Stabilization on A Class of Networked Control Systems with Random Communication Delay

Transcription

1 Milano (Italy) August 8 - Septeber, Stabilization on A Class of Networked Control Systes with Rando Counication Delay Tao Sun, Hongye Su, Zhengguang Wu State Key Laboratory of Industrial Control Technology, Institute of Cyber-Systes and Control, Zheiang University, Hangzhou, 7,PRChina(e-ail: suntaohe479@gailco; hysu@iipczueducn; nashwzhg@6co) Abstract: This paper is concerned with the stabilization proble of wireless networked control systes with network-induced delays of tie-varying Markovian characterization The effects of the saple rate changing on the network induced delay is the ain topic At first place, the tie delay atrices changing stochastically and arbitrarily are considered A state feedback controller is designed to guarantee the resulting close-loop syste stochastically stable Following that, a bridge is built to give out a unifor worka nuerical exaple is presented to illustrate the effectiveness and potential of the developed theoretical results INTRODUCTION As wireless network technology is becoing ore and ore widespread, industrialists and researchers are finding the potentials of wireless network control systes (WNCS)[,, ] However, this kind of networks are very vulnerable to environental disturbances When a node enters a wireless counication region, the variance of the sending frequency of this node will change other nodes delay distribution under the 854 situation[4] This kind of frae contention builds up the network induced delay, because fraes ay reain longer in the buffer bas the other nodes copetition Many researchers would like to odel this kind of delay as a hoogeneous Markov chain, eg[5], [6], which is ostly appeared in real counication systes, the current tie delay is usually related to the previous tie delays[7] In [5] a controller was designed to stabilize a discrete-tie arkovian up linear syste via a delayed network Efforts like deadbands[8] and odel-based feedback[9] are introduced to construct counication control strategies This kind of control strategies ay prevent network fro being saturated and buffer queues building up Saple rate adaption is another for of reduced counication control, in which the saple rate is varied on the basis of either control perforance, network perforance or a cobination of both[] Being independent with controller and protocol akes saple rate adaptation ore suitable for integration with the current network control technology This work is supported by the National Creative Research Groups Science Foundation of China (NCRGSFC: 676); National Natural Science Foundation of PR China(NSFC: 676); The National High Technology Research and Developent Progra of China (86 Progra 8AA49);Open Proect of the State Key Laboratory of Industrial Control Technology under grant No ICT Fig A wireless counication situation Consider a counication situation shown in figure using saple rate adaption The coputer takes the part as the controller, plant and the sapler Two serial ports of the coputer connect to two wireless nodes Node is connected to controller, and node is connected to the sapler Such kind of WNCS configuration gives proinent to the wireless link part easily The coputer changes the saple rate according to a certain rules If there is a hoogeneous Markov transition atrix(tm) π is used to describe the delay transition fro node to node at saple rate f >, when the saple rate changes to soe other value, ie, f = f/c, c >,c, because this will change the frae arrival frequency fro node to node, which would vary the contention situation at node So it is ore reasonable to use different TM to describe the transission delay of the WNCS under different saple rates This paper focuses on the effects brought in the wireless counication contention when the saple rate is changing In order to describe the vary of the saple rate rules, a hoogeneous Markov TM is eployed to describe it, firstly Second, the saple rate changing arbitrarily is taken into consideration To bridge the gaps between the first and the second case, the eleents of the transition atrix in the first case is partially unknown is considered In the following of this paper, a ethod to stabilize a plant sapled by saple rate adaptation and controlled over a WNCS with different delay TM(DTM) for different saple Copyright by the International Federation of Autoatic Control (IFAC) 45

