Solving nonlinear time delay control systems using Fourier series

Mohammad Hadi Farahi Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 5), June 2014, pp.217-226
www.ijera.com 217 | P a g e
Solving Nonlinear Time Delay Control Systems by Fourier series
Mohammad Hadi Farahi1
and Mahmood Dadkhah 2
1
Department of Applied Mathematics, Ferdowsi university of Mashhad, Mashhad, Iran.
2
Department of Mathematics, Payame Noor University, Tehran, Iran,
Abstract: In this paper we present a method to find the solution of time-delay optimal control systems using
Fourier series. The method is based upon expanding various time functions in the system as their truncated
Fourier series. Operational matrices of integration and delay are presented and are utilized to reduce the
solution of time-delay control systems to the solution of algebraic equations. Illustrative examples are included
to demonstrate the validity and applicability of the technique.
Keywords: Fourier series, Time delay system, Operational matrix, Nonlinear systems.
I. Introduction
The control of systems with time delay has been of considerable concern. Delays occur frequently in
biological, chemical, transportation, electronic, communication, manufacturing and power systems [5]. Time-
delay and multi-delay control systems are therefore very important classes of systems whose control and
optimization have been of interest to many investigators [2-6]. Orthogonal functions (OFs) and polynomial series
have received considerable attentions in dealing with various problems of dynamic systems. Much progress has
been made towards the solution of delay systems. The approach is that of converting the delay-differential
equation govering the dynamical systems to an algebraic form through the use of an operational matrix of
integration . The matrix can be uniquely determined based on the particular OFs. Special attentions has been
given to applications of Walsh functions [3], block-pulse functions[12], Laguerre polynomials [6], Legendre
polynomials [7], Chebyshev polynomials[4] and Fourier series [9]. The available sets of OFs can be divided into
three classes. The first includes a set of piecewise constant basis functions (PCBFs) (e.g. Walsh, block-pulse,
etc.). The second consists of a set of orthogonal polynomials (OPs) (e.g. Laguerre, Legendre, Chebyshev, etc).
The third is the widely used set of sine-cosine functions (SCFs) in the form of Fourier series. In this paper we use
Fourier series method to solve time delay control systems. The method consists of reducing the delay problem to
a set of algebraic equations by first expanding the candidate function as a Fourier series with unknown
coefficients. These Fourier series are first introduced. The operational matrices of integration, delay and product
are given. These matrices are then used to evaluate the coefficients of the Fourier series for the solution of time
delay control systems.
II. Fourier series and their properties:
2.1. Expansion by Fourier series:
A function )(tf belongs to the apace ][0,2
LL may be expanded by Fourier series as follows[11]:
},
22
{=)( *
1=
0
L
tn
sina
L
tn
cosaatf nn
n

 

(1)
where
,)(
1
=
0
0 dttf
L
a
L

,1,2,3,=,
2
)(
2
=
0
ndt
L
tn
costf
L
a
L
n


.1,2,3,=,
2
)(
2
=
0
*
ndt
L
tn
sintf
L
a
L
n


By truncating the series (1) up to 1)(2 r th term we can abtain an approximation for )(tf as follows:
RESEARCH ARTICLE OPEN ACCESS

)(=)}()({)( **
1=
0 tAtataatf T
nnnn
r
n
   (2)
where
,],,,,,,,,[= **
2
*
1210
T
rr aaaaaaaA 
,)](,),(),(),(,),(),(),([=)( **
2
*
1210
T
rr tttttttt   (3)
and
,,0,1,2,=,
2
=)( rn
L
tn
costn 


.,1,2,=,
2
=)(*
rn
L
tn
sintn 


It can be easily seen that the elements of )(t in interval )(0,L are orthogonal.
2.2. Operational matrices of integration, product and delay:
The integration of )(t in (3) can be approximated by )(t as follows:
)()(
0
tPdss
t
 
where P is the operational matrix of integration of order 1)(21)(2  rr and is given by[8,10]:
.
0000
2
1
000
2
1
000000
4
1
0
4
1
0000000
2
1
2
1
2
1
00000000
0
1)2(
1
0000000
000
2
1
00000
1
1)(
1
2
11
0000
2
1
=

































































rr
r
r
rr
LP
Furthermore we have:
,=)()(
2**
1
**
1
*
0
*
**
1
2*
1
*
11
*
10
*
1
**
1
2
10
*
1
*
111
2
101
*
0
*
10010
2
0






















rrrrrr
rr
rrrrrr
rr
rr
T
tt













(4)

