{ij}$ are defined in (32).
Through the utilisation of these aforementioned notations, the term
$\mathbf{NL}$ is altered as:

$\mathbf{NL}=\mathbb{U}^{\top}\mathbb{V}+\mathbb{V}^{\top}\mathbb{U}+\mathbb{M}^{\top}\mathbb{N}+\mathbb{N}^{\top}\mathbb{M}.$
(30)

Furthermore, the implementation of the inequality (5) on (30) yields:

$\displaystyle\begin{split}\mathbb{U}^{\top}\mathbb{V}+\mathbb{V}^{\top}\mathbb{U}&\leq(\mathbb{U}+\mathbb{X}\mathbb{V})^{\top}(2\mathbb{X})^{-1}(\mathbb{U}+\mathbb{X}\mathbb{V}),\\\
\mathbb{M}^{\top}\mathbb{N}+\mathbb{N}^{\top}\mathbb{M}&\leq(\mathbb{M}+\mathbb{Y}\mathbb{N})^{\top}(2\mathbb{Y})^{-1}(\mathbb{M}+\mathbb{Y}\mathbb{N}),\end{split}$
(31)

where $\mathbb{X}$ and $\mathbb{Y}$ are defined in (37) and (38),
respectively.

$\begin{split}\mathbf{NL}&=\sum_{i,j=1}^{m,\bar{n}}\begin{pmatrix}\underbrace{\begin{bmatrix}\mathbb{O}\\\
\mathbb{O}\\\
PG\mathcal{H}_{ij}\end{bmatrix}}_{\mathbb{U}_{ij}^{\top}}\underbrace{f_{ij}\overbrace{\begin{bmatrix}H_{i}&\mathbb{O}&\mathbb{O}\end{bmatrix}}^{\mathbb{H}_{i}}}_{\mathbb{V}_{ij}}+\mathbb{V}_{ij}^{\top}\mathbb{U}_{ij}\end{pmatrix}+\sum_{i,j=1}^{r,\bar{p}}\begin{pmatrix}\underbrace{\begin{bmatrix}\mathbb{O}\\\
\mathbb{O}\\\
-PLF\mathcal{G}_{ij}\end{bmatrix}}_{\mathbb{M}_{ij}^{\top}}\underbrace{g_{ij}\overbrace{\begin{bmatrix}G_{i}&\mathbb{O}&\mathbb{O}\end{bmatrix}}^{\mathbb{G}_{i}}}_{\mathbb{N}_{ij}}+\mathbb{N}_{ij}^{\top}\mathbb{M}_{ij}\end{pmatrix}.\end{split}$
(32)

$\begin{split}\min~{}\mu~{}\text{subject~{}to}&\\\
\begin{bmatrix}\Sigma&(\mathbb{U}+\mathbb{X}\mathbb{V})^{\top}&(\mathbb{M}+\mathbb{Y}\mathbb{N})^{\top}\\\
\star&-2\mathbb{X}&\mathbb{O}\\\
\star&\star&-2\mathbb{Y}\end{bmatrix}&<0,\,\,{\forall\mathbb{V}\in\mathcal{F}_{{H}_{m}},\forall\mathbb{N}\in\mathcal{G}_{{H}_{m}}},\end{split}$
(33)

where

$\Sigma=\begin{bmatrix}\begin{bmatrix}\mathbb{I}-P&\mathbb{O}\\\
\mathbb{O}&-\mu\mathbb{I}\end{bmatrix}&\begin{bmatrix}A^{\top}P-C^{\top}R\\\
E^{\top}P-D^{\top}R\end{bmatrix}\\\ \star&-P\end{bmatrix},$ (34)

and $\mathbb{U}$, $\mathbb{V}$, $\mathbb{M}$ and $\mathbb{N}$ are described in
(26), (27), (28) and (29), respectively.

From (13), (14), (27) and (29), one can easily notice that each element
$f_{ij}$ and $g_{ij}$ inside $\mathbb{V}$ and $\mathbb{N}$, respectively, are
bounded and belongs to its respective convex sets and their set of vertices
are described as follows:

$\small\begin{split}\mathcal{F}_{{H}_{m}}&=\bigg{\\{}\\{\mathcal{F}_{11},\ldots,\mathcal{F}_{1\bar{n}},\ldots,\mathcal{F}_{m1},\ldots,\mathcal{F}_{m\bar{n}}\\}:\mathcal{F}_{ij}\in[0,f_{b_{ij}}]\bigg{\\}},\\\
\mathcal{G}_{{H}_{m}}&=\bigg{\\{}\\{\mathcal{F}_{11},\ldots,\mathcal{F}_{1\bar{p}},\ldots,\mathcal{F}_{r1},\ldots,\mathcal{F}_{r\bar{p}}\\}:\mathcal{F}_{ij}\in[0,g_{b_{ij}}]\bigg{\\}}.\end{split}$

It leads to

$\small\begin{split}\mathbb{U}^{\top}\mathbb{V}+\mathbb{V}^{\top}\mathbb{U}&\leq\Big{[}(\mathbb{U}+\mathbb{X}\mathbb{V})^{\top}(2\mathbb{X})^{-1}(\mathbb{U}+\mathbb{V})\Big{]}_{\Phi\in\mathcal{F}_{{H}_{m}}},\\\
\mathbb{M}^{\top}\mathbb{N}+\mathbb{N}^{\top}\mathbb{M}&\leq\Big{[}(\mathbb{M}+\mathbb{Y}\mathbb{N})^{\top}(2\mathbb{Y})^{-1}(\mathbb{M}+\mathbb{Y}\mathbb{N})\Big{]}_{\Psi\in\mathcal{G}_{{H}_{m}}}.\end{split}$
(35)

Therefore, the inequality (23) is modified as:

$\begin{split}\Sigma_{1}&+\Big{[}(\mathbb{U}+\mathbb{X}\mathbb{V})^{\top}(2\mathbb{X})^{-1}(\mathbb{U}+\mathbb{V})\Big{]}_{\mathbb{V}\in\mathcal{F}_{{H}_{m}}}\\\
&+\Big{[}(\mathbb{M}+\mathbb{Y}\mathbb{N})^{\top}(2\mathbb{Y})^{-1}(\mathbb{M}+\mathbb{Y}\mathbb{N})\Big{]}_{\mathbb{N}\in\mathcal{G}_{{H}_{m}}}\leq
0.\end{split}$ (36)

Now, we are ready to propose the main outcome of this letter through the
following theorem.

######  Theorem 1

If there exist positive definite matrices $\mathbb{X}$ and $\mathbb{Y}$ under
the form (37) and (38), respectively, along with matrices
$P>0\in\mathbf{S}^{n}$, $R\in\mathbb{R}^{p\times n}$ such that the
optimisation problem (33) is solvable. Then, the error dynamic (15) is
$\mathcal{H}_{\infty}$ asymptotically stable. The gain matrix $L$ is computed
by utilising $L=P^{-1}R^{\top}$.

###### Proof:

The LMI (33) is derived by substituting $R^{\top}=PL$ in the term
$\Sigma_{1}$, and by implementing the Schur lemma on (36). From convexity
principle which is proposed in [15], the error dynamic (15) holds
$\mathcal{H}_{\infty}$ criterion (16) if the LMI (33) is solved for all
$\mathbb{V}\in\mathcal{F}_{{H}_{m}}$ and $\mathbb{N}\in\mathcal{G}_{{H}_{m}}$.
∎

###### Remark 1

At $\omega=0$, one can reformulate the inequality (19) into

$\Delta V_{k}+||e_{k}||^{2}\leq 0.$

It yields the exponential stability condition

$\Delta V_{k}\leq-\sigma V(e_{k}),$

along with $\sigma=\frac{1}{\lambda_{\max}(P)}>0$. Hence, the proposed LMI
ensures the exponential stability of the error dynamic (15) when $\omega=0$.

$\mathbb{X}=\begin{bmatrix}\mathbb{X}_{1}&\mathbb{X}_{b^{1}_{2}}&\ldots&\mathbb{X}_{b^{1}_{m}}\\\
\star&\mathbb{X}_{2}&\ldots&\mathbb{X}_{b^{2}_{m}}\\\
\star&\star&\ddots&\vdots\\\
\star&\star&\ldots&\mathbb{X}_{m}\end{bmatrix},\text{where}~{}\mathbb{X}_{i}=\begin{bmatrix}{X}_{i1}&{X}_{a^{1}_{i2}}&\ldots&{X}_{a^{1}_{i\bar{n}}}\\\
\star&{X}_{i2}&\ldots&{X}_{a^{2}_{i\bar{n}}}\\\ \star&\star&\ddots&\vdots\\\
\star&\star&\ldots&{X}_{i\bar{n}}\end{bmatrix}\,\forall
i\in\\{1,\ldots,m\\},~{}\mathbb{X}_{b^{j}_{i}}=\begin{bmatrix}{X}_{b^{j1}_{i1}}&{X}_{b_{i2}^{j1}}&\ldots&{X}_{{b}_{i\bar{n}}^{j1}}\\\
{X}_{b_{i1}^{j2}}&{X}_{b^{j2}_{i2}}&\ldots&{X}_{{b}_{i\bar{n}}^{j2}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{X}_{{b}_{i1}^{j\bar{n}}}&{X}_{{b}_{i2}^{j\bar{n}}}&\ldots&{X}_{b_{i\bar{n}}^{j\bar{n}}}\end{bmatrix}\,\forall
i\in\\{2,\ldots,m\\},j\in\\{1,\ldots,m-1\\},$ (37)

where $X_{ij},X_{a^{k}_{ij}}\in\mathbf{S}^{\bar{n}}\forall
i,k\in\\{1,\ldots,m\\},\&j\in\\{1,\ldots,\bar{n}\\}$;$~{}X_{b^{kj}_{ij}}\in\mathbf{S}^{\bar{n}},\forall
i\in\\{2,\ldots,m\\},k\in\\{1,\ldots,m-1\\},\&j\in\\{1,\ldots,\bar{n}\\}$ such
that $\mathbb{X}>0$.

