PURE TIME DELAY ANALYSIS FORDECENTRALIZED NETWORKED CONTROL SYSTEMS

Ahmad F. TahaPh.D. Candidate

School of Electrical and Computer EngineeringPurdue University

West Lafayette, IN 47906Email: tahaa@purdue.edu

Ahmed ElmahdiResearch Associate

School of Aeronautics and AstronauticsPurdue University

West Lafayette, Indiana 47907Email: aelmahdi@purdue.edu

Jitesh H. PanchalAssistant Professor

School of Mechanical EngineeringPurdue University

West Lafayette, Indiana 47907Email: panchal@purdue.edu

Dengfeng SunAssistant Professor

School of Aeronautics and AstronauticsPurdue University

West Lafayette, Indiana 47907Email: dsun@purdue.edu

ABSTRACT

The network disturbance effect can be considered as eithera perturbation or as a pure time delay for the exchanged signals.The network-induced time delay is one of the main challengeswhen a network is inserted in the feedback loops of a control sys-tem. In this paper, our objective is to improve the behavior ofa Networked Control System (NCS) by analyzing the time-delaygiven that the decentralized control design method is adopted.First, we review an observer-based control method for decen-tralized control systems. Second, we establish a map between thedecentralized non-networked system, and the typical NCS state-space representation. The main idea the mapping is to put theDecentralized Networked Control System (DNCS) in a generalform and then map the resulting system to the typical NCS form.Third, we derive the global dynamics of the DNCS. Fourth, anupper bound for the time-delay is derived that guarantees thestability of LTI DNCSs. Finally, we present a numerical examplethat illustrates the applicability of the derived bound.

1 INTRODUCTION

Recent advances in integrated circuits and computer net-works technologies in addition to the integration of physicalsystems with communication networks have led to the emer-gence of inexpensive large-scale Cyber Physical Systems (CPS).Various CPSs are monitored and controlled by Networked Con-trol Systems (NCS) [1]. Wireless distributed robotic systems,power networks, and automotive vehicles are some of the mostcomplex large-scale NCSs that are inherently decentralized anddistributed. These systems integrate complex distributed controlsystems and modern communication networks. Hence, they areviewed as Decentralized Networked Control Systems (DNCS),which is an important sub-class of NCSs. In addition to thestakeholders’ decision making aspect of these systems, theoverall performance and resilience of CPSs and DNCSs aresignificantly influenced by the network communication andcontrol aspect for such systems.

NCSs technologies have great impact in industrial systemswhere all levels of system components are being integrated

1 Copyright © 2014 by ASME

Proceedings of the ASME 2014 Dynamic Systems and Control Conference DSCC2014

October 22-24, 2014, San Antonio, TX, USA

DSCC2014-5925

through different types of communication networks. The in-crease in the usage of NCS-based technologies have highlightedthe need to do more research in this area, specifically to tacklethe challenges that are caused by the existence of the communi-cation network in the feedback loops of control systems. Someof these challenges include but are not limited to: time-induceddelay, network perturbation and disturbance, cyber-attacks, over-all system stability, and packets loss. Furthermore, NCSs arefound in a variety of CPSs and applications such as: passengercars, trucks and buses, aircraft and aerospace electronics, factoryautomation, medical equipment, mobile sensor networks andmany more [3].

The growing interest in NCSs is motivated by many bene-fits they offer such as the ease of maintenance and installation,systems’ weight reduction, increase in reliability, the largeflexibility and the low cost [2]. Sampling, encoding, decoding,medium access scheduling, data packet dropouts and the finitebandwidth of the network, are all factors that challenge the in-sertion of communication networks in the feedback loops of anycontrol system. All these factors cause a network-induced delaywhich in turn causes a deterioration of the system performanceor can even destabilize it.

In this paper, we address the network-induced delay chal-lenge. In particular, we derive an upper bound on the time-delayinduced by the communication network. In addition, thedetermination of an upper bound on the networked-inducedtime-delay is crucial in the design of an NCS so that a suitablesampling period is chosen.

1.1 Motivation and Problem DescriptionAs mentioned in the previous section, one of the main

challenges in NCS design and analysis is the problem of thenetwork-induced delay. There are two approaches to control thenetwork-induced delay: either through the “control of network”actions, or through “control over network” strategies.

