Robust Stabilization of Networked Control Systems withMultiple-packet Transmission via Jump System Approach

Wang Haixiao 1, Yu Mei1, Zhang Weiliang1

1. School of Control and Computer Engineering,North China Electric Power University Beijing,102206, P. R. ChinaE-mail: meiyu@ncepu.edu.cn

Abstract: A jump system approach to stabilization and robust stabilization of networked control systems (NCSs) with multiple-packet transmissions are addressed. We focus our attention on the case that the packets are transmitted via limited capacitycommunication channels. Sufficient conditions on the mean square stabilization of NCSs are obtained in terms of linear matrixinequalities (LMIs). Non-fragile state feedback controller can be constructed directly via LMIs. A numerical example is workedout to demonstrate the effectiveness of the proposed method.

Key Words: Networked control systems, multiple-packet transmission, jump system, robust stabilization, LMIs

1 Introduction

In recent years, networked control systems (NCSs) has re-ceived increasing attention in control field [9],[15]. NCSsare feedback control systems with the control system com-ponents (sensors, controller, actuator, etc) connected via areal-time network. NCSs have many attractive advantages,such as reduce the cost of cable and power, ease of sys-tem diagnosis and maintenance, and increase the reliabil-ity. However, the insertion of communication network in thefeedback control loop complicates the analysis and design ofan NCS because many ideal assumptions made in the tradi-tional control theory can not be applied to NCSs directly. Itshould be noted that there are some inevitable unfavorableeffects, such as communication constraints [7]-[9], uncer-tainty caused by network transmission [6], multiple packetstransmissions and packet dropouts [16, 20]. These problemsmay deteriorate the control performance or even destroy thestability of the system. So how to control such NCSs is a bigproblem.

Reference [19] models networked control system withmultiple-packet transmission as a linear switching modelwith certain switching rules. Output feedback controller isconstructed in terms of linear matrix inequalities (LMIs). In[21], multiple-packet transmission and short time delay isconsidered, where the packet transmission sequence is de-terministic. But in the real networked control system withmultiple-packet transmission, the packet transmission se-quence is generally non-deterministic. A lot of work hasbeen committed to a class of stochastic linear systems sub-ject to variations governed by a Markov process. Refer-ence [2] studies stochastic stability, stabilizability and andH∞ disturbance attenuation of discrete-time linear time-delay systems with Markovian jumping parameters using thestochastic Lyapunov function approach. However, multiple-packet transmission is not considered in this paper. [4]considers the case of multiple-packet transmission underdynamic scheduling strategy. It is modeled as a class ofstochastic system, the stability of the system is analyzed byusing the method of stochastic system. None of them has

*This work is supported by the National Natural Science Foundationof China under grants ( No.61174096 and No.61104141), the FundamentalResearch Funds for the Central Universities.

considered the effects of the uncertainties.Uncertainty is common in control systems and it in-

evitably exists in system modeling due to the complexityof the system itself, exogenous disturbance, measurementerrors and so on. Some results have studied the effects ofthe uncertainties for NCSs. [5] concerns the problem ofH∞ output tracking for NCSs. Both network-induced de-lays and data packet dropouts have been taken into con-sideration. [12] discusses robust H∞ control problems forNCSs with time delays and subject to norm-bounded param-eter uncertainties. In [17], robust H∞ control is consideredfor a class of networked systems with random communica-tion packet losses and norm-bounded parameter uncertain-ties. An observer-based feedback controller is designed torobustly exponentially stabilize the system by solving cer-tain LMIs. However, none of them has considered the effectof the uncertainties in the controller and the effect of limitedcommunication. It is necessary to introduce some uncertainparameters in the modeling of NCSs, and design robust sta-bilizing controllers.

Motivated by the references above, this paper propose ajump system approach to stabilize NCSs with uncertaintiesand multiple-packet transmission. The main contribution ofthis paper is to develop non-fragile stabilizing controllersfor such NCSs. Constant non-fragile stabilizing controllersare developed for the NCSs in terms of linear matrix in-equalities, which can be easily solved by Matlab toolbox[1]. Moreover, the constant controllers are often effectiveand are easy to implement and maintain by plant personnel.For simplicity, we consider the sensors are clock-driven, thecontroller and the actuator are event-driven. The controllerwill use the old state measurement if there is no new dataupdating.

This paper is structured as follows. Section 2 models un-certain NCSs with multi-packet transmission as jump linearsystems. Section 3 presents robust stabilization and non-fragile state feedback controller design results for state feed-back case. Section 4 presents a numerical simulation to il-lustrate the efficiency of our approach. Section 5 concludesthis paper.