$\mathbb{Y}=\begin{bmatrix}\mathbb{Y}_{1}&\mathbb{Y}_{b^{1}_{2}}&\ldots&\mathbb{Y}_{b^{1}_{r}}\\\
\star&\mathbb{Y}_{2}&\ldots&\mathbb{Y}_{b^{2}_{r}}\\\
\star&\star&\ddots&\vdots\\\
\star&\star&\ldots&\mathbb{Y}_{r}\end{bmatrix},\text{where}~{}\mathbb{Y}_{i}=\begin{bmatrix}{Y}_{i1}&{Y}_{a^{1}_{i2}}&\ldots&{Y}_{a^{1}_{i\bar{p}}}\\\
\star&{Y}_{i2}&\ldots&{Y}_{a^{2}_{i\bar{p}}}\\\ \star&\star&\ddots&\vdots\\\
\star&\star&\ldots&{Y}_{i\bar{p}}\end{bmatrix}\,\forall
i\in\\{1,\ldots,r\\},~{}\mathbb{Y}_{b^{j}_{i}}=\begin{bmatrix}{Y}_{b^{j1}_{i1}}&{Y}_{b_{i2}^{j1}}&\ldots&{Y}_{{b}_{i\bar{p}}^{j1}}\\\
{Y}_{b_{i1}^{j2}}&{Y}_{b^{j2}_{i2}}&\ldots&{Y}_{{b}_{i\bar{p}}^{j2}}\\\
\vdots&\vdots&\ddots&\vdots\\\
{Y}_{{b}_{i1}^{j\bar{p}}}&{Y}_{{b}_{i2}^{j\bar{p}}}&\ldots&{Y}_{b_{i\bar{n}}^{j\bar{p}}}\end{bmatrix}\,\forall
i\in\\{2,\ldots,r\\},j\in\\{1,\ldots,r-1\\},$ (38)

where $Y_{ij},~{}Y_{a^{k}_{ij}}\in\mathbf{S}^{\bar{p}}\forall
i,k\in\\{1,\ldots,r\\},\&j\in\\{1,\ldots,\bar{p}\\}$;$~{}Y_{b^{kj}_{ij}}\in\mathbf{S}^{\bar{p}},\forall
i\in\\{2,\ldots,r\\},k\in\\{1,\ldots,r-1\\},\&j\in\\{1,\ldots,\bar{p}\\}$ so
that $\mathbb{Y}>0$.

###  III-C Special Case: Nonlinear Systems with Linear Outputs

This section encompasses the observer design for the nonlinear systems having
linear outputs in the presence of disturbances/noise. Such type of system is
described as follows:

$\displaystyle\begin{split}x_{k+1}&=Ax_{k}+Gf(x_{k})+Bu_{k}+E\omega_{k},\\\
y_{k}&=Cx_{k}+D\omega_{k},\end{split}$ (39)

where all variables and parameters remain consistent with those specified in
(6). The nonlinear function $f(\cdot)$ is presumed to be globally Lipschitz
and fulfils the detailed form (7). Analogous to the previous Section III-A,
the states of the system (39) are estimated by deploying the ensuing observer:

$\hat{x}_{k+1}=A\hat{x}_{k}+Gf(\hat{x}_{k})+Bu_{k}+L\big{(}y_{k}-C\hat{x}_{k}\big{)},$
(40)

where all the parameters and variables are the same as the ones illustrated in
(8). If one follows the steps (9)- (15) showcased in Section III-A, it is easy
to obtain the subsequent error dynamics of the observer (40):

$\begin{split}e_{k+1}=(A-LC)e_{k}+\sum_{i,j=1}^{m,\bar{n}}f_{ij}G\mathcal{H}_{ij}F_{i}e_{k}+(E-LD)\omega_{k}.\end{split}$
(41)

The succeeding theorem presents a new LMI condition which guarantees the
$\mathcal{H}_{\infty}$ stability of the closed-loop system (41).

######  Theorem 2

Let us consider the positive definite matrix $\mathbb{X}$, as defined in (37),
along with the existence of matrices $P>0\in\mathbf{S}^{n}$ and
$R\in\mathbf{S}^{p\times n}$. If the ensuing optimisation problem is solvable:
$\begin{split}\min~{}\mu~{}\text{subject~{}to}&\\\
\begin{bmatrix}\Sigma_{1}&(\mathbb{U}+\mathbb{X}\mathbb{V})^{\top}\\\
\star&-2\mathbb{X}\end{bmatrix}&<0~{}{\forall\mathbb{V}\in\mathcal{F}_{{H}_{m}}},\end{split}$
(42) where $\Sigma_{1}$, $\mathbb{U}$ and $\mathbb{V}$ are described in (34),
(26) and (27), respectively. Then, the estimation error dynamic (41) is
$\mathcal{H}_{\infty}$ asymptotically stable. The observer parameter $L$ is
evaluated by employing $L=P^{-1}R^{\top}$.

###### Proof:

The proof of this theorem is very straightforward. If one follows the proof of
Theorem 1, it is easy to prove the theorem above. ∎

In the next part, the effectiveness of the established approach is verified
through numerical examples.

##  IV Illustrative examples

This section is dedicated to the analysis of the established LMI-based
observer methodology. The first part of this segment emphasises the
superiority of the proposed LMI approach through a numerical example. Later
on, the performance of the observer is demonstrated by applying it to SoC
estimation in Li-ion batteries.

###  IV-A Numerical example 1

Let us consider a numerical example represented in the form (39) with the
ensuing parameters: $A=\begin{bmatrix}0&1&0\\\ 0&-1&1\\\ 0&-1&1\end{bmatrix}$,
$G=\begin{bmatrix}1&0\\\ 0&0\\\ 0&1\end{bmatrix}$, $C=\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix}^{\top}$, $B=E=\begin{bmatrix}1\\\ 1\\\ 1\end{bmatrix}$ and
$D=1$. In addition to this, $f(x)=\begin{bmatrix}\sin(\theta_{1}x_{1})\\\
\cos(\theta_{2}x_{2})\end{bmatrix}$ along with $H_{1}=\begin{bmatrix}1&0&-1\\\
-1&0&1\\\ 1&0&0\end{bmatrix}$ and $H_{2}=\begin{bmatrix}0&-1&1\\\ 1&1&0\\\
-1&0&0\end{bmatrix}$.
Hence, we get $m=2$ and $\bar{n}=3$.

Further, the developed LMI condition (42) is solved using MATLAB toolbox to
compute the observer parameter $L$ and the optimal noise attenuation index
$\sqrt{\mu}$ for different values of $\theta_{1}$ and $\theta_{2}$. The
obtained minimised value $\sqrt{\mu}$ is outlined in Table I. The proposed LMI
provides a better noise attenuation level than the one obtained from [9,
Corollary (2)] for all considered values of $\theta_{1}$ and $\theta_{2}$. It
implies that the noise mitigation achieved by the developed LMI-based observer
is more efficient than the existing methods. Hence, it emphaises that the
established LMI condition gives better results than the LMI of [9]. Therefore,
the effectiveness of the derived LMI condition is highlighted through this
comparison.

TABLE I: Portraying the optimal values of $\sqrt{\mu}$ in several cases LMI approach |  | $\theta_{1}=0.1$
---
$\theta_{2}=0.1$
| $\theta_{1}=0.3$
---
$\theta_{2}=0.2$
| $\theta_{1}=0.3$
---
$\theta_{2}=0.3$
| $\theta_{1}=0.2$
---
$\theta_{2}=0.4$
LMI (42) | 2.1677 | 3.4913 | 5.8411 | 9.6774
[9, Corollary (2)] | 2.1950 | 3.7433 | 6.3776 | 10.0382

In the sequel, the validation of observer performance is showcased.