The proposed methods and formulations in this paper arerelated to the “control over network” design strategy. Generally,there are several types of control and various methods to designcontrollers; not all of them are applicable for networked systems.For example, control systems are classified as either centralizedor decentralized. In complex dynamical systems, centralizedcontrol can be optimal, but in many cases it is neither robustnor scalable. This is due to the high computational complexityof employing such centralized controllers. In addition, thecentralized control for a distributed system over a large ge-ographical region needs a large amount of information to beexchanged through a communication network. Accordingly, thiscauses long delays and loss of data that degrades the quality

of the data transmitted and received. Furthermore, adaptingcentralized control makes it harder to apply physical changes tosystems. Moreover, large-scale systems are generally composedof smaller subsystems. When the communication between thesmaller subsystems is poorly modeled, the centralized controlmethod becomes ineffective. On the other hand, the decentral-ized control is more suitable for NCSs because it reduces thetraffic to be exchanged, which in turn reduces the network trafficand the expected time-delay for each network user.

In control theory, there is rich literature on decentralizedcontrol covering various methods of design and analysis ofcontrollers. Most of the decentralized control theory had beenintroduced before the emergence of networked systems thatclose the feedback loops through communication networks. Asa result, the area of DNCS has recently emerged. In this paper,we introduce a general framework that maps the framework ofdecentralized control of non-networked systems to the frame-work of NCS. This mapping process is important to conducta research that applies the theory of decentralized control ofnon-networked systems into modern networked control systems.

1.2 Literature ReviewIn modern control theory, one of the fundamental problems

that has been a subject of significant research, is the devel-opment of methods for control of dynamical systems that arecomposed of complex interconnected subsystems [4]– [11].Generally, the controller design problem has been addressed forlarge-scale dynamical systems by considering either centralizedor decentralized control. The robust design of the decentralizedcontrol strategies has been introduced in [12]– [14]. In [15],the authors proposed an observer-based control algorithm forlinear systems where the design uses low-order linear functionalobservers. The individual subsystem states are estimated in [16]and [17] by using a dynamic observer. Observer-based controldesign for non-linear systems is introduced in [18]– [21]. Thekey feature of the design proposed in [18] is that the separationprinciple for linear systems holds in their design of non-linearsystems. Another approach for controlling large-scale systems isthe quasi-decentralized control which compromises between thecomplex centralized control and decentralized control that hasperformance limitations [19]– [21]. Quasi-decentralized controlcan be considered as a distributed control strategy in the sensethat most control signals are collected and processed locally. Inthe mean time, some signals are transferred between the localunits and controllers to properly account for the interactions andreduce the propagation of process failures.

In general, there is a rich literature that deals with the con-trol of large-scale systems, whether these systems adapt thecentralized, decentralized or mixed as in the quasi-decentralized

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control methods. Nonetheless, most of the aforementionedapproaches are introduced for non-networked systems. Thereare relatively few efforts [22]– [24] that tackle the decentralizedcontrol problem for large-scale systems that is concerned withnetworked dynamical systems. In [27], we analyzed the networkeffect as perturbation to the signals exchanged through thenetwork. In this paper, we provide a general framework thatmaps the general framework of decentralized control of non-networked systems to the framework of NCS. This frameworkis then used to design and analyze an observer-based controlfor decentralized networked control systems, assuming that thenetwork effect is modeled as a pure time-delay.

This paper is organized as follows. In Section 2 we re-view the mathematical formulation of the observer-based controldesign. In Section 3 we present the DNCS configuration andmap the DNCS setup to the typical NCS setup. The stabilityanalysis of the proposed design is introduced in Section 4.Numerical examples and simulation results are discussed inSection 5. Section 6 is dedicated for closing remarks andconclusions.

2 OBSERVER BASED CONTROL DESIGN FORMULA-TION REVIEWIn this section we review the observer-based design of the

controller of the non-networked system, which is followed bythe DNCS configuration in Section 3. In this paper, we are con-sidering the observer based control design from [15]. Considera large-scale system where the plant dynamics are described asfollows: xxxp = AAApxxxp +

N

∑i=1

BBBiuuui

yyyi =CCCixxxp, i = 1,2, . . . ,N(1)

where xxx ∈ Rn is the state vector of the plant of the system,uuui ∈ Rmi is the input vector of the ith subsystem and yyyi ∈ Rpi

is the output vector of the ith subsystem. AAAp ∈ Rn×n,BBBi ∈Rn×mi ,and CCCi ∈ Rpi×n are real constant matrices. Without lossof generality, AAAp could be a block-diagonal concatination of Ndifferent sub-systems or plants, assuming that there is no inter-action between the states of each individual sub-system. Further-more, if there are interdependencies between the subsystems, AAApwon’t be a block-diagonal matrix. Let

{uuu =

[uuu>1 . . . uuu>N

]> , yyy =[yyy>1 . . . yyy>N

]>BBBp =

[BBB1 . . . BBBN

], CCCp =

[CCC>1 . . . CCC>N

]>.

Then the plant dynamics can be written in the following compactform:

xxxp = AAApxxxp +BBBpuuu

yyy = CCCpxxxp.

