Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 1, No. 3, 2022 

 

31 

Fault	Detection	of	Discrete‐time	LPV	System	Based	on	
Event	Triggering	Mechanism	
Shuai Mao 

School of Electrical Engineering and Information, Northeast Petroleum University, Heilongjiang Province, Daqing 163318, China 

 

Abstract: For discrete-time linear parameter varying systems with distributed time delays, the problem of fault detection under 
event triggering mechanism is studied. Based on the parameter dependent Lyapunov function method, the sufficient conditions 
for the system to be asymptotically stable and meet the H∞ performance criterion are given. The parameter matrix of fault 
detection filter is obtained by solving LMI technology. In addition, the event triggering strategy is introduced to save network 
resources. 

Keywords: Fault detection, Discrete-time LPV system, H∞ performance, LMI. 
 

1. Introduction 
As a kind of time-varying system with uncertain 

parameters, linear parameter variation (LPV) system has been 
widely used in aerospace, industrial manufacturing, wind 
power generation and other fields [1-3]. No matter in which 
field, the operation of the control system is inseparable from 
the network environment. On the one hand, due to some 
disadvantages of the network control system itself, on the 
other hand, due to the influence of external complex working 
conditions, various faults will appear in the operation process 
of the control system [4-5]. If these faults can not be found 
and solved in time, it will either lead to the temporary collapse 
of the system, affect the working process, or lead to the empty 
of human and financial resources. Therefore, it is very 
meaningful for the system to detect the fault information as 
soon as possible and take necessary measures before the fault 
occurs. However, the network control system is limited by the 
network bandwidth, and a large number of data will be 
congested in the transmission process [6-8]. In addition, the 
previous control strategy based on time trigger requires the 
system to sample in each cycle. If the sampling cycle interval 
is too small, the system will sample and send data frequently, 
which can not effectively filter useful information. In this 
regard, a new control strategy based on event triggering is 
introduced to make the system execute tasks under specific 
triggering conditions, the required information is transmitted 
and the useless information is filtered [9]. Based on the mode 
dependent switching strategy and the Lyapunov stability 
theory of dual dependence of parameters and modes, 
reference [10] studies the design of asynchronous H∞ fixed 
order filters for LPV time-delay switched systems under the 
mode dependent mean residence time method. Literature [11] 
gives sufficient conditions for the existence of filters for 
discrete LPV systems by using linear matrix inequality 
technology and parameter dependent quadratic Lyapunov 
function method. On this basis, literature [12] and literature 
[13] give H∞ filter design methods for parameter bounded 
discrete LPV systems and observer design methods for 
discrete LPV systems with unknown inputs respectively, The 
former also verifies the effectiveness of the proposed filter 
design scheme through a numerical simulation. 

2. Problem Statement 
Consider the following form of discrete-time switched 

LPV system 
 

                           

                

1 2 1 2
1

1 2

1        

   






       


   

 i
i

x k A k x k A k x k k A k x k i B k k B k f k

y k C k x k D k k D k f k
(1) 

 
where   xnx k is the system state vector,   yny k is 

the measurement output vector of controlled object,
  fnf k  is the fault signal of system,    nk  is 

disturbance input variable in  2 0,L . For the convenience 

of description, we use k   for   k   and  kA  ,  1 kA  ,

 2 kA  ,  1 kB  ,  2 kB  ,  kC  ,  1 kD  ,  2 kD  are the 

functions of time-varying parameter k  with its parameter 

variation rate   l k  bounded in ,   l l  , 1,2, , l s  . The 

value of parameter k can be measured online.   k is the 

discrete time-varying delay, which meets  1 20     k  . 

The parameter  0 1,2,   i i and
1 1

  
 

 

    i i
i i

i . 

A fault detection filter in the form of formula (2) is 
designed to obtain the residual signal of the system and detect 
the fault of the system  

 

 
         

         
1  

 

   


 

f f k f f k

f k f f k

x k A x k B y k

r k C x k D y k
      (2) 

 
where   xn

fx k  is the state of the designed filter,

  rnr k  is the residual signal. The parameter-varying 

matrices  f kA  ,  f kB  ,  f kC  ,  f kD   are represent 

the parameter matrix of fault detection filter to be solved.  
 