###  IV-B Application: SoC estimation of Li-ion batteries

In this segment, the authors have demonstrated the effectiveness of the
established observer methodology through the deployment of the developed
observer for the State-of-Charge (SoC) estimation of the Li-ion battery model.
Let us consider the subsequent $2^{\text{nd}}$ order equivalent circuit model
proposed in [16]:

$\begin{split}\dot{V}_{1}&=\frac{1}{R_{1}C_{1}}V_{1}+\frac{1}{C_{1}}I,\\\
\dot{V}_{2}&=\frac{1}{R_{2}C_{2}}V_{2}+\frac{1}{C_{2}}I,\\\
\dot{s}&=-\frac{1}{C_{n}}s,\\\ V_{t}&=OCV(s)-V_{1}-V_{2}+R_{0}I\end{split}$
(43)

where $V_{1}$ and $V_{2}$ represent voltages across polarisation resistances
$R_{1}$ and $R_{2}$, respectively, whereas $s$ indicates the SoC of the
battery. $I$ infers the current flowing through the load. The terms $C_{1}$
and $C_{2}$ denote polarisation capacitors, however, $C_{n}$ depict the
capacity of the battery. The terminal voltage ($V_{t}$) is considered as the
output of the model. The term $OCV(s)$ depicts the open circuit voltage (OCV)
of the battery, and it is illustrated as:

$OCV(s)=0.9206\cdot s^{3}-1.3781\cdot s^{2}+1.3905\cdot s+3.2416.$ (44)

The details of the remaining parameters of the model are as follows:

  * •

Battery capacity:
$C_{n}=5~{}$\mathrm{A}\cdot\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}$$; Battery
resistance: $R_{0}=0.0314~{}$\Omega$$; Sampling time:
$T_{s}=0.02~{}$\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}$$;

  * •

Polarisation resistances:
$R_{1}=0.0181~{}$\Omega$,\,\,R_{2}=0.0281~{}$\Omega$$;

  * •

Polarisation capacitors:
$C_{1}=1712~{}$\mathrm{F}$,\,\,C_{2}=55257~{}$\mathrm{F}$$;

Through the utilisation of the Euler forward method, the system (43) is
transformed into the discrete-time model represented (6), whose parameters are
showcased as: $x(k)=\begin{bmatrix}V_{1}(k)\\\ V_{2}(k)\\\
s(k)\end{bmatrix},\,A=\begin{bmatrix}1-\frac{T_{s}}{R_{1}C_{1}}&0&0\\\
0&1-\frac{T_{s}}{R_{2}C_{2}}&0\\\
0&0&1\end{bmatrix},\,\,B_{1}=\begin{bmatrix}\frac{T_{s}}{C_{1}}\\\
\frac{T_{s}}{C_{2}}\\\
-\frac{T_{s}}{C_{n}}\end{bmatrix},\,\,B_{2}=-R_{0},\,\,F=1,\,\,g(x(k))=OCV(x_{3}(k))$,
$E=B_{1}$ and $D=1$. Let us assume that the system dynamics and outputs are
corrupted with the Gaussian noise $\omega\leadsto(0,0.1)$.
Since $0\leq x_{3}(k)\leq 1$, it is easy to infer that the partial derivative
of $g(x)$ satisfies (14).

The developed LMI (33) is solved by using YALMIP toolbox [17], and we obtain:

$\sqrt{\mu}=1.6502\times 10^{-6}~{}\text{and}~{}L=\begin{bmatrix}-0.0005\\\
-0.0000\\\ 0.1769\end{bmatrix}.$

Figure 1: Graph of estimated and actual SoC

By using the aforementioned matrix $L$, the proposed observer (8) is
implemented in MATLAB environment for the state estimation purpose. The plot
of estimated SoC and actual SoC is shown in Figure 1. It shows that the
developed observer performs the efficient estimation of SoC along with optimal
noise attenuation. Further, for the same task, the extended Kalman filter
(EKF) methodology of [18] is implemented in MATLAB. The RMSE values of the
estimation error obtained from EKF and the established observer are summarised
in Table II. It emphasises that the accuracy of the SoC obtained from the
proposed approach is relatively better than the one achieved from EKF. Thus,
the effectiveness of the new LMI-based observer over the well-known EKF
technique is highlighted.

TABLE II: Comparison of RMSE values of the estimation error Methodology | $\tilde{x}_{1}$ | $\tilde{x}_{2}$ |  $\tilde{x}_{3}$ (SoC)
---|---|---|---
Proposed observer (8) | $7.71\times 10^{-4}$ | $1.97\times 10^{-7}$ | 0.0014
EKF approach [18] | $7.78\times 10^{-4}$ | $7.92\times 10^{-4}$ | 0.0028

##  V Conclusion

In this paper, the problem of nonlinear observer design for discrete-time
systems is addressed. It is tackled by formulating a new LMI condition which
provides the observer gain and ensures the $\mathcal{H}_{\infty}$ stability of
the estimation error of the proposed observer. The established matrix-
multiplier-based LMI is derived by incorporating the reformulated Lipschitz
property, a new variant of Young inequality and the well-known Linear
Parameter Varying (LPV) approach. The obtained LMIs encompass some additional
decision variables as compared to the existing ones due to the deliberate use
of matrix multipliers and a new variant of Young inequality. It resulted in an
improvement in LMI feasibility. Thus, the introduction of generalised matrix
multipliers inside LMIs plays a vital role in their enhancement. Further, the
performance of the observer and the efficacy of the LMI are demonstrated by
using numerical examples. From a future perspective, the authors plan to
implement the proposed strategy for the stabilisation of the same class of
systems used in this article.

## References

  * [1] Y. Shen, “A robust method for state of charge estimation of lithium-ion batteries using adaptive nonlinear neural observer,” _Journal of Energy Storage_ , vol. 72, p. 108480, 2023.
  * [2] A. Zemouche, R. Rajamani, G. Phanomchoeng, B. Boulkroune, H. Rafaralahy, and M. Zasadzinski, “Circle criterion-based $\mathcal{H}_{\infty}$ observer design for Lipschitz and monotonic nonlinear systems-enhanced LMI conditions and constructive discussions,” _Automatica_ , vol. 85, pp. 412–425, 2017.
  * [3] J. Gauthier, H. Hammouri, and S. Othman, “A simple observer for nonlinear systems applications to bioreactors,” _IEEE Transactions on Automatic Control_ , vol. 37, no. 6, pp. 875–880, 1992.
  * [4] J. H. Ahrens and H. K. Khalil, “High-gain observers in the presence of measurement noise: A switched-gain approach,” _Automatica_ , vol. 45, no. 4, pp. 936–943, 2009.
  * [5] S. K. Spurgeon, “Sliding mode observers: a survey,” _International Journal of Systems Science_ , vol. 39, no. 8, pp. 751–764, 2008.
  * [6] J. Hu, Z. Wang, Y. Niu, and L. K. Stergioulas, “$\mathcal{H}_{\infty}$ sliding mode observer design for a class of nonlinear discrete time-delay systems: A delay-fractioning approach,” _International Journal of Robust and Nonlinear Control_ , vol. 22, no. 16, pp. 1806–1826, 2012.
  * [7] S. Ibrir, “Circle-criterion approach to discrete-time nonlinear observer design,” _Automatica_ , vol. 43, no. 8, pp. 1432–1441, 2007.
  * [8] O. Hernández, T. Menard, M. Farza, C.-M. Astorga-Zaragoza, and G.-V. Guerrero-Ramírez, “High gain observer for a class of nonlinear systems with sampled measurements: Application to a free-piston stirling engine,” in _2015 16th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA)_. IEEE, 2015, pp. 805–810.
  * [9] K. Chaib Draa, A. Zemouche, M. Alma, H. Voos, and M. Darouach, “A discrete-time nonlinear state observer for the anaerobic digestion process,” _International Journal of Robust and Nonlinear Control_ , vol. 29, no. 5, pp. 1279–1301, 2019.
  * [10] A. Zemouche and M. Boutayeb, “Observer design for lipschitz nonlinear systems: the discrete-time case,” _IEEE Transactions on Circuits and Systems II: Express Briefs_ , vol. 53, no. 8, pp. 777–781, 2006.
  * [11] M. Abbaszadeh and H. J. Marquez, “LMI optimization approach to robust $\mathcal{H}_{\infty}$ observer design and static output feedback stabilization for discrete-time nonlinear uncertain systems,” _International Journal of Robust and Nonlinear Control: IFAC-Affiliated Journal_ , vol. 19, no. 3, pp. 313–340, 2009.
  * [12] S. Mohite, M. Alma, and A. Zemouche, “LMI-based $\mathcal{H}_{\infty}$ observer design for nonlinear Lipschitz system,” in _22nd IFAC World Congress, IFAC 2023_ , Yokohama, Japan, Jul. 2023.
  * [13] A. Zemouche and M. Boutayeb, “On LMI conditions to design observers for Lipschitz nonlinear systems,” _Automatica_ , vol. 49, no. 2, pp. 585–591, 2013.
  * [14] X. Chu and M. Li, “$\mathcal{H}_{\infty}$ observer-based event-triggered sliding mode control for a class of discrete-time nonlinear networked systems with quantizations,” _Isa Transactions_ , vol. 79, pp. 13–26, 2018.
  * [15] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, _Linear matrix inequalities in system and control theory_. SIAM, 1994.
  * [16] Y. Tian, D. Li, J. Tian, and B. Xia, “State of charge estimation of lithium-ion batteries using an optimal adaptive gain nonlinear observer,” _Electrochimica Acta_ , vol. 225, pp. 225–234, 2017.
  * [17] J. Lofberg, “Yalmip : a toolbox for modeling and optimization in matlab,” in _2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508)_ , 2004, pp. 284–289.
  * [18] Z. Chen, Y. Fu, and C. C. Mi, “State of charge estimation of lithium-ion batteries in electric drive vehicles using extended kalman filtering,” _IEEE Transactions on Vehicular Technology_ , vol. 62, no. 3, pp. 1020–1030, 2013.


