Networked Control Systems: A Time-Delay Approach

Kun Liu, Emilia Fridman and Laurentiu Hetel

Abstract— Networked control systems (NCSs) are spatiallydistributed systems for which the communication betweensensors, actuators, and controllers is supported by a sharedcommunication network. This paper gives an overview of thetime-delay approach to NCSs. Firstly, the modelling on theNetworked Control Systems (NCSs) with one sensor node isrecalled. Then, for the NCSs with distributed sensor nodesand Round-Robin (RR) protocol, the closed-loop system can bepresented as a switched system with multiple ordered delays.Next, a continuous-time hybrid system model for the closed-loopsystem, where time-varying delays appear in the dynamics andin the reset equations, is introduced to NCSs under RR andTry-Once-Discard (TOD) protocols. Lastly, the hybrid systemmodelling for the closed-loop system is extended to the discrete-time case. Differently from other approaches to NCSs in thepresence of scheduling protocols (in the frameworks of hybridand discrete-time systems), the time-delay approach allows thecommunication delays to be greater than the sampling intervals.

Keywords: Networked control systems, time-delay ap-proach, scheduling, Lyapunov-Krasovskii method.

I. INTRODUCTION

Networked Control Systems (NCSs) have received consid-erable attention in recent years (see e.g., [1]). In many suchsystems, only one node is allowed to use the communicationchannel at a time. The communication is orchestrated by ascheduling rule called a protocol. Three recent approachesfor NCSs are based on discrete-time systems [2] [6], impul-sive/hybrid systems [10] [20] and time-delay systems [3] [4][7].

Differently from the existing results on NCSs in the pres-ence of scheduling protocols (in the frameworks of hybridand discrete-time systems), the time-delay approach allowsthe communication delays to be greater than the samplingintervals.

This paper gives an overview of the time-delay approachto NCSs. Firstly, the modelling on the NCSs with onesensor node is recalled. Then, for the NCSs with distributedsensor nodes and Round-Robin (RR) protocol, the closed-loop system can be presented as a switched system withmultiple ordered delays. Next, a continuous-time hybridsystem model for the closed-loop system, where time-varyingdelays appear in the dynamics and in the reset equations, isintroduced to NCSs under RR and Try-Once-Discard (TOD)

Kun Liu is with the ACCESS Linnaeus Centre and School of ElectricalEngineering, KTH Royal Institute of Technology, SE-100 44, Stockholm,Sweden. kunliu@kth.se

Emilia Fridman is with the School of Electrical Engineering, Tel-AvivUniversity, 69978, Tel-Aviv, Israel. emilia@eng.tau.ac.il

Laurentiu Hetel is with University Lille Nord de France, LAGIS, FRECNRS 3303, Ecole Centrale de Lille, Cite Scientifique, BP 48, 59651 Vil-leneuve d’Ascq cedex, France. laurentiu.hetel@ec-lille.fr

Communication Network Delay

( )u tPlant

Controller

Delay

Sampler

ZOH

( ) ( ( ))k k ku t Kq y t ! "

( )ky s( )y t

Quantizer

( ( ))kq y s

Fig. 1. Networked static output-feedback control system

protocols. Lastly, the hybrid system modelling for the closed-loop system is extended to the discrete-time case.

Notation: Throughout the paper, the superscript ‘T ’ standsfor matrix transposition, Rn denotes the n dimensionalEuclidean space with vector norm | · |, Rn×m is the set ofall n × m real matrices, and the notation P > 0, for P ∈Rn×n means that P is symmetric and positive definite. Thesymmetric elements of the symmetric matrix will be denotedby ∗. Z+ and N denote the set of non-negative integersand positive integers, respectively. The symbol ∥ω[t0, t]∥∞stands for the essential supremum of the Euclidean norm|ω[t0, t]|, where ω : [t0, t] → Rnω . MATI and MADdenote the Maximum Allowable Transmission Interval andthe Maximum Allowable Delay, respectively.

II. TIME-DELAY SYSTEM MODEL

A. The description of NCS

Consider the system architecture in Figure 1 with plant

x(t) = Ax(t) +Bu(t), (1)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rm is thecontrol input, A and B are system matrices with appropriatedimensions. These matrices can be uncertain with polytopictype uncertainties. The initial condition is given by x(0) =x0.

