On stability for hybrid system under stochastic perturbations

tion protocols, considered for nonlinear hybrid system under stochastic 
 perturbations. By using the Lyapunov-type function, we proved that the almost surely exponential 
 stability remain be guaranteed as long as a bound on the maximum allowable transfer interval 
 (MATI) is satisfied. 
 Keywords: Networked Control System, almost surely exponential stability, maximum allowable 
 transfer interval, Lyapunov function. 
1. Introduction 
 In recent years, Networked Control Systems (NCS) were addressed strongly in the control 
community because of its extensive applications in wireless as well as wireline. The pioneering papers 
were proposed by Walsh, Beldiman and Bushnell [10, 11, 12]. They introduced about stability of control 
systems with deterministic protocol. More recently, quite many articles and literatures referred to 
study stability of hybrid systems by specifically showing the Lyapunov-type function and bounds on 
the maximum allowable transfer interval (MATI), see [1, 2, 3, 4, 8, 6, 9, 13] for more details. This 
paper is divided into two sections. Beside Introduction, we state Preliminary and main problem in 
the second section. 
________ 
 Corresponding author. 
 Email address: caotanbinh@qnu.edu.vn 
 https//doi.org/ 10.25073/2588-1124/vnumap.4522 
 82 
 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 83 
 In [5], the authors solved entirely for researching the stable types of solution of hybrid systems, 
modelled as follows: 
 x() t f ((),()), x t e t t (, tkk t 1 ), (1a )
 e() t g ((),()), x t e t t (, tkk t 1 ), (1b )
  (t ) 1, t ( tkk , t 1 ), (1c )
  (tk ) 0, (1d )
 (1e )
 x( tkk ) x ( t ),
 (1f )
 et(k ) bhket k ( , ( k )) (1 betk k ) ( k ), 0,1,2,...
 Remind here that the variable bk belongs to the set 0,1 . If bk 1 then transmission is successful, 
and the protocol h determines the updated error. While if bk 0 then the error remains unchanged 
at the tk . We get a sequence ()bkk . Let S : 0,1 and the probability space (,,)SFPb with 
the sequence space 
 S:():, bk k b k S  k  
 SS
where the σ-algebra Fb : 2 2 ... and the probability P satisfying 
 P( b S : bk 1) p ,  k . 
We also assume that the random variables bk are independently and identically distributed. 
 Motivated from this paper, we concern to hybrid system in which exogenously stochastic 
perturbation is a Wiener process. This is, up to now, one of proposed problems remain have not been 
solved yet. To solve the problem, we make use of tools as introduced in [5] by defining MATI or 
choosing the Lyapunov function W for protocol. We also, of course, use other tools for stochastic 
stability from [7] in order to support our proof. 
2. Preliminary and main result 
 Let us now consider the perturbed hybrid system that is of form 
 dxt() fxtetdt1 ((),()) fxtetd 2 ((),())w(), tt (, ttkk 1 ), (2a )
 det() gxtetdt1 ((),()) gxtetd 2 ((),())w(), tt (, ttkk 1 ), (2b )
  (t ) 1, t ( tkk , t 1 ), (2c )
  (tk ) 0, (2d )
 x( tkk ) x ( t ), (2e )
 (2f )
 et(k ) bhket k ( , ( k )) (1 betk k ) ( k ), 0,1,2,...
where x n is the state of the system, e n is the error at the controller, h is the update function 
that models the particular protocol, τ is a timer to constrain both the transmission interval and the 
84 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 
transmission delay, and w(t) is a Wiener process. In this paper, suppose that f1 , f2 , g1 and g2 satisfy 
Lipschitz and linear growth conditions which guarantee the existence and uniqueness of the solution 
of (2). Assume furthermore that f1(0,0) f 2 (0,0) g 1 (0,0) g 2 (0,0) and hk( ,0) 0 for all 
 k . So system (2) has the solution ξ(t) := (x(t); e(t)) = (0,0) corresponding to the initial value 
  ***: (xe , ) (0,0) . 
Now, we introduce the concept of almost surely exponential stability, which can be found in 
Mao [7]. 
Definition 1 Consider the system (2). The solution  *** (xe , ) (0,0) of (1) is called almost surely 
exponentially stable, if for all 0 
 1
 limsup log (tb ,0,0 , ) 0 , almost surely. 
 t t
We need the following assumptions for the stability of network and system. 
Assumption (A1) The probability p (0,1) of successful transmission of the k-th sampling time 
is identical for all k and independent of k . 
