FOMCON: a MATLAB Toolbox for Fractional-order System Identification and Control Aleksei Tepljakov, Eduard Petlenkov, and Juri Belikov Abstract—FOMCON is a new fractional-order modeling and control toolbox for MATLAB. It offers a set of tools for re- searchers in the field of fractional-order control. In this paper, we present an overview of the toolbox, motivation for its development and relation to other toolboxes devoted to fractional calculus. We discuss all of the major modules of the FOMCON toolbox as well as relevant mathematical concepts. Three modules are presented. The main module is used for fractional-order system analysis. The identification module allows identifying a fractional system from either time or frequency domain data. The control module focuses on fractional-order PID controller design, tuning and optimization, but also has basic support for design of fractional lead-lag compensators and TID controllers. Finally, a Simulink blockset is presented. It allows more sophisticated modeling tasks to be carried out. Index Terms—fractional calculus, matlab toolbox, automatic control, pid controller, identification, control system design I. I NTRODUCTION In recent years fractional-order calculus has gained a lot of attention, especially in the field of system theory and control systems design due to more accurate modeling and control enhancement possibilities [1], [2]. Several tools have been developed for fractional order system analysis, modeling and controller synthesis. Among these tools are MATLAB toolboxes CRONE [3], developed by the CRONE team, and NINTEGER [4], developed by Duarte Valério. The FOMCON toolbox for MATLAB [5] is an extension to the mini toolbox introduced in [6], [7], [8], providing graphical user interfaces (GUIs), convenience functions, means of model identification in both time and frequency domains and fractional PID controller design and optimization and a Simulink block set. The goal of the toolbox is to provide an easy-to-use, convenient and useful toolset for a wide range of users. It is especially suitable for beginners in fractional order control because of the availability of GUIs, encompassing nearly every toolbox feature, applied workflow considerations and the ability to get practical results quickly. This work was supported by the Estonian Doctoral School in Information and Communication Technology under interdisciplinary project FOMCON, the Governmental funding project no. SF0140113As08 and the Estonian Science Foundation Grant no. 8738. A. Tepljakov is with Department of Computer Control, Tallinn Univer- sity of Technology Ehitajate tee 5, 19086, Tallinn, Estonia (e-mail: alek- [email protected]) E. Petlenkov is with Department of Computer Control, Tallinn Univer- sity of Technology Ehitajate tee 5, 19086, Tallinn, Estonia (e-mail: ed- [email protected]) J. Belikov is with Institute of Cybernetics, Tallinn University of Tech- nology, Akadeemia tee 21, 12618, Tallinn, Estonia, e-mail: (e-mail: jbe- [email protected]) In this paper we present an overview of the FOMCON toolbox and its functions with a summary of used theoretical aspects as well as illustrative examples. The paper is organized as follows. In Section II the reader is introduced to some basic concepts of fractional-order calculus used in control. In Section III an overview of FOMCON toolbox and its features is presented. In Section IV the main module and main GUI facility used for fractional-order system analysis are introduced. Then, the fractional-order identification toolset is presented and discussed in Section V. An overview of the fractional controllers follows in Section VI with particular focus on the PI λ D μ control design and optimization. Section VII is devoted to an overview of the provided Simulink blockset which can be used for more sophisticated fractional- order system modeling. In Section VIII some of the current limitations of the toolbox are outlined. Finally, in Section IX conclusions are drawn. II. AN I NTRODUCTION TO FRACTIONAL CALCULUS Fractional calculus is a generalization of integration and differentiation to non-integer order operator a D α t , where a and t denote the limits of the operation and α denotes the fractional order such that a D α t = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ d α dt α (α) > 0, 1 (α)=0, t a (dt) −α (α) < 0, (1) where generally it is assumed, that α ∈ R, but it may also be a complex number [7]. There exist multiple definitions of the fractional differintegral. The Riemann-Liouville differintegral is a commonly used definition [8] a D α t f (t)= 1 Γ(m − α) d d t m t a f (τ ) (t − τ ) α−m+1 dτ (2) for m − 1 < α < m, m ∈ N, where Γ(·) is Euler’s gamma function. Consider also the Grünwald-Letnikov definition a D α t f (t) = lim h→0 1 h α [ t−a h ] j=0 (−1) j α j f (t − jh) , (3) where [·] denotes the integer part. The Laplace transform of an α-th derivative with α ∈ R + of a signal x(t) relaxed at t =0 (assuming zero initial conditions) is given by L D α x (t) = s α X (s) . (4)
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FOMCON: a MATLAB Toolbox forFractional-order System Identification and Control
Aleksei Tepljakov, Eduard Petlenkov, and Juri Belikov
Abstract—FOMCON is a new fractional-order modeling andcontrol toolbox for MATLAB. It offers a set of tools for re-searchers in the field of fractional-order control. In this paper, wepresent an overview of the toolbox, motivation for its developmentand relation to other toolboxes devoted to fractional calculus. Wediscuss all of the major modules of the FOMCON toolbox as wellas relevant mathematical concepts. Three modules are presented.The main module is used for fractional-order system analysis.The identification module allows identifying a fractional systemfrom either time or frequency domain data. The control modulefocuses on fractional-order PID controller design, tuning andoptimization, but also has basic support for design of fractionallead-lag compensators and TID controllers. Finally, a Simulinkblockset is presented. It allows more sophisticated modeling tasksto be carried out.