The sampler is time-driven, whereas the controller andthe Zero-Order Hold (ZOH) are event-driven (in the sensethat the controller and the ZOH update their outputs assoon as they receive a new sample). We assume that themeasurement output y(sk) = Cx(sk) ∈ Rny is available atdiscrete sampling instants

0 = s0 < s1 < · · · < sk < · · · , limk→∞ sk = ∞,sk+1 − sk ≤ MATI, k ∈ Z+,

(2)

We consider data packet dropouts by allowing the sam-pling to be nonuniform. Thus, in our formulation y(sk), k ∈Z+ correspond to the measurements that are not lost.

2014 European Control Conference (ECC)June 24-27, 2014. Strasbourg, France

978-3-9524269-1-3 © EUCA 1434

We take into account the quantization effect. Let z ∈ Rny

be the vector being quantized. The quantizer is piecewiseconstant functions q: Rny → D, where D is a countablesubset. We assume that there exist real number ∆ > 0 suchthat |q(z)− z| ≤ ∆ holds. We are interested in the situationwhere only quantized measurement q(y(sk)) of the outputy(sk) ∈ Rny is available at the controller.

Denote by tk the updating instant time of the ZOH,and suppose that the updating signal at the instant tk hasexperienced a signal transmission delay ηk. Following [19],we allow the delays ηk to grow larger than sk+1 − sk,provided that the sequence of input update times tk remainsstrictly increasing. This means that if an old sample getsto the destination after the most recent one, it should bedropped.

Note that the first updating time t0 corresponds to the timeinstant when the first data is received by the actuator, whichmeans that u(t) = 0, t ∈ [0, t0). Then for t ∈ [0, t0), (1) isgiven by

x(t) = Ax(t), x(0) = x0, t ∈ [0, t0). (3)

The static output-feedback controller has a form u(tk) =Kq(y(tk − ηk)), where K is the controller gain. Thus,considering the behavior of the ZOH, we have

u(t) = Kq(y(tk − ηk)), t ∈ [tk, tk+1), k ∈ Z+ (4)

with tk+1 being the next updating instant time of the ZOHafter tk.

As in [19], assume that the network-induced delay ηk andthe time span between the updating and the most recentsampling instants are bounded:

tk+1−tk+ηk ≤ τM , 0 ≤ ηm ≤ ηk ≤ ηM = MAD, k ∈ Z+,(5)

where τM denotes the maximum time span between the time

sk = tk − ηk

at which the state is sampled and the time tk+1 at which thenext update arrives at the destination. Here ηm and ηM areknown bounds and τM = MATI+ηM .

Defining

τ(t) = t− tk + ηk, tk ≤ t < tk+1, (6)

we obtain the following closed-loop system (1), (4):

x(t) = Ax(t) +A1x(t− τ(t)) +B1ω(t), t ∈ [tk, tk+1),(7)

where A1 = BKC, B1 = BK, and

ω(t) = q(y(sk))− y(sk)

represents the quantization error, which satisfies |ω(t)| ≤ ∆,for t ∈ [tk, tk+1). Under (5) and (6), we have ηm ≤ τ(t) ≤tk+1 − tk + ηk < τM and τ(t) = 1 for t = tk.

Definition 1: The system (7) with essentially boundeddisturbance ω is Input-to-State Stable (ISS) if there exist

constants b > 0, α > 0 and c > 0 such that the followingholds for t ≥ t0

|x(t)|2 ≤ be−α(t−t0)∥xt0∥2W + c∥ω[t0, t]∥2∞

for the solutions of the system initialized with xt0 = ϕ ∈W [−τM , 0].

A standard time-independent Lyapunov-Krasovskii Func-tional (LKF) for ISS of (7) with fast varying delay τ(t) ∈[ηm, τM ] can be applied (see, e.g. [5], [9], [21]). In a recentpaper [3], time-dependent LKFs have been introduced forthe analysis of sampled-data systems in the framework oftime-delay approach. The introduced time-dependent termsof LKFs lead to qualitatively new results, taking into accountthe sawtooth evolution of the delays induced by sample-and-hold. The above discontinuous LKF constructions have beenextended to NCSs (see, e.g. [12], [13]).