Setting the event-triggered condition 
 

         1 min    T T
i i y yk k k e k e k y k y k      (3) 

 
where    is the event-triggered threshold, 



 

32 

      y ie k y k y k , 1,  i ik k k . 

Combining (1)-(3), we can get 
 
                      

             

1 2 1
1

1

1          

   






       


   

k k k i k y k
i

k f k y k

k A k A H k k A H k i B e k B v k

e k C k D e k D v k
(4) 

 

where ( ) ( ) ( )    
TT T

fk x k x k  ,       e k r k f k  ,

 0H I ,    
     

0


  
 

  
 

k
k

f k k f k

A
A

B C A
, 

   1
1 0




 
  
 

k
k

A
A  ,    2

2 0




 
  
 

k
k

A
A  ,    

0



 

  
 

k
f k

B
B

 ,

     
       

1 2
1

1 2

 


   
 

  
 

k k
k

f k k f k k

B B
B

B D B D
, 

             k f k k f kC D C C  ,

         1 1 2       k f k k f k kD D D D D I . 

Definition 1. The switched LPV system (1) has H∞ 

performance, if there exist event-triggered condition (3), filter 
(2) such that the following conditions are satisfied  

1. When   0 k , the system (4) is asymptotically stable. 

2. Under the zero initial condition, for all nonzero 
   2 0,  k L  , the filtering error system and the control 

system satisfies        2

0 0


 

 

 T T

k k

e k e k v k v k . 

3. Main Results 
For the fault detection system (4), by selecting the 

appropriate Lyapunov Krasinski functional, a sufficient 
condition is proposed to make the fault detection system (4) 
asymptotically stable and meet the H∞ performance criterion 
under the event trigger mechanism, which provides a 
theoretical basis for the later design of fault detection filter. 

 
Theorem 1. If there exist positive definite matrix functions
  0 kP , 0Q  and 0M , such that the following matrix 

inequalities hold 
 

 
     
   

   

     
     

 

1

1 1

2 1

2
1 1 1

1

1

0 0 0 0

* 0 0 0 0 0

1
* * 0 0 0 0

0* * * 0

* * * * 0

* * * * * 0 0

* * * * * * 0

* * * * * * *

   
 

 


    
  















 
 

 
 

 
 
  
 

 
  
 
  





T T T
k k k

T
k k

T
k k

T T T
k k k

T T
k k f k

k

A P C C

Q A P

M A P

I B P D D

I B P D

P

I

I

(5) 

 
where        T T

kP H QH H MH ， 2 1 1     ，

  0   
kC C ，    1 2    

k kD D D  , Then the fault 

detection system (4) is asymptotically stable, and its H∞ 

performance is guaranteed. 
Proof. Choosing the multiple parameter-dependent 

Lyapunov functions as 
              1 2 3 4, , , , ,       V k k V k k V k k V k k V k k

,       
1

4
1

,   
 

  

  
k

T T
i

i j k i

V k k j H MH j , 

        1 ,    T
kV k k k P k ,

      
 

1

2 ,


  


 

 
k

T T

i k k

V k k i H QH i ,

      
1

2

1 1

3
2 1

,




  
  

    

  
k

T T

j i k j

V k k i H QH i . 

The forward difference along the system (4) can be 
obtained 

              1 1, 1 1          T T
k kV k k k P k k P k  

                
1

2

2
1

,




        


  

      
k

T T T T T T

i k

V k k k H QH k k k H QH k k i H QH i
 

            
1

2

3 2 1
1

,




      


  

    
k

T T T T

i k

V k k k H QH k i H QH i  

          4
1 1

1
,      



 

 

      
 
 

T

T T T
i i

i i

V k k k H MH k k i H MH k i  

Defining 

            
1

     




    
 


