Discretizing of linear systems with time-delay Using method of Euler’s and Tustin’s approximations

Bemri H’mida et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 3, ( Part -2) March 2015, pp.83-89
www.ijera.com 83 | P a g e
Discretizing of linear systems with time-delay Using method of
Euler’s and Tustin’s approximations
Bemri H’mida1
, Mezlini Sahbi2
and Soudani Dhaou3
Tunis El Manar University
Laboratory of Research on Automatic (LARA)
National Engineers School of Tunis
BP 37, Belvedere 1002 Tunis. Tunisia.
ABSTRACT
Delays deteriorate the control performance and could destabilize the overall system in the theory of discrete-
time signals and dynamic systems. Whenever a computer is used in measurement, signal processing or control
applications, the data as seen from the computer and systems involved are naturally discrete-time because a
computer executes program code at discrete points of time. Theory of discrete-time dynamic signals and systems
is useful in design and analysis of control systems, signal filters, state estimators and model estimation from
time-series of process data system identification. In this paper, a new approximated discretization method and
digital design for control systems with delays is proposed. System is transformed to a discrete-time model with
time delays. To implement the digital modeling, we used the z-transfer functions matrix which is a useful model
type of discrete-time systems, being analogous to the Laplace-transform for continuous-time systems. The most
important use of the z-transform is for defining z-transfer functions matrix is employed to obtain an extended
discrete-time. The proposed method can closely approximate the step response of the original continuous time-
delayed control system by choosing various of energy loss level. Illustrative example is simulated to demonstrate
the effectiveness of the developed method.
Keywords - Discretization, Time-delay systems, Z-transform.
I. INTRODUCTION
Sampling a continuous-time system is a
fundamental problem in a variety of scientific
areas, such as computer control, system
identification and signal processing. It is
becoming even more conspicuous in light of the
huge success of computer-aided processing and
networking [9, 12]. There are many intriguing
problems related to sampling. Time delay is one
of the key factors influencing the overall system
stability and performance. In particular, as the
different effects of actuator, sensor and
controller exist in control systems, delays are
often formulated as state time delays, input
time delays as well as output time delays in a
continuous-time or discrete-time framework [8,
14]. To digitally simulate and design a
continuous-time delayed control system, it is
often required to obtain an equivalent discrete-
time model. The digital modeling of
continuous-time systems with input delays can
be found in a standard textbook .To improve the
performance of a continuous-time system with
multiple time delays, several advanced control
theories and practical design techniques have
been proposed [7,10]. Most control systems are
formulated in a continuous-time framework, for
which many analysis tools and control
methodologies are well-established. With the
rapid advances in digital technology and
computers, digital control provides various
advantages over its analog counterpart for better
reliability, lower cost, smaller size, more
flexibility and better performance. The resulting
digitally controlled continuous time system
becomes a sampled-data system.
Systems including time delay, due to the
system dynamics, are widely present in the
industry which imposes a lot of constraints that
make the control and computer programming of
such system difficult [1].Then the control of a
system with a time delay is generally difficult;
due to the constraints imposed by the time
delay. These constraints can cause performance
deterioration that leads the process to instability
especially when operating in closed loop [2, 3,
4, 11].Most physical systems, a macroscopic
point of view, are continuous. In modern
control systems, information is digitally
processed which requires sampling signals [3,
RESEARCH ARTICLE OPEN ACCESS