III. ROUND-ROBIN PROTOCOL AND THE SWITCHEDSYSTEM MODEL

In this section, it is assumed that the NCS has distributedsensors, a controller node and an actuator node, which areconnected via networks. In many such systems, only onenode is allowed to use communication channel at a time.For the sake of simplicity, we consider (1) with two sensornodes yi(t) = Cix(t), i = 1, 2 and we denote C =

[C1C2

],

y(t) =[

y1(t)y2(t)

]∈ Rny . The results can be easily extended

to any finite number of sensors.We take into account the quantization effect. Let zi ∈

Rni , i = 1, 2, be the vectors being quantized. The quantizersare piecewise constant functions qi: Rni → Di, where Di isa countable subset, i = 1, 2. We assume that there exist realnumber ∆i > 0 such that |qi(zi)− zi| ≤ ∆i holds, i = 1, 2.

Consider a sequence of sampling instants (2). At eachsampling instant sk, one of the outputs yi(t) ∈ Rni(n1 +n2 = ny) is sampled, quantized and transmitted via thenetwork. We consider the RR scheduling protocol for thechoice of the active output node: the outputs are transmittedone after another, i.e., qi(yi(t)) is transmitted only at thesampling instant t = s2p+i−1, p ∈ Z+, i = 1, 2. Aftereach transmission and reception, the values in qi(yi(t)) areupdated with the newly received values, while the valuesof qj(yj(t)) for j = i remain the same, as no additionalinformation is received. This leads to the constrained dataexchange expressed as

qi(yik) =

qi(yi(sk)), k = 2p+ i− 1,qi(y

ik−1), k = 2p+ i− 1,

p ∈ Z+.

It is assumed that no packet dropouts and no packetdisorders will happen during the data transmission overthe network. The transmission of the information over thenetwork is subject to a variable delay ηk. Then tk = sk+ηkis the updating time instant. As in the previous section, weallow the delays to be non-small (larger than the samplingintervals) provided that the old sample cannot get to thedestination (to the controller or to the actuator) after thecurrent one. Assume that the network-induced delay ηk and

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the time span between the updating and the current samplinginstants satisfy (5). Then

tk+1 − tk ≤ τM − ηm,

tk+1 − tk−1 + ηk−1 ≤ 2τM − ηm∆= τM .

(8)

In [16], a time-delay approach was developed for the sta-bility of continuous-time NCSs with RR scheduling. Supposethat there exists a matrix K = [K1 K2], K1 ∈ Rm×n1 ,K2 ∈Rm×n2 such that A+BKC is Hurwitz. Consider the staticoutput feedback of the form:

u(t) = uk = K1q1(y1k) +K2q2(y

2k), t ∈ [tk, tk+1), k ∈ N,

and u0 = K1q1(y10). The closed-loop system can be pre-

sented in the form of the switched system:

x(t) = Ax(t) +A1x(tk−1 − ηk−1) +A2x(tk − ηk)+B1ω1(t) +B2ω2(t), t ∈ [tk, tk+1),

x(t) = Ax(t) +A1x(tk+1 − ηk+1) +A2x(tk − ηk)+B1ω1(t) +B2ω2(t), t ∈ [tk+1, tk+2),

(9)

where k = 2p− 1, p ∈ N, Ai = BKiCi, Bi = BKi, i =1, 2, and

ω1(t)=

q1(C1x(sk−1))− C1x(sk−1), t ∈ [tk, tk+1),q1(C1x(sk+1)− C1x(sk+1), t ∈ [tk+1, tk+2),

ω2(t) = q2(C2x(sk))− C2x(sk), t ∈ [tk, tk+2).

From (5), (8), we have

t ∈ [tk, tk+1) ⇒ t− tk + ηk ∈ [ηm, τM ],t− tk−1 + ηk−1 ∈ [ηm, τM ],

t ∈ [tk+1, tk+2) ⇒ t− tk + ηk ∈ [ηm, τM ],t− tk+1 + ηk+1 ∈ [ηm, τM ].

(10)

Moreover,

t− tk + ηk < t− tk−1 + ηk−1, t ∈ [tk, tk+1),t− tk + ηk > t− tk+1 + ηk+1, t ∈ [tk+1, tk+2).

For t ∈ [tk, tk+1) we can represent tk − ηk = t −τ1(t), tk−1 − ηk−1 = t− τ2(t), where

τ1(t) = t− tk + ηk < τ2(t) = t− tk−1 + ηk−1,τ1(t) ∈ [ηm, τM ], τ2(t) ∈ [ηm, τM ], t ∈ [tk, tk+1).