Index Terms—fractional calculus, matlab toolbox, automaticcontrol, pid controller, identification, control system design
I. INTRODUCTION
In recent years fractional-order calculus has gained a lot
of attention, especially in the field of system theory and
control systems design due to more accurate modeling and
control enhancement possibilities [1], [2]. Several tools have
been developed for fractional order system analysis, modeling
and controller synthesis. Among these tools are MATLABtoolboxes CRONE [3], developed by the CRONE team, and
NINTEGER [4], developed by Duarte Valério.
The FOMCON toolbox for MATLAB [5] is an extension
to the mini toolbox introduced in [6], [7], [8], providing
graphical user interfaces (GUIs), convenience functions, means
of model identification in both time and frequency domains
and fractional PID controller design and optimization and a
Simulink block set. The goal of the toolbox is to provide an
easy-to-use, convenient and useful toolset for a wide range of
users. It is especially suitable for beginners in fractional order
control because of the availability of GUIs, encompassing
nearly every toolbox feature, applied workflow considerations
and the ability to get practical results quickly.
This work was supported by the Estonian Doctoral School in Informationand Communication Technology under interdisciplinary project FOMCON,the Governmental funding project no. SF0140113As08 and the EstonianScience Foundation Grant no. 8738.
A. Tepljakov is with Department of Computer Control, Tallinn Univer-sity of Technology Ehitajate tee 5, 19086, Tallinn, Estonia (e-mail: [email protected])
E. Petlenkov is with Department of Computer Control, Tallinn Univer-sity of Technology Ehitajate tee 5, 19086, Tallinn, Estonia (e-mail: [email protected])
J. Belikov is with Institute of Cybernetics, Tallinn University of Tech-nology, Akadeemia tee 21, 12618, Tallinn, Estonia, e-mail: (e-mail: [email protected])
In this paper we present an overview of the FOMCON
toolbox and its functions with a summary of used theoretical
aspects as well as illustrative examples. The paper is organized
as follows. In Section II the reader is introduced to some
basic concepts of fractional-order calculus used in control.
In Section III an overview of FOMCON toolbox and its
features is presented. In Section IV the main module and
main GUI facility used for fractional-order system analysis are
introduced. Then, the fractional-order identification toolset is
presented and discussed in Section V. An overview of the
fractional controllers follows in Section VI with particular
focus on the PIλDμ control design and optimization. Section
VII is devoted to an overview of the provided Simulink
blockset which can be used for more sophisticated fractional-
order system modeling. In Section VIII some of the current
limitations of the toolbox are outlined. Finally, in Section IX
conclusions are drawn.