2 Milano (Italy) August 8 - Septeber, rates is given The rest of this paper is organized as follows Proble forulation is presented in section The proble of stability analysis is introduced in section Section 4 gives the controller design A siulation result verifies the effectiveness of the presented ethod is acquired in section 5 Section 6 concludes all the work PROBLEM FORMULATION It is assued that there are at ost α different sapling rates for the saple rate adaptation ethod A index set S {,,, α}, here is used to represent different sapling rate with different nuber,,,, α The sequence θ(k) is used to indicate the index nuber of the sapling rate of the plant at tie k For exaple, if the sapling rate at tie k is the ith eleent in S, then we have θ(k) = i Consider a continuous tie linear syste which is sapled by the saple rate θ(k) as x(k + ) = Ã(θ(k))x(k) + B(θ(k))u(k) () where, x(k) R nx,u(k) R nu Ã(θ(k)), B(θ(k)) are constant atrices with appropriate diensions Next we assue the network induced delay takes values in set T {,,,, d} Under sapling rate θ(k), the delay ter is described by τ θ(k) = where is the th eleent in set T So takes values in T {,,, d+} We assue that the rando counication delay under sapling rate θ(k + ) in this work obeys the following hoogeneous DTM, π θ(k+) = π θ(k+), π θ(k+),d+ π θ(k+) d+, π θ(k+) d+,d+ () In order to obtain a better control result, θ(k) and x(k) are wrapped in one packet and sent to the controller over the delayed wireless network Then the delayed state feedback controller can be described as u(k) = K(θ(k τ θ(k) ))x(k τθ(k) ) () so () can be equivalently rewritten as x(k + ) = Ã(θ(k))x(k) + B(θ(k)) K(θ(k τ θ(k) ))x(k τθ(k) ) (4) By defining, θ k =[ θ(k),θ(k ),, θ(k d) ], K(θ k ) =[ K T (θ(k)),k T (θ(k )),, K T (θ(k d)) ] T, ξ(k) =[ x T (k),x T (k ),, x T (k d) ] T, R () =[ nu n u,, nu n u,i nu n u, nu n u,, nu n u ] R nu (d+)nu, R () =[ nx n x,, nx n x,i nx n x, nx n x,, nx n x ] R nx (d+)nx (5) where R () and R () have all eleents being zeros except for the th block being identity (4) is equivalent to ξ(k + ) = [A(θ k ) + B(θ k )R ()K(θ k )R ()]ξ(k) (6) with, Ã(θ(k)) A(θ k ) = I,B(θ k ) = I n B(θ(k)) (7) First, the changing rule of θ(k) governed by a hoogeneous syste TM(STM) Λ is considered, where, Pr{θ(k + ) = n θ(k) = } = λ n After that, the following two cases are considered as deductions, θ(k) changes arbitrarily and θ(k) changes stochastically with partial eleents unknown λ λ α Λ = (8) λ α λ αα Now we are in a position to give the definition of stochastic stability For ore details, we refer the readers to [] and the references therein Definition The closed-loop syste (6) is said to be stochastically stable if E( k= ξ(k) ξ ) < for the initial condition ξ ξ() R (τ+) n STOCHASTIC STABILITY ANALYSIS Lea [7] The closed-loop syste (6) possesses (d + )α d+ odes If θ(k) varies stochastically, the transition probability for θ k is γ s,t = λ i, δ(i, )δ(i, ) δ(i d+, d ) Where, θ k = [i,i,, i d ], θ k+ = [,,, d ], s = Φ(θ k ) = i + i α + i α + + i d α d, t = Φ(θ k+ ) = + α + α + + d α d, γ s,t = Pr{Φ(θ k+ ) = t Φ(θ k ) = s} Case : Stochastic variation The following theore solves the stabilization proble for (6) when the STM changes in the sense of stochastic variation Theore Syste (6) with piecewise hoogeneous DTM changing in the sense of stochastic variation is stochastically stable if there exist atrices P(s,) > such that the following atrix inequalities are satisfied, [ P P [A(s) + (B(s)K R ] i ())] < (9) where, A(s) = A(θ k ),B(s) = B(θ k ),K i = K(i ), P S λ i,, with n T π,np(η+, n), η = (i )α + (i )α + + (i d+ )α d Proof Consider a Lyapunov candidate as V (ξ(k),θ k,) = ξ T (k)p(φ(θ k ),)ξ(k) () We set θ k = [i,i,, i d ], Φ(θ k ) = s, = as the starting point At k +, we have θ k+ = [,,, d ], Φ(θ k+ ) = t, σ(k + ) = n Then with the transition probability atrices (), (8), we have (see the equation on the top of next page) 46