Therefore, (9) for t ∈ [tk, tk+1) can be considered as asystem with two time-varying interval delays, where τ1(t) <τ2(t). Similarly, for t ∈ [tk+1, tk+2) (9) is a system with twotime-varying delays, one of which is less than another.

For t ∈ [0, t0), the system is given by (3). For t ∈ [t0, t1)the closed-loop system is reduced to the following form

x(t)=Ax(t)+A1x(t0 − η0) +B1ω(t), t ∈ [t0, t1),

where ω(t) = q1(C1x0)− C1x0.The stability analysis of (9) follows from the standard

arguments for the switched and the time-delay systems (seee.g., [11], [3]), see more details in [16].

IV. TOD/RR PROTOCOL AND A HYBRID TIME-DELAYMODEL: CONTINUOUS-TIME CASE

In this section, we consider continuous-time NCSs underTOD or RR scheduling via a novel hybrid delayed model.The description of NCS follows from Section III.

A. A hybrid time-delay modelDenote by y(sk) = [yT1 (sk) yT2 (sk)]

T ∈ Rny the outputinformation submitted to the scheduling protocol. At eachsampling instant sk, one of the system nodes i ∈ 1, 2is active, that is only one of yi(sk) values is updated withthe recent output yi(sk). Let i∗k ∈ 1, 2 denote the activeoutput node at the sampling instant sk, which will be chosenaccording to TOD or RR protocol.

Following [15], consider the error between the systemoutput y(sk) and the last available information y(sk−1):

e(t) = cole1(t), e2(t) ≡ y(sk−1)− y(sk),

t ∈ [tk, tk+1), k ∈ Z+, y(s−1)∆= 0, e(t) ∈ Rny .

(11)The most recent output information at the controller level isdenoted by q(y(sk)) = [qT1 (y1(sk)) q2(y2(sk))]

T ∈ Rny .Assume that there exists a matrix K = [K1 K2], Ki ∈Rm×ni such that A + BKC is Hurwitz. Then, the staticoutput feedback controller has a form

u(t) = Ki∗kqi∗k(yi∗k(tk − ηk))

+Kiqi(yi(tk−1 − ηk−1))|i =i∗k,

t ∈ [tk, tk+1), k ∈ Z+,(12)

where i∗k is the index of the active node at sk and ηk iscommunication delay. We obtain thus the impulsive closed-loop model with the following continuous dynamics:

x(t) = Ax(t) +A1x(tk − ηk) +Biei(t)|i =i∗k

+∑2

i=1 Biωi(t),e(t) = 0, t ∈ [tk, tk+1),

(13)

where A1 = BKC, Bi = BKi, i = 1, 2, and

ωi(t) = qi(yi(sk))− yi(sk), i = i∗k,ωi(t) = qi(yi(sk−1))− yi(sk−1), i = i∗k,

(14)

represent the quantization errors, which satisfy |ωi(t)| ≤∆i, i = 1, 2, for t ∈ [tk, tk+1).

Taking into account (11), we obtain

ei(tk+1) = yi(sk)− yi(sk+1) = yi(sk)− yi(sk+1)= Cix(sk)− Cix(sk+1), i = i∗k,

andei(tk+1) = yi(sk)− yi(sk+1) = yi(sk−1)− yi(sk+1)

= yi(sk−1)− yi(sk) + yi(sk)− yi(sk+1)= ei(tk) + Ci[x(sk)− x(sk+1)], i = i∗k.

Thus, the delayed reset system is given by

x(tk+1) = x(t−k+1),ei(tk+1) = Ci[x(tk − ηk)−x(tk+1 − ηk+1)], i = i∗k,ei(tk+1) = ei(tk) + Ci[x(tk − ηk)− x(tk+1 − ηk+1)],

i = i∗k.(15)

Therefore, (13)–(15) is the hybrid model of the NCS. Sincex(tk − ηk) = x(t − τ(t)) for t ∈ [tk, tk+1) with τ(t) =t − tk + ηk ∈ [ηm, τM ] (cf. (5)), the hybrid system model(13)–(15) contains the piecewise-continuous delay τ(t) in thecontinuous-time dynamics (15). Even for ηk = 0 we have thedelayed state x(tk) = x(t − τ(t)) with τ(t) = t − tk. Theinitial conditions for (13)–(15) are given by (3) and e(t0) =−Cx(t0 − η0) = −Cx0.