one can esealy show that )(
~
=)()( tAAtt T
 where A
~
is called the product operational matrix for the so
called vector A in (3) and is given:
.
2
1
2
1
0
2
1
2
1
2
1
2
1
2
1
)(
2
1
2
1
2
1
)(
2
1
2
1
2
1
)(
2
1
2
1
2
1
)(
2
1
2
1
2
1
0
2
1
2
1
2
1
2
1
2
1
2
1
2
1
)(
2
1
2
1
2
1
)(
2
1
2
1
2
1
)(
2
1
2
1
2
1
)(
2
1
2
1
2
1
=
~
1)(21)(2
021
*
2
*
1
*
24031
*
2
*
4
*
1
*
3
*
2
13120
*
1
*
1
*
3
*
2
*
1
*
2
*
1021
*
2
*
4
*
1
*
3240312
*
1
*
3
*
1
*
2131201
**
2
*
1210


















































rr
rrrrr
rr
rr
rrrrr
rr
rr
rr
aaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaaaaa
aaaaaaaaaa
aaaaaaaaaa
aaaaaaa
A









By integrating of (4) and considering the orthogonality components of )(t we have:
,)()(=
0
dtttE T
L
 (5)
where
.
1/2
0
1/2
01/2
1
=
1)(21)(2 
















rr
LE

(6)
Now we are going to derive operational delay matrix. From calculus we know that:
,=)(  sincoscossinsin 
,=)(  sinsincoscoscos 
so if  is any delay(or lag) for the functions )(t and )(*
t then we have:
),()
2
()()
2
(=)( *
t
L
tn
sint
L
tn
cost nnn 



 
).()
2
()()
2
(=)( **
t
L
tn
sint
L
tn
cost nnn 



 
Now it is esealy to show that ),(=)( tDt   where D is delay operational matrix and have the
following form:

.
)
2
(00)
2
(000
0)
4
(00)
4
(00
00)
2
(00)
2
(0
)
2
(00)
2
(000
0)
4
(00)
4
(00
00)
2
(00)
2
(0
0000001
=









































L
r
cos
L
r
sin
L
cos
L
sin
L
cos
L
sin
L
r
sin
L
r
cos
L
sin
L
cos
L
sin
L
cos
D
















III.Problem statement
Consider the following quadratic time-independent delay control system:
dttRututQxtxtSxtxJ TTft
ff
T
)}()()()({
2
1
)()(
2
1
=min
0
  (7)
,0),()()()(=)(. LttDutCutBxtAxtxts   (8)
,=(0) 0xx (9)
0,<),(=)( tttx  (10)
0,<),(=)( tttu  (11)
(For the time being, assume that the matrices DCBA ,,, are constant but the result can be extended to time
varying systems by appropriate changes) where R is symmetric positive definite and SQ, are positive
semidefinite matrices,
l
Rtx )( ,
q
Rtu )( are state and control vectores respectively and DCBA ,,, are
matrices of appropriate dimensions, 0x is a constant specified vector, and )(),( tt  are arbitrary known
functions. The problem is to find )(tx and )(tu , Lt 0 , satisfying (8)-(11) while minimizing (7).
Assume that
,],,,,,,,,[=),(=)( **
2
*
1210
T
rr
T
xxxxxxxXwheretXtx 
,],,,,,,,,[=),(=)( **
2
*
1210
T
rr
T
uuuuuuuUwheretUtu 
.,0],0,0,(0),0,0,[=),(=(0) 00
TT
xXwheretXx 
Due to (10), in 0 t we have )(=)( ttx  so for  t0 and consequently for 0  t it
is true to have ),(=)(=)( tGttx T
  where
T
G is the Fourier series coefficient of )(  t . Now
we have






LttDXtX
ttGt
tx TT
T



 ),(=)(
0),(=)(
=)( (12)
and by the same approach with respect to (11)






LttDUtU
ttHt
tu TT
T



 ),(=)(
0),(=)(
=)( (13)
The integration of (8) from 0 to t and using of (9) gives

,)()()()(=)(
00000
dssuDdssuCdssxBdssxAdssx
ttttt
    (14)
or equivalently
dssxBdssxBdssxAxtx
tt
)()()(=(0))(
00



 
.)()()(
00
dssuDdssuDdssuC
tt



  (15)
Now, from (12) and (13) we have:
dssDXdssDXdssDXdssx TT
t
T
tt
)()(=)(=)(
00
 