5]. One speaks in this case of sampled or
discrete systems. If necessary, it is always
possible to obtain the state equations from the
transfer-function matrix [13]. Another
advantage of using the transfer-function matrix
is that one can decompose the multivariable
system into p subsystems, each with one output
and m inputs. For this reason we need the
discretization of continuous time-delay systems.
The objective of this paper is to extend and
analyze the ideas in the just cited references; we
consider three methods for obtaining the
discrete-time delay approximation. These are:
(i) Backward difference method.
(ii) Forward difference method.
(iii) Bilinear z-transformation method.
The paper is organized as follows: The next
section discusses discretization of systems with
external point delay; Section 3 provides
numerical examples; Section 4 includes a
comparison between the three methods
followed by a conclusion in the end of the
paper.
II. DISCRETIZATION OF SYSTEMS WITH
EXTERNAL POINT DELAY
In a digital computer, time cannot flow
continuously as it is perceived in the physical
world. The time is defined on a discrete set of
times, which are separated by a regular time
interval known by one sampling period. It is
therefore necessary to define new mathematical
tools adapted to discrete time, to represent the
sampled signals and systems and to adapt tools
and methods for automatic analog continuous
time in the design of digital controllers.
Then our problem may be stated as the
determination of a discrete time approximation
corresponding to the following state-space
equations:
    


0 1
.
( ) ( ) ( ) ( )
( ) ( )
t A x t A x t h Bu t
y t Cx t
x (1)
Where:
0
( )A x t : Original Term state.
1
( )A x t h : Delayed Term state.
With h qT : a multiple delay the sampling
period is an integer q .
T : The sampling period assumed chosen
suitably.
,x u andy Respectively are the state vector, the
vector and the control vector output.
0 1
, ,A A BandC are matrices of suitable
dimensions.
Calculating the Laplace transform of the
system (1), we get the following equation
model:

    

 
0 1
( ) X(0) A ( ) ( ) B ( )
( ) ( ) ; X(0) 0
hp
pX p X p A e X p U p
Y p CX p
(2)
This is equivalent to this equation:

    0 1
( ) ( )hp
X p pI A A e BU p (3)
This equation can be written as:


    
1
0 1
( ) ( )hp
X p pI A A e BU p (4)
Then:


     
1
0 1
( ) ( ) ( )hp
Y p CX p C pI A A e BU p (5)
From (5) we can get:


     
1
0 1
( )
( )
( )
hpY p
H p C pI A A e B
U p
(6)
Implementation of continuous-time control
and filtering functions in a computer program,
in some cases we need to find a discrete-time z-
transfer function matrix from a given
continuous-time p -transfer function. In
accurate model based design of a discrete
controller for a process originally in the form of
a continuous-time p -transfer function,
( )H p .The latter should be discredited to get a
discrete-time process model before the design is
started. There are several methods for
discretization of a p -transfer function. The
discretization can be realized in several ways by
calculating the z-transfer function matrix for
( )H p .The methods can be categorized as
follows, and they are described in the following
section:
A. Forward difference method
This method is also called Forward
rectangular rule. Here we apply the technique of
the first order approximation by linear
transformation. These approximations on the z-
transform function matrix exploit the

relationship: pT
z e .The idea is to approximate
this relationship by a linear relationship
between z and p .
When the sampling period T is small, the
linear approximation of the first order function
exponential gives: 1pT
z e pT   , so we can
write:
1
1
1 1z z
p
T Tz


 
  (7)
This technique of discretization results from
the approximation of the derivative of two
sampling instants:
 1 1( ) ( ) ( ) 1
( ) ( )
dx t x t T x t z
L pX p z X z
dt T T
    
    
 
(8)
It can be shown that the discrete-time
transfer function matrix of the transfer function
matrix given in (6) is obtained by using the
Forward difference method after substituting
for
1z
p
T

 :

  
  
 
   
    
   
1
1
0 1
1
( )
z
h
Tz
H z C I A A e B
T
(9)
From (7) we can write:

  
  
 
   
    
   
1
1
0 1
1
( )
z
h
Tz
H z C I A A e B
T
(10)
By applying an approximation Taylor
expansion near 0 of the term
  
  
 
1z
h
T
e to the first
order approximation, we can write:
  
  
  
 
1
1
1
z
h
T z
e h
T
(11)
The expression of ( )H z becomes:

  
      
1
1 1
1 0 1 1( ) ( )H z CTz I A h I A T A T A h z B (12)
B. Backward difference method
This method is also called backward
rectangular rule. To simulate the entire system
as a discrete-time system one needs to find the
discrete equivalent. Here we apply the
technique of the first order approximation. This
Backward discretization results from
approximation (8) of the derivative that can be
made between two sampling times:
 1 1( ) ( ) ( ) 1
( ) ( )
dx t x t x t T z
L pX p z X z
dt T zT
   
