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Engineering, Technology & Applied Science Research Vol. 10, No. 2, 2020, 5487-5491 5487 
 

www.etasr.com Amara et al.: Sliding Mode Observer and Event Triggering Mechanism Co-design 

 

Sliding Mode Observer and Event Triggering 

Mechanism Co-design 
 

Houssem Eddine Amara  

Electrical Engineering Department 

University Ferhat Abbas Setif 1 
Setif, Algeria 

houssem_am@hotmail.fr 

Mohamed Amine Sid 

Electrical Engineering Department 

University Ferhat Abbas Setif 1 
Setif, Algeria 

amineatweb@yahoo.fr 

Samia Latreche 

Electrical Engineering Department 

University Ferhat Abbas Setif 1 
Setif, Algeria  

ksamia2002@yahoo.fr 

Mabrouk Khemliche 

Electrical Engineering Department 

University Ferhat Abbas Setif 1 
Setif, Algeria  

mabroukkhemliche@univ-setif.dz 
 

 

Abstract—Sliding mode observer and event triggering mechanism 

co-design is the subject of this paper. The synthesis of a sliding 

mode observer under any event triggering mechanism is given, 

where the proposed observer can be viewed as a special structure 

of the Kalman filter. Then, the event triggering condition is 

updated according to the coverage index value computed with the 

help of the reachability set representation. A numerical example 
illustrates the effectiveness of the proposed approach. 

Keywords-sliding mode observer; event-based scheduling; co-

design; estimation performance; communication design 

optimization 

I. INTRODUCTION  

Many estimator design methods have been developed under 
the assumption that the measurement vector is periodically 
available. However, sensors may not feed-back the estimator 
periodically, due to several reasons such as limited network 
bandwidth and high energy consumption of the radio unit for 
wireless sensors. In this case, the classic periodic sampling can 
be replaced by an event-based technique, which insures 
acceptable estimation performance with limited network 
resources. Achieving a desired balance between network state 
estimator quality and sensor to estimator communication rate 
has led to the use of robust observer methods and in this 
context sliding mode observer is a very popular tool. The 
sliding mode observers are proposed with the idea to drive the 
dynamics of a system to an sliding manifold, that is an integral 
manifold with finite reaching time [1]. The first sliding mode 
has been discussed in [2-4]. In addition, some SMO have 
attractive properties similar to those of the Kalman filter (i.e. 
noise resilience) [5], but with a simpler implementation [6]. 
Sometimes this design can be performed by applying an 
equivalent control method [7, 8], allowing the proposal of 
robust to noise observers, since the equivalent control is 
slightly affected by noisy measurements. A key feature in the 
Utkin observer [9] is the introduction of a switching function in 

the observer to achieve a sliding mode and also stable error 
dynamics. For better utilization of shared communication and 
processing resources or reduction of hardware costs we deal 
with event-based designs. The area of event-based control and 
estimation has substantially grown during the last decades [10-
12]. In [13 ],  an event-triggered approach was used to trigger 
the data transmission from a sensor to a remote observer. In 
[14], a sensor data scheduling problem was considered and a 
feedback policy to choose the transmission times which 
provides a trade-off between the communication rate and the 
estimation error was used. The objective in [15] was the 
development of predictive triggering mechanisms for event-
based state estimation. 

The main contribution of this paper is the introduction of an 
event-based technique and a communication design 
optimization with a corresponding estimation by adopting a 
sliding mode from which we obtain high estimation quality 
with low energy consumption by minimizing the number of 
sensors. 

II. PROBLEM FORMULATION 

The event-based sliding mode state estimation process is 
illustrated in Figure 1.  

 

 
Fig. 1.  Event-based sliding mode state estimation 

Consider initially a nominal linear system as described in 
(1): 

Corresponding author: Mabrouk Khemliche



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�x� � = Ax� + Bu�
� = Cx�													    (1) 
where: x ∈ ���,
 ∈ ��� ,u ∈ ��� , and the matrices A, B, C 
have the appropriate dimensions. Without loss of generality, we 
assume that C ∈ ℛ��×�� has full row rank which means each of 
the measured outputs is independent.  