Assumption (A2) The stochastic perturbations b and w are mutually independent. Put Fb is 
the σ-algebra generated by ()b , and F is the σ-algebra generated by w(t) . The system (2) 
 kk w t 0
defined on a probability space (Ω, F, P) where FFF  b w . Hereafter, we use notation Eb (.) 
instead of EFb (.|b ) and Ew (.) instead of EFww(.| ) . 
Assumption (A3) Lyapunov functions for the protocol and the perturbed system. 
 n
 (i) There exist constants 0, aa12, 0 < λ < 1 such that for all e : 
 22
 a12 e W(e) a e (3) 
 W( h ( k , e ))  W ( e ) . (4) 
(ii) The evolution of Lyapunov function W is bounded in the sense that there exist a constant 
 n n
  0, and a continuous function H : such that for all xe, : 
 WW
 .(,)g x e ,(,)2 gT x e  W () e H () x (5) 
 ee11
 2
(iii) There exist a C Lyapunov function V and constants b1, b 2 , b 3 0 such that for all 
 22
 b12 x V() x b x (6) 
 VV1 2
 LVx(): .(,) fxe fxeT (,). .(,) fxe bVx () , (7) 
 xx12 22 2 3
where 
 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 85 
 Tn (1) ( ) n
 fi(,) x e f i f i is the transpose of fi ( x , e ) , i 1,2 
 Tn (1) ( ) n
 gi(,) x e g i g i is the transpose of gi ( x , e ) , i 1,2 
 22VV 22VV
 x1  x 1  x 1  xn x1  e 1  x 1  en
 2V 22VV 
 V : , : 
 x2 xx x  e  e  x 
 22VV 22VV
 x  x  x  x x  e  x  e
 n1 n n nn n1 n n nn 
 VVV   VVT VVT
 , .(,),(,)f11 x e f x e , .(,),(,)g11 x e g x e . 
 x  x1  xn xxee
Here,  MATI follows from the equation 
  2   ( 21 1),  (0)  . (8) 
We choose τ(η) such that for all  0,  (  ) we have 
 1
  (),    , (9) 
see [5] for more details. 
Theorem 2 Consider the system (2). Assume that (A1), (A2) and (A3) hold. If there exist 
 (0,1) and γ > 0 as defined in (8) satisfying 
 2W
 gxeT (,). .(,)2(2 gxe   bWe )()-  Hx () for almost all xe, n (10) 
 2e2 2 3
then the solution  * (0,0) of system (2) is almost surely exponentially stable. 
Proof: We first assume that system (2a), (2b) is almost surely exponentially stable. 
Consider Lyapunov-type function 
 U(,)(,,):()()()  U x e  V x   W e . (11) 
It follows that 
 2 2 2 2 1
 b x V() x b x , a e W(e)a e ,  ()  . 
 1 2 1 2 
We yield 
 2 2 2 1 2
 bx1  aeUxeVx 1 (,,)  ()  ()W(e)  bx 2  ae 2 
and 
 m2 mxe(,)(,,)(,) 2 Uxe  Mxe 2 M  2 (12) 
 1
where m min b1 , a 1 , M max b 2 ,  a 2 . 
By Ito’s formula and Assumption (A3), we can derive that 
86 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 
 22
 UUUUU  1 TT  
 dUxe(,,) dx de d fxe2 (,).22 .(,) fxegxe 2 2 (,). .(,) gxedt 2
 x  e  2  x  e
 VW
  fxedt(,)(,)()(,)(,) fxedw    gxedt gxedw
 xe1 2 1 2
 22
 1 TTVW
 ()() Wedt  fxe2 (,).2 .(,) fxe 2  ()(,).  gxe 2 2 .(,)g2 x e dt (13) 
 2 x e
 2
 WW1 T
 LVxdt()  ()  .(,) gxe1  ()()  We  ()(,).  gxe 22 .(,) gxedt 2
 ee2
 VW
 .(,)().(,)f22 x e  g x e dw
 xe
and 
 WW1 2
 ()  .(,)gxe  ()()  We  ()(,).  gxeT .(,) gxe
 ee12 22 2
 WW1 2
 () .(,)gxe   2() (()1)2 We ()  ()(,). gxeT .(,) gxe
 ee1 2 22 2
 (5),(8),(10)
 22
  ()2 W () e  H ()2 x   ()()- W e  ()1 W () e
 () (2  b3 ) W ( e )  H ( x ) (14) 
 2  ()()W e  ()()2 H x   ()() W e 2 2 ()() W e  2 W () e
 ()()()()    b3 W e  H x 
 (9)
 (  )b3 W ( e ).