T

T T T T T T T
i y

i

k k k k H k i H v k e k  , 

we can get 
                         2 2 2 22 2, ,                T T T

y yV k k e k v k V k k e k v k y k y k e k e k k k

. 
The performance index in the form of formula (6) is 

introduced 
 

        2

0






    T T

k

J e k e k v k v k         (6) 

 

Under zero initial conditions,   
0

, 0



k
V k k  , 

  , 0



k
V k k , we can get 

 

             

              

2

0
0

2

0 0

, ,

,

  

   



 


 

 

     

       



 

T T

k k
k

T T T

k k

J e k e k v k v k V k k V k k

e k e k v k v k V k k k k
(7) 

 
According to theorem 1, for any non-zero vector  v k  , 

0J , then        2

0

0




    T T

k

e k e k v k v k , 

so we can get    2 22

0 0


 

 

 
k k

e k v k . 

Theorem 2. If there exist positive definite matrix functions
 kP  , Q  , M  and U  , such that the following matrix 

inequalities hold 
 

   
 

 

   
   

 

1

2

2
1 1

1

0 0 0 0

* 0 0 0 0 0

1
* * 0 0 0 0

0* * * 0

* * * * 0

* * * * * 0 0

* * * * * * 0

* * * * * * *

  





   
 

 

 
 

 
 

 
 
  
 

 
    
 
  





T T T
k k

T
k

T
k

T T T
k k

T T
k f k

T
k

A U C C

Q A U

M A U

I B U D D

I B U D

U U P

I

I

(8) 

 
then the fault detection system (4) is asymptotically stable, 

and its H∞ performance is guaranteed. 
Proof. It is necessary to prove that inequality (5) is 



 

33 

equivalent to inequality (8). If inequality (5) holds, defining 
 1  T

kV V P  , substitute in (5), so we can get (8). 

Therefore, if (8) holds, according to  1 0    T
kV V P and

 1 0  kP ,we have 

  1
1 1 1( ) ( )( ( )) 0  
    T

k k kV P P V P  , 
1

1 1( ) ( ) 
     T T
k kV P V V V P . 

So 

1
2

1
1

( ) 0 0 ( ) ( )

0 ( ) 0

0( ) 0

( ) 0

   


 




  
   
    
 

    
      

 



T T T
k k k

T
k

T
k

T
k

P H QH A V L

Q A V

I B V

V P V

I

 

Assuming V reversibility, using   1, , , ,


T
kdiag I I I V P I  

congruent transformation of inequality, we can get (8), 
theorem proving. 

 
Theorem 3. If there exist positive definite matrix functions
 kP  , Q  , M  and filter parameters  f kA ,  f kB , 

 f kC ,  f kD  , such that the following matrix 

inequalities hold 
 

       
   

 

11 12 17

22 27

37

47

2
57 58 59

67

77

0 0 0 0

* 0 0 0 0 0

* * 0 0 0 0 0

1
* * * 0 0 0 0

0
* * * * 0

* * * * * 0

* * * * * * 0 0

* * * * * * * 0

* * * * * * * *

    
 






   
 

  
  
 
   

     
  
 

 
 
 

  

T T T
k k f k k

T
k f k

T
f k

P C D C

P C

Q

M

I

I D

I

I

(9) 

 
 

 
   

 
11 12

22

0
*

 


 
 

 

k k

k

P P

P
,              (10) 

 
where  11 11       kP Q M ,

   27      
T T
f k f kA A ,    37 1 1     

T T
k kA X A Y ,  

           17           
T T T T T T

k k f k k k f kA X C B A Y C B

，    47 2 2     
T T

k kA X A Y ,  

   67      
T T
f k f kB B ,

           
           

1 1 1 1
57

2 2 2 2

     
     

  
     

T T T T T T
k k f k k k f k

T T T T T T
k k f k k k f k

B X D B B Y D B

B X D B B Y D B
,   

   
   
1

58
2

 
 

 
    

T T
k f k

T T
k f k

D D

D D I
, 

 
 

1
59

2

 

 

 
   

  

T
k

T
k

D

D
,

   
 

11 1 12 1
77

22 1*

 


 



      
      