We assume that the assumptions on the system dynamicsfrom [15] are also valid in this paper (system controllability, ob-servability, . . . ). As in [15], we also assume that a global statefeedback control exists such that uuu = −FFFxxx, where FFF ∈ Rm×n.The global state feedback control gain FFF can be obtained by us-ing any standard state feedback control method. Partitioning theglobal controller uuu, we get,

uuu1uuu2...

uuuN

=−

FFF1FFF2...

FFFN

xxx.

The authors in [15] proposed the following decentralized con-troller:

uuui =−FFF ixxx≈−(KKKiLLLi +WWW iCCCi)xxx≈−KKKizzzi−WWW iyyyi,

where zzzi ∈Roi is an estimate of the weighted plant state (zzzi tracksLLLixxx) that has the following dynamics:

zzzi = EEE izzzi +LLLiBBBiuuui +GGGiyyyi, (2)

where

EEE i ∈ Roi×oi ,Li ∈ Roi×n,KKKi ∈ Rmi×oi ,WWW i ∈ Rmi×pi

and GGGi ∈ Roi×pi are all real matrices that represent the controllerdesign parameters [15], and

FFF i ≈ KKKiLLLi +WWW iCCCi. (3)

The observation error vector is defined as:

eeeoi = zzzi−LLLixxx, i = 1,2, . . . ,N.

Therefore, the observation error dynamics are:

eeeoi = zzzi−LLLixxx.

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After some simple manipulations, we obtain the following equa-tion:

eeeoi= EEE ieeeoi +(GGGiCCCi−LLLiAAA+EEE iLLLi)xxx−LLLiBBBriuuur. (4)

BBBri is a partition of BBB, BBB =[BBBi BBBri

], where BBBri ∈Rn×(m−mi) is the

input matrix for uuur(t) which contains (N−1) input vectors of theremaining (N− 1) subsystems. With this particular partition ofthe input matrix B, the dynamics of the plant states are:

xxx = AAAxxx+BBBiuuui +BBBriuuur, i = 1,2, . . . ,N.

Choosing EEE i to be asymptotically stable, (2) can be viewed as adecentralized linear observer if LLLi and GGGi fulfill the following setof constraints:

LLLiBBBri = O (5)KKKiLLLi +WWW iCCCi = FFF i (6)

GGGiCCCi−LLLiAAA+EEE iLLLi = O , (7)

Ha and Trinh in [15] and Elmahdi et. al in [27] proposed twodifferent methods to solve the above system of equations. Aftersolving for the system design unknowns (GGGi,LLLi,KKKi,WWW i), we nowhave all the design parameters for the observer-based controller.In the next section, we will map the observer-based control dy-namics’ parameters to the typical NCS formation.

3 PROBLEM FORMULATION AND DNCS SETUPTo analyze the DNCS design, we map the DNCS formation

to the equivalent NCS setup. In other words, we start with a con-troller design method for the non-networked system of decentral-ized control, then we map the closed-loop non-networked systemto its equivalent configuration in networked dynamical system inorder to apply the stability analysis tools. The general setup of aLarge Scale System (LSS) DNCS is shown in Figure 1.

3.1 Mapping the DNCS to the NCS SetupWe now convert the DNCS setup to the general setup of the

NCS. For simplicity, we consider the case of a lumped delay be-tween the sensor and the controller as shown in Figure 2. Thecontroller’s output (uuu(t)) and input (yyy(t)) are defined as:

uuu(t) = CCCcxxxc(t)+DDDcCCCpxxxp(t− τ)

yyy(t) = yyy(t− τ) =CCCpxxxp(t− τ).

In our discussion we analyze the behavior of the system between

FIGURE 1: DNCS State-Space Configuration.

FIGURE 2: One Lumped Delay Between the Sensor and the Con-troller.

transmission times. The plant and controller state dynamics canbe written as:

xxxp(t) = AAApxxxp(t)+BBBpCCCcxxxc(t)+BBBpDDDcCCCpxxxp(t− τ)

xxxc(t) = AAAcxxxc(t)+BBBcCCCpxxxp(t− τ).

To analyze stability, we need to combine the nominal system withthe perturbation in one state. The perturbation represents the er-ror due the network existence in the feedback loops of the sys-tem, and the observation error. From the observer-based controldesign we have the following representation:

Plant:{

xxxp(t) = AAApxxxp(t)+BBBpuuu(t)yyyp(t) =CCCpxxxp(t)

(8)

Controller/Observer:

{zzz(t) = EEEzzz(t)+LLLBBBpuuu(t)+GGGyyy(t)uuu(t) =−KKKzzz(t)−WWWyyy(t), (9)

4 Copyright © 2014 by ASME

where EEE,LLL,GGG,KKK and WWW are the observer-based control designparameters in compact form.

