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Automatica 39 (2003) 1309–1312

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Technical Communique

Robust stability analysis of simple systems controlled overcommunication networks�

Qing-Chang Zhong∗

Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London, SW7 2BT, UK

Received 23 April 2002; received in revised form 21 October 2002; accepted 20 March 2003

Abstract

There is increasing interest in controlling systems over communication networks. Using a simple method called dual-locus diagram,this communique proposes complete stability criteria for a mass-spring-damper system controlled over the network. The stability regionis divided into a delay-dependent stability region and a delay-independent stability region, which o.ers a nice graphical view on theconservativeness of the delay-independent stability criteria.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Stability analysis; Time-delay systems; Communication networks; Dual-locus diagram; Control-via-internet

1. Introduction

In recent years, there has been increasing interest incontrolling systems over communication networks becausecommunication networks (Varaiya & Walrand, 1996) areamong the fastest-growing areas in engineering. Thanks tohigh-speed networks, control-via-internet is now available(Safaric, Jezernik, Calkin, & Parkin, 1999; Malinowski,Booth, Grady, & Huggins, 2001). These systems arefrequently modeled from the control point of view astime-delay systems because of the inherent propagationdelays; see, for example, Izmailov (1996) and Mascolo(1999). These delays are crucial to the system stability andthe quality-of-service (QoS). As is well known, the pres-ence of delays makes the control design and system analysismuch more complicated. In this communique, we focuson the stability analysis of a mass-spring-damper systemcontrolled over the network, which is studied in Chen andMoore (2002). For details on the control of communicationnetworks, see Izmailov (1996), Mascolo (1999), Quetet al. (2002) and G@emez-Stern, Forn@es, and Rubio (2002)and the references therein.

� This paper was not presented at any IFAC conference. This articlewas recommended for publication in revised form by Associate EditorRick Middleton under the direction of Editor Paul Van den Hof.

∗ Tel.: +44-20-759-46295; fax: +44-20-759-46282.E-mail address: zhongqc@ic.ac.uk (Q.-C. Zhong).URL: http://members.fortunecity.com/zhongqc

The robust stability analysis of time-delay systems isnot well established and has become a very active researchHeld in recent years. Current e.orts can be divided into twocategories: delay-dependent stability criteria (Gu, 1997;Niculescu, 1999; Park, 1999) and delay-independent stabil-ity criteria (Kokame, Kobayashi, & Mori, 1998). Althoughdelay-dependent stability criteria are in general less con-servative than delay-independent criteria, they may stillbe quite conservative. One reason is that delay-dependentstability criteria were frequently obtained by using a modeltransformation, which introduces additional dynamics(Gu & Niculescu, 2000, 2001). An interesting case, wherethe delay-independent stability criteria are not conservativeat all, will be shown.A mass-spring-damper system, controlled over the net-

work using a simple proportional controller, can be de-scribed by the following second-order delay di.erentialequation:

Ky(t) + 2��y(t) + �2y(t)− Kpy(t − �) = 0; (1)

where y(t) is the position of the mass, � ¿ 0 is the damp-ing ratio, � ¿ 0 is the natural frequency, Kp ¿ 0 is theproportional control gain and � ¿ 0 is the network commu-nication delay. Chen and Moore (2002) used the LambertW function (Corless, Gonnet, Hare, Je.rey, & Knuth, 1996;Zhong & Mirkin, 2002; Chen & Moore, 2001) to calcu-late the closed-loop pole of the system and proposed ana-lytical stability criteria for it. The approach is e.ective for

0005-1098/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0005-1098(03)00110-9

1310 Q.-C. Zhong /Automatica 39 (2003) 1309–1312

some high-order systems with a delay as well. However, theassumption � = 1 (equivalent to the condition bm = a2m=4in the paper) or the assumption having repeating poles forhigh-order systems considerably limits the applicability ofthe results.In this communique, the stability analysis of this sys-

tem is reconsidered using the dual-locus diagram (alsocalled Satche diagram) method (Satche, 1949; Smith, 1958;Olgac & Holm-Hansen, 1995; Zhong, 2003). The assump-tion � = 1 has been removed and thus the result can beapplied to general second-order systems having various �and/or �. The stability region of the system is divided intoa delay-dependent stability region and a delay-independentstability region on the parametric plane (Kp=�2) − �. Thiso.ers a nice graphical view on the conservativeness of thedelay-independent stability criteria: the delay-independentstability criteria are not conservative at all when �¿ 1√

2.

It is worth noting that this method also leads to the sameresult for the system studied in Niculescu (2002).