# Reduction in Circulating Current with Improved Secondary Side Modulation in
Isolated
Current-Fed Half Bridge AC-DC Converter

Manish Kumar, Sumit Pramanick,
and B K Panigrahi

###### Abstract

Current-fed half bridge converter with bidirectional switches on ac side and a
full bridge converter on dc side of a high frequency transformer is an optimal
topology for single stage galvanically isolated ac-dc converter for onboard
vehicle charging application. AC side switches are actively commutated to
achieve zero current switching (ZCS) using single phase shift modulation
(SPSM) and discontinuous current phase shift modulation (DCPSM). Furthermore,
zero voltage turn-on (ZVS) is achieved for dc side switches. Compared to SPSM,
DCPSM maintains a constant peak current in the converter throughout the grid
cycle of ac mains voltage. However, constant peak current contributes to a
high circulating current near the zero crossings of ac mains voltage and also
at light load conditions. This paper proposes an improved discontinuous
current phase shift modulation (IDCPSM) to increase the efficiency of the
converter across different loading conditions. A dual control variable is
adopted to actively reduce the circulating current while maintaining soft
switching of both ac and dc side switches across the grid cycle of ac mains
voltage. A 1.5 kW laboratory prototype has been developed to experimentally
validate the analysis, design and improvement in performance for different
loading conditions.

###### Index Terms:

Current-fed half bridge, ac-dc converter, single stage, improved discontinuous
current phase shift modulation (IDCPSM)

##  I Introduction

Electric vehicles (EVs) have gained tremendous popularity against their
internal combustion engine (ICE) counterparts over the past decade. As EVs are
becoming more common, ac-dc power converters are being explored for onboard
charging applications (OBCs). OBCs provides the user with the convenience of
using an ac utility outlet to charge the EVs battery. The design of an OBC is
mainly driven by electrical and volumetric efficiency [1]. These ac-dc power
converters should meet power quality standards on the grid side as per IEEE
519 [2]. Such converters should also have galvanic isolation between the ac
side (grid) and the dc side (vehicle battery) as per Underwriters Laboratories
(UL) 2202 safety standard [3].

The most general approach to attain the above requirements is a two-stage
solution. An active power factor correction circuit as the first stage
followed by an isolated dc-dc converter as second stage[4, 5]. The isolated
DC-DC converter is realized by a dual active bridge (DAB) in [4] and series
resonant converter in [5]. However, the two-stage approach increases device
count and losses, contributing to lower efficiency, along with increase in
control complexity and volume. To overcome the disadvantages of the two-stage
solution, several single-stage approaches, which could be either voltage-fed
or current-fed, have been reported in the literature[6, 7, 8, 9].

Current-fed isolated converters have some distinct advantages over the
voltage-fed isolated converters, such as lower input current ripple, inherent
short circuit protection, wide soft-switching operating range and ease of
current control [10, 11]. Work presented in [12] exhaustively compared the
various isolated current-fed (ICF) topologies in terms of electrical and
volumetric efficiency and reported that the L-L type isolated current-fed
half-bridge (ICFHB) is an optimal topology among the different ICF topologies.
Thus, an L-L type single stage single phase ICFHB ac-dc converter using
bidirectional switches proposed in [13] is the focus of this paper. This
converter shown in Fig. 1 is conceptually derived from an L-L type ICFHB dc-dc
converter [14].

However, one major drawback of such converters is that a huge voltage spike
appears at the switching node caused by an abrupt change in the current
through the high frequency transformer (HFT) winding. It occurs at the instant
when the boost inductor current finds a path through the HFT windings. A
common technique to mitigate the problem of voltage spike is to use an
additional active clamp circuit. An active clamp circuit consisting of two
active switches and a clamping capacitor was proposed in [15] for L-L type
ICFHB. This clamp circuit assisted in zero voltage switching (ZVS) of all the
switches. Another active clamp circuit consisting of a single switch, two
diodes, a clamping capacitor and a resonating inductor was proposed in [16]
that ensures zero current switching (ZCS) of the main primary side switches.
This increased component count in the auxiliary clamp circuit attributes to
higher losses.

Another technique is the modulation based approach also known as secondary
side modulation (SSM) [14, 17, 18, 19, 20]. The secondary side voltage-fed
full bridge (VFFB) is used for active current commutation at the primary side
current-fed half bridge (CFHB). This is achieved by using the reflected output
voltage to completely transfer the boost inductor current from the commutating
leg to the non-commuting leg through the HFT. The reflected output voltage is
applied across the primary side of the HFT for a sufficiently long duration to
force the current through the commutating leg in the reverse direction,
through the body diode. When the body diode current naturally reaches zero,
the switch is turned-off achieving ZCS. Also, body diode conduction in the
secondary side ensures ZVS turn-on of secondary side switches. The merit of
this technique is that it does not require any additional circuits.

A single phase shift modulation (SPSM) for an L-L type ICFHB is proposed in
[14]. The phase shift $\phi$ is between the primary and secondary side bridge
gating signals. Soft switching is achieved in both bridge switches for a wide
operating range. However, when implemented in the case of the converter shown
in Fig. 1 with ac input, the circulating current flowing through the switches
and the HFT is substantially high near the zero crossing of the ac mains
voltage and at light load conditions, attributing to poor efficiency of the
converter.

A discontinuous current single phase shift modulation (DCPSM) is proposed in
[17] where the current through the HFT drops to zero and a resonance occurs
between the output capacitance of the dc side switches and the leakage
inductance of the transformer. The circulating current flowing through the
devices is still relatively large near the zero crossing of the ac mains
voltage. A discontinuous current dual phase shift modulation (DCDPSM) for an
L-L type ICFHB dc-dc converter is proposed in [18], where unsymmetrical
switching pattern is applied to dc side switches to avoid resonance. However,
a five control variable were adopted to reduce the circulating current value
at light load conditions making the control architecture complex.

This paper proposes an improved discontinuous current single phase shift
modulation (IDCPSM) with a simpler unsymmetrical switching pattern. The
objective is to minimize the circulating current, flowing through the switches
and the HFT across the grid cycle of ac mains voltage and also at different
loading conditions. This is achieved by modulating the duty of the dc side
VFFB in every switching interval. The duty of ac side CFHB is modulated to
maintain unity power factor (UPF). This dual control variable approach keeps
the control architecture simple and easy to implement. ZCS turn-off for ac
side switches and ZVS turn-on for dc side switches is maintained across the
grid cycle. Reduced RMS current through ac and dc side switches as well as
through the HFT is achieved contributing to lower conduction losses with
significant improvement in the efficiency of the converter.

This paper is organized as follows. Section II provides a detailed description
of the schematic of ICFHB ac-dc converter. Operational stages with the
proposed IDCPSM are described and compared with the conventional SSM in
Section III. Section IV shows the experimental results of the implemented
IDCPSM on the developed laboratory prototype and the comparative performance
evaluation among the different SSM. Lastly, Section V concludes this paper.

##  II ICFHB AC-DC Converter Description

The schematic of a single phase single stage galvanically isolated current-fed
dual active bridge ac-dc converter is shown in Fig. 1. It consists of a
current-fed half bridge converter with bidirectional switches on the ac side
and a full bridge converter on the dc side, connected through a high frequency
transformer (HFT). $L_{1}$ and $L_{2}$ are the two boost inductors. $L_{lk}$
is the leakage inductance of the HFT. An external series inductor $L_{s}$ is
connected on the dc side to achieve the required total series inductance
$L_{t}=L_{lk}+L_{s}/n^{2}$. $C_{o}$ is the output filter capacitor.

The ac side switches are annotated as $S_{xyg}$, where $x$ denotes leg number
and $y=a$ or $b$. $S_{xag}$ are switched at high frequency in the positive
half cycle of grid voltage and remains continuously on in the negative half
cycle. While $S_{xbg}$ are switched at high frequency in the negative half
cycle of grid voltage and remains continuously on in the positive half cycle.
The gate signals between ac side leg $1$ and leg $2$ switches are phase-
shifted by $180^{\circ}$. $S_{xyg}$ is switched at a duty ratio $d_{1}$ always
greater than $0.5$, ensuring conduction overlap between the legs. Battery side
switches are annotated as $S_{p}$, where $p=A,B,C$ or $D$. $S_{A}$ and $S_{B}$
are switched at a fixed duty of 0.5. $S_{C}$ and $S_{D}$ are switched at a
duty ratio of $d_{2}$. $S_{p}$ turn-off is synchronised with the turn-off of
$S_{xyg}$ in the following way. In the positive half cycle of the grid
voltage, $S_{C,D}$ and $S_{A,B}$ turn off is synchronised with $S_{1ag}$ and
$S_{2ag}$ respectively. In the negative half cycle, $S_{C,D}$ and $S_{A,B}$
turn off is synchronised with $S_{2bg}$ and $S_{1bg}$ respectively. During the
turn-off duration of ac side switches, a resonance occurs between its output
device capacitance and the total series inductance, $L_{t}$. In order to
suppress this parasitic ringing, HFT is designed to have very minimum leakage
inductance $L_{lk}$. Diode clamps $D_{c1}$ and $D_{c2}$ are connected between
the HFT terminals and external series inductor at dc side, thereby clamping
the ac side switch voltage to $V_{o}/n$. These diode clamps incur lower losses
than the damping resistor used in [13].