Let ℰ� ≜ 
� − 
�� , where ��  denotes the previous time 
instant when the measurement of sensor is transmitted defined 
as follows: 

�� =	���
�	��	γ� = 0!	��	γ� = 1		 , $%&		�'

� 	= 0    (2) 
where ��� denotes the left hand limit of �( at the instant t. The 
main advantage of using the send-on-delta conditions is that 
feedback communication from the estimator is not needed as 
the conditions only depend on the sensor measurements. This 
fact reduces hardware and energy consumption cost and 
facilitates the implantation of event-triggering schemes. The 
objective is to obtain an estimate of the state x� based only on 
the knowledge of the quantities y��and u�. In the simplest form 
of the sliding mode observer, instead of feeding back the output 
error between the observer and the system in a linear fashion, 
the output error is fed back via a discontinuous switched signal. 
Furthermore, we consider the following EB sensor data 
scheduler: 

γ� =	�0,							if‖ϵ�‖∞ ≤ ζ,1,						otherwise     (3) 
where ζ ≤ ∞ is a predefined threshold and ϵ� the output error. 

The objective of this article is to find or approximate an 
optimal state estimate of EB sensor data scheduler (3), which 
we call nonlinear EB state estimation or EB state estimation 
problem interchangeably. In the next section, a sliding mode 
framework of the EB state estimation problem is presented. 

III. EVENT BASED SLIDING MODE OBSERVER 

A. Similarity Transformation 

Consider the transformation x = T7�8x8 associated with the 
invertible matrix: 

T7 = 9N;<C =    (4) 
where N7 ∈ ℛ��×(	�����)  is a full rank column matrix 
composed with the null space basis of C. By substitution we 
obtain: 

@x�8(t) =	AAx8(t) +	BAu(t)
(t) = CBx8(t)     (5) 
where: 

AA = T7AT7�8,BA =	T7B, and CB =	CT7�8 = C0						IEF (6) 
An observer for (1) was proposed as follows [9]: 

@xG� � = AxG� + Bu� + G�v
Jt = CxG�     (7) 

where (xG,
J) are the estimates of x, 
 and v is a discontinuous 
injection term. e� =	xG� − x�	and	eNO(t) =	
Jt − 
�� are defined 
as the state estimation and output estimation errors 
respectively. The term v is defined component-wise as: 

vP = ρsignSeT
,PU, i = 1,… . ,p    (8) 
where ρ is a positive scalar and eTO,P represents the ith component 
of eT
. The term v is designed to be discontinuous with respect 
to the sliding surface S =	Ze:Ce = 0\ to force the trajectories 
of e(t) into S  in finite time. Assuming, without loss of 
generality, that the system is already in the coordinate 
associated with (6), then the gain G�  has the following 
structure: 

G� = 9 L−IE=    (9) 
where L ∈ ℛ(��E)×E represents the design freedom, following 
the definition of e(t) and (1). The error system is given by: 

é(t) = Ae(t) +	G�v    (10) 
From the structure of the output distribution matrix C in (6) 

the state estimation error can be partitioned as e = col	(e8, e
), 
where e8 ∈ ℛ��E. Consequently the error system from (10) can 
be written in the form: 

é8(t) =	A`8e8(t) +	A8`e
(t) + Lv    (11) 
é
(t) =	A`8e8(t) +	A8`e
(t) − v    (12) 

Furthermore, (12) can be written component-wise as: 

é
,P(t) =	A`8,Pe8(t) +	A``,Pe
(t) − psign(eT
,P)    (13) 
where A`8,P  and A``,P  represent the ith rows of A`8  and A`` 
respectively. To develop the conditions under which the sliding 
will take place, the reachability condition will be tested. 

