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Feedback-Linearization-Based Nonlinear Controlfor PEM Fuel Cells

Woon Ki Na, Student Member, IEEE, and Bei Gou, Member, IEEE

AbstractThis paper presents a dynamic nonlinear model forpolymer electrolyte membrane fuel cells (PEMFCs). A nonlinearcontroller is designed based on the proposed model to prolong thestack life of the PEM fuel cells. Since it is known that large devi-ations between hydrogen and oxygen partial pressures can causesevere membrane damage in the fuel cell, feedback linearizationis applied to the PEM fuel cell system so that the deviation can bekept as small as possible during disturbances or load variations. Adynamic PEM fuel cell model is proposed as a nonlinear, multiple-input multiple-output system so that feedback linearization canbe directly utilized. During the control design, hydrogen and oxy-gen inlet flow rates are defined as the control variables, and thepressures of hydrogen and oxygen are appropriately defined as the

control objectives. The details of the design of the control schemeare provided in the paper. The proposed dynamic model was testedby comparing the simulation results with the experimental datapreviously published. The simulation results show that PEMFCsequipped with the proposed nonlinear controls have better tran-sient performances than those with linear controls.

Index TermsExact linearization, nonlinear dynamic model,polymer electrolyte membrane fuel cells.

I. INTRODUCTION

A

FUEL cell is an electrochemical energy device that con-

verts the chemical energy of fuel directly into electricity

and heat with water as a by product of the reaction. As a renew-able energy source, the fuel cell is widely regarded as one of the

most promising energy sources because of its high energy effi-

ciency, extremely low emission of oxides of nitrogen and sulfur,

and very low noise, as well as the cleanness of its energy pro-

duction. Based on the currently used types of electrolytes, fuel

cells are divided into polymer electrolyte membrane fuel cells

(PEMFCs), solid oxide fuel cells (SOFCs), phosphoric acid fuel

cells (PAFCs), molten carbonate fuel cells (MCFCs), alkaline

fuel cells (AFCs), direct methanol fuel cells (DMFCs), zinc air

fuel cells (ZAFCs), and photonic ceramic fuel cells (PCFCs) [1].

In order to generate a reliable andefficient powerresponseand

to prevent membrane damage as well as detrimental degradationof the stack voltage and oxygen depletion, it is necessary to de-

sign a better control scheme to achieve optimal air and hydrogen

inlet flow ratesi.e., controls that can perform air and hydrogen

pressure regulation and heat/water management precisely based

on the current drawn from the fuel cell system [3], [4].

Manuscript received March 24, 2007; revised June 4, 2007. Paper no.TEC-00091-2006.

The authors are withthe Energy System Research Center,University of Texas,Arlington, TX 76019 USA (e-mail: bgou@uta.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEC.2007.914160

Fig. 1. Polarization curve (Ballard Mark V PEMFC at 70

C) [1].

First of all, an PEMFC system must be accurately modeled in

order to apply a suitable nonlinear control scheme. Models have

been reported so far in the literature for the PEMFC, ranging

from stationary and dynamic models [3][9], [13][17], [20],

[24] for the control design applied to a fuel cell vehicle and a

distributed generation system [11], [12].

Unfortunately, those models are mainly for experimental ver-

ifications other than control design [1][3] or for prediction of

the fuel cell phenomenon by analyzing an electrochemical re-

action, the thermodynamics, and the fluid mechanics. Recently,Purkrushpan et al. [3] developed a control-oriented PEMFC

model that includes flow characteristics and dynamics of the

compressor and the manifold (anode and cathode), reactant par-

tial pressures, and membrane humidity. However, because of

the nonlinear relationship between stack voltage and load cur-

rent shown in Fig. 1 [1] and the state equations [3], [20], it is

a challenge to develop a nonlinear controller for the PEMFC.

Because of the operational parametric uncertainties such as the

parametric coefficients for each cell on kinetic, thermodynamic,

and electrochemical foundations, and the resistivity of the mem-

brane for the electron flow, the linear PEMFC models proposed

by Purkrushpan and coworkers [3][5] and Chiu et al. [20]

using Jacobian linearization via a Taylor series expansion at

the nominal operating point cannot easily achieve satisfactory

dynamic performance under large disturbances. An accurate

nonlinear dynamic model needs to be developed for the fuel

cell system as well as an advanced controller design technique,

considering the nonlinearity and uncertainty that need to be

proposed.