Figure 1: L-L type current fed half bridge single stage ac-dc converter.

(a)

(b)

(c)

Figure 2: Operating waveform of ICFHB ac-dc converter in a switching interval
and in the positive half cycle of ac mains voltage (a) at
$\omega\tau=90^{\circ}$ peak of ac mains voltage, (b) at
$\omega\tau=10^{\circ}$ near zero crossing of ac mains voltage, (c) ac side
switch current $i_{Sxyg}$ and dc side switch current $i_{Sp}$ at
$\omega\tau=10^{\circ}$ near zero crossing of ac mains voltage

##  III Operation stages with IDCPSM

To simplify analysis, it has been assumed that the converter is operating at
unity power factor (UPF). Further, input ac mains voltage $v_{g}^{(k)}(\tau)$
is assumed constant in a switching interval $k$. Variable with superscript $k$
is changing every next switching interval and variable without the superscript
$k$ is assumed constant throughout the converter operation.

$\displaystyle v_{g}^{(k)}(\tau)=V_{m}sin(\omega\tau)$ (1) $\displaystyle
i_{g}^{(k)}(\tau)=\frac{2P_{o}}{V_{m}}sin(\omega\tau)$ (2)

AC side duty cycle is given by

$\displaystyle d_{1}^{(k)}=\frac{V_{o}-n|v_{g}^{(k)}(t)|}{V_{o}}$ (3)

where, $\tau=k*T_{s}$ and $T_{s}$ is the time period of a switching cycle,
$\omega=2\pi/t_{g}$, $t_{g}$ is the time period of a grid cycle, $P_{o}$ is
the output power and $V_{o}$ is the output voltage. Fig. 2a and Fig. 2b, shows
the operating waveform of the converter at the peak and near the zero crossing
in the positive half cycle of ac mains voltage respectively. Fig. 2c, shows
the ac side switch current $i_{Sxyg}$ and dc side switch current $i_{Sp}$ near
the zero crossing in the positive half cycle of ac mains voltage. Inductor
current $i_{L1}^{(k)}(t)$ is given by Eq. 4.

$\begin{split}&i_{L1}^{(k)}(t)=\\\
&\left\\{\begin{matrix}[l]\frac{1}{2}(i_{g}^{(k)}-\frac{v_{g}^{(k)}}{L_{1}}d_{1}^{(k)}T_{s})+\\\
\frac{v_{g}^{(k)}}{L_{1}}[t-(k-1)T_{s}]&(k-1)T_{s}<t\leq(k-1+d_{1}^{(k)})T_{s}\\\
\\\ \frac{1}{2}(i_{g}^{(k)}+\frac{v_{g}^{(k)}}{L_{1}}d_{1}^{(k)}T_{s})+\\\
\frac{v_{g}^{(k)}}{L_{1}}[t-(k-1+d_{1}^{(k)})T_{s}]\\\
-\frac{V_{o}}{nL_{1}}[t-(k-1+d_{1}^{(k)})T_{s}]&(k-1+d_{1}^{(k)})T_{s}<t<kT_{s}\\\
\\\
\frac{1}{2}(i_{g}^{(k+1)}+\frac{v_{g}^{(k+1)}}{L_{1}}d_{1}^{(k+1)}T_{s})&t=kT_{s}\end{matrix}\right.\end{split}$
(4)

Stage 1 ($t_{0},t_{1}$): Fig. 3a shows the equivalent circuit during this
stage. At $t_{0}$, the HFT ac side winding current $i_{lk}^{(k)}(t)$ equals
boost inductor current $i_{L2}^{(k)}(t)$. The current through $S_{2ag}$ body
diode reaches zero and naturally turns-off achieving ZCS. The output parasitic
capacitance of $S_{2ag}$ charges to $V_{o}/n$, therefore
$v_{pq}^{(k)}=-V_{o}/n$. $S_{1ag}$ is on in this stage. $S_{C}$ and $S_{D}$
are turned on at $t_{0}$ and their respective body diode current transfers to
the channel of the devices thereby achieving ZVS. $S_{A}$ and $S_{B}$ are off
during this stage and their respective device output capacitance are charged
to $V_{o}$. The equation of $i_{lk}^{(k)}(t)$ during stage 1 is given by (5)
and at $t_{0}$ is given by (6). The equation of ac side switch current and dc
side switch current during this stage is give by (7) and (8) respectively.

$\displaystyle i_{lk}^{(k)}(t)$
$\displaystyle=-\frac{(v_{g}^{(k)}-\frac{V_{o}}{n})}{L_{2}+L_{t}}(t-t_{0})+i_{lk}^{(k)}(t_{0})$
(5) $\displaystyle i_{lk}^{(k)}(t_{0})$
$\displaystyle=-\frac{1}{2}(i_{g}^{(k)}+\frac{v_{g}^{(k)}}{L_{2}}d_{1}^{(k)}T_{s})$
(6) $\displaystyle i_{S_{1ag}}^{(k)}(t)$
$\displaystyle=i_{L1}^{(k)}(t)+i_{lk}^{(k)}(t)$ (7) $\displaystyle
i_{S_{C}}^{(k)}(t)$
$\displaystyle=i_{S_{D}}^{(k)}(t)=\frac{i_{lk}^{(k)}(t)}{n}$ (8)

Stage 2 ($t_{1},t_{2}$): At $t_{1}$, $S_{2ag}$ is turned on, as a result, its
device output capacitance completely discharges in short duration. $S_{1ag}$
is already on from stage 1, therefore $v_{pq}^{(k)}=0$. $S_{C}$ and $S_{D}$ is
also kept on from stage 1 and thus $v_{rs}^{(k)}=-V_{o}$. Fig. 3b shows the
circuit condition during this stage. The ac side HFT winding current gets
transferred to switch $S_{2ag}$ at a constant slope defined by the total
series inductor $L_{t}$ and $v_{rs}^{(k)}$. The equation of $i_{lk}^{(k)}(t)$
during this stage is given by (9) and (10) gives the value of
$i_{lk}^{(k)}(t)$ at $t_{1}$. AC side leg 1 switch current follows the same
equation as (7) and leg 2 switch current is as per (11). DC side switch
current in stage 2 is same as in stage 1.

$\displaystyle i_{lk}^{(k)}(t)$
$\displaystyle=\frac{V_{o}}{nL_{t}}(t-t_{1})+i_{lk}^{(k)}(t_{1})$ (9)
$\displaystyle i_{lk}^{(k)}(t_{1})$
$\displaystyle=-\frac{1}{2}(i_{g}^{(k)}-\frac{v_{g}^{(k)}}{L_{2}}d_{1}^{(k)}T_{s})$
(10) $\displaystyle i_{S_{2ag}}^{(k)}(t)$ $\displaystyle=-i_{lk}^{(k)}(t)$
(11)

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Equivalent circuits of ICFHB ac-dc converter in a switching interval
at various stages (a) Stage 1 ($t_{0},t_{1}$), (b) Stage 2 ($t_{1},t_{2}$),
(c) Stage 3 ($t_{2},t_{3}$), (d) Stage 4 ($t_{3},t_{4}$), (e) Stage 5
($t_{4},t_{5}$), (f) Stage 6 ($t_{5},t_{6}$).

The duration of this stage is denoted as $\alpha^{(k)}T_{s}$ as shown in Fig.
2a and is given by

$\displaystyle\alpha^{(k)}T_{s}=\frac{nL_{t}|i_{lk}^{(k)}(t_{1})|}{V_{o}}$
(12)