Notice that e
,P =	
P − 
aP  and eT
,P =	
�,P − 
aP . Then e
,P − eT
,P =	
P − 
�,P and from (3) we get: 
be
,Pb − ζ ≤ beT
,Pb    (14) 

From (13) and (14) we obtain: 

e
,Pe�
,P =	e
,PSA`8,Pe8 +	A``,Pe
U − ρb(eT
,P)b    (15) 
≤ e
,PSA`8,Pe8 +	A``,Pe
U − ρbSe
,PUb + ρζ    (16) 

< −bSe
,PUb(ρ − bSA`8,Pe8 +	A``,Pe
Ub − deb(f
,g)b    (17) 
Provided that the scalar p is chosen such (large enough) 

that: 

p > bSA`8,Pe8 +	A` ,̀PeOUb − debSf
,gUb + n    (18) 
where the scalar n ∈ ℜj. Then : 

e
,Pe�
,P < −nbe
,Pb    (19) 
This is the eta-reachability condition and implies that e
,P 

will converge to zero in finite time. When every component of 



Engineering, Technology & Applied Science Research Vol. 10, No. 2, 2020, 5487-5491 5489 
 

www.etasr.com Amara et al.: Sliding Mode Observer and Event Triggering Mechanism Co-design 

 

e
>t? has converged to zero, a sliding motion takes place on 
the surface S. 

Remark: Note that this is not a global result. For any 
givenρ , there will be an initial condition of the observer 
(typically representing the very poor estimation of the initial 
conditions of the plant) so that (16) is not satisfied.  

During sliding mode e
>t? �	e�
>t? � 0 , and the error 
system defined by (11) and (12) can be written in a collapsed 
form as:  

e�8>t? � SA88e8>t? �	LvfkU    (20) 
0 � SA`8e8>t? �	vfkU    (21) 

where vfk is the so-called equivalent output error injection that 
is required to maintain the sliding motion. This is the natural 

analogue of the equivalent control. Substituting vfk from (18) 
and (19) yields the following expression for the reduced-order 
sliding motion: 

	e� 8>t? �		>A88 �	LA`8?e8>t?    (22) 
This represents the reduced-order motion of order n � p 

that governs the sliding mode dynamics. It can be shown that if 
(A, C) is observable then >A88A`8? is also observable, and a 
matrix L can always be chosen to ensure that the reduced-order 
motion in (20) is stable. 

B. Communication Rate and Threshold Design 

The average sensor communication rate is defined as: 

γ �	lim	sup<→j∞ n 8<j8op γ�dt
<
�q'     (23) 

The average rate r  gives a general description of the 
network utilization. Below we consider the choice of the 
threshold s issue, given in (3), in order to achieve a desirable 
trade-off between communication rate and estimation quality. 

C. Communication Design Optimization 

In this section, the tradeoff between the estimation quality 
and the network bandwidth use as a constrained optimization 
problem is described. For this aim, the following cost functions 
�8 and �̀  are defined: 

�8>ζ? � 	γ    (24) 
This function is related to number of the transmitted sensor 

measurement packets. 

�̀ >ζ? �	lim<→j∞ t 8<j8 p eO>t?`dt
<
�q'     (25) 

where �̀  is an estimation error RMS performance index.  
Finally, both functions can be used to formulate a general 

performance index as following: 

�>ζ? � Q�8>ζ? � R�̀ >ζ?    (26) 
where, Q and R are weighting matrices used to penalize the 
number of transmitted packets and the estimation quality 
respectively. Thus the co-design problem can be formulated as 
follows: 

ζ∗ � argminf>ζ?    (27) 
IV. SIMULATION RESULTS 

In this section, the effectiveness of the proposed co-design 
approach is illustrated with a numerical example. For 
simulation purposes, let us consider a nominal linear system 
described by (1) and modeled by: 

A �	x 0 1�2 0z,B � x
0
1z, C �	{1				1|    (28) 

which represents a simple harmonic oscillator. For simplicity, 
we assume u>t? � 0,D � 0, and ρ � 	1, Q � 5, R � 1. The 
system’s initial state is x' �	{0.5	;	�0.8|<	, and the observer 
has zero initial condition. Figure 2 shows the simulation 
diagram using Matlab/Simulink version R15.a. using a variable 
step ode45, Dormand Prince solver with a processor Intel(R) 
Core(TM) i5-8250U CPU. 