1436

B. Scheduling protocols

TOD protocol: Let Q1 > 0, Q2 > 0 be some weightingmatrices. At the sampling instant sk, the weighted TODprotocol is a protocol for which the active output node isdefined as any index i∗k that satisfies

|√Qi∗k

ei∗k(t)|2 ≥ |

√Qiei(t)|2, t ∈ [tk, tk+1),

k ∈ Z+, i = 1, 2.(16)

A possible choice of i∗k is given by

i∗k = minarg maxi∈1,2

|√

Qi (yi(sk−1)− yi(sk)) |2.

RR protocol: The active output node is chosen periodi-cally:

i∗k = i∗k+2, for all k ∈ Z+, i∗0 = i∗1. (17)

C. ISS of the continuous-time hybrid delayed system underTOD/RR protocol

Definition 2: The hybrid system (13)–(15) with essen-tially bounded disturbances ω1, ω2 is said to be partiallyISS with respect to x (or x-ISS) if there exist constantsb > 0, α > 0 and c > 0 such that the following holdsfor t ≥ t0

|x(t)|2 ≤ be−α(t−t0)[∥xt0∥2W + |e(t0)|2

]+ c∥ω[t0, t]∥2∞

for the solutions of the hybrid system initialized with (3) ande(t0) ∈ Rny , where ω = colω1, ω2.

Note that the differential equation for x given by (13)depends on ei(t) = ei(tk), t ∈ [tk, tk+1) with i = i∗k only.Consider the following LKF:

Ve(t) = V (t, xt, xt) +tk+1−tτM−ηm

eTi (t)Qiei(t)|i =i∗k,

V (t, xt, xt) = V (t, xt, xt) + VQ,

VQ =∑2

i=1(τM−ηm)∫ t

ske2α(s−t)e2ατM |

√QiCix(s)|2ds,

V (t, xt, xt) = xT (t)Px(t)

+∫ t

t−ηme2α(s−t)xT (s)S0x(s)ds

+∫ t−ηm

t−τMe2α(s−t)xT (s)S1x(s)ds

+ηm∫ 0

−ηm

∫ t

t+θe2α(s−t)xT (s)R0x(s)dsdθ

+(τM − ηm)∫ −ηm

−τM

∫ t

t+θe2α(s−t)xT (s)R1x(s)dsdθ,

P > 0, Sj > 0, Rj > 0, α > 0, j = 0, 1,Q1 > 0, Q2 > 0, t ∈ [tk, tk+1), k ∈ Z+,

(18)where i∗k ∈ 1, 2, xt(θ)

∆= x(t + θ), θ ∈ [−τM , 0] and

where we define (for simplicity) x(t) = x0, t < 0. Theterm tk+1−t

τM−ηmeTi (tk)Qiei(tk) is inspired by the similar

construction of Lyapunov functionals for the sampled-datasystems [3], [18]. The following statement holds:

Lemma 1: if there exist positive constants α, b1, b2 andVe(t) of (18) such that along (13), (15), (16) ((13), (15),(17)) the following inequality holds

Ve(t) + 2αVe(t)−∑2

i=1 bi|ωi(t)|2 ≤ 0, t ∈ [tk, tk+1).

Then Ve(t) does not grow in the jumps along (13), (15), (16)((13), (15), (17)), where

Θ∆= Ve(tk+1)− Ve(t

−k+1) ≤ 0.

Moreover, the following bounds hold for a solution of(13), (15), (16) ((13), (15), (17)) initialized by xt0 ∈W [−τM , 0], e(t0) ∈ Rny :

V (t, xt, xt) ≤ e−2α(t−t0)Ve(t0) +12α

∑2i=1 bi∆

2i , t ≥ t0,

Ve(t0) ≤ V (t0, xt0 , xt0) + mini=1,2|√Qiei(t0)|2,

implying the ISS of (13), (15), (16) ((13), (15), (17)) withrespect to x.

The LMI conditions for x-ISS of (13), (15), (16) ((13),(15), (17)) can be derived by using Lemma 1 and the standardarguments for the delay-dependent analysis [21].

V. TOD/RR PROTOCOL AND A HYBRID TIME-DELAYMODEL: DISCRETE-TIME CASE

In this section, we present the discrete-time counterpart ofthe results obtained in the previous one.