 
 
  
   (13)
Then, we can write:
  1
1 1z z
p
zT T

 
  (14)
backward difference method after substituting
for
 1z
p
zT

 :
 
  
 


  
  
   

  
1
1
0 1
1
( )
z
h
zTz
H z C I A A e B
zT
(15)
  
  
 
1z
h
zT
e to the first
  
  
  
 
1
1
1
z
h
zT z
e h
zT
(16)


  
       
   
1
11 1
0 1
( )
A h A hI I
H z C A A z B
T T T T
(17)
C. Bilinear z-transformation method
This method is also called Tustin’s method
also the trapezoid rule in digital control
community, there is a good discrete-time
approximation for a continuous-time linear
system is obtained through the bilinear z-
transformation if the sampling interval T is
selected suitably so that 0.5wT  , where w is
the magnitude of the pole of the transfer
function of the continuous-time system far from
the origin of the P-plane [6]. However, while
the bilinear z-transformation leads to a
realization in the form of a transfer-function
matrix, the method based on the trapezoidal rule
leads directly to a state space realization, more
suitable for digital simulation.
We apply a second-order approximation by
bilinear transformation homographic: Tustin
discretization. The linear approximation of the
second order function exponential gives:
1
2
1
2
T
P
z
T
P



(18)
Therefore:
1
1
2 1 2 1
1 1
z z
p
T z T z


 
 
 
(19)
This approximation results from the
approximate numerical integration by the

trapezoidal rule, between two sampling times,
indeed, either:
1
( ) ( ) ( ) ( )y t x t dt Y p X p
p
   (20)
From where:
(p)
( )
X
p
Y p
 (21)
By approximating between two seconds
sampling, we get:
1
1( )
2
k k
k k
X X
y y kT y T


   (22)
The z-transform function gives then:
   1 1
1 ( ) 1 ( )
2
T
z Y z z X z 
   (23)
From where:
( ) 2 1
( ) 1
X z z
Y z T z



(24)
Then, we can write:
1
1
2 1 2 1
1 1
z z
p
T z T z


 
 
 
(25)
bilinear transformation method after
substituting for
1
1
2 1
1
z
p
T z





:



 


 
   
  
1
1
1
2 11
1
0 11
2 1
( )
1
h z
T z
z
H z C I A A e B
T z
(26)





1
1
2 1
1
h z
T z
e to the first


 



 

1
1
2 1 1
1
1
2 1
1
1
h z
T z
z
e h
T z
(27)

           
 
1
1 1( ) (1 ) (I )
1 0 1 1 0 12 2 2 2 2
T T T T T
H z C z I A h A A A h A A z B (28)
III. NUMERICAL EXAMPLES
In this section an example of application of
the proposed approach are presented. This
example shows the validity of those
methodologies. As the system is considered by
this equation:
 
0 1 0 0 0
x( ) ( ) ( 0.2) ( )
2 3 1 2 1
( ) 1 1 ( )
t x t x t u t
y t x t
   
  

      
      
     


(29)
In this section we are treating an example of
a continuous linear system. A delay is only in
the state which was chosen a sampling period
T=0.2s and applying the Forward difference
method, the following discrete time-delay
system is obtained:

  
      
1
1 1
1 0 1 1( ) ( )H z CTz I A h I A T A T A h z B (30)
With:
 
0 1
0 1 0 0
; ;
2 3 1 2
0
; 1 1 0.2
1
A A
B C and h qT s
   
    
     
 
    