  
)()(= tZDXtPDX TT
 
dssDUdssDUdssDUdssu TT
t
T
tt
)()(=)(=)(
00
 

  
)()(= tZDUtPDU TT
  
)(=)(
0
tZdtt 

 (16)
where
.
0000))
2
((1
2
0000))
2
((1
2
0000)
2
(
2
0000)
2
(
2
0000
=





































L
r
cos
r
L
L
cos
L
L
r
sin
r
L
L
sin
L
Z









(17)
Thus (15) reduces to
)()()()(=)()( 0 tZDBXtPDBXtZBGtPAXtXtX TTTTTT
  
).()()()( tZDDUtPDDUtZDHtPCU TTTT
   (18)
By deleting )(t from both sides of (18) and ordering we conclude that:
ZDBXPDBXZBGPAXXXC TTTTTT
  0
*
=
0=ZDDUPDDUZDHPCU TTTT
  (19)
By substituting the Fourier series in ( J ) we have
dttURUttXQXttXSXtJ TTTTft
f
T
f
T
)}()()()({
2
1
)()(
2
1
=
0
   (20)
The optimal control problem has now been reduced to a parameter optimization problem which can be stated as
follows. Find X and U so that ),( UXJ is minimized subject to the constraints in Eq. (19).
Let
**
),(=),,( CUXJUXJ T
  (21)
where the vector  represents the unknown Lagrange multipliers, then the necessary conditions for stationary
are given by
0=),,(*
UXJ
X


0=),,(*
UXJ
U

0=),,(*


UXJ


(22)
Remark1. Note that if delays in state and control vectors aren’t the same, then solving the system is the same as
previous but we have two or more delay operational matrices.
IV. Illustrative Examples:
In this section two examples are given to demonstrate the applicability, efficiency and accuracy of our
proposed method.
4.1. Example 1:
Consider the following system [13]
dttutxJ )}()({
2
1
=min 22
2
0
 (23)
2,0),(1)(=)(.  ttutxtxts  (24)
0.11,=)(  ttx
Here, we solve the same problem by using of Fourier series. Note that in third example delay is applied on state
only, and 1= . Suppose that
),(=(0)),(=)(),(=)( 0 tXxtUtutXtx TTT

where 0),(,, XtUX TT
 are defined previously. If we integrate (24) from 0 to t and use (12)-(13)we have
dssudssxdssx
ttt
)(1)(=)(
000  
,)(1)(1)(=
01
1
0
dssudssxdssx
tt
 
and we have
,1)(1)(=1)(
1
001
dssxdssxdssx
tt
 
so we obtain
).()()()(=)()( 0 tPUtZDXtPDXtZGtXtX TTTTTT
  
By deleting )(t from both sides and ordering we conclude that:
0== 0
*
PUZDXPDXZGXXC TTTTTT
 
By substitution the Fourier series in (23) for J we have
dtUttUXttXJ TTTT
})()()()({=
1
0
 
,= EUUEXX TT

where E is defined in (6). Thus we have redused the system as follows
EUUEXXJ TT
=min
0==. 0
*
PUZDXPDXZGXXCtoS TTTTTT
 
Approximate solutions of )(),( tutx with 25=r are shown in Fig.1. The value of J is 1.62421313.

Figure 1: State vector x(t) and control u(t) for r=25 in Example 1
4.2. Example 2:
Consider the following system [8,13]
dttutxJ )}(
2
1
)({
2
1
=min 22
1
0
 (25)
1,02/3),(
2
1
)(1/3)()(=)(.  ttututxtxtxts  (26)
0,11,=)(  ttx
0.<2/30,=)( ttu 
Here we have different delays in state( 1/3=1 ) and control( 2/3=2 ). Suppose that
),(=(0)),(=)(),(=)( 0 tXxtUtutXtx TTT

 are defined previously. If we integrate (26) from 0 to t and use (12)-(13) we have
dssudssudssxdssxdssx
ttttt
)
3
2
(
2
1
)()
3
1
()(=)(
00000
  
dssudssxdssxdssxxtx
ttt
)()
3
1
()
3
1
()(=(0))(
0
3
1
3
1
00  
,)
3
2
(
2
1
)
3
2
(
2
1
3
2
3
2
0
dssudssu
t
 
so we obtain
)()()()(=)()( 111100 tZDXtPDXtZXtPXtXtX TTTTTT
  
)(
2
1
)(
2
1
)( 222
tZDUtPDUtPU TTT
  
By deleting )(t from both sides and ordering we conclude that:
111100
*
= ZDXPDXZXPXXXC TTTTT
 
0,=
2
1
2
1
222
ZDUPDUPU TTT
 
By substitution the Fourier series in (23) for J we have

dtUttUXttXJ TTTT
})()(
2
1
)()({
2
1
=
1
0
 
),
2
1
(
2
1
= EUUEXX TT

where E was defined in (6). now we have redused system as follows
)
2
1
(
2
1
=min EUUEXXJ TT