Notation: We use standard notations throughout this es-say. Denote AT the transpose of a matrix A. A > 0(A < 0)means that A is positive definite (negative definite). I is the

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- x(k + 1) = (A + ∆A)x(k) + Bu(k)

X1(k)

Xd(k)

ppp

p p pq q

network

¶¶/S1 Sdeeq

¾ ¾registerx(k)

F + ∆F

Fig. 1: An NCS with multiple-packet transmission

identify matrix of appropriate dimension.

2 System ModelingIn distributed NCSs, due to the wide location of sensors

whose information length may surpass the capacity of net-work, so multiple-packet transmission is necessary. We con-sider the case that all the nodes communicate via a limitedbandwidth communication channel. State information of thesystem is split into different packets and only one packet canbe transmitted at a time. The packet transmission process isgoverned by a Markov chain [10] with finite state-space.

An NCS with multiple-packet transmission can be de-scribed in Fig.1. The uncertain NCS is expressed by

x(k + 1) = (A + ∆A)x(k) + Bu(k), (1)

and the controlled input is

u(k) = (F + ∆F )x(k) k = 1, 2, · · · , (2)

where x(k) ∈ Rn, u(k) ∈ Rp are the plant state and theplant input, respectively. A, B are known real constant ma-trices with appropriate dimensions. F is the feedback gain tobe designed. x(k) is the content of the register, ∆F is varia-tions of the controller gain and ∆A characterizes the uncer-tainty in the system, which satisfy the following assumption.

∆A = EΓ(k)H, (3)

∆F = GΓ1(k)J, (4)

with E, H , G and J being known real constant matrix ofappropriate dimensions. Γ(k),Γ1(k) are unknown matrixfunction with Lebesgue-measurable elements which satisfiesΓ(k)T Γ(k) ≤ I,Γ1(k)T Γ1(k) ≤ I,∀k.

We assume that the state is split into d pack-ets x(k) = [XT

1 (k), · · · , XTd (k)], where Xi(k) =

[xri−1+1(k), · · · , xri(k)]T . The controller will use thecontent of the register x(k) = [XT

1 (k), · · · , XTd (k)], where

Xi(k) =

Xi(k), if the packet containing Xi(k) istransmitted;

Xi(k − 1), otherwise.

The packet transmission sequence of sensor nodes is non-deterministic under dynamic scheduling strategy. The statesof markov chain expressed by Sk are utilized to model thepacket transmission process, which takes values in a finiteset S = 0, 1, · · · , d, where

Sk =

i, If the ith packet’s state information is transmitted.0, otherwise

The markov chain with a transition probability matrixP = bPijc is given by

Pij = ProbSk+1 = j|Sk = i, Pij ≥ 0,d∑

j=0

Pij = 1, i,j ∈ 0,· · · ,d. (5)

Define diagonal matrix Λsk=

diag(δsk,1, δsk,2, · · · , δsk,d), where

δsk,j =

Ir×r, Sk = j,0, Sk 6= j (j = 0, 1,· · · ,d).

thenx(k) = Λsk

x(k) + (I − Λsk)x(k − 1),

and the controller is

u(k) = (F + ∆F )[Λskx(k) + (I − Λsk

)x(k − 1)].

Now, we can write the evolution of the closed-loop systemas

x(k + 1) = ((A + ∆A) + B(F + ∆F )Λsk)x(k)

+B(F + ∆F )(I − Λsk)x(k − 1). (6)

DefineZ(k) =

[xT (k) xT (k − 1)

]T,

A = A + ∆A, F = F + ∆F.

The generalized jump linear system is written as

Z(k + 1) = ΦskZ(k), (7)

where

Φsk=

[A + BFΛsk

BF (I − Λsk)

ΛskI − Λsk

].

3 Stability Analysis and Stabilization Result

The NCS (7) is a general jump linear sys-tem with the Markovian integer jump parameterΦsk

∈ Φ0,Φ1, · · · ,Φd. Clearly, the original system(1)-(2) is stable if system (7) is stable. So we only need to provethat system (7) is stable.

First, we study the stabilization of the NCS (7) withoutuncertainties. We know the closed-loop system without un-certainties can be described as:

Z(k + 1) = ΨskZ(k), (8)

where

Ψsk=

[A + BFΛsk

BF (I − Λsk)

ΛskI − Λsk

].

Before proceeding, we need the following definition.