 
The operator 1
z is here a time-step delay
operator, and it can be regarded as an operator
of the time-step delay, 1
z is also the transfer
function of a time-step delay.
The simulation result in the step
responses of the continuous system and
approximate discrete system is depicted in
figure1.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Amplitude
Step response of the continuous and discrete time delay system
Continuous
forward Discretization
Fig. 1. Step response of the continuous and discrete
time delay system.
This method is not recommended because
the discrete-time equivalent may become
unstable; which is commonly used in
developing simple simulators, higher distortion
with the forward rule.
Applying the method based on Backward
difference method to the system considered in
the state-space equations (29), and keeping the
same sampling period T=0.2s. The following
discrete time-delay system is obtained:


  
       
   
1
11 1
0 1
( )
A h A hI I
H z C A A z B
T T T T
(31)

With:
 
0 1
0 1 0 0
; ;
2 3 1 2
0
; 1 1 0.2
1
A A
B C and h qT s
   
    
     
 
    
 
The simulation result in the step responses
of the continuous system and approximate
discrete system is depicted in figure2.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Amplitude
Continuous
backward Discretization
Fig. 2. Step response of the continuous and discrete
time delay system.
The forward rectangular rule could cause a
stable continuous filter to be mapped into an
unstable digital filter, contrary to the rule of
backward rectangular rule that keeps the system
stable conditions except that the answer discrete
time not following the continuous time.
Applying the method based on Bilinear z-
transformation method to the system considered
in the state-space equations (29), and keeping
the same sampling period T=0.2s.The following
discrete time-delay system is obtained:

 
        
 
  
1
1 1
( ) (1 ) (I )
1 0 1 1 0 12 2 2 2 2
T T T T T
H z C z I A h A A A h A A z B (32)
With:
 
0 1
0 1 0 0
; ;
2 3 1 2
0
; 1 1 0.2
1
A A
B C and h qT s
   
    
     
 
    
 
The simulation result in the step responses of
the continuous system and approximate discrete
system is depicted in figure3.
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
1.2
Time (sec)
Amplitude
Continuous
Tustin Discretization
Fig. 3. Step response of the continuous and
discrete time delay system.
It is clear that the value of the delay is shown
by the simulation by the three discretization
methods knowing that it did not affect the
evolution of the system. The evolution of the
response the field below presents a delay that
does not degrade the performance of the
system.
The bilinear transformation is motivated by
considering the trapezoidal approximation of an
integrator. It allows the passage of a Laplace
transform in a z-transform. It uses the trapezoid
method to calculate integral. This is a first-order
approximation of the natural logarithm function
that is an exact mapping of the z-plane to the P-
plane. When the Laplace transform is
performed on a discrete-time signal with each
element of the discrete-time sequence attached
to a correspondingly delayed unit impulse. The
trapezoid rule maps the stable region in the P-
plane exactly into the stable region of the z-
plane. This method also offers the most
accurate phase relative to the continuous
system. It is clear that both digital controllers
designed by emulation perform slightly worse
than the continuous controller. As expected
however, the trapezoidal integration method
performs better than the forward and the
backward rectangular method.
IV. COMPARISON BETWEEN EULER AND
TUSTIN’S APPROXIMATIONS
We have discussed three different methods
for obtaining discrete time approximations for
continuous-time delay systems. Note how the
discrete equivalent model predicts the
continuous output at the sample instances. The
facts that the second and third graph lines
generate the same result show that the formula