Recall that the observer-based controller’s goal is to track a linearweighted combination of the plant state, (zzz(t)→ LLLxxxp(t−τ)), andthe observation error can be written as: eeeo(t) = zzz(t)− LLLxxxp(t),while the plant output is yyy(t) = yyy(t − τ) = CCCpxxxp(t − τ). Thecontroller’s output uuu(t) can be found as follows:

uuu(t) = −KKKzzz(t)−WWWyyy(t)

= −KKK (LLLxxxp(t− τ)+ eee0(t))−WWWCCCpxxxp(t− τ)

= − [(KKKLLL+WWWCCCp)− (FFF− (KKKLLL+WWWCCCp)]xxxp(t− τ)

−KKKeeeo(t),

where FFF ≈KKKLLL+WWWCCCp and ∆FFF =FFF−(KKKLLL+WWWCCCp). We can writeuuu(t) as:

uuu(t) =−FFFxxxp(t− τ)+∆FFFxxxp(t− τ)−KKKeeeo(t). (10)

Substituting (10) in the plant dynamics (xxxp(t)) equation, we get:

xxxp(t) = AAApxxxp(t)+(BBBp∆FFF−BBBpFFF)xxxp(t− τ)−BBBpKKKeeeo(t). (11)

The controller dynamics (xxxc(t) or zzz(t)) can be written as:

zzz(t) = EEEzzz(t)+LLLBBBpuuu(t)+GGGyyy(t)

= EEEzzz(t)+LLLBBBp(−KKKzzz(t)−WWWyyy(t))+GGGyyy(t)

= (EEE−LLLBBBpKKK)zzz(t)+(GGG−LLLBBBpWWW )yyy(t).

Letting zzz(t) = xxxc(t), we can rewrite the controller dynamics as:

xxxc(t) = AAAcxxxc(t)+BBBcyyy(t) = AAAcxxxc(t)+BBBcCCCpxxxp(t− τ), (12)

where AAAc = EEE − LLLBBBpKKK, BBBc = GGG− LLLBBBpWWW , and from uuu(t) =−KKKzzz(t)−WWWyyy(t), we get CCCc =−KKK and DDDc =−WWW .

3.2 Observation Error Dynamics and Closed-LoopStates Augmentation

The observation error dynamics can be formulated as:

eeeo(t) = EEEzzz(t)+LLLBBBpuuu(t)+GGGyyy(t)−LLLAAApxxxp(t)

−LLLBBBpuuu(t)+EEELLLxxxp(t)−EEELLLxxxp(t)

= EEE (zzz(t)−LLLxxxp(t))+GGGCCCpxxxp(t− τ)

−LLLAAApxxxp(t)+EEELLLxxxp(t)

= EEEeeeo(t)+GGGCCCpxxxp(t− τ)−LLLAAApxxxp(t)

+EEELLLxxxp(t)+GGGCCCpxxxp(t)−GGGCCCpxxxp(t)

eeeo(t) = EEEeeeo(t)+(GGGCCCp−LLLAAAp +EEELLL)xxxp(t)

+GGGCCCpxxxp(t− τ)−GGGCCCpxxxp(t)

Hence,

eeeo(t) = EEEeeeo(t)+(∆MMM−GGGCCCp)xxxp(t)+GGGCCCpxxxp(t− τ), (13)

where ∆MMM = GGGCCCp−LLLAAAp +EEELLL. Using the previous derivationsin (12), (11), and (13), we can augment the general state xxx(t)with the observation error as follows:xxxp(t)

xxxc(t)eeeo(t)

=

AAApxxxp(t)+(BBBp∆FFF−BBBpFFF)xxxp(t− τ)−BBBpKKKeeeo(t)AAAcxxxc(t)+BBBcCCCpxxxp(t− τ)

(∆MMM−GGGCCCp)xxxp(t)+GGGCCCpxxxp(t− τ)+EEEeeeo(t)

=

AAAp O BBBpCCCcO AAAc O

∆MMM−GGGCCCp O EEE

xxxp(t)xxxc(t)eeeo(t)

+−BBBpFFF O O

BBBcCCCp O OGGGCCCp O O

xxxp(t− τ)xxxc(t− τ)eeeo(t− τ)

.

Let www =

xxxp(t)xxxc(t)eeeo(t)

, AAA0 =

AAAp O BBBpCCCcO AAAc O

∆MMM−GGGCCCp O EEE

and

AAA1 =

−BBBpFFF O OBBBcCCCp O OGGGCCCp O O

.Hence, we can write the augmented state dynamics as,

www(t) = AAA0www(t)+AAA1www(t− τ). (14)

4 STABILITY ANALYSISIn this section, we analyze the stability of the DNCS given

that the network disturbance effect is treated as pure time-delay.