Definition 1. [11] NCS (1)-(2) is mean square (MS)asymptotically stable, if for all initial state (Z0, S0),limk→∞Eb‖Z(k)‖2c = 0 holds, where E is the statisticalexpectation operator.

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Below, we present necessary and sufficient matrix in-equality conditions on MS stable of the system (8), whichhas been proved in [3].

Lemma 1. [3] Jump linear system (8) is said to be MSstable, if there exist symmetric positive definite matricesQm, (m = 0, 1, · · · , d) satisfying the following condition:

ΨTm(

d∑n=0

PmnQn)Ψm < Qm, m=0,1,· · · ,d (9)

With a freedom matrix S introduced, we will develop newsufficient conditions on the stabilization of the NCS. The fol-lowing lemma is a different form of Theorem 1 in [13].

We present the following MS stabilization result on NCS(8).

Lemma 2. Jump linear system in (8) is said to be MSstable, if there exist symmetric positive definite matricesQm, (m = 0, 1, · · · , d), and matrices K, Y, S satisfyingSB = BK and the following LMI:

Qm√

pm0WT √

pm1WT · · · √

pmdW T

∗ S + ST − Q0 0 · · · 0

∗ ∗ S + ST − Q1 · · · 0...

......

. . ....

∗ ∗ ∗ · · · S + ST − Qd

> 0,

(10)then NCS (8) can be MS stabilized with the state feedback

gain

F = K−1Y, (11)

where

W =[

SA + BY ΛskBY (I − Λsk

)SΛsk

S(I − Λsk)

],

Qm =[

Qm 00 Qm

], S =

[S 00 S

]. (12)

The following lemma will play a key rule to give a suffi-cient condition on the robust mean square stability of NCS(1).

Lemma 3. [14] Let M , N , Λ be given matrices of compati-ble dimensions with Λ satisfying ΛT Λ ≤ I, then the follow-ing inequality holds for any positive scalar ε > 0

MΛN + NT ΛT MT ≤ εMMT + ε−1NT N

In the following part we consider the robust mean squarestabilization of NCS(1).

Theorem 1. NCS (1)-(2) is robust MS stable if there existsymmetric positive definite matrices Qm(m = 0, 1, · · · , d),positive scalars ε , ε and matrices K, Y, S satisfying SB =BK and

−Ω1 ¯MT ¯N ¯LT ¯V¯M −Ω2 0 0 0¯NT 0 −Ω3 0 0¯L 0 0 −Ω4 0

¯V T 0 0 0 −Ω5

< 0, (13)

then NCS can be MS stabilized with the state feedback gain

F = K−1Y, (14)

where

W =[

SA + BY ΛskBY (I − Λsk

)SΛsk

S(I − Λsk)

],

Ω1 =

Qm√

pm0WT √

pm1WT

∗ S + ST − Q0 0∗ ∗ S + ST − Q1

......

...∗ ∗ ∗

· · · √pmdW

T

· · · 0· · · 0. . .

...· · · S + ST − Qd

,

Ω2 = Ω3 =

εI 0 · · · 00 εI · · · 0...

.... . .

...0 0 · · · εI

,

Ω4 = Ω5 =

εI 0 · · · 00 εI · · · 0...

.... . .

...0 0 · · · εI

,

¯M =

εM 0 · · · 00 0 · · · 0...

.... . .

...0 0 · · · 0

,M =

[ −HT 00 0

],

¯L =

εL εL · · · εL0 0 · · · 0...

.... . .

...0 0 · · · 0

,

L =[ −(JΛsk

)T 0−(J(I − Λsk

))T 0

],

N = ¯N =

0 Nm0 · · · Nmd

0 0 · · · 0...

.... . .

...0 0 · · · 0

,

Nmn =[ √

pmnET ST 00 0

],

¯V =

0 Vm0 · · · Vmd

0 0 · · · 0...

.... . .

...0 0 · · · 0

,

Vmn =[ √

pmn(SBG)T 00 0

], (n = 0, · · · , d).

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Proof. From lemma 2, system (7) is robust mean squarestable if there exist symmetric positive definite matricesQm(m = 0, 1, · · · , d) satisfy:

Qm√

pm0(SΦsk)T √pm1(SΦsk)T · · · √

pmd(SΦsk)T

∗ S + ST − Q0 0 · · · 0

∗ ∗ S + ST − Q1 · · · 0...

......

. . ....

∗ ∗ ∗ · · · S + ST − Qd

> 0,(15)

where Qm, S are defined by (12). Inequality (15) can berewritten as:

Qm√

pm0WT √

pm1WT · · · √

pmdW T

∗ S + ST − Q0 0 · · · 0

∗ ∗ S + ST − Q1 · · · 0...