we have derived to calculate the discrete
equivalent transfer function model is correct,
Euler’s backward differentiation method, which
is commonly used in discretizing simple signal
filters and industrial controllers. The backward
differentiation method may give problems since
it results in an implicit equation for the output
variable, while the Forward differentiation
method always gives an explicit equation. The
Forward differentiation method is somewhat
less accurate than the backward differentiation
method, but it is simpler to use. In signal
processing one is interested in making the
difference between the original and the
reconstructed signal as small as possible.
For continuous systems, the Laplace
transform has played a major role. In fact, this
linear transformation allows algebraically treat
the linear operators, in the case of sampled
systems; it has to introduce a transformation,
called z, which has similar properties. The z-
transfer function matrix will substitute for the
p-transfer function matrix end of the Laplace
transform. We can see very clearly the
importance of the sampling period; a lower
sampling period provides monitoring of the
response of faithful continuous system with a
longer period. The consequence of this is that
the system with the lowest sampling period will
be the one that will reach stability soon. So for
the most accurate results possible relative to the
continuous time it will be beneficial to not take
too much sampling period. For better
monitoring of the output in continuous time
then choose a sampling period that will not be
too great: but keep in mind that too little time
require great precision the calculator. In signal
processing literature the Tustin’s method is
frequently denoted the bilinear transformation
method. The term bilinear is related to the fact
that the imaginary axis in the complex P-plane
for continuous-time systems is mapped or
transformed onto the unity circle for the
corresponding discrete-time system. In addition,
the poles are transformed so that the stability
property is preserved; this is not guaranteed in
the Euler’s Forward and Backward methods.
Tustin’s method is the most accurate of these
three methods, so I suggest it is the default
choice. However, typically it is not much
difference between the Tustin’s method and the
Euler’s backward method.
The Euler’s forward method is the least
accurate method. Tustin’s method and Euler’s
backward methods are implicit methods since
the output variable, appears on both the left and
the right side of the discretized expression, and
it is in general necessary to solve the discretized
expression for at each time-step, and this may
be impractical.
V. CONCLUSION
In this paper we have proposed three methods
for the discrete time approximation of
continuous-time delay systems, using an
integral numerical approximation, typically
Euler’s forward method, Euler’s backward
method or Tustin’s method. These methods
should be used when a continuous-time
controller transfer function or filter transfer
function are discretized to get an algorithm
ready for programming. In such cases the input
signal is a discrete-time signal with no holding.
The discretization is based on some numerical
method so that the behaviour of the discrete-
time transfer function is similar to that of the
continuous-time transfer function.
Usually we try to choose the sampling
period an integer sub multiple of delay in
addition to its forced choice for dynamic
systems without delay. Finally it appears that
the model using the Tustin’s method is best
suited for numerical simulation in the case of
the continuous system is defined by a state
model and in the case of a suitable choice of the
sampling period. In a next job I am interested to
study the discretization of these types of
systems by the same methods taking into
consideration the delay this time is manifold
which appears in the state and control.
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Authors’ information
Bemri H'mida was born in Tunisia, in
January1979.He received the scientific
degree in mathematics, physics and
computer science within the Faculty of
Sciences of Tunis (FST) in 2003.He received
the degree of MA within the Faculty of
Sciences of Tunis (FST) in 2005.He received
the certificate of Aptitude Teaching of
Second Degree (CAPES) in 2008.He has been working as a
teacher in 2 Mars1934 secondary school in Siliana since 2008.
He received the degree of Master of Research in automatic and
signals processing (ATS) within the National Engineering
School of Tunis (ENIT) in 2013. His research interests include
modeling and control systems sampled to delays.
Mezlini Sahbi was born in Tunisia, in
March 1972.He received the scientific
degree in mathematics and physics
preparatory cycle of the National School
of Engineers of Gabes (ENIG) in 2000.He
received the engineering degree in
electrical engineering in the National
School of Engineers of Monastir (ENIM)
in 2003. He received the degree of Master of research in
automatic and signal processing (ATS) within the National
Engineering School of Tunis (ENIT) in 2005. His research
interests include the command by internal model of discrete
system.
Dhaou Soudani was born in Tunisia, in July 1954. He received the Master
degree in Electrical and Electronic Engineering and the “Diplôme des
Etudes Approfondies” in Automatic Control from the “Ecole Normale
Supérieure de l’Enseignement Technique” Tunisia in 1982 and 1984,
respectively. He obtained both the Doctorat in 1997, and the “Habilitation
Universitaire” in 2007 in Electrical Engineering from the “Ecole Nationale
d’Ingénieurs de Tunis” (ENIT) Tunisia. He is currently a professor in
Automatic Control and a member of the unit research “Laboratoire de
Recherche en Automatique” (LA.R.A).

Discretizing of linear systems with time-delay Using method of Euler’s and Tustin’s approximations

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