2. Preliminary: dual-locus diagram

The dual-locus diagram method is an extension or a vari-ant of the well-known Nyquist diagram (Nyquist, 1932).They are based on the same reasoning, i.e., the celebratedargument principle cited below. The dual-locus diagramwas proposed in Satche (1949) and then developed inSmith (1958). Applications can be found in Olgac andHolm-Hansen (1995), Zhong and Mirkin (2002) and Zhong(2003). The dual-locus diagram is quite e.ective for stabil-ity analysis of time-delay systems when the delays appearin only one of the locus. The advantages consist in thesimplicity of the approach and the ease of understanding.

Lemma 1 ((Chen, 2002) Argument principle). Supposethat a function f is meromorphic in a simply connecteddomain D. Suppose further that C is a Jordan curve inD, followed in the positive (anticlockwise) direction, andthat f has no poles or zeros on C. If Z and P denote thenumber of the zeros and poles, respectively, of f in theinterior of C, counted with multiplities, then the variationof the argument of f(s) along the Jordan curve C is

var(argf(s); C) = 2�(Z − P)

and the winding number of f(s) round the point 0, i.e. thenumber of times f(s) winds round the origin, is

n(f(s); 0) = Z − P:

Generally speaking, the characteristic equation of a(closed-loop) system can be written in the form

1 + L(s) = 0; (2)

where L(s) is the loop transfer function. The Nyquist criteriauses the second statement in the argument principle whilethe dual-locus diagram uses the Hrst statement. Here, the

Fig. 1. Dual-locus diagram of the system.

Jordan curve C is the Nyquist contour. If the characteristicequation (2) is rearranged as

L1(s) = L2(s);

then the dual-locus diagram is obtained when s traversesthe Nyquist contour. The argument of L1(s) − L2(s) is theangle between the vector joining the corresponding pointson loci L1(s) and L2(s) and the real axis. If P = 0, then thesystem is stable (i.e. Z = 0) if and only if the variation ofthe argument of L1(s)− L2(s) is naught.

3. Stability analysis of the system

System (1) can be represented in s-domain ase−�s = (s2 + 2��s + �2)=Kp.The dual-locus diagram of this system is shown in Fig. 1.

When ! increases from 0 to +∞, the locus L1 = e−�s is theclockwise unity circle starting at (1; 0) and the locus L2 =(s2 + 2��s + �2=Kp) is a parabola originating at �2=(Kp; 0),which is at the right side of the unity circle, and extendingtowards the left. The part corresponding to !=−∞ ∼ 0 issymmetric with respect to the real axis and is thus omitted.It is assumed that � ¿ 0, � ¿ 0 and Kp ¿ 0 as in Chen andMoore (2002) but � is not limited to 1.When decreasing Kp and/or increasing �; L2 (denoted as

L′2 in Fig. 1) moves towards the outside of the unity circle

in parallel and no longer intersects with L1 when �2=Kp islarge enough. Hence, for some large �2=Kp, L2 always staysat the right side of L1 and the system is delay-independentlystable. When increasing the damping ratio �, the intersectionof L2 (denoted as L′′

2 in Fig. 1) with the imaginary axismoves up, but the starting point of L2 remains still. Hence,the system is delay-independently stable for large �.If Kp ¿ �2, then the starting point (�2=Kp; 0) of L2 lies

inside L1 and the system is unstable because the variation ofthe argument of L1(s)−L2(s) can no longer be 0. Hence, theproportional gain guaranteeing the system stability is limited

Q.-C. Zhong /Automatica 39 (2003) 1309–1312 1311

Fig. 2. The stability region of the system.

by �2, i.e. Kp ¡ �2: In order to simplify later expositions,the proportional control gain Kp is normalized as

� =Kp

�2 ;

and it is assumed that 0¡ � ¡ 1 in the sequel.The parabola L2 may have two intersections or no inter-

section with L1, corresponding to the solution condition ofthe following equation with respect to !:

(�2 − !2)2 + (2��!)2

K2p

= 1:

Assuming that there exist two positive solutions !A and !B,which are actually the corresponding frequencies on L2 atthe two intersections A and B, respectively, then !A and !B

can be solved from the last equation as

!A = �√1− 2�2 −

√�2 − 4�2 + 4�4; (3)

!B = �√1− 2�2 +

√�2 − 4�2 + 4�4: (4)

The conditions on the existence of !A and !B willnow be analyzed to derive the delay-dependent and/ordelay-independent stability criteria:(i) If �¿ 1, then �2 − 4�2 + 4�4 is always posi-

tive. However, 1 − 2�2 −√

�2 − 4�2 + 4�4 ¡ 0 and1− 2�2 +

√�2 − 4�2 + 4�4 ¡ 0 for any 0¡ � ¡ 1. Hence,

either !A or !B does not exist (hereafter, “to exist” meansthe existence of a positive solution) and the system isdelay-independently stable. The corresponding stabilityregion is denoted as RC in Fig. 2.(ii) If 0¡ � ¡ 1 and � ¡ 2�

√1− �2, then �2 − 4�2 +

4�4 ¡ 0. Either !A or !B does not exist and the system isdelay-independently stable. This stability region is denotedas RA in Fig. 2. When �= 1√

2the delay-independent stability

region reaches the maximum because L2 never intersectswith L1 and the system is stable for any 0¡ � ¡ 1.(iii) If 0¡ � ¡ 1 and �¿ 2�

√1− �2 (and � ¡ 1 by as-

sumption), �2 − 4�2 + 4�4¿ 0. Either !A or !B does notexist when 1√

2¡ � ¡ 1 but both exist when 0¡ � ¡ 1√

2.