......

. . ....

∗ ∗ ∗ · · · S + ST − Qd

+

0√

pm0ΥT √pm1ΥT · · · √

pmdΥT

∗ 0 0 · · · 0

∗ ∗ 0 · · · 0...

......

. . ....

∗ ∗ ∗ · · · 0

> 0,

(16)

where Υ =[

S∆A + SB∆FΛsk SB∆F (I − Λsk )0 0

].

Then LMI (16) can be rewritten as:

−Ω1+ ¯M ¯Γ(k)N+NT ¯Γ(k)T ¯MT +¯L ¯Γ1(k)V +V T ¯Γ1(k)T ¯LT < 0,

where ¯Γ(k) = diagΓ(k), Γ(k), · · · , Γ(k), Γ =diagΓ(k),Γ(k), · · · ,Γ(k), ¯Γ1(k) = diagΓ1(k), Γ1(k), · · · , Γ1(k),Γ1 = diagΓ1(k),Γ1(k), · · · ,Γ1(k).

From lemma 3, we know LMI (16) is satisfied if thereexist positive scalars ε , ε satisfie:

−Ω1+ε ¯M ¯MT +ε−1 ¯NT ¯N +ε ¯L ¯LT +ε−1 ¯V T ¯V < 0, (17)

By Schur complements, we know inequality (17) is equiv-alent to (16), then NCS (7) can be robust MS stabilized withthe state feedback gain

F = K−1Y.

4 Simulation ExamplesTo illustrate the efficiency of the proposed method, we

give the following example.Example 1. Consider the following NCS:

x(k + 1) = (

[ −0.1 30 −1.01

]+

[0.001−0.002

]

∗Γ(k)[

0.002 0.003])x(k) +

[01

]u(k), (18)

u(k) = (F +[

0.02 0.03]Γ1(k)0.01)x(k). (19)

The probability of packet transmission is expressed by aMarkov transition probability matrix P with

P =

0.48 0.51 0.010.499 0.499 0.0020.48 0.51 0.01

.

Case (1): let Γ(k) = 0 ,Γ1(k) = 0 which means there isno uncertainty in the system.

Solving the LMIs in lemma 2 with LMI toolbox [1], weget

S =[

0.2116 0.04810.0481 0.6983

], Q0 =

[ −0.5836 0.07970.0797 0.3651

],

Q1 =[ −0.5904 0.0787

0.0787 0.3567

], Q2 =

[0.2197 0.04730.0473 0.8868

],

and the feedback gain F = [0 0.8127]. With the initialcondition x(0) = [10 − 10]T , the state values of the NCSwith multiple packet transmission is shown in Fig.2.

0 20 40 60 80 100−40

−30

−20

−10

0

10

20

30

40

Step number

Sta

te v

alue

s

Fig. 2: The state values of the NCS in case (1)

Case (2): let Γ(k) = sin(20k), Γ1(k) = sin(50k) whichmeans there is a certain degree of uncertainty in system.

Solving the LMIs in Theorem 1 with LMItoolbox [1], we obtain ε = 108.2585, ε =

108.2515, Q1 =[ −98.7895 2.8417

2.84174 29.3105

], Q2 =

[31.1067 4.77384.7738 139.6529

], and the mode-independent

controller F = [0.0014 0.9247]. With the initial conditionx(0) = [10 − 10]T , the state values of the NCS withmultiple packets transmission is shown in Fig.3.

It can be seen from the figures that the NCS is robustMS stable. This example illustrates that the jump systemapproach proposed in this paper leads to effective results.It only requires plant state measurements be transmittedsparsely and the packet transmission sequence in the NCSis statistic.

5 Conclusion

In this paper, we dealt with stability and stabilizationof uncertain NCS with multiple packet transmitted over a

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0 20 40 60 80 100−30

−20

−10

0

10

20

30

40

Step number

Sta

te v

alue

s

Fig. 3: The state values of the NCS in case (2)

shared channel. It was modeled as a class of discrete-timejump system. The packets transition sequence of the sys-tems was governed by a finite state Markov chain. Suffi-cient condition on robust stability of the NCS was derivedand a state feedback controller design method was also ob-tained. Finally, we gave an numerical example to illustratethe feasibility and effectiveness of our approach. The re-sults suggested that data packet could be transmitted sparselyto save network bandwidth while preserving the stability ofthe NCS. The packet transmission sequence in the NCS wasnon-deterministic. This was of practical interest in the appli-cation of NCSs.

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