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Equation (14) represents the dynamics of the overall system,combining the nominal system dynamics and the perturbationdue to the network. We can obtain an expression for www(t − τ)by using the Taylor series expansion formula as follows,

www(t− τ) =∞

∑n=0

(−1)n τn

n!www(n)(t)

= www(t)− τwww(t)+R2(www,τ).

Neglecting the higher order terms, we get the following approx-imation for www(t− τ):

www(t− τ)≈ www(t)− τwww(t). (15)

Substituting (15) into (14), we get:

www(t) = AAA0www(t)+AAA1www(t)− τAAA1www(t).

Rearranging,

www(t) = (III + τAAA1)−1 (AAA0 +AAA1)︸ ︷︷ ︸

AAA

www(t). (16)

If there is no network in the feedback-loops of the control system,then the system representation will be reduced to the nominalnon-networked system which is

www(t) = (AAA0 +AAA1)www(t) ⇒ www(t) = AAAwww(t).

In what follows, we provide a bound on the time-delay (τ) thatguarantees the asymptotic stability of the global DNCS state dy-namics (www(t)).

Theorem 1. For the DNCS represented in (8,9), let the glob-ally exponentially stable equilibrium point of the closed-loopnon-networked system be www = 0, and V (www) = www>PPPwww be a Lya-punov function that satisfies AAA>PPP+ PPPAAA = −2QQQ, where PPP is asymmetric positive definite matrix. If the origin is a globallyexponentially stable equilibrium point of the DNCS, then thenetwork-induced time delay τ satisfies:

τ2‖PPPAAA2

1AAA‖− τ‖PPPAAA1AAA‖−λmax(QQQ)< 0. (17)

Proof. We have the Lyapunov function V (www) = www>PPPwww. Takingthe derivative, we get,

V (www) = 2www>PPPwww = www>PPPwww+www>PPPwww. (18)

Substituting (16) into (18), we get:

V (www) = www>AAA>(III + τAAA1)−>PPPwww+www>PPP(III + τAAA1)

−1AAAwww.

Let HHH = (III+τAAA1)−1 ⇒ V (www) = www>AAA>HHH>PPPwww+www>PPPHHHAAAwww. We

can write V (www) as:

V (www) = www>AAA>PPPPPP−1HHH>PPPwww+www>PPPHHHPPP−1PPPAAAwww

−www>(AAA>PPP+PPPAAA)www+www>(AAA>PPP+PPPAAA)www

= www>AAA>(PPPPPP−1HHH)>PPPwww+www>PPPHHHPPP−1PPPAAAwww

−www>AAA>PPPwww−www>PPPAAAwww−2www>QQQwww

= www>AAA>(PPPPPP−1HHH>PPP− III)www

+www>(PPPHHHPPP−1− III)PPPAAAwww−2www>QQQwww

= 2www>(PPPHHHPPP−1− III)PPPAAAwww−2www>QQQwww

= 2www>PPP(HHH− III)AAAwww−2www>QQQwww (19)

Using the Neumann series formula for the inverse of the sum oftwo matrices,

HHH = (III + τAAA1)−1 = III− τAAA1 + τ

2AAA21− τ

3AAA31 + . . .

= III− τAAA1 + τ2AAA2

1 (20)

Substituting (20) into (19),

V (www) = 2www>PPP(III− τAAA1 + τ2AAA2

1− III)AAAwww−2www>QQQwww

= 2www>PPP(τ2AAA21− τAAA1)AAAwww−2www>QQQwww

= 2τ2www>PPPAAA2

1AAAwww−2τwww>PPPAAA1AAAwww−2www>QQQwww.

In addition, for any symmetric matrix QQQ we have:

λmin(QQQ)‖xxx‖22 ≤ xxx>QQQxxx≤ λmax(QQQ)‖xxx‖2

2.

We can now upper bound the derivative of the Lyapunov candi-date function V (www) as follows:

V (www) ≤ 2τ2‖PPPAAA2

1AAA‖‖w‖2−2τ‖PPPAAA1AAA‖‖www‖2

−2λmax(QQQ)‖www‖2

V (www) ≤ −2(−τ

2‖PPPAAA21AAA‖+ τ‖PPPAAA1AAA‖+λmax(QQQ)

)‖www‖2.

From the above, when the origin is a globally exponentially sta-ble equilibrium point of the DNCS, then

τ2‖PPPAAA2

1AAA‖− τ‖PPPAAA1AAA‖−λmax(QQQ)< 0. �

6 Copyright © 2014 by ASME

5 NUMERICAL SIMULATIONSIn this section, and to further asses the applicability of the

derived bound on the maximum time delay (τ), we introduce anumerical example to analyze the effect of the pure time-delayconsidering the observer-based decentralized control.