Hence, when 0¡ � ¡ 1√2and �¿ 2�

√1− �2, there are two

intersections. As can be seen later, this provides the uniquedelay-dependent stability region, denoted as RD in Fig. 2.When 1√

2¡ � ¡ 1 and �¿ 2�

√1− �2, there is no intersec-

tion and the system is delay-independently stable, denotedas RB in Fig. 2.It is trivial that the system is stable when Kp = 0 be-

cause the open-loop system is stable. The � − � planeshown in Fig. 2 is then divided into an unstable re-gion (RU), a delay-dependent stability region (RD) and adelay-independent stability region (including RA, RB andRC). This o.ers a nice view on the conservativeness ofthe delay-independent stability criteria. For 1√

2¡ � ¡ 1,

the delay-independent region consists of two complemen-tary parts: RB and the right portion of RA. When �¿ 1√

2,

the delay-independent stability criteria are not conserva-tive at all: the system is stable for all possible gains in0¡ Kp ¡ �2.In region RD, 0¡ � ¡ 1√

2and 2�

√1− �26 � ¡ 1, L2

intersects with L1 at points A and B. The system is stableif L2 arrives at B before L1 (Smith, 1958; Zhong, 2003). Inother words, the phase shift of L1 should be less than thephase angle of point B on L2. This provides the major delaybound as

06 � ¡ 1!B

(3�2

+ arctan�2 − !2

B

2��!B

):

In fact, L1 may have already travelled several cycles alongthe unity circle before L2 arrives at A, then the followingcondition is required for the stability:

�!A ¿ − �2+ arctan

�2 − !2A

2��!A+ 2i�;

�!B ¡3�2

+ arctan�2 − !2

B

2��!B+ 2i�:

This provides the following theorem:

Theorem 2. If 0¡ � ¡ 1√2and 2�

√1− �26 � ¡ 1, sys-

tem (1) is delay-dependently stable. The stability delaybounds (�¿ 0) are given by1

!A

(−�2+ arctan

�2 − !2A

2��!A+ 2i�

)

¡ � ¡1

!B

(3�2

+ arctan�2 − !2

B

2��!B+ 2i�

);

where i = 0; 1; 2; : : : ; until the right side is no longer largerthan the left, and !A and !B are given in (3) and (4)respectively.

The delay-independently criteria can be summarized as:

Theorem 3. System (1) is delay-independently stable for:

(i) 0¡ Kp ¡ �2 if �¿ 1√2;

(ii) 0¡ Kp ¡ 2��2√1− �2 if 0¡ � ¡ 1√

2.

1312 Q.-C. Zhong /Automatica 39 (2003) 1309–1312

Remarks. (i) The damping ratio � is a crucial parameterfor the system stability. The larger the damping ratio �, thebetter the stability (but, of course, the slower the system).If � is too small, then the delay-independent stability regionis very narrow. For example, the damping ratio in the sys-tem studied in (Niculescu, 2002) is 0. Hence, there is nodelay-independent stability region and the control interval ris (very likely, has to be) artiHcially introduced to stabilizethe system.(ii) The network communication delays impose very strict

limitations on the system performance. The control gainis considerably limited (although the system may still bedelay-independently stable). This is one of the reasons whycontrol-via-internet requires a reliable high-speed commu-nication network.(iii) When 0¡ � ¡ 1√

2, the delay-dependent stability

criteria o.er a larger control gain and, hence, a better dy-namic performance but the allowable communication delaysare limited. On the other hand, the delay-independent sta-bility criteria o.er a smaller control gain but allow a broadrange of communication delays (theoretically, 0 ∼ +∞).Hence, the compromise between the network requirementsand the control performance requirements is necessarywhen designing a system to be controlled via network. Thedelay-dependent stability criteria are suitable for the controlover a high-performance communication network, e.g. alocal area network (LAN), and the delay-independent sta-bility criteria are suitable for the control over a less reliableand/or low-speed communication network, e.g. the Internet.

4. Conclusions

In this communique, the stability analysis of a mass-spring-damper system controlled over the network isre-considered using the dual-locus diagram method. Bothdelay-independent and delay-dependent stability criteria arederived. A nice graphical view on the conservativeness ofthe delay-independent stability criteria is obtained. In somecases (�¿ 1√

2), the delay-independent stability criteria are

not conservative at all. It is revealed that the dual-locusdiagram method is very e.ective in the robust stabilityanalysis of simple time-delay systems.