3 Milano (Italy) August 8 - Septeber, E[ V (ξ(k), θ k, )] =E[V (ξ(k + ), θ k+, σ(k + )) ξ(k), θ k, ] V (ξ(k), θ k, ) =ξ T (k + ) P(η +, n)pr(θ k+ =, σ(k + ) = n θ k = i, = )ξ(k + ) V (ξ(k), θ k, ) =ξ T (k + ) n T, S P(η +, n)pr(θ k+ = σ(k + ) = n, θ k = i, = )Pr(σ(k + ) = n n T, S θ k = i, = )ξ(k + ) V (ξ(k), θ k, ) =ξ T (k + ) λ i πnp(η +, n)ξ(k + ) V (ξ(k), θ k, ) S n T =ξ T (k)[(a + BK i R ()) T P (A + BK i R ()) P(s, )]ξ(k) If [A + BK i R ()] T P [A + BK i R ()] P(s,) < then (6) is stochastically stable, which is equal to (9) by Shur copleent Case : θ(k) Changing arbitrarily Corollary 4 Consider the syste (6),with the piecewise hoogeneous DTM atrix in the sense of arbitrary variation If there exist atrices P(s,) >, such that [A(s) + (B(s)K i R ())] < () where is defined in theore, the syste (6) is stochastically stable Proof This is oitted here, because it is siilar to the proof of theore Case : Λ known partially The above two cases are two extree exaples where all eleents of Λ are unknown, and all are known This sparks us to consider the following proble with partial eleents of Λ is known, aybe as?? λ λ λ λ??? where? represents the unknown eleents in Λ For i S, we define Q = Q i K Q i UK with Q i K { : λ i is known}, Q i UK { : λ i is uknown} () () Moreover, if Q i K, it is further described as Q i K {K i,,k i v}, v α (4) where K i v represents the vth known eleent in the ith row of atrix Λ We also denote λ i K Q λ i i for later K usage Reark 5 Fro the description above we see that case and case are two special cases of case when Q i K =, Q i UK = Q and Qi K = Q, Qi UK = Corollary 6 Consider the syste (6) with uping atrices () and () If there exist atrices P(s,) >,V (s,) >, i S, T such that Ω Ω[A + BKi R ()] < (5) holds, the syste (6) is stochastically stable Where if λ i K =, then Ω = Υ, other wise { Ω = Υ, Q i UK Ω = Υ i,otherwise with Υ, Υ i, P λ i K i, i P K Q λ i i K K (6) Proof According to theore, if (9) holds, the syste (6) is stochastically stable under the copletely known Λ If λ i K, the left side of (9) can be written as Ξ i = i i P K P K [A(s) + B(s)K i R ()] λ K P(p,s) + λ [A(s) + B(s)K i R ()] i, P(p,s) Q i UK (7) Therefor, if [ i i P K P K [A(s) + B(s)K R ] i ()] λ < (8) K P(p,s) and [ ] [A(s) + B(s)K i R ()] < (9) P(p,s) Then the syste (6) is stochastically stable under (6) 4 CONTROLLER DESIGN Theore 7 The closed-loop syste (6) is stochastically stable with a state feedback control law () and stochastically varying θ(k), if there exist atrices P(i,) > and K i with appropriate diensions, and the following LMI holds [ P I A(s) + (B(s)K R ] i ()) < () Proof The LMI () can be obtained fro (9) by ultiplying ( P ) at both leftside and rightside Then using the fact that X X I, we can get (), where X is any syetric positive atrix Corollary 8 The closed-loop syste (6) is stochastically stable under the state feedback control law () and arbitrarily varying θ(k), if there exist atrices P(i,) > and K i with appropriate diensions, and the following LMI [ holds ] I A(s) + (B(s)K i R ()) < () I 47

4 Milano (Italy) August 8 - Septeber, And, the closed-loop syste (6) is stochastically stable with a state feedback control law () and a θ(k) is governed by a partially known STM Λ, if there exist atrices P(i,) > and K i with appropriate diensions, and the following LMI holds Ω I A + BKi R () < () 4 x x 5 NUMERICAL EXAMPLE In this section, we present a nuerical exaple to show the application of the developed theory on the STM changes stochastically situation The other two cases presented in this paper can also be verified in the sae way Three discrete systes and two delay [ ters ] considered are described as follows, A =,B 6 8 =,A =,B 4 =,A =,B 7 =, τ θ(k) {,} In[ this situation, ] the [ transition ] atrices [ are] taken as π =,π 4 6 =, π 4 6 =,Λ = Fig 4 States traectory Assue the initial condition of the syste () to be x( ) = x( ) = x() = [,4] T By Theore, we can obtain a ode dependent controller K = [ 9 48],K = [ 867 ],K = [ 84] Soe siulation results of the resulting closed-loop syste are introduced Fig, Fig shows one of the possible realizations of the Markovian uping odes θ(k) and Under these ode sequences the corresponding state traectories of the closed-loop syste(6) is shown is Fig4 It is shown that the closed-loop syste is stochastically stable τ θ(k) τ(k) 6 CONCLUSION Fig Rando sequence of θ(k) Fig Rando sequence of τ θ(k) In this paper, the stochastically stabilization proble for a class of WNCS is considered The ethod eployed here can give out effective controllers with the changed delay distribution due to the saple rate adaption adaption adopted taken into consideration to stabilize the target plant The siulation result shows the effectiveness of the algorith given above REFERENCES [] Willing,A, Matheus, K, and Wolisz, A (5) Wireless technology in industrial networks Proc IEEE, 9, -5 [] Xiao,L,Johannsson, M,Hindi,H,Boyd,S,and Goldsith,A (5Joint optiization of wireless counication and networked control systes Lecture Notes Coput Sci, 55, 48-7 [] Thopson,HA(4)Wireless and internet counication technologies for onitoring and control Control Eng Pract,,78-79 [4] Iyappan Raachandran, Arinda K Das, Suit Roy (7) Analysis of the contenntion access period of IEEE 854 MACACM Transactions on Sensor Networks,, Article 4 [5] Junlin Xiong,Jaes La (6)Stabilization of discrete-tie Markovian up linear systes via tie-delayed controllers Autoatica, 4, [6] Xiao He, Zidong Wang, DHZhou (9) Robust fault detection for networked systes with counication delay and data issing Autoatica, 45, [7] Ming Liu, Daniel WC Ho, Yuyang Niu (9)Stabilization of Markovian up linear syste over networks with rando counication delay Autoatica, 45,

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