Consider a fourth order system with the following plantstate space representation:

{xxxp(t) = AAApxxxp(t)+BBBpuuu(t)yyyp(t) =CCCpxxxp(t),

(21)

where,

AAAp =

1 2 3 −45 6 7 −89 10 11 −1213 14 15 −16

∈ R4×4,BBBp =

1 0 0 11 1 −1 22 1 4 33 1 2 5

∈ R4×4,

and CCCp =

1 1 0 02 −1 1 00 0 0 10 1 0 00 0 1 00 0 0 1

∈ R6×4.

The eigenvalues of AAAp are:

eig(AAAp) = {1.00+5.56i,1.00−5.56i,0,0}.

Hence, the non-networked non-controlled plant is initiallyunstable. In addition, the plant dynamics satisfies the assump-tions in [15]. We now start with the design of the observer baseddecentarlized controller by solving the system of equations (5–7) as in [15]. The algorithm used to design the non-networkedcontroller is highlighted in [15, p. 725]. We assume that the firsttwo columns of BBBp are equivalent to BBB1, while the rest are BBBr1.Also, the first three rows of CCCp constitute CCC1. In this example,we use a full-order observer of size four (i.e., xxxc(t) = zzz(t) ∈ R4)and that the observation error (eo(t)) is neglected. Choosing EEEto be Hurwitz, we can find the observer-based controller param-eters (FFF ,LLL,KKK,WWW ,GGG), and write the controller’s dynamics as

C:{

zzz(t) = EEEzzz(t)+LLLBBBpuuu(t)+GGGyyy(t)uuu(t) =−KKKzzz(t)−WWWyyy(t),

where,

EEE =−

0.1 0 0 00 0.2 0 00 0 0.1 00 0 0 0.5

,LLL =

0.81 0.19 0.23 −0.38−0.44 0.65 0.35 −0.380.81 0.19 0.23 −0.38−0.44 0.65 0.35 −0.38

,

GGG =

−0.53 −0.09 −1.57 0 0 01.04 0.15 −1.63 0 0 0

0 0 0 −0.17 0.69 −1.570 0 0 1.52 1.55 −1.75

,

KKK =

−0.03 −0.45 0 00.27 1.45 0 0

0 0 0.92 0.520 0 −0.07 −0.65

,

and WWW =

0.21 −0.01 0.26 0 0 00.14 0.30 −0.25 0 0 0

0 0 0 0.21 0.94 −0.660 0 0 0.06 −0.44 0.87

.

After designing the observer-based controller for the system,we can write the networked dynamics of the overall systemas in (16) where AAA0 and AAA1 are computed based on the non-networked system dynamics parameters. The dynamics of thenon-networked (τ = 0) controlled system can be written as

www(t) = (AAA0 +AAA1)www(t) = AAAwww(t).

The eigenvalues of the non-networked closed loop system are:

eig(AAA) = {−1.76±5.52i,−1.53,−1.20,−0.40,

−0.11,−0.50,−0.10}.

Hence, the designed controller stabilizes the overall closed-loopsystem, as AAA is Hurwitz. Figure 3 shows the stable behavior ofthe plant for random inputs.

To analyze the effect of the time-delay, we apply Theorem 1 byfinding a Lyapunov function V (www) = www>PPPwww and a symmetricpositive definite matrix PPP that satisfies AAA>PPP + PPPAAA = −2QQQ,where QQQ = QQQ> � 0. As mentioned in the previous section, theobjective of this numerical simulation is to test the applicabilityof the derived time-delay bound derived in Theorem 1, suchthat the networked system with the observer-based controller isglobally stable.

7 Copyright © 2014 by ASME

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−6

−5

−4

−3

−2

−1

0

1

2

Time (seconds)

x1(t),x2(t),x3(t),x4(t)

Non-Networked Stable Plant State Trajectories (Random Input, Random ICs)

x1(t)

x2(t)

x3(t)

x4(t)

FIGURE 3: Non-Networked Plant States Trajectories for τ = 0, Random Inputs (uuu(t)) and Random Initial Conditions (www(0)).

Recall that for the origin to be a globally exponentiallystable equilibrium point of the DNCS, then

τ2‖PPPAAA2

1AAA‖− τ‖PPPAAA1AAA‖−λmax(QQQ)< 0.