Acknowledgements

This research was supported by the EPSRC (Grant No.GR/N38190/1).

References

Chen, W. W. L. (2002). Introduction to complex analysis. Lecturenotes, available at http://www.maths.mq.edu.au/∼wchen/lnicafolder/lnica.html.

Chen, Y. Q., & Moore, K. L. (2001). Analytical stability bound for aclass of delayed fractional-order dynamic systems. In Proceedings ofthe 40th IEEE conference on decision and control, Orlando, FL, USA,Vol. 2 (pp. 1421–1426).

Chen, Y. Q., & Moore, K. L. (2002). Analytical stability bound for delayedsecond-order systems with repeating poles using Lambert function W.Automatica, 38(5), 891–895.

Corless, R. M., Gonnet, G. H., Hare, D. E. G., Je.rey, D. J., & Knuth, D.E. (1996). On the Lambert W function. Advances in ComputationalMathematics, 5, 329–359.

G@emez-Stern, F., Forn@es, J. M., & Rubio, F. R. (2002). Dead-timecompensation for ABR traSc control over ATM networks. ControlEngineering Practice, 10(5), 481–491.

Gu, K. (1997). Discretized LMI set in the stability problem of linearuncertain time-delay systems. International Journal of Control, 68,923–934.

Gu, K., & Niculescu, S.-I. (2000). Additional dynamics in transformedtime-delay systems. IEEE Transactions on Automatic Control, 45(3),572–575.

Gu, K., & Niculescu, S.-I. (2001). Further remarks on additional dynamicsin various model transformations of linear delay systems. IEEETransactions on Automatic Control, 46(3), 497–500.

Izmailov, R. (1996). Analysis and optimization of feed-back controlalgorithms for data transfers in high-speed networks. SIAM Journalon Control and Optimization, 34(5), 1767–1780.

Kokame, H., Kobayashi, H., & Mori, T. (1998). Robust H∞ performancefor linear delay-di.erential systems with time-varying uncertainties.IEEE Transactions on Automatic Control, 43(2), 223–226.

Malinowski, A., Booth, T., Grady, S., & Huggins, B. (2001). Real timecontrol of a robotic manipulator via unreliable internet connection.In The 27th annual conference of the IEEE industrial electronicssociety, IECON’01, Denver, Colorado, USA, Vol. 1 (pp. 170–175).

Mascolo, S. (1999). Congestion control in high-speed communicationnetworks using the Smith principle. Automatica, 35(12), 1921–1935.

Niculescu, S.-I. (1999). A model transformation class for delay-dependentstability analysis. In Proceedings of the ACC, Vol. 1. San Diego (pp.314 –318).

Niculescu, S.-I. (2002). On delay robustness analysis of a simple controlalgorithm in high-speed networks. Automatica, 38(5), 885–889.

Nyquist, H. (1932). Regeneration theory. Bell System Technical Journal,11(1), 126–147.

Olgac, N., & Holm-Hansen, B. (1995). Design considerations fordelayed-resonator vibration absorbers. ASCE Journal of EngineeringMechanics, 121(1), 80–89.

Park, P. G. (1999). A delay-dependent stability criterion for systems withuncertain time-invariant delays. IEEE Transactions on AutomaticControl, 44(4), 876–877.

Quet, P.-F., Atalar, B., Iftar, A., KOzbay, H., Kalyanaraman, S., & Kang, T.(2002). Rate-based Tow controllers for communication networks in thepresence of uncertain time-varying multiple time-delays. Automatica,38(6), 917–928.

Safaric, R., Jezernik, K., Calkin, D. W., & Parkin, R. M. (1999). Telerobotcontrol via internet. In Proceedings of the IEEE InternationalSymposium on Industrial Electronics, Vol. 1 (pp. 298–303).

Satche, M. (1949). Discussion on ‘stability of linear oscillating systemswith constant time lag’. Journal of Applied Mechanics, Transactionsof the ASME, 16(4), 419–420.

Smith, O. J. M. (1958). Feedback Control Systems. New York:McGraw-Hill Book Company, Inc.

Varaiya, P., & Walrand, J. (1996). High-performance communicationnetworks. San Francisco, CA: Morgan Kaufmann Publishers.

Zhong, Q.-C. (2003). Control of integral processes with dead-time. Part3: Deadbeat disturbance response. IEEE Transactions on AutomaticControl, 48(1), 153–159.

Zhong, Q.-C., & Mirkin, L. (2002). Control of integral processes withdead-time. Part 2: Quantitative analysis. IEE Proceedings—ControlTheory and Applications, 149(4), 291–296.