TABLE I: Current Expressions Parameters | SPSM [14] | DCPSM[17] | Proposed IDCPSM
---|---|---|---
$i_{lk,rms}^{(k)}$ |  | $\begin{aligned} \frac{i_{g}^{(k)}}{2}\sqrt{\frac{5}{3}-\frac{4d_{1}^{(k)}}{3}}+i_{lk,pk}^{(k)}\end{aligned}$
---
$\begin{aligned}
\sqrt{\frac{2d_{1}^{(k)}}{3}-\frac{1}{3}+\frac{i_{g}^{(k)}}{i_{lk,pk}^{(k)}}\left(\frac{1}{6}-\frac{d_{1}^{(k)}}{3}+\frac{2\beta^{(k)}}{3}\right)}\end{aligned}$
| $\begin{aligned}
\frac{i_{g}^{(k)}}{\sqrt{2}}\sqrt{1-d_{1}^{(k)}+\frac{\alpha^{(k)}}{3}+\frac{\beta^{(k)}}{3}}\end{aligned}$
---
$\begin{aligned}
+I_{max}\sqrt{\frac{2d_{2}}{3}+\frac{2\beta^{(k)}}{3}+\frac{i_{g}^{(k)}}{I_{max}}\frac{\beta^{(k)}}{3}}\end{aligned}$
| $\begin{aligned}
\frac{i_{g}^{(k)}}{\sqrt{2}}\end{aligned}\sqrt{\begin{aligned}
1&-d_{1}^{(k)}+\frac{4d_{2}^{(k)}}{3}\\\
&+\frac{\alpha^{(k)}}{3}+\frac{7\beta^{(k)}}{3}\end{aligned}}$
---
$i_{S_{xyg,rms}}^{(k)}$ |  | $\begin{aligned} \frac{i_{g}^{(k)}}{2}\sqrt{\frac{5}{3}-\frac{4d_{1}^{(k)}}{3}}+i_{lk,pk}^{(k)}\end{aligned}$
---
$\begin{aligned}
\sqrt{\frac{2d_{1}^{(k)}}{3}-\frac{1}{3}-\frac{\beta^{(k)}}{3}+\frac{i_{g}^{(k)}}{i_{lk,pk}^{(k)}}\left(\frac{\beta^{(k)}}{3}-\frac{d_{1}^{(k)}}{3}-\frac{1}{6}\right)}\end{aligned}$
| $\begin{aligned}
\frac{i_{g}^{(k)}}{2}\sqrt{\frac{11}{3}-\frac{10d_{1}^{(k)}}{3}+\frac{4d_{2}^{(k)}}{3}+\frac{7\alpha^{(k)}}{3}+\frac{5\beta^{(k)}}{3}}\end{aligned}$
---
$\begin{aligned}
+I_{max}\sqrt{\frac{2d_{2}}{3}+\frac{\beta^{(k)}}{3}+\frac{i_{g}^{(k)}}{I_{max}}\frac{2\beta^{(k)}}{3}}\end{aligned}$
| $\begin{aligned} \frac{i_{g}^{(k)}}{2}\end{aligned}\sqrt{\begin{aligned}
\frac{11}{3}&-\frac{10d_{1}^{(k)}}{3}+4d_{2}^{(k)}\\\
&+\frac{7\alpha^{(k)}}{3}+\frac{17\beta^{(k)}}{3}\end{aligned}}$
---
$i_{D_{xyg,avg}}^{(k)}$ |  | $\begin{aligned} \left(i_{lk,pk}^{(k)}-\frac{i_{g}^{(k)}}{2}\right)\frac{\beta^{(k)}}{2}\end{aligned}$
---
| $\begin{aligned}
\left(I_{max}-\frac{i_{g}^{(k)}}{2}\right)\frac{\beta^{(k)}}{2}\end{aligned}$
---
| $\begin{aligned} \frac{i_{g}^{(k)}}{2}\frac{\beta^{(k)}}{2}\end{aligned}$
---
$i_{S_{p,rms}}^{(k)}$ |  | $\begin{aligned} \frac{i_{g}^{(k)}}{2n}\sqrt{\frac{5}{6}-\frac{2d_{1}^{(k)}}{3}}+i_{lk,pk}^{(k)}\end{aligned}$
---
$\begin{aligned}
\sqrt{\frac{d_{1}^{(k)}}{3}-\frac{1}{6}+\frac{i_{g}^{(k)}}{i_{lk,pk}^{(k)}}\left(\frac{1}{12}-\frac{d_{1}^{(k)}}{6}+\frac{\beta^{(k)}}{3}\right)}\end{aligned}$
| $\begin{aligned} \frac{I_{max}}{n}\sqrt{\frac{d_{2}}{3}}\end{aligned}$
---
| $\begin{aligned} \frac{i_{g}^{(k)}}{2n}\end{aligned}\sqrt{\begin{aligned}
1&-d_{1}^{(k)}+\frac{4d_{2}^{(k)}}{3}\\\
&+\frac{\alpha^{(k)}}{3}+\frac{7\beta^{(k)}}{3}\end{aligned}}$
---
$i_{D_{p,avg}}^{(k)}$ | 0 |  | $\begin{aligned} \frac{i_{g}^{(k)}}{2n}\left(1-d_{1}^{(k)}+\frac{\alpha^{(k)}}{2}+\frac{\beta^{(k)}}{2}\right)+\frac{I_{max}}{n}\frac{\beta^{(k)}}{2}\end{aligned}$
---
0

(a)  (b)  (c)  (d)

Figure 4: Variation of switching cycle RMS and average value of current in a
half grid cycle for different modulation scheme at $P_{o}=1.5\>kW$ and
$V_{o}=345\>V$ (a) RMS current through the ac side winding of HFT, (b) RMS
current of ac side switch, (c) RMS current of dc side switch, (d) average
current of ac side body diode.

Stage 3 ($t_{2},t_{3}$): At $t_{2}$, HFT winding current $i_{lk}$ reaches zero
as the line inductor current $i_{L2}$ gets completely transferred to switch
$S_{2ag}$. Since both $S_{1ag}$ and $S_{2ag}$ is on, $v_{pq}^{(k)}=0$. $S_{D}$
is kept on and gate pulse to $S_{C}$ is removed. Since $v_{rs}^{(k)}=-V_{o}$
at $t_{2}$, this voltage is kept applied across the winding of HFT for a short
duration even after $t_{2}$. This rises the HFT winding current to a minimum
value which in turn discharges the output device capacitance of $S_{B}$ and
charges the the output device capacitance of $S_{C}$. The body diode of
$S_{B}$ gets forward biased and the HFT winding current is maintained at that
minimum value as $v_{rs}^{(k)}=0$. This minimum current value is very small
compared to the total load current and hence it is assumed to be zero. Fig. 3c
shows the circuit condition during this stage.

Stage 4 ($t_{3},t_{4}$): Fig. 3d shows the equivalent circuit during this
stage. $S_{1ag}$ and $S_{2ag}$ is kept on and $v_{pq}^{(k)}=0$. At $t_{3}$,
$S_{C}$ is turned on and its output device capacitance discharges and at the
same time output device capacitance of $S_{B}$ charges to $V_{o}$. Thus,
$v_{rs}^{(k)}=-V_{o}$ and is applied across the dc side winding of HFT, rising
the current through it at a constant slope. The current through $S_{1ag}$
falls at the same constant slope and through $S_{2ag}$ rises at the same
constant slope. This duration is kept sufficiently long enough for the current
to go negative through $S_{1ag}$ as shown in Fig. 2c. The negative current
flowing through the loop of ac side switches and the HFT is the circulating
current $i_{cir}^{(k)}$ as shown in Fig. 3e. The equation of $i_{lk}^{(k)}(t)$
during this stage is given by (13). AC side leg 1 switch current and leg 2
switch current is as per (14) and (15) respectively. DC side switch current
equation in stage 4 is same as in stage 1.

$\displaystyle i_{lk}^{(k)}(t)$ $\displaystyle=\frac{V_{o}}{nL_{t}}(t-t_{3})$
(13) $\displaystyle i_{S_{1ag}}^{(k)}(t)$
$\displaystyle=i_{L1}^{(k)}(t)-i_{lk}^{(k)}(t)$ (14) $\displaystyle
i_{S_{2ag}}^{(k)}(t)$ $\displaystyle=i_{L2}^{(k)}(t)+i_{lk}^{(k)}(t)$ (15)

From Fig. 2a it can be noted that the duration of this stage is the secondary
side duty $d_{2}^{(k)}T_{s}$ and is given by

$\displaystyle d_{2}^{(k)}T_{s}=\frac{nL_{t}|i_{lk,pk}^{(k)}|}{V_{o}}$ (16)

It is clear from (16) that the peak value of ac side HFT winding current
$i_{lk,pk}^{(k)}$ is dependent on $d_{2}^{(k)}$. In the existing modulation
scheme, the value of $d_{2}^{(k)}$ was decided for the maximum output power
and was kept constant irrespective of the operating conditions [17]. Thus,
$i_{lk,pk}=I_{max}$ remained constant across the entire grid cycle of ac mains
voltage and in different load conditions. This can be observed from Fig. 2a
that at the peak of ac mains voltage i.e at $\omega\tau=90^{\circ}$, the
winding current reaches $I_{max}$ during $d_{2}T_{s}$. For this fixed value of
$d_{2}^{(k)}$, the peak current value remains constant near the zero crossing
of ac mains voltage i.e at $\omega\tau=10^{\circ}$ as shown by dotted lines
$(t_{4}^{\prime}-t_{3}^{\prime})T_{s}$ in Fig. 2b. Fig. 2c shows the current
through the ac side and dc side switches near the zero crossing. It can be
observed in the figure, as shown by dotted lines, that the peak current
through the switches remains constant across the grid cycle. From (17), it is
evident that for a fixed value of $i_{lk,pk}=I_{max}$, the circulating current
flowing through the converter is substantially high near the zero crossing of
ac mains voltage. This high circulating current attributes to increased
conduction losses in both ac and dc side switches and in the HFT.