After finding PPP and calculating AAA0,AAA1 and AAA=AAA0+AAA1, the max-imum value of the time-delay that would keep the networkedsystem stable is τmax = 0.2309sec (computed by evaluating thequadratic bound on τ from Theorem 1). We now set τ = τmax−ε

and simulate the networked dynamics of the system (a refer-ence input is added), where ε is a small positive constant sinceτ < τmax. For τ = τmax− ε = 0.202sec. The eigenvalues of

AAA = (III + τAAA1)−1(AAA0 +AAA1)

are: {−13.4±16.6i,−2.1,−1.2,−0.4,−0.1,−0.5,−0.1}.

Checking the stability of (16) for τ = τmax, we find that the sys-tem becomes unstable with the following eigenvalues of AAA:

{249.5,−19.6,−2.2,−1.2,−0.4,−0.5,−0.1,−0.1}.

Therefore, the derived τ bound is very accurate. Starting fromrandom initial conditions and random inputs, Figures 4 shows thestable behavior of the plant for τ = τmax− ε = 0.202sec assum-ing random initial conditions and random inputs. The networked-controlled plant states are stable and converging within few sec-onds, as shown in Figure 4.

6 CONCLUSIONS

The network-induced disturbances is one of the main de-merits of installing communication networks in modern controlsystems. The networked-induced time-delay is an example ofnetworked-induced disturbances. In this paper, we analyze theeffect of modeling the network as a pure time-delay on thelinearized time-invariant dynamics of any plant. In particu-lar, we study how to improve the behavior of DecentralizedNetworked Control Systems by establishing realistic bound onthe time-delay induced by the communication network. Thisbound would lead, if satisfied, to the overall closed-loop stability.

First, we review an observer-based decentralized controlmethod for decentralized control systems. Second, we establisha map between the decentralized non-networked system, andthe typical NCS state-space representation. The main idea ofour design is to put the DNCS in the general form and thenmap the resulted system to the general form of the NCS. Thiswould facilitate deriving the maximum time-delay bound for theobserver-based control of the NCS. Third, treating the networkeffect as pure time-delay, we derive the global dynamics of theDNCS. Fourth, an upper bound for the time-delay is derived,that guarantees the global exponential stability of any lineartime-invariant DNCS. Finally, we present a numerical examplethat illustrates the applicability of the proposed upper bound ofthe time-delay. Results show that the overall closed-loop systemis stable, assuming that the networked induced delay is withinthe derived bound.

As mentioned in Section 1, the determination of an upper

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4

−2

0

2

4

6

8

10

12

Time (seconds)

x1(t),x2(t),x3(t),x4(t)

Stable Plant State Trajectories (Random Input, Random ICs)

x1(t)

x2(t)

x3(t)

x4(t)

FIGURE 4: Plant States Trajectories for τ = 0.202sec, Random Inputs (uuu(t)) and Random Initial Conditions (www(0)).

bound on the networked-induced time-delay is very crucialin the design of an NCS so that a suitable sampling period ischosen. When the time-delay is larger than the sampling periodin an NCS, then the global stability of the overall NCS can notbe guaranteed. Therefore, having an accurate bound on thetime-delay for DNCS applications is crucial for both, the designand stability of such systems. In future work, the bound derivedcan be used to analyze the effect of this time-delay schedulingprotocols in NCS applications.

ACKNOWLEDGMENTThe authors gratefully acknowledge the financial support

from the National Science Foundation through NSF CMMIGrants 1201114 and 1265622.

REFERENCES[1] J. Bai, E.P. Eyisi, F. Qiu, Y. Xue and X.D. Koutsoukos,

“Optimal Cross-Layer Design of Sampling Rate Adap-tation and Network Scheduling for Wireless NetworkedControl Systems,” Cyber-Physical Systems (ICCPS), 2012IEEE/ACM Third International Conference on, vol., no.,pp.107,116, 17-19 April 2012.

[2] A. Bemporad, W.P.M.H. Heemels and M. Johansson, “Net-worked Control Systems”, Lecture notes in control andinformation sciences, Vol. 406, Springer-Verlag, London,2010.

[3] T.-C. Yang, “Networked control system: a brief sur-

vey,” Control Theory and Applications, IEE Proceedings,vol. 153, no. 4, pp.403–412, 10 July 2006.

[4] D. D. Siljak, Decentralized Control of Complex Systems,Academic Publisher, Boston, 1991.

[5] J. Lunze, “Feedback Control of Large Scale Systems,”U.K.: Prentice-Hall, 1992.

[6] E. Camponogara, D. Jia, B. H. Krogh, and S. Talukdar,“Distributed model predictive control,” IEEE Contr. Syst.Mag., vol. 22, pp. 44-52, 2002.

[7] N. Venkat, J. B. Rawlings, and S. J. Wright, “Stability andoptimality of distributed model predictive control,” Deci-sion and Control, 2005 and 2005 European Control Confer-ence. CDC-ECC ’05. 44th IEEE Conference on, vol., no.,pp. 6680–6685, 12–15 December 2005.