A fuzzy control system for a boost dc/dc converter of a fuel

cell system was developed in [18]. Neural optimal control was

presented for the PEMFC by using an artificial neural network

(ANN) in [19]. However, instead of controlling the PEM fuel

cell system, the neural optimal control is mainly used to derive a

new architecture to synthesize an approximated optimal control

0885-8969/$25.00 2008 IEEE

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by means of the ANN, where the PEM fuel cell was chosen as

a test bed.

In this paper, feedback linearization, a well-known nonlinear

approach, is applied to design a controller, based directly on

the nonlinear dynamic fuel cell model, to achieve more robust

transient behavior. Furthermore, the fuel cell stack life can be

prolonged and stack systems can be protected by minimizing thedeviations between the hydrogen and oxygen partial pressures

[2], [4].

In the last few years, feedback linearization for nonlinear dy-

namic models has been widely used [26][29], [31]. Feedback

linearization uses a nonlinear transformation to transform an

original nonlinear dynamic model into a linear model by diffeo-

morphism mapping [26][29], [31]. An optimal control theory

is also applied to obtain a linear control that is transformed back

to the original space by using the nonlinear mapping.

In this paper, the nonlinear dynamic model developed in

[3][5] and the small signal model of an PEMFC in [20] are

considered together to obtain a new dynamic nonlinear model

that is appropriate for developing a nonlinear controller. Theproposed controller, which is expected to perform rapid tran-

sient responses under load variations, is tested in MATLAB

simulink environment.

The paper is organized as follows. Section II gives a brief

introduction to feedback linearization. The PEMFC dynamic

model based on [3][5] and [20] is proposed in Section III, and

Section IV addresses the design of a nonlinear controller for

an PEMFC. Section V provides the simulation results for the

proposed controller, and Section VI concludes the paper.

II. NONLINEAR CONTROL BY FEEDBACK LINEARIZATION

For decades, significant progress has been made in control

designs based on nonlinear concepts. In particular, nonlinear

control theory developed from differential geometry, known as

exact linearization or feedback linearization, has become more

and more attractive for chemical process control because many

chemical processes are basically of high nonlinearity [29], [34].

Hence, one of the main motivations of utilizing feedback lin-

earization for a fuel cell system is that the operation of PEMFC

is inherently a nonlinear chemical process. In this section, feed-

back linearization of nonlinear systems is briefly introduced.

More details of nonlinear control based on differential geome-

try are available in [26][28], [34].

A. Feedback or Exact Linearization

Consider a single-input single-output (SISO) nonlinear sys-

tem described by the state equation

x = f(x) + g(x)u

y = h(x) (1)

where x is an n-dimensional state vector that is assumed to be

measurable, u is a scalar input, and y is a scalar output.

The objective of feedback linearization is to create a linear

differential relation between the output y and a newly defined

input v. The notation and concepts of differential geometry are

essential to understand this approach.

The Lie derivative of a scalar function h(x) with respect to a

vector function f(x) is defined as

Lfh(x) = hf =h(x)

xf(x). (2)

Repeated Lie derivatives can be defined recursively as

Lkfh(x) = Lf

Lk1f h

=

Lk1f h

f

L0fh(x) = h(x) (3)

fork = 1, 2, . . .Similarly, in case of another vector field g

Lg Lfh = (Lfh)g. (4)

The output needs to be differentiated for r times until it is

directly related to the input u. The numberris called the relative

degree of the system.The system is said to have a relative rat a point x0 if1) Lg L

kfh(x) = 0 for all xin the neighborhood ofx0 and for

k < r 1;2) Lg L

r1f h(x0 ) = 0.

Thus, according to the aforementioned condition, with a de-

fined relative degree, the r time derivatives of y are described

as

y(k ) = Lkf(x)h(x), fork = 0, 1, . . . , r 1,

y(r ) = Lrf(x)h(x) + Lg Lr1f h(x)u. (5)

The control law is

u =1

Lg Lr1f h(x)

Lrfh(x) + v

(6)

where y(r ) = v. This control law can transform the nonlinearsystem into a linear one. In addition, a nonlinear transformation

of a coordinate in the state space

z = (x) (7)

is called a local diffeomorphism, in which the map between

the new input v and the output is exactly linear for all x in the

neighborhood ofx0 .This feedback linearization theory can be used to design

multiple-input multiple-output (MIMO) nonlinear system:

x = f(x) +m

i=1

gi (x)ui

yi = h(x), i = 1, 2, . . . , m (8)

where x is an n-dimensional state vector and u and y are m-

dimensional input and output vectors. The system is said to

have a vector relative degree {r1 , r2 , . . . , rm } at a point x0 if1) Lgj L

kfhi (x) = 0 for all 1 j m, all 1 i m, k