$\displaystyle i_{cir}^{(k)}=i_{lk,pk}^{(k)}-\frac{1}{2}i_{g}^{(k)}$ (17)

In the proposed IDCPSM, the ratio between $i_{g}^{(k)}$ and $i_{lk,pk}^{(k)}$
is kept fixed in the entire grid cycle of ac mains voltage i.e
$i_{lk,pk}^{(k)}$ follows the grid current sinusoidal envelope. This is
achieved by varying $d_{2}^{(k)}$ in every switching interval. This can
observed from Fig. 2b that near the zero crossing of ac mains voltage,
$d_{2}^{(k)}$, is adjusted as per the fixed ratio between $i_{g}^{(k)}$ and
$i_{lk,pk}^{(k)}$. This substantially reduces the peak current value through
the HFT winding as shown by bold lines. The merits of the proposed IDCPSM can
also be seen in Fig. 2c. The peak current through the ac and dc side switches
has also reduced substantially as shown by bold lines near the zero crossing
of ac mains voltage. As per (17) the circulating current will also follow the
sinusoidal envelope across the grid cycle. This reduced peak and circulating
currents through the switches and the HFT contributes to lower conduction
losses, significantly improving the efficiency of the converter.

Stage 5 ($t_{4},t_{5}$): At $t_{4}$, $S_{1ag}$ is turned off and the negative
current flowing through it shifts to its body diode. $S_{2ag}$ continues to
stay on from the previous stage therefore $v_{pq}^{(k)}=0$. $S_{C}$ and
$S_{D}$ are turned off and its output parasitic capacitance charges to $V_{o}$
and at the same time output parasitic capacitance of $S_{A}$ and $S_{B}$
discharges and their body diode starts conducting. Thus, $v_{rs}^{(k)}=V_{o}$
and is applied across the dc side winding of HFT and current through it falls
at a constant slope. The equation of $i_{lk}^{(k)}(t)$ during this stage is
given by (18) at $t_{4}$ is given by (19). AC side switch current follows the
same equation as per stage 4. DC side body diode current equation is given by
(20).

$\displaystyle i_{lk}^{(k)}(t)$
$\displaystyle=\frac{-V_{o}}{nL_{t}}(t-t_{4})+i_{lk}^{(k)}(t_{4})$ (18)
$\displaystyle i_{lk}^{(k)}(t_{4})$
$\displaystyle=\frac{V_{o}d_{2}^{(k)}T_{s}}{nL_{t}}=i_{lk,pk}^{(k)}$ (19)
$\displaystyle i_{D_{A}}^{(k)}(t)$
$\displaystyle=i_{D_{B}}^{(k)}(t)=-\frac{i_{lk}^{(k)}(t)}{n}$ (20)

As represented in Fig. 2a, this duration is $\beta T_{s}$ and is given by

$\displaystyle\beta^{(k)}T_{s}=\frac{nL_{t}|i_{lk}^{(k)}(t_{5})-i_{lk}^{(k)}(t_{4})|}{V_{o}}$
(21)

Stage 6 ($t_{5},t_{6}$): At $t_{5}$, the current flowing through the body
diode of $S_{1ag}$ goes to zero, naturally turning off the device and
achieving ZCS. $S_{2ag}$ continues to stay on therefore
$v_{pq}^{(k)}=V_{o}/n$. $S_{A}$ and $S_{B}$ are turned on and the current
through their body diode shifts to channel undergoing ZVS.
$v_{rs}^{(k)}=V_{o}$. The equation of $i_{lk}^{(k)}(t)$ during this stage is
given by (22) and at $t_{5}$ is given by (23). AC side leg 2 switch current
equation in given by (24) and dc side switch current equation is given by
(25).

$\displaystyle i_{lk}^{(k)}(t)$
$\displaystyle=\frac{(v_{g}^{(k)}-\frac{V_{o}}{n})}{L_{1}+L_{t}}(t-t_{5})+i_{lk}^{(k)}(t_{5})$
(22) $\displaystyle i_{lk}^{(k)}(t_{5})$
$\displaystyle=\frac{1}{2}(i_{g}^{(k)}+\frac{v_{g}^{(k)}}{L_{1}}d_{1}^{(k)}T_{s})$
(23) $\displaystyle i_{S_{2ag}}^{(k)}(t)$
$\displaystyle=i_{L2}^{(k)}(t)+i_{lk}^{(k)}(t)$ (24) $\displaystyle
i_{S_{A}}^{(k)}(t)$
$\displaystyle=i_{S_{B}}^{(k)}(t)=-\frac{i_{lk}^{(k)}(t)}{n}$ (25)

Stage 1 to stage 5 describes the converter operation in a half switching
interval. The other half is symmetrical to the above described stages as
evident from stage 6.

Table I presents the expressions of rms current through ac side HFT winding
$i_{lk,rms}^{(k)}$, rms current through ac side switch $i_{S_{xyg,rms}}^{(k)}$
and dc side switch $i_{S_{p,rms}}^{(k)}$, average current through the body
diode of ac side switch $i_{D_{xyg,avg}}^{(k)}$ and dc side switch
$i_{D_{p,avg}}^{(k)}$ in a switching interval $k$ and Fig. 4 shows their
variation in a half grid cycle for SPSM, DCPSM and proposed IDCPSM. Fig. 4a
shows that the ac side HFT winding rms current is maximum in switching
intervals near the zero crossing of ac mains voltage for SPSM and is
substantially low for DCPSM. In the proposed IDCPSM, this value is further
reduced as the peak current through the windings of HFT is significantly
reduced by modulating $d_{2}^{(k)}$ across the grid cycle. Reduction can also
be seen in rms current through ac and dc side switches for the proposed IDCPSM
as depicted in Fig. 4b-4d. This amounts to lower conduction losses with
considerable improvement in the efficiency of the converter.

Figure 5: Laboratory prototype of a L-L type ICFHB ac-dc converter. TABLE II: Parameters of the implemented hardware prototype AC mains voltage | $V_{g}$ | 230 Vrms
---|---|---
AC mains frequency | $f_{g}$ | 50 Hz
Output Voltage | $v_{o}$ | 300 V…400 V
Output Power | $P_{o}$ | 1.5 kW
Switching Frequency | $f_{sw}$ | 100 kHz
Transformer | $HFT$ |  | Transformer ratio: 0.38
---
Prim: 26 turns, 135 wires, #40 SWG
Sec: 10 turns, 260 wires, #40 SWG
Core: Ferrite EE80/20 (2 stacked)
$L_{lk}$ = 600 nH
Boost Inductor |  $L_{1}$, $L_{2}$ |  | 740 $\mu$H
---
82 turns, 135 wires, #40 SWG
Core: Kool M$\mu$ 77620
Series Inductor | $L_{s}$ |  | 7.5 $\mu$H
---
10 turns, 260 wires, #40 SWG
Core: Kool M$\mu$ 77071
Output Capacitor | $C_{o}$ |  | 940 $\mu$F
---
4 parallel leg
Each leg: 2 series connected 470 $\mu$F
(Electrolytic 400 V)
AC side SiC Mosfets | $S_{xyg}$ | Cree C2M0080170P
DC side SiC Mosfets | $S_{p}$ | UnitedSiC UF3C065030K4S

##  IV Experimental Results

A 1.5 kW laboratory prototype was developed and is shown in Fig. 5. The
performance of the converter with proposed modulation scheme is validated at
different loading conditions. Table II lists the parameters of the implemented
prototype. The control scheme is implemented in a TI TMS320F28397D launchpad
and the combinational logic of gate pulses is generated using Xilinx XC6SLX4
based FPGA board.

(a)

(b)

(c)

Figure 6: Measured grid voltage $v_{g}$, grid current $i_{g}$, output voltage
$V_{o}$ and ac side HFT winding current $i_{lk}$ for DCPSM in (a) grid cycle,
(b) zoomed near zero crossing of ac mains voltage.
($P_{o}=1.5\>kW,v_{g}=230\>V_{rms}$ and $V_{o}=345\>V$)

(a)

(b)

(c)

Figure 7: Measured grid voltage $v_{g}$, grid current $i_{g}$, output voltage
$V_{o}$ and ac side HFT winding current $i_{lk}$ for IDCPSM in (a) grid cycle,
(b) zoomed near peak of ac mains voltage, (c) zoomed near zero crossing of ac
mains voltage. ($P_{o}=1.5\>kW,v_{g}=230\>V_{rms}$ and $V_{o}=345\>V$) Figure
8: Zoomed-in measured gate source voltage $v_{gs-S_{1ag}}$, drain source
voltage $v_{ds-S_{1ag}}$ and the ac side HFT voltage $v_{pq}$

###  IV-A Experimental Waveform

Fig. 6a shows the measured ac grid voltage $v_{g}$ along with the input grid
current $i_{g}$, ac side HFT winding current $i_{lk}$ and output voltage
$V_{o}$ for DCPSM [17]. It can be observed that the peak value of the winding
current is constant in the entire grid cycle of ac mains voltage. This is
further depicted in Fig. 6b and 6c showing the zoomed waveform at the peak and
near the zero crossing of the ac mains voltage respectively. The peak value of
the winding current is 10.2 A. This higher circulating current near the zero
crossing of ac mains voltage, attributing to higher conduction losses and
deteriorating efficiency.