II. AN INTRODUCTION TO FRACTIONAL CALCULUS
Fractional calculus is a generalization of integration and
differentiation to non-integer order operator aDαt , where a and
t denote the limits of the operation and α denotes the fractional
order such that
aDαt =
⎧⎪⎪⎨⎪⎪⎩dα
dtα �(α) > 0,1 �(α) = 0,∫ ta(dt)−α �(α) < 0,
(1)
where generally it is assumed, that α ∈ R, but it may also be
a complex number [7]. There exist multiple definitions of the
fractional differintegral. The Riemann-Liouville differintegral
is a commonly used definition [8]
aDαt f (t) =
1
Γ (m− α)
(d
d t
)mt∫
a
f (τ)
(t− τ)α−m+1 dτ (2)
for m− 1 < α < m, m ∈ N, where Γ(·) is Euler’s gamma
function. Consider also the Grünwald-Letnikov definition
aDαt f (t) = lim
h→01
hα
[ t−ah ]∑j=0
(−1)j(α
j
)f (t− jh) , (3)
where [·] denotes the integer part.
The Laplace transform of an α-th derivative with α ∈ R+ of
a signal x(t) relaxed at t = 0 (assuming zero initial conditions)
Let us build the corresponding model in Simulink, using
the above blockset. The resulting model is given in Fig. 19.
A saturation block is added, limiting the control signal within
an interval Ulim = [−100; 100] and adding a band-limited
white noise block for simulating disturbance in the system
with power of P = 10−9, sample time of T = 0.01 and seed
value of 23341. System simulation result is given in Fig. 20.
VIII. DISCUSSION
The FOMCON toolbox was developed and tested in MAT-
LAB v. 7.7. However, most of the features are backwards-
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.5
0
0.5
1
1.5
Pro
cess
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−50
0
50
Time [s]
Con
trol s
igna
l
Fig. 20. Fractional-order control system simulation result
compatible and were tested with earlier releases of MAT-
LAB (versions 7.4-7.6). FOMCON requires System Control
toolbox for general functionality and Optimization toolbox for
model identification in the time domain.
Further, we discuss some of the current limitations of the
FOMCON toolbox.
• The PID optimization tool lacks complete control over
control system gain and phase margins. The algorithm
can only guarantee that the minimum given specifications
are met by evaluating the open-loop control system
at every optimization step when the Strict option is
checked. However, the initial fractional PID parameters
should strictly satisfy the minimum specifications or else
optimization will not be carried out and an error will be
issued.
• More design specifications settings are required for PID
tuning, including minimum and maximum allowed value
settings for the control effort.
• Both the identification and optimization tools work with
numbers at a fixed accuracy of four decimal places.
• Time domain identification tool does not yet identify the
system lag parameter.
• There are no automatic tuning algorithms implemented
for the fractional lead-lag compensator and TID con-
troller.
• While the order of the fractional derivative block in
Simulink can have an order α > 1, the accuracy of the
simulation will be reduced with higher orders.
The current limitations of the FOMCON toolbox will be
the subject of further development and will be gradually
eliminated in future releases.
IX. CONCLUSIONS
In this paper, we presented a MATLAB toolbox containing
the necessary tools to work with a class of fractional-order
models in control. Theoretical aspects behind the tools were
also covered with illustrative examples. A set of graphical
user interfaces was introduced with relevant comments. We
have discussed fractional-order system analysis, identification
in both time and frequency domains and a set of fractional-
order controllers, focusing on tuning and optimization of the
fractional PID controller. The performance of the latter was
found to be superior to an integer-order PID obtained during
the same tuning procedure. A Simulink blockset was also
presented in the paper.
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Aleksei Tepljakov was born in 1987 in Tallinn. He received his B.Sc and M.Scin computer and systems engineering from Tallinn University of Technology.He is currently working at the Department of Comupter Control at TallinnUniversity of Technology. His main research interests include fractional-ordercontrol of complex systems and fractional filter based signal processing.
Eduard Petlenkov Eduard Petlenkov was born in 1979. He recieved hisB.Sc, M.Sc and PhD degrees in computer and systems engineering fromTallinn University of Technology. He is an Associate Professor in theDepartment of Computer Control at Tallinn University of Technology. Hismain research interests lie in the domain of nonlinear control, system analysisand computational intelligence.
Juri Belikov was born in 1985. He recieved his B.Sc degree in mathematicsfrom Tallinn University, and his M.Sc in computer and systems engineeringfrom Tallinn University of Technology. He joined the the Institute of Cybernet-ics and Department of Computer Control at Tallinn University of Technology.His main research interests lie in the domain of nonlinear control theory.