Stability Analysis of Network Controlled Temperature Control System with Additive Delays

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1 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem wih Addiive Delays V. Venkaachalam, * and D. Prabhakaran Absrac: his paper presens, using Lyapunov-Krasovskii funcional echnique combined wih reciprocal convex lemma is considered for a neworked conrol emperaure conrol sysem wih addiive ime-varying-delays. In he sabiliy analysis, a new LK funcional is assumed, and ake he ime-derivaive of he (LK) funcional, using reciprocal convex combinaion echnique was employed o obain less conservaive sabiliy crieria. Finally, he proposed sabiliy analysis culminaes ino a sabiliy crierion in he LMI (linear marix inequaliies) framework. he resuls obained are in accordance wih he heoreically obained in he emperaure conrol sysem and hey are closer o he sandard benchmark emperaure-conrol sysem. Keywords: Delay-dependen sabiliy, ime-varying addiive delays, emperaure conrol, hea exchanger, lyapunov-krasovskii funcional, linear marix inequaliy, reciprocal convex lemma. Inroducion In his paper, sabiliy of nework conrolled hea exchanger sysem, also known as emperaure conrol sysem ha mainains oupu emperaure of a fluid wihin specified limis is analyzed. When hea exchanger is employed for ransfer of hea beween one or more fluids, closed loop conrol is very essenial and sabiliy of he closed loop sysem is significan. he fluids in a ypical hea exchanger sysem may be separaed by a solid wall o preven mixing or hey may be in direc conac also [Kakac and Liu (); Saunders (9)]. he hea exchangers are widely used in space heaing, refrigeraion, air condiioning, power generaions, chemical indusries, perochemical and peroleum indusries, Oil-indusries, gas-indusries and sewage reamen [Kakac and Liu (); Saunders (9); Sephanopoulas ()]. If he hea exchanger sysem is conrolled hrough a maser conrol cener locaed a remoe, hen he feedback loop of he closedloop conrol scheme will be compleed hrough communicaion neworks. Such a ypical nework conrolled emperaure conrol scheme will consis of hea exchanger, sensor, valve, and a regulaor (embedding PI conrol logic) placed a he remoe conrol cener [Sharma, Gupa and Kumar ()]. he emperaure conrol scheme will be pressed ino service in he even of any unexpeced deviaion in he emperaure conrol sysem oupu variable like sudden excessive wihdrawal of seam by oupu loads. Under such Deparmen of Chemical Engineering, Coimbaore Insiue of echnology, Coimbaore-6, India. * Corresponding auhor: V. Venkaachalam. vikranh64@gmail.com. CMES. doi:.97/cmes..4.