A. Discrete-time hybrid time-delay modelThe discrete-time plant is given by

x(t+ 1) = Ax(t) +Bu(t), t ∈ Z+, (19)

where x(t) ∈ Rn is the state vector, u(t) ∈ Rnu is the controlinput, A and B are (probably, uncertain) system matriceswith appropriate dimensions. The initial condition is givenby x(0) = x0.

The NCS has several nodes (distributed sensors, a con-troller node and an actuator node) which are connected vianetworks. For the sake of simplicity, we consider two sensornodes yi(t) = Cix(t), i = 1, 2 and we denote C =

[C1C2

],

y(t) =[

y1(t)y2(t)

]∈ Rny , t ∈ Z+. We let sk denote

the unbounded and monotonously increasing sequence ofsampling instants, i.e.,

0 = s0 < s1 < · · · < sk < · · · , limk→∞ sk = ∞,sk+1 − sk ≤ MATI, k ∈ Z+,

(20)

where s0, s1, s2, . . . is a subsequence of 0, 1, 2, . . . .At each sampling instant sk, one of the outputs yi(t) ∈Rni(n1 + n2 = ny) is sampled, quantized and transmittedvia the network.

As in the previous Sections III and IV, it is assumed thatno packet dropouts and no packet disorders will happen dur-ing the data transmission over the network. The transmissionof the information over the network is subject to a variabledelay ηk ∈ Z+, which is allowed to be non-small. Thentk = sk+ηk is the updating time instant of the ZOH device.

Assume that the network-induced delay ηk and the timespan between the updating and the current sampling instantsare bounded:

tk+1 − 1− tk +ηk ≤ τM , 0 ≤ ηm ≤ ηk ≤ ηM , k ∈ Z+,

where τM , ηm and ηM are known non-negative integers.Then

(tk+1 − 1)− sk = sk+1 − sk + ηk+1 − 1≤ MATI+ηM − 1 = τM ,

(tk+1 − 1)− sk−1 = sk+1 − sk−1 + ηk+1 − 1≤ 2MATI+ηM − 1

= 2τM − ηM + 1∆= τM ,

tk+1 − tk ≤ τM − ηm + 1.

1437

Following (11), consider the error between the systemoutput y(sk) and the last available information y(sk−1):

e(t) = cole1(t), e2(t) ≡ y(sk−1)− y(sk),t ∈ [tk, tk+1 − 1], t ∈ Z+, k ∈ Z+,

y(s−1)∆= 0, e(t) ∈ Rny .

The control signal to be applied to the system (19) is givenby

u(t) = Ki∗kqi∗k(yi∗k(tk − ηk)) +Kiqi(yi(tk−1 − ηk−1))|i =i∗k

,

t ∈ [tk, tk+1 − 1], t ∈ Z+,

where K = [K1 K2], K1 ∈ Rnu×n1 ,K2 ∈ Rnu×n2 suchthat A + BKC is Schur. The closed-loop system can bepresented as

x(t+ 1) = Ax(t) +A1x(tk − ηk) +Biei(t)|i =i∗k,

+∑2

i=1 Biωi(t),e(t+ 1) = e(t), t ∈ [tk, tk+1 − 2], t ∈ Z+,

(21)

with the delayed reset system for t = tk+1 − 1

x(tk+1) = Ax(tk+1 − 1) +A1x(tk − ηk) +Biei(tk)|i =i∗k,

ei(tk+1) = Ci[x(tk − ηk)− x(tk+1 − ηk+1)], i = i∗k,ei(tk+1) = ei(tk) + Ci[x(tk − ηk)− x(tk+1 − ηk+1)],

i = i∗k,(22)

where A1 = BKC, Bi = BKi, K = [K1 K2], i = 1, 2,and ωi(t), i = 1, 2 have the form of (14) with t ∈ [tk, tk+1−2], t ∈ Z+. The initial condition for (21)–(22) has the formof e(t0) = −Cx(t0 − η0) = −Cx0 and

x(t+ 1) = Ax(t), t = 0, 1, . . ., t0 − 1, t ∈ Z+. (23)

We will consider ISS of the discrete-time hybrid system(21)–(22) under TOD/RR protocol. The TOD and RR pro-tocols are given by (16) with t ∈ [tk, tk+1 − 1], t ∈ Z+ and(17), respectively.