Fig. 7a shows the measured ac grid voltage $v_{g}$, grid current $i_{g}$, ac
side HFT winding current $i_{lk}$ and output voltage $V_{o}$ for proposed
IDCPSM. It is clear from the figure that converter is operating at 0.999 power
factor. The total series inductor current now has a sinusoidal envelope with
twice the ac mains voltage frequency. Its peak current value has reduced by
almost $55\%$ from the peak Fig. 7b to near the zero crossing Fig. 7c of the
ac mains voltage.

Fig. 8 shows the zoomed waveform of gate source voltage $v_{gs-S_{1ag}}$ and
drain source voltage $v_{ds-S_{1ag}}$ of ac side switch $S_{1ag}$, along with
the ac side HFT voltage $v_{pq}$. It can be observed that (dotted region) even
though the gate voltage is at -5V, the switch voltage is zero, indicating that
its body diode is conducting as discussed in section II. The switch naturally
turns-off as the current through its body diode goes to zero achieving ZCS.

Figure 9: Efficiency curve for different SSM at $V_{o}=345\>V$

###  IV-B Efficiency

Fig. 9 shows the efficiency curve for different SSM at nominal output voltage
$V_{o}=345\>V$. Power analyser module of Tektronix MDO3104 DSO is used for
measuring power. From the figure it can be concluded that the proposed IDCPSM
has the best performance with peak efficiency of $94.4\%$. Furthermore, the
efficiency curve for IDCPSM remains nearly flat across different loading
conditions. An improvement of around $3\%$ is seen at lighter loading
condition $P_{o}=1\>kW$ and $1.5\%$ at full load $P_{o}=1.5\>kW$ from DCPSM.
This improvement in performance is due to active control of the peak
circulating across the gird cycle of the ac mains voltage and at different
loading conditions. The SPSM is the least efficient among the three SSM. Fig.
10 shows the loss distributions among the components. The sum of switching and
conduction loss is depicted as ac or dc side switch loss. HFT loss shows the
total loss occurred in the HFT and the series inductor. The reduction in peak
value of circulating current with the proposed IDCPSM reduces the conduction
losses in ac side switch as depicted in the Fig. 10. The total loss in the dc
side switch is reduced by almost $42\%$ from DCPSM. This is because both
switching and conduction loss are reduced in the dc side switch. Furthermore,
the total loss in the HFT is reduced by almost $20\%$ with the proposed IDCPSM
as both core loss and copper loss reduces in the HFT and the series inductor.

Figure 10: Loss distribution among the components of L-L type ICFHB ac-dc
converter at $P_{o}=1.5\>kW$and $V_{o}=345\>V$ for different SSM

##  V Conclusion

This paper proposed a IDCPSM for an L-L type ICFHB ac-dc converter. A dual
control variable approach has been developed where in ac side CFHB duty is
modulated to maintain UPF and dc side VFFB duty is modulated to reduce the
peak circulating current in a grid cycle of ac mains voltage and also at light
load conditions. ZCS turn-off of ac side switches and ZVS turn-on of dc side
switches are maintained throughout the operating condition. This has
significantly reduced the conduction and switching losses in both ac and dc
side switches. A comparative performance analysis with different SSM reported
in literature is also presented. The proposed IDCPSM provides the best
performance with peak efficiency of $94.4\%$ and almost a flat efficiency
profile across different loading conditions.

## References

  * [1] A. Khaligh and M. D’Antonio, “Global Trends in High-Power On-Board Chargers for Electric Vehicles,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 4, pp. 3306–3324, Apr. 2019.
  * [2] “IEEE Recommended Practice and Requirements for Harmonic Control in Electric Power Systems,” _IEEE Std 519-2014 Revis. IEEE Std 519-1992_ , pp. 1–29, Jun. 2014.
  * [3] “UL - 2202 Standard for Electric Vehicle (EV) Charging System Equipment — Standards Catalog.”
  * [4] L. Xue, Z. Shen, D. Boroyevich, P. Mattavelli, and D. Diaz, “Dual Active Bridge-Based Battery Charger for Plug-in Hybrid Electric Vehicle With Charging Current Containing Low Frequency Ripple,” _IEEE Trans. Power Electron._ , vol. 30, no. 12, pp. 7299–7307, Dec. 2015.
  * [5] M. Kwon and S. Choi, “An Electrolytic Capacitorless Bidirectional EV Charger for V2G and V2H Applications,” _IEEE Trans. Power Electron._ , vol. 32, no. 9, pp. 6792–6799, Sep. 2017.
  * [6] J. Everts, F. Krismer, J. V. den Keybus, J. Driesen, and J. W. Kolar, “Optimal ZVS Modulation of Single-Phase Single-Stage Bidirectional DAB AC–DC Converters,” _IEEE Trans. Power Electron._ , vol. 29, no. 8, pp. 3954–3970, Aug. 2014.
  * [7] N. D. Weise, G. Castelino, K. Basu, and N. Mohan, “A Single-Stage Dual-Active-Bridge-Based Soft Switched AC–DC Converter With Open-Loop Power Factor Correction and Other Advanced Features,” _IEEE Trans. Power Electron._ , vol. 29, no. 8, pp. 4007–4016, Aug. 2014.
  * [8] F. Jauch and J. Biela, “Combined Phase-Shift and Frequency Modulation of a Dual-Active-Bridge AC–DC Converter With PFC,” _IEEE Trans. Power Electron._ , vol. 31, no. 12, pp. 8387–8397, Dec. 2016.
  * [9] A. Tausif, H. Jung, and S. Choi, “Single-stage isolated electrolytic capacitor-less ev onboard charger with power decoupling,” _CPSS Trans. Power Electron. Appl._ , vol. 4, no. 1, pp. 30–39, Mar. 2019.
  * [10] K. Gnanasambandam, A. K. Rathore, A. Edpuganti, D. Srinivasan, and J. Rodriguez, “Current-Fed Multilevel Converters: An Overview of Circuit Topologies, Modulation Techniques, and Applications,” _IEEE Trans. Power Electron._ , vol. 32, no. 5, pp. 3382–3401, May 2017.
  * [11] F. Wu, S. Fan, X. Li, and S. Luo, “Bidirectional Buck–Boost Current-Fed Isolated DC–DC Converter and Its Modulation,” _IEEE Trans. Power Electron._ , vol. 35, no. 5, pp. 5506–5516, May 2020.
  * [12] X. Pan, H. Li, Y. Liu, T. Zhao, C. Ju, and A. K. Rathore, “An Overview and Comprehensive Comparative Evaluation of Current-Fed-Isolated-Bidirectional DC/DC Converter,” _IEEE Trans. Power Electron._ , vol. 35, no. 3, pp. 2737–2763, Mar. 2020\.
  * [13] U. R. Prasanna, A. K. Singh, and K. Rajashekara, “Novel Bidirectional Single-phase Single-Stage Isolated AC–DC Converter With PFC for Charging of Electric Vehicles,” _IEEE Trans. Transp. Electrification_ , vol. 3, no. 3, pp. 536–544, Sep. 2017.
  * [14] A. K. Rathore and P. U.R, “Analysis, Design, and Experimental Results of Novel Snubberless Bidirectional Naturally Clamped ZCS/ZVS Current-Fed Half-Bridge DC/DC Converter for Fuel Cell Vehicles,” _IEEE Trans. Ind. Electron._ , vol. 60, no. 10, pp. 4482–4491, Oct. 2013.
  * [15] A. K. Rathore, A. K. S. Bhat, and R. Oruganti, “Analysis, Design and Experimental Results of Wide Range ZVS Active-Clamped L-L Type Current-Fed DC/DC Converter for Fuel Cells to Utility Interface,” _IEEE Trans. Ind. Electron._ , vol. 59, no. 1, pp. 473–485, Jan. 2012.
  * [16] C. F. Moraes, E. G. Carati, J. P. da Costa, R. Cardoso, and C. M. d. O. Stein, “Active-Clamped Zero-Current Switching Current-Fed Half-Bridge Converter,” _IEEE Trans. Power Electron._ , vol. 35, no. 7, pp. 7100–7109, Jul. 2020.
  * [17] U. R. Prasanna, A. K. Rathore, and S. K. Mazumder, “Novel Zero-Current-Switching Current-Fed Half-Bridge Isolated DC/DC Converter for Fuel-Cell-Based Applications,” _IEEE Trans. Ind. Appl._ , vol. 49, no. 4, pp. 1658–1668, Jul. 2013.
  * [18] X. Pan, F. Yang, L. Li, R. Zhang, and C. Wang, “An Improved Modulation Scheme of Active Commutated Current-Fed Bidirectional DC/DC Converter,” _IEEE J. Emerg. Sel. Top. Power Electron._ , pp. 1–1, 2020\.
  * [19] S. Bal, D. B. Yelaverthi, A. K. Rathore, and D. Srinivasan, “Improved Modulation Strategy Using Dual Phase Shift Modulation for Active Commutated Current-Fed Dual Active Bridge,” _IEEE Trans. Power Electron._ , vol. 33, no. 9, pp. 7359–7375, Sep. 2018.
  * [20] Y. Zhang, Z. Wang, Y. W. Li, N. Hou, and M. Cheng, “Decoupled Dual-PWM Control for Naturally Commutated Current-Fed Dual-Active-Bridge DC/DC Converter,” _IEEE J. Emerg. Sel. Top. Power Electron._ , pp. 1–1, 2019.