T T
k k

T
k

X X P Y Z P

Z Z P
 

Then the fault detection system (4) is asymptotically stable, 
and its H∞ performance is guaranteed. The parameters of fault 
detection filter are 

 
1( ) ( ) ( ) ( )0

( ) ( ) ( ) ( )0

   
   

     
     
    

f k f k f k f k

f k f k f k f k

A B A BZ
C D C DI

   (11) 

 

Proof. Defining 11 12

22

( ) ( )
( )

* ( )

 



 

  
 

k k
k

k

P P
P

P
 , 1 2

3 4

 
  
 

V V
V

V V
 , 

1
4 3

0

0 

 
   

 

I

V V
 , using  , , , , , , , diag I I I I I I   to make 

congruent transformation of inequality (8), and defining  
 
   
           

1 1
1 2 4 3 3 4 3

1 1
3 4 3 3 4 3

, 

     

 

 

     

  

， ， ,

， ，

T T
k k

T T
f k f k f k f k f k f k

P P X V Y V V V Z V V V

A V A V V B V B C C V V
 (12) 

 
inequality (9) can be obtained, that is, theorem 3 can be 

proved. 
Since the matrix ( )kP  is positive definite, it can be 

obtained ( ) 0  T
kP , that equation (10) holds. Substitute 

(12) into the following formula and simplify it appropriately 
 

        

        

        

1

11 1
3 4 3 3 4 3

11 1

   

   

   



   

 

  

  

  

f fz y f k f k f k f k

T T
f k f k f k f k

f k f k f k f k

T C sI A B D

C V V sI V A V V V B D

C sI Z A Z B D

  (13) 

 
thus, the filter parameter matrix with form (11) can be 

obtained. The certificate is completed. 
In this section, numerical simulation will be used to verify 

the effectiveness of the fault detection scheme of discrete 
LPV system under the designed event triggering mechanism. 
Considering the LPV system with mixed delay (1), the system 
parameter matrix is given as follows

0.02 0.08

0.12 0.05 0.02
 

     k
A  , 1

0.14 0.1 0.3

0.1 0.06

  
   

kA  ,

2

0.12 0.14

0.14 0.1 0.12
 

    k

A , 1

0.5

0.2

 
   

B , 2

0.1

0.3

 
  
 

B , 

0.03 0.02

0.02 0.08 0.05
 

   k
C  , 1

0.4

0.8

 
  
 

D  , 2

0.2

0.4

 
   

D  . 

Interference signal    0.15 ( 0 1 0.5)  kk e rand  , the fault 

signal is set to 
2sin( ), 40,41,...,60

( )
0,


 


 k k
f k

else
. 

Select the basis function 1( ) 1 f , 2 ( ) ( ) f k , and set 

the event trigger threshold 0.3   at the same time. The H∞ 
performance index min 0.9577    is solved by convex 

optimization, and the parameters of fault detection filter are 
 

 
0.0076 0.0221 0.0001 0.0654

0.0229 0.0043 0.0770 0.0668


    
        

fA k  , 

 
0.4202 0.4720 0.1979 0.0025

0.5426 0.0487 0.0724 0.4144


     
        

fB k , 

    0.0150 0.0249 0.0073 0.0109    fC k  , 

    0.3522 0.2532 0.2649 0.0776   fD k . 

 



 

34 

 
Figure 1. Residual function under event triggering 

mechanism 
 

 
Figure 2. Residual evaluation function and threshold 

under event triggering mechanism 

 
Figure 3. Time and interval of event triggered data 

transmission 
 

Figure 1 is the residual function under the event trigger 
mechanism, and Figure 2 is the residual evaluation function 
and threshold under the event trigger mechanism. The fault 
detection threshold 0.3724thJ   can be obtained from the 

residual evaluation function and threshold formula. It can be 
seen from Figure 2 that when 40k  the system starts to fail, 
the system detects the threshold of the fault signal is 
 40 0.3951 0.3724 J , which shows that the fault detection 

filter designed in this chapter can accurately detect the fault 
signal, and the fault detection filter is effective. Figure 3 

shows the transmission time and interval of event triggered 
data. The time length of analog signal is set to 120. After 
adopting the event trigger strategy, only 97 valid data are 
successfully sent, which effectively saves network resources. 