B. ISS of the discrete-time hybrid delayed system underTOD/RR protocol

Consider the LKF of the form:

Ve(t) = V (t) + tk+1−tτM−ηm+1e

Ti (tk)Qiei(tk)|i =i∗k

,

V (t) = V (t) + VQ(t),

VQ(t) = (τM − ηm)∑t−1

s=tk−ηkλt−s−1ζT (s)Qζ(s),

V (t) = xT (t)Px(t) +∑t−1

s=t−ηmλt−s−1xT (s)S0x(s)

+∑t−ηm−1

s=t−τMλt−s−1xT (s)S1x(s)

+ηm∑−1

j=−ηm

∑t−1s=t+j λ

t−s−1ζT (s)R0ζ(s)

+(τM − ηm)∑−ηm−1

j=−τM

∑t−1s=t+j λ

t−s−1ζT (s)R1ζ(s),

ζ(t+ 1) = x(t+ 1)− x(t),P > 0, Si > 0, Ri > 0, Q > 0, Qj > 0, 0 < λ < 1,i = 0, 1, j = 1, 2, t ∈ [tk, tk+1 − 1], t ∈ Z+, k ∈ Z+,

(24)where we define x(t) = x0, t ≤ 0.

Our objective is to guarantee that

Ve(t+ 1)− λVe(t)−∑2

i=1 bi|ωi(t)|2 ≤ 0,t ∈ [tk, tk+1 − 1], t ∈ Z+,

(25)

holds along (21)–(22). The inequality (25) implies the fol-lowing bound

V (t) ≤ Ve(t) ≤ λt−t0Ve(t0) +1

1−λ

∑2i=1 bi∆

2i ,

Ve(t0) ≤ V (t0) + mini=1,2|√Qiei(t0)|2, t ≥ t0, t ∈ Z+,

(26)for the solution of (21)–(22) with the initial condition (23)and e(t0) ∈ Rny . Here we took into account that for the caseof two sensor nodes

|√Qiei(t0)|2|i =i∗k

= mini=1,2

|√Qiei(t0)|2.

From (26), it follows that the system (21)–(22) is ISS withrespect to x.

The novel term VQ(t) of LKF is inserted to cope with thedelays in the reset conditions:

VQ(tk+1)− λVQ(tk+1 − 1)

= (τM − ηm)∑tk+1−1

s=tk+1−ηk+1λtk+1−s−1ζT (s)Qζ(s)

−(τM − ηm)∑tk+1−2

s=tk−ηkλtk+1−s−1ζT (s)Qζ(s)

≤ (τM − ηm)ζT (tk+1 − 1)Qζ(tk+1 − 1)

−(τM − ηm)λτM∑tk+1−ηk+1−1

s=tk−ηkζT (s)Qζ(s)

≤ (τM − ηm)ζT (tk+1 − 1)Qζ(tk+1 − 1)−λτM |

√Q[x(tk+1 − ηk+1)− x(tk − ηk)]|2,

where we applied Jensen’s inequality (see e.g., [8]). Theterm tk+1−t

τM−ηm+1eTi (tk)Qiei(tk) is inspired by the similar

construction of LKF for the sampled-data systems [3].We have

Ve(tk+1)− λVe(tk+1 − 1) = V (tk+1)− λV (tk+1 − 1)

+ tk+2−tk+1

τM−ηm+1eTi (tk+1)Qiei(tk+1)|i =i∗k+1

− λτM−ηm+1e

Ti (tk)Qiei(tk)|i =i∗k

+(τM − ηm)ζT (tk+1 − 1)Qζ(tk+1 − 1)−λτM |

√Q[x(tk+1 − ηk+1)− x(tk − ηk)]|2.

(27)Note that under TOD protocol for i∗k+1 = i∗k

eTi (tk+1)Qiei(tk+1)|i=i∗k+1≤ eTi∗k

(tk+1)Qi∗kei∗k(tk+1),

(28)whereas for i∗k+1 = i∗k the latter relation holds with theequality. Under RR protocol we have i∗k+1 = i∗k meaningthat (28) holds with the equality. Hence

tk+2−tk+1

τM−ηm+1eTi (tk+1)Qiei(tk+1)|i =i∗k+1

≤ eTi∗k(tk+1)Qiei∗k(tk+1)

= |√Qi∗k

Ci∗k[x(tk+1 − ηk+1)− x(tk − ηk)]|2.