 

 
Fig. 2.  Simulation diagram 

In Table I the simulation results for different arbitrary 
values of ζ are summarized along with the optimal one for 
giving priority to the minimisation of energy consumption. 

TABLE I.  OPTIMIZATION OF ENERGY CONSUMPTION 

Case no 

Different values of � with priority to energy minimization 
� rms(ey) Number of 

transmissions 

Objective 

function 

1 0.1 0.0801 115 0.0806 

2 0.50 0.1059 22 0.1060 

3 1.50 0.3139 3 0.3139 

Optimal 0.9409 0.2291 9 0.2586 
 

Figures 3-6 show the system output yt, the transmitted 

output y�� and the output estimation yestim for different values of ζ. Figure 6 is the optimal case. Excellent tracking of the output 
occurs after approximately 1.3s and the output estimation error 
ey after approximately 1.0s becomes zero, and the transmitted 
outputs are minimized. This is indicative of a high estimation 
quality with minimization of energy consumption. If we give 
priority to estimation rather than energy minimization, 
then	Q	 � 1,R	 � 	5. Table II exhibits the results of simulation 
for different values of ζ. 



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Fig. 3.  1
st
 case: The output yt, the transmitted output y�� , the output 

estimation error ey, and the output estimation yestim  

 

Fig. 4.  2
nd
 case: The output yt, the transmitted output y�� , the output 

estimation error ey, and the output estimation yestim  

 

Fig. 5.  3
rd
 case: The output yt, the transmitted output y�� , the output 

estimation error ey, and the output estimation yestim  

 

Fig. 6.  Optimal case: The output yt, the transmitted output y��, the output 
estimation error ey, and the output estimation yestim  

TABLE II.  OPTIMIZATION OF ESTIMATION QUALITY 

Case no 

Different values of � with priority to estimation 
� rms(ey) Number of 

transmissions 

Objective 

function 

1 0.20 0.0801 57 0.4004 

2 1.00 0.2585 10 1.2926 

3 1.50 0.3139 3 1.5693 

Optimal 0.70 0.1564 14 0.7821 

 

Figures 7-10 plot the system output yt, the transmitted 
output y��, and the output estimation yestim for different values 
of ζ . Figure 10 shows the optimal case which clearly 
demonstrates a high estimation quality always with 
minimization of the transmitted output y��. 

 

 
Fig. 7.  1

st
 case: The output yt, the transmitted output y�� , the output 

estimation error ey, and the output estimation yestim  

 
Fig. 8.  2

nd
 case: The output yt, the transmitted output y�� , the output 

estimation error ey, and the output estimation yestim  

 

Fig. 9.  3
rd
 case: The output yt, the transmitted output y�� , the output 

estimation error ey, and the output estimation yestim  



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Fig. 10.  Optimal case: The output yt, the transmitted output y��, the output 
estimation error ey, and the output estimation yestim  

V. CONCLUSIONS 

Because of the verified benefits in terms of reduced energy 
consumption and better resource utilization compared to 
traditional designs, event-triggered estimation methods can be 
widely adopted in industrial applications. For instance, in 
achieving optimal state estimation over a shared network we 
use a sliding mode observer associated with an event triggering 
mechanism. It was shown that by the introduction of an event 
based technique and the communication design optimization 
with a corresponding estimator by adopting sliding mode, 
reduction of the estimation error and of the sensor transmission 
can be achieved. The simulation results illustrate the 
effectiveness of the proposed sliding mode observer in terms of 
decreased estimation error and reduced sensor transmitted 
information. 

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