4. Conclusion 
In this paper, the problem of fault detection under event 

triggering mechanism is studied for discrete-time linear 
parameter varying systems with distributed delay. Based on 
the parameter dependent Lyapunov function method, the 
sufficient conditions for the system to be asymptotically 
stable and meet the H∞ performance criterion are given. The 
threshold logic method is used to detect the fault signal. The 
coupling between the system matrix and the Lyapunov 
function matrix is decoupled, and the parameter matrix of the 
fault detection filter is solved by LMI technology. Finally, the 
simulation results show that the fault detection filter designed 
in this chapter can accurately detect the fault signal of the 
system and achieve the expected goal. In addition, after 
introducing the event triggering strategy, it effectively avoids 
the transmission of some useless data, which not only saves 
network resources, but also improves the efficiency of 
network transmission. 

References 
[1] Gao Z, Fu J. Robust LPV modeling and control of aircraft 

flying through wind disturbance[J]. Chinese Journal of 
Aeronautics, 2019, 32(7): 1588-1602. 

[2] Ni S, Wu J, Shan J, et al. LPV modeling of the missile attitude 
control system based on small deviation equation[J]. Lecture 
Notes in Electrical Engineering, 2012, 138: 1837-1844. 

[3] Apkarian P, Gahinet P. A convex characterization of gain-
scheduled controllers[J]. IEEE Trans Automatic Control, 1995, 
40(5): 853-864. 

[4] Feng H, Liu W, Chen H. Robust distributed filters design for 
discrete-time spatially interconnected systems with LFT 
uncertainties[J]. Journal of Circuits, Systems and Computers, 
2021, 30(09): 2150159. 

[5] Zhu Y, Zhong Z, Zheng W X, et al. HMM-based H∞ filtering 
for discrete-time Markov jump LPV systems over unreliable 
communication channels [J]. IEEE Transactions on Systems, 
Man, and Cybernetics: Systems, 2018, 48(12): 2035-2046. 

[6] Velni J M, Grigoriadis K M. Delay-dependent H∞ filtering for 
time-delayed LPV systems[J]. Systems & control letters, 2008, 
57(4): 290-299. 

[7] Nejem I, Bouazizi M H, Bouani F. H∞ based state feedback 
control of LPV time-delay systems via parameter-dependent 
Lyapunov-Krasovskii functionals[C]// International 
Conference on Electrical Engineering and Control 
Applications. Springer, Cham, 2017: 60-71. 

[8] Wu L, Shi P, Wang C, et al. Delay-dependent robust H∞ and 
L2-L∞ filtering for LPV systems with both discrete and 
distributed delays[J]. IEE Proceedings-Control Theory and 
Applications, 2006, 153(4): 483-492. 

[9] Karimi H R, Lohmann B, Buskens C. A delay-dependent 
approach to robust filtering for LPV systems with discrete and 
distributed delays using PPDQ functions[J]. International 
Journal of Control, Automation, and Systems, 2007, 5(2): 170-
183. 

[10] Li Y, Bo P, Qi J. Asynchronous H∞ fixed-order filtering for 
LPV switched delay systems with mode-dependent average 
dwell time[J]. Journal of the Franklin Institute, 2019, 356(18): 
11792-11816. 



 

35 

[11] Pandey A P, de Oliveira M C. On the necessity of LMI-based 
design conditions for discrete time LPV filters[J]. IEEE 
Transactions on Automatic Control, 2018, 63(9): 3187-3188. 

[12] de Oliveira M S, Pereira R L. LMI-based filter design 
conditions for discrete-time LPV systems with bounded 

parameter variation[J]. IEEE Transactions on Automatic 
Control, 2020, 66(2): 910-915. 

[13] de Oliveira M S, Pereira R L. Improved LMI conditions for 
unknown input observer design of discrete-time LPV 
systems[J]. International Journal of Control, Automation and 
Systems, 2020, 18(10): 2543-2551.