Therefore 
 (7),(14) VW
dUxe(,,)()()().(,)().(,) bVxdt3   bWedt 3 fxe 2   gxedw 2
 xe
 (15) 
 VW
 bUxedt3(,,).(,)().(,). fxe 2   gxedw 2
 xe
This implies 
 dEUxeww (,,)(,,) bEUxe3   dt . (16) 
For each k = 1,2,, integrating both sides of (16) from tk 1 to tk , we get 
 t
 k
 EUxtbetb ((,),(,),()) t EUxt (( ,),( bet ,),( b  t )) E bUxe (,,)   dt
 w k k k w k 1 k 1 k 1 t w 3
 k 1 (17) 
 Ew U(( x t k 1 ,),( b e t k 1 ,),( b t k 1 )).
If at time transmission is successful, i.e. if bk 1, then 
 2
 Uxtbetb((,),(,),())k k t k Vxtb ((,)) k   (())  tWetb k ((,)). k 
 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 87 
On the other hand if transmission fails, i.e. if bk 0 then 
 2
 Uxtbetb((,),(,),())k k t k Vxtb ((,)) k   (())  tWetb k ((,)). k 
These give 
 EEUxtbetb (( ,),( ,),( t ))(( xtbetb ,),( ,))
 b w k k k k k 
 2
 pEVxtb w ((,)) k   (())  tEWetb k w ((,)) k 
 (18) 
 2
 (1 pEVxtb ) w ((,)) k   (())  tEWetb k w ((,)) k 
 EUxtbetbw ((,),(,),()) k k t k  EWetb w ((,)) k 
where : 1 (1 pp  )  2 . 
From (16) it follows that 
 b3 () t tk 
 EUw ((,),()) tb  t EUxtbetb w ((,),(,),())  t e EU w ((,),()).  tb k  t k 
Taking expectation in b, we obtain 
 b33 t b tk 
 eEEUtbb w ((,),())  t eEEUtb b w ((,),())  k  t k  (19) 
and 
 0 eEEUtbtb33 tkk ((,),())  eEEUtbt b t ((,),())(,)    tb
 b w k k b w k k k 
 (17),(18) k (20) 
 2
 bt3 i
 MEw0  e E b E w W( e ( t i , b )) .
 i 0
From (12), (19) and (20), it follows that 
 b33 t b t
 meEEb w(,) tb eEEU b w  ((,),())  tb  t 
 2 
 bt3 k 
 e Eb E w U( ( t k , b ),  ( t k )) ME w 0 .
Hence 
 2
 M bt3
 Eb E w( t , b ) E w 0 e ,  t 0. (21) 
 m
From the system (2), we have 
 tt
 xt()()() xt fds fdws 
 k tt12
 kk
and 
 tt
 et()()() et gds gdws . 
 k tt12
 kk
In addition, the conditions f1(0,0) f 2 (0,0) g 1 (0,0) g 2 (0,0) lead to exist a positive 
constant K such that 
 2 2 2
 f12(,)(,)(,) x e f x e K x e
 (22) 
 2 2 2 
 g12(,)(,)(,) x e g x e K x e
88 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 
Therefore, we obtain 
 tt
 2 2 2 2
 Extw()3 Ext w () k E w ( fds12 ) E w ( fdws ())
 ttkk
 tt
 3 Ext2 ( ) ( tt ) Efds 2 Efds 2
 w k k tt w12 w
 kk 
 (22) tt
 2 22
 3Extw ( k )  KE w  ( sdsKE ) w  ( sds )
 ttkk
 t
 2 2
 3Ew x ( t k ) ( 1) K E w ( s ) ds
 tk
and 
 t
 22 2
 Eetw()3 Eet w ()(1) k  KE w () sds . (23) 
 tk
As a result 
 2 2 2 2 2t 2
 Ew Exe w(,) Ext w () Eet w ()3 Et w  ()2(1) k  KEsds w  () . 
 tk
Hence 
 2 2tk 1 2
 Esup Etb ( , ) 3 EEtb  ( , ) 2(  1) KEEsbds  ( , )
 b w b w k t b w
 k
 tkk t t 1 
 M 22tk 1
 3E e b33 tk 2(  1) K E  e b s ds
 ww00 t
 m k 
 2
 2 b3 M b 3 tk 1
 3 1 K ( 1)(1 e ) Ew 0 e
 bm3
 Ce bt31k ,
where 
 2
 2 b3 M
 C 3 1 K ( 1)(1 e ) Ew 0 . 
 bm3
Applying Chebyshev’s inequality, we get 
 2
 Esup E ( t , b )
 b3 bw 
 2 tk 1 t t t
 P b: sup E ( t , b ) e 2 kk 1
 w b3
 t t t tk 1 
 kk 1 e 2
 b
 3 t
 Ce 2 k 1 .
