The advantageous use of fractional calculus (FC) in the modeling and control of many dynamical systems has been recognized. In this paper, we study the control of a heat diffusion system based on the application of the FC concepts. Several algorithms are investigated and compared, when integrated within a Smith predictor control structure. Simulations are presented assessing the performance of the proposed fractional algorithms.

Issue Section:
Research Papers
Keywords:
calculus, diffusion, heat systems, predictive control
Topics:
Algorithms, Closed loop systems, Control equipment, Optimization, Thermal diffusion, Simulation, Heat, Diffusion (Physics), Dynamic systems
1.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974,
The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order
,
University of Michigan Press
,
New York
.
2.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
University of Michigan Press
,
San Diego, CA
.
3.
Battaglia
,
J. L.
,
Cois
,
O.
,
Puigsegur
,
L.
, and
Oustaloup
,
A.
, 2001, “
Solving an Inverse Heat Conduction Problem Using a Non-Integer Identified Model
,”
Int. J. Heat Mass Transfer
0017-9310,
44
, pp.
2671
–
2680
.
Crossref
Search ADS
 
4.
Vinagre
,
B. M.
,
Chen
,
Y.
, and
Petráš
,
I.
, 2003, “
Two Direct Tustin Discretization Methods for Fractional-Order Differentiator/Integrator
,”
J. Franklin Inst.
0016-0032,
340
(
5
), pp.
349
–
362
.
Crossref
Search ADS
 
5.
Jesus
,
I. S.
,
Barbosa
,
R. S.
,
Machado
,
J. A. T.
, and
Cunha
,
J. B.
, 2006, “
Strategies for the Control of Heat Diffusion Systems Based on Fractional Calculus
,”
IEEE International Conference on Computational Cybernetics
.
6.
Courant
,
R.
, and
Hilbert
,
D.
, 1962,
Methods of Mathematical Physics, Partial Differential Equations
,
Interscience Publishers
,
New York
.
7.
Zhuang
,
M.
, and
Atherton
,
D. P.
, 1993, “
Automatic Tuning of Optimum PID Controllers
,”
IEE Proc.-D: Control Theory Appl.
0143-7054,
140
(
3
), pp.
216
–
224
.
Crossref
Search ADS
 
8.
Podlubny
,
I.
, 1999, “
Fractional-Order Systems and piλdμ-Controllers
,”
IEEE Trans. Autom. Control
0018-9286,
44
(
1
), pp.
208
–
214
.
Crossref
Search ADS
 
9.
Petrás
,
I.
, and
Vinagre
,
B. M.
, 2002, “
Practical Application of Digital Fractional-Order Controller to Temperature Control
,”
Acta Montanistica Slovaca
1335-1788,
7
(
2
), pp.
131
–
137
.
10.
Barbosa
,
R. S.
,
Machado
,
J. A. T.
, and
Ferreira
,
I. M.
, 2004, “
Tuning of PID Controllers Based on Bode’s Ideal Transfer Function
,”
Nonlinear Dyn.
0924-090X,
38
(
1–4
), pp.
305
–
321
.
11.
Chen
,
Y.
,
Vinagre
,
B. M.
, and
Podlubny
,
I.
, 2004, “
Continued Fraction Expansion to Discretize Fractional Order Derivatives—An Expository Review
,”
Nonlinear Dyn.
0924-090X,
38
(
1–4
), pp.
155
–
170
.
12.
Barbosa
,
R. S.
,
Machado
,
J. A. T.
, and
Silva
,
M. F.
, 2006, “
Time Domain Design of Fractional Differintegrators Using Least-Squares
,”
Signal Process.
0165-1684,
86
(
10
), pp.
2567
–
2581
.
Crossref
Search ADS
 
13.
Chen
,
Y.
, and
Moore
,
K. L.
, 2005, “
Relay Feedback Tuning of Robust PID Controllers With Iso-Damping Property
,”
IEEE Trans. Syst., Man, Cybern., Part B: Cybern.
1083-4419,
35
(
1
), pp.
23
–
31
.
Crossref
Search ADS
 
14.
Machado
,
J. T.
,
Jesus
,
I.
,
Cunha
,
J. B.
, and
Tar
,
J. K.
, 2006, Fractional Dynamics and Control of Distributed Parameter Systems, No. 2 in Intelligent Systems at the Service of Mankind.
15.
Crank
,
J.
, 1956,
The Mathematics of Diffusion
,
Oxford University Press
,
London
.
16.
Gerald
,
C. F.
, and
Wheatley
,
P. O.
, 1999,
Applied Numerical Analysis
,
Addison-Wesley
,
Reading, MA
.
17.
Farlow
,
S. J.
, 1993,
Partial Differential Equations for Scientists and Engineers
,
Wiley
,
New York
.
18.
Jesus
,
I. S.
, and
Machado
,
J. A. T.
, 2007, “
Application of Fractional Calculus in the Control of Heat Systems
,”
Journal of Advanced Computational Intelligence and Intelligent Informatics
1343-0130,
11
(
9
), pp.
1086
–
1091
.
Crossref
Search ADS
 
19.
Jesus
,
I. S.
,
Machado
,
J. T.
, and
Barbosa
,
R. S.
, 2008,
On the Fractional Order Control of Heat Systems
,
Intelligent Engineering Systems and Computational Cybernetics
,
The Netherlands
.
20.
Jesus
,
I. S.
, and
Machado
,
J. A. T.
, 2008, “
Fractional Control of Heat Diffusion Systems
,”
Nonlinear Dyn.
0924-090X,
54
(
3
), pp.
263
–
282
.
Crossref
Search ADS
 
21.
Chen
,
Y.
, 2006, “
Ubiquitous Fractional Order Controls?
,”
The Second IFAC Symposium on Fractional Derivatives and Applications
.
22.
Smith
,
O. J. M.
, 1957, “
Closed Control of Loops With Dead Time
,”
Chem. Eng. Process.
0255-2701,
53
, pp.
217
–
219
.
23.
Jesus
,
I. S.
, and
Machado
,
J. T.
, 2009, “
Smith predictor embedded with fractional algorithms for the control of a heat diffusion system
,”
ASME-IDETC/CIE2009—International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
.
24.
Graham
,
D.
, and
Lathrop
,
R. C.
, 1953, “
The Synthesis of Optimum Response: Criteria and Standard Forms—Part 2
,”
Trans. AIEE
0731-5996,
72
, pp.
273
–
288
.
25.
Astrom
,
K. J.
, 1996,
The Control Handbook
,
IEEE
,
New York
.
26.
Feng
,
W.
, and
Li
,
Y.
, 1999, “
Performance Indices in Evolutionary CACSD Automation With Application to Batch PID Generation
,”
IEEE International Symposium on Computer Aided Control System Design
.
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