[8] M. Baldea, P. Daoutidis, and A. Kumar, “Dynamics andcontrol of integrated networks with purge streams, AIChEJournal, vol. 52, pp. 1460–1472, 2006.

[9] K. Jillson and B. Ydstie, “Process networks with decentral-ized inventory and flow control,” J. Proc. Contr., vol. 17,pp.399-413, 2007.

[10] M. D. Tetiker, A. Artel, F. Teymour, and A. Cinar, “Con-trol of grade transitions in distributed chemical reactor net-works: An agent-based approach,” Computers & ChemicalEngineering, vol. 32, pp. 1984-1994, 2008.

[11] J. Liu, D. M. de la Pena, and P. D. Christofides, “Dis-tributed model predictive control of nonlinear systems sub-ject to asynchronous and delayed measurements,” Auto-matica, vol. 46, pp. 52-61, 2009.

[12] A. I. Zecevic, G. Neskovic, and D. D. Siljak, “Robust

9 Copyright © 2014 by ASME

decentralized exciter control with linear feedback,” IEEETransactions on Power Systems, Vol. 19, No. 2, pp. 1096–1103, 2004.

[13] Z. Gong, “Decentralized robust control of uncertain inter-connected systems with prescribed degree of exponentialconvergence,” IEEE Transactions on Automatic Control,Vol. 40, No. 4, pp. 704–707, 1995.

[14] A. Swarnakar, H. J. Marquez, and T. Chen, “Robust sta-bilization of nonlinear interconnected systems with appli-cation to an industrial utility boiler”, Control EngineeringPractice, Vol. 15, No. 6, pp. 639–654, 2007.

[15] Q. Ha and H. Trinh, “Observer-based control of multi-agentsystems under decentralized information structure,” Int. J.Sys. Science, Vol. 35, No. 12, pp.719–728, October 2004.

[16] N. Viswanadham and A. Ramakrishna, “Decentralized esti-mation and control of interconnected systems,” Large ScaleSystems, Vol. 3, pp. 255–266, 1982.

[17] P. R. Pagilla, E. O. King, L. H. Drienhoefer, and S. S.Garimella, “Robust observer-based control of an aluminumstrip processing line,” IEEE Tran. on Industry Applications,Vol. 36, pp. 865–870, 2000.

[18] K. Kalsi, J. Lian, and S. H. Zak, ”Reduced-order observer-based decentralized control of non-linear interconnectedsystems,” International Journal of Control, Vol. 82 No. 6,pp. 1157–1166, 2009.

[19] Y. Sun and, and N. H. El-Farra, “Quasi-decentralizedmodel-based networked control of process systems,” Com-puter & Chemical Engineering, vol. 32, pp. 2016-2029,2008.

[20] S. Yulei, and N. H. El-Farra, “Resource-aware quasi-decentralized control of nonlinear plants over communica-tion networks,” American Control Conference, ACC 2009,vol., no., pp.154–159, 10–12 June 2009.

[21] S. Yulei, and N. H. El-Farra, “A Quasi-decentralized Ap-proach for Networked State Estimation and Control of Pro-cess Systems,” Industrial & Engineering Chemistry Re-search, vol. 49, no. 17, pp. 7957–7971, 2010.

[22] Lo Chun-Hao, and N. Ansari, “Decentralized Controls andCommunications for Autonomous Distribution Networksin Smart Grid,”IEEE Transactions on Smart Grids, vol. 4,no. 1, pp.66–77, March 2013.

[23] H. Changchun and S. Ding, “Decentralized NetworkedControl System Design Using TS Fuzzy Approach,” FuzzySystems, IEEE Transactions on, vol. 20, no. 1, pp.9–21,February 2012.

[24] S. Yulei and N. H. El-Farra, “Dynamic quasi-decentralizedcontrol of networked process systems with limited mea-surements,” American Control Conference (ACC) 2011,pp.3152–3157, June 29—July 1, 2011.

[25] A. Nguyen, Q. P. Ha, S. Huang, and H. Trinh, “Observer-based decentralized approach to robotic formation control,”Proceedings of the 2004 Australian Conference of Robotics

and Automation, pp. 1-8, Australia, December 2004.[26] W. Zhang, M.S. Branicky and S.M. Phillips, “Stability of

networked control systems,” IEEE Control Systems Maga-zine, vol. 21, no. 1, pp. 84–99, February 2001.

[27] A. Elmahdi, A.F. Taha, D. Sun and J.H. Panchal, “Decen-tralized Control Framework and Stability Analysis for Net-worked Control Systems”, ASME Journal of Dynamic Sys-tems, Measurement, and Control. February 2014. UnderReview.

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