Assume that

λτMQ > CTi QiCi, i = 1, 2. (29)

Then for t = tk+1 − 1 we obtain

Ve(t+ 1)− λVe(t) ≤ V (t+ 1)− λV (t)+(τM − ηm)ζT (t)Qζ(t)− λ

τM−ηm+1eTi (tk)Qiei(tk)|i =i∗k

.

Furthermore, due to

− λτM−ηm+1 = − 1

τM−ηm+1 + 1−λτM−ηm+1

≤ − 1τM−ηm+1 + 1− λ,

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for t = tk+1 − 1 we arrive at

Ve(t+1)−λVe(t)≤ V (t+1)−λV (t)+(τM−ηm)ζT (t)Qζ(t)

−[ 1τM−ηm+1 − (1− λ)]eTi (tk)Qiei(tk)|i =i∗k

∆= Ψ(t).

(30)For t ∈ [tk, tk+1 − 2] we have

Ve(t+1)−λVe(t)≤ V (t+1)−λV (t)+(τM−ηm)ζT (t)Qζ(t)

+[

tk+1−t−1τM−ηm+1 − λ tk+1−t

τM−ηm+1

]eTi (tk)Qiei(tk)|i =i∗k

.

Sincetk+1−t−1τM−ηm+1 − λ tk+1−t

τM−ηm+1 = − 1τM−ηm+1 + (1− λ) tk+1−t

τM−ηm+1

≤ − 1τM−ηm+1 + 1− λ,

we conclude that (30) is valid also for t ∈ [tk, tk+1 − 2].Therefore, (25) holds if

Ψ(t)−2∑

i=1

bi|ωi(t)|2 ≤ 0, t ∈ [tk, tk+1 − 1]. (31)

Note that i = i∗k for i = 1, 2 is the same as i = 3 − i∗k.By using the standard arguments for the delay-dependentanalysis [21], we derive the following conditions for (31)(and, thus for (26)):

Theorem 1: Given scalar 0 < λ < 1, positive integers0 ≤ ηm ≤ ηM < τM , and K1,K2, if there exist positivescalars b1, b2, n × n matrices P > 0, Q > 0, Sj > 0,Rj > 0(j = 0, 1), S12, ni × ni matrices Qi > 0(i = 1, 2)such that (29) and

Ω =

[R1 S12

∗ R1

]≥ 0,

FT0 PF0+Σ+FT

1 WF1−ληmFT2 R0F2−λτMFTΩF <0,

are feasible, where

F0 = [A 0n×n A1 0n×n B3−i B1 B2],F1 = [A−In 0n×n A1 0n×n B3−i B1 B2],F2 = [In − In 0n×(2n+2ny−ni)],

F =

[0n×n In −In 0n×n 0n×(2ny−ni)

0n×n 0n×n In −In 0n×(2ny−ni)

],

Σ = diagS0 − λP,−ληm(S0 − S1), 0n×n,−λτMS1, φ,−b1In1 ,−b2In2,

φ = −[ 1τM−ηm+1

− (1− λ)]Q3−i,

W = η2mR0 + (τM − ηm)2R1 + (τM − ηm)Q, i = 1, 2.

Then solutions of the hybrid system (21)–(22) satisfy thebound (26), implying ISS of (21)–(22) with respect to x.Moreover, if the above inequalities are feasible with λ = 1,then the bound (26) holds with λ = 1 − ε, where ε > 0 issmall enough.

Remark 1: The inequality Ve(t) ≤ λt−t0Ve(t0) +1

1−λ

∑2i=1 bi∆

2i , t ≥ t0, t ∈ Z+ in (26) guarantees that

tk+1 − tkτM − ηm + 1

eTi (tk)Qiei(tk)|i =i∗k

is bounded, and it does not guarantee that e(tk) is bounded.That is why (26) implies only partial ISS with respect to x.

For application of the presented approach in this section todiscrete-time NCSs under scheduling protocols and actuatorconstraints, see [14].

VI. CONCLUSIONS

In this paper, we presented an overview of time-delayapproach to NCSs with non-small communications delays.Under RR and TOD protocols, it is assumed that no packetdropouts will happen during the data transmission overthe network. Note that the time-delay approach has beendeveloped to NCSs under stochastic protocol in [17], wherethe network-induced delays are allowed to be larger than thesampling intervals in the presence of collisions.

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