111100
*
=. ZDXPDXZXPXXXCts TTTTT
 
0=
2
1
2
1
222
ZDUPDUPU TTT
 
Approximate solutions of )(),( tutx with 25=r are shown in Fig.2. the value of J is 0.35944042 (In [3],
the value of J is 0.3731).
4.3. Example 3:
Consider the following system[1,3]
dttutxJ )}()({=min 22
3
0
 (27)
2),(1)(=)(.  tutxtxts  (28)
0,11,=)(  ttx
0,20,=)(  ttu
Here, the delays are 1=1 for state and 2=2 for control vectors, respectively. Suppose that
),(=(0)),(=)(),(=)( 0 tXxtUtutXtx TTT

 are defined previously. With respect to (12) and (13) we have







31),(=1)(
10),(=)(=1)(
=1)(
1
0
ttDXtX
ttXtGt
tx TT
T










32),(=2)(
200,=2)(
=2)(
2
ttDUtU
tt
tu TT



If we integrate (28) from 0 to t then
dssusxdssx
tt
2)(1)(=)(
00
 
dssusxdssusxdssusx
t
2)(1)(2)(1)(2)(1)(=
2
2
1
1
0
 
By substitution the Fourier series and simplifying, we obtain

,)]()([)0]([)0]([=)()(
2121
2
1
0
1
0
0 dssDUsDXdssDXdssXtXtX TT
t
TTTT
   
since we have 200,=2)(  ttu . Now we have
dssDUsDXtXtX TT
t
TT
)]()([=)()(
212
0  
dssDUsDXdssDUsDX TTTT
t
)]()([)]()([=
21
2
0210
   
dsUDssDXdsUDssDX TTTTT
t
T
])()([])()([=
2
2
01201     (29)
Assume that 12
= UUDT
 , thus (29) reduces to
0=
~~
= 11110
*
ZUDXPUDXXXC TTTT
  (30)
By substitution the Fourier series for J we have
dtUttUXttXJ TTTT
})()()()({=
1
0
  ,= EUUEXX TT
 (31)
where E is defined in (6). So the delay optimal control (27-28) now is reduced to the following optimazation
problem
EUUEXXJ TT
=min
0=
~~
=. 11110
*
ZUDXPUDXXXCts TTTT
 
Approximate solutions of )(),( tutx with 30=r are shown in Fig.3. the value of J is 2.3496 while the
exact value is
1
13
2
2


e
e
.
V. Conclusions
Using Fourier series, a simple and computational method for solving time delay optimal problems is
considered. The method is based upon reducing a nonlinear time delay optimal control problem to an nonlinear
programming problem. The unity of the function of orthogonality for Fourier series and the simplicity of
applying delays in Fouries series are great merits that make the approach very attractive and easy to use.
Although the method is simple, by solving various examples, accuracy in comparison of the other methods can
be found.

References
[1] A.J. Koshkouei , M.H. Farahi , K.J. Burnham., An almost optimal control design method for nonlinear
time-delay systems, International Journal of Control,Vol. 85, No. 2, (2012), 147-158
[2] Chen, C. K. and Yang, C. Y., Analysis and parameter identification of time-delay systems via
polynomial series, International Journal of Control 46 ( 1987), 111-127.
[3] Chen, W. L. and Shih, Y. P., Shift Walsh matrix and delay differential equations, IEEE Transactions on
Automatic Control 23(1978), 265-280.
[4] Horng, I. R. and Chou, J. H., Analysis, parameter estimation and optimal control of time-delay systems
via Chebyshev series, International Journal of Control 441(1985), 1221-1234.
[5] Jamshidi, M. and Wang, C. M., A computational algorithm for large-scale nonlinear time-delays
systems, IEEE Transactions on Systems, Man, and Cybernetics 14(1984), 2-9.
[6] Kung, F. C. and Lee, H., Solution and parameter estimation of linear time-invariant delay systems
using Laguerre polynomial expansion, Journal on Dynamic Systems, Measurement, and Control 105
(1983), 297-301.
[7] Lee, H. and Kung, F. C., Shifted Legendre series solution and parameter estimation of linear delayed
systems, International Journal of Systems Science 16 (1985), 1249-1256.
[8] Marzban, H., Razzaghi, M., Optimal Control of Linear Delay Systems via Hybrid of Block-Pulse and
Legendre Polynomials, Journal of the Franklin Institute, 341 (2004), 279-293
[9] Mouroutsos, S. G. and Sparis, P. D., Shift and product Fourier matrices and linear-delay differential
equations, International Journal of Systems Science 17 (1986), 1335-1348.
[10] Razzaghi, M. and Razzaghi, M., Taylor series analysis of time-varying multi-delay systems,
International Journal of Control 50 (1988), 183-192.
[11] Richard A. Bernatz. Fourier series and numerical methods for partial defferential equations. John
Wiley & Sons, Inc., New York, 2010.
[12] Shih, Y. P., Hwang, C., and Chia, W. K., Parameter estimation of delay systems via block pulse
functions, Journal on Dynamic Systems, Measurement, and Control 102 (1980), 159-162.
[13] Wang, X., T, Numerical solutions of optimal control for time delay systems by hybrid of block-pulse
functions and Legendre polynomials Applied Mathematics and Computation bf 184 (2007), 849-856.

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