Since t0 0 and 0  ttkk 1 , it is clear that 
 b b b
 3t 3 t 3 t
 e2k 1 e 2 1 e 2 2 
 k 0 
 bb
 33()()t t t t t t
 ee221 0 2 1 1 0 .
 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 89 
Using Borel-Cantelli’s lemma argument (see Mao [7]) to conclude that there exist a set 1 with 
 P( 1 ) 1 and an integer-value random variable k0 such that for every b 1 we have 
 b3
 2 tk 1
 2
 supEw  ( t , b ) e ,  k k0 ( b ). (24) 
 tkk t t 1
That means 
 b3
 2 tk 1
 2
 Ew(,) t b e ,  t (,), t k t k 10  k k (). b 
Similarly to argument as above, using Borel-Cantelli’s lemma again, there exist a set 2 with P( 2 ) 1 
and an integer-value random variable k1 such that for every w 2 we have 
 b3
 2 tk 1
 2
 (,)t b e ,  t (,), tkk t 11  k k (). w (25) 
Let kc max k01 , k  , 0  1  2 , we have P( 0 ) 1 and 
 b3
 2 tk 1
 2
 (,)tb e ,  ttt (,),k k 10  kkwbw c (),(,)  . (26) 
Consequently 
 1 b
 limsup log ( (tb ), ) 3 0. (27) 
 t t 8
The proof is completed. 
Remark 3 The inequalities (5) and (10) are existent. In fact, we choose g12(,)(,) x e g x e e , 
 221/ 2
W() e e e12 e and  0 . Then we have 
 WWT 221/ 2
 .(,)g1 x e ,(,) g 1 x e e 1 e 2 2(), W e  0, 
 ee
Moreover, 
 2We()
 gT . . g 0 2(2 b ) W ( e ) , 
 2e2 2 3
as long as 2 b3 0. 
References 
 [1] D. Carnevale, A. R. Teel, D. Nesic, Further results on stability of networked control systems: 
 a lyapunov approach, In 2007 American Control Conference, IEEE. (2007) 1741-1746. 
 [2] D. Carnevale, A. R. Teel, D. Nesic, A Lyapunov proof of an improved maximum allowable transfer interval for 
 networked control systems, IEEE Transactions on Automatic Control. 
 52 (2007) 892-897. 
 [3] D. Christmann, On the behavior of black bursts in tick-synchronized networks, Techn. Ber. 
 337 (2010). 
90 C.T. Binh, T.C. Son / VNU Journal of Science: Mathematics – Physics, Vol. 37, No. 1 (2021) 82-90 
 [4] M. B. Cloosterman, N. Van de Wouw, W. P. M. H. Heemels, H. Nijmeijer, Stability of 
 networked control systems with uncertain time-varying delays, IEEE Transactions on Automatic 
 Controll. 54 (2009) 1575-1580. 
 [5] L. H. Duc, D. Christmann, R. Gotzhein, S. Siegmund, F. Wirth, The stability of try-oncediscard for stochastic 
 communication channels: Theory and validation, In 2015 54th IEEE 
 Conference on Decision and Control (CDC). (2015) 4170-4175. 
 [6] W. P. M. H. Heemels, D. Nesic, A. R. Teel, N. Van de Wouw, Networked and quantized control 
 systems with communication delays, In Proceedings of the 48h IEEE Conference on Decision 
 and Control (CDC) held jointly with 2009 28th Chinese Control Conference. (2009) 7929-7935. 
 [7] X. Mao, Stochastic differential equations and applications, Elsevier, 2007. 
 [8] D. Nesic, D. Liberzon, A unified framework for design and analysis of networked and quantized 
 control systems, IEEE Trans. Automatic Control. 54 (2009) 732-747. 
 [9] P. Naghshtabrizi, J. P. Hespanha, A. R. Teel, Stability of delay impulsive systems with application to networked 
 control systems, Transactions of the Institute of Measurement and Control. 
 32 (2010), 511-528. 
[10] G. C. Walsh, O. Beldiman, L. G. Bushnell, Asymptotic behavior of nonlinear networked 
 control systems, IEEE transactions on automatic control. 46 (2001) 1093-1097. 
[11] G. C. Walsh, O. Beldiman, L. G. Bushnell, Error encoding algorithms for networked control 
 systems, Automatica. 38 (2002) 261-267. 
[12] G. C. Walsh, H. Ye, L. G. Bushnell, Stability analysis of networked control systems, IEEE 
 transactions on control systems technology. 10 (2002) 438-446. 
[13] L. Zhang, Y. Shi, T. Chen, B. Huang, A new method for stabilization of networked control 
 systems with random delays, IEEE Transactions on automatic control. 50 (2005) 1177-1181.