2 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, perurbed operaing condiion, he emperaure sensor sense he oupu emperaure. his signal is feedback o he conroller uni hrough a communicaion nework (as digially processed daa packes). In he conroller, he error beween he reference signal and acual signal (delayed by communicaion nework) is compued and processed by proporional-inegral (PI) conrol scheme. he PI conroller subsequenly offers a suiable inpu signal hrough he communicaion nework o he final conrol elemen (he valve) hereby bringing he perurbed hea exchanger erminal emperaure back o he equilibrium condiion. he hear of he hea exchanger conrol scheme, he PI conroller, is designed apriori assuming zero nework delays. However, he use of communicaion nework for daa (signal) ransfer in he closed-loop sysem inroduces ime-delays in he feedback pah. his, in urn, will enable one o achieve opimal performance from he ime-delayed sysem [Baisha and Joa (4)]. Delay dependen sabiliy crieria are employed o compue he sable margin for he nework delays wihin which he emperaure conrol sysem remains asympoically sable. o he bes of auhor s knowledge, he delay-dependen sabiliy of hea exchanger [Yeroglu (5)]. hese wo feedback-loop delays, popularly referred o as nework induced delays, are pracically unavoidable in a disribued conrol sysem scenario where he plan and conroller are conneced hrough a communicaion channel in which he daa (conrol or/and measured) experience buffering, processing and propagaion a various inernodes; refer Zhao e al. [Zhao, Liu and Rees ()]. As a resul, he nework-inroduced wo addiive delays are ime-varying in naure and have dissimilar characerisics [Lam, Goa and Wang (7)]. In general case, hese nework induced delays are ineviably pose serious limiaions o achievable performance of he closed loop sysem, and in some cases, may even lead o desabilizaion of he emperaure conrol sysem. Hence, sable delay margin (i.e. maximum delay bound which he neworked conrolled sysem can accommodae wihou losing asympoic sabiliy) for such emperaure conrol sysem mus be compued for various subses of conroller parameers (proporional gain K p and inegral gain K I ) hrough delay-dependen sabiliy analysis procedure so ha hey serve as a pracical guide-line for fine uning of conroller parameers a he implemenaion sage. Sysem under neworked conrol environmen has no been repored in lieraure so far. In his paper, a generalized crierion (sufficien condiion) is presened for asceraining delay dependen sabiliy of emperaure conrol sysems wih ime-varying addiive feedback loop delays. In he proposed approach, he mahemaical model of he PI conrolled emperaure conrol sysem wih ime-varying feedback loop delays is firs developed as a linear rearded delay-differenial coninuous-ime equaion. his sae-space model is a generalized modelling framework for dynamical sysems wih ime-delays. Subsequenly, based on Lyapunov-Krasovskii funcional approach combined wih Reciprocal convex combinaion [Ko and Park ()], a delay-dependen sabiliy crierion is presened in LMI framework o compue he maximum value bound of he ime-delay wihin which he closed-loop hea exchanger (emperaure conrol) sysem remains asympoically sable in he sense of Lyapunov. he paper concludes wih he presenaion of simulaion resuls of subsaniae he analyical resuls on delay-dependen sabiliy. he proposed sabiliy crierion is expressed as a se of solvable linear marix inequaliies (LMIs) ha

3 Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem can be solved using sandard numerical packages [Gahine, Nemirovskii, Laub e al. (995)] by casing i as a convex opimizaion problem. hese adequae condiions generally indicaed as sabiliy crieria are expressed as a se of solvable LMI condiions [Xu and Lam ()]. he less conservaism of new delaydependen sabiliy crieria mainly depends on he selecion of he LK funcional used in he analysis, and many useful resuls have been repored in lieraure in recen imes [Lam, Goa and Wang (7); Xu and Lam (); Gao, Chng and Lam (); Wu, Liao, Feng e al. (9); Ramakrishnan and Ray (9, b); Dey, Ray, Ghosh e al. (); Shao and Han (); Shao, Zhu, Zhang e al. (); Li and ian (); Ko, Lee and Park (); Jiao (); Chaibi, isser and Hmamed (); Hamed, Chaabane and Kacem ()]. However, many useful resuls exising in he lieraure, o he bes of our knowledge, here are no resuls available for he delay-dependen sabiliy of emperaure conrol sysem wih addiive ime-varying delays. emperaure conrol model wih addiive ime-varying delays he linear model of emperaure conrol sysem is used o analyze he sysem dynamics and o design a conroller [Sharma, Gupa and Kumar ()]. Fig. shows he block diagram of a emperaure conrol wih including nework-induced ime-delays. Now ha each componen of he sysem namely hea exchanger, sensor, valve, he conroller is modeled by a firs-order ransfer funcion [Sharma, Gupa and Kumar ()]. hey repeaed here for convenience: G V (s) = K V ; G V s+ H(s) = K H ; G H s+ F(s) = K F ; () F s+ where K V, K H and K F are he gains of a valve, hea exchanger and sensor respecively, and V, H and F are he corresponding ime consans. he ransfer funcion of he PI conroller is given below: G C (s) = K P + K I () s where K P and K I are he PI conroller gains respecively; he proporional erm conrols he rae of emperaure rise afer iniial ransiens. he inegral conroller gain adds a pole a he origin and increases he sysem ype by one and hence reduces he seady-sae error [Ramakrishnan and Ray (6)]. he combined effec of he PI conroller gains will srucure he response of he emperaure conrol sysem wih reduced overshoo and faser seling ime.

4 4 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, ref (s) e (s) Conroller PI Conroller Law Conroller o Acuaor delay k k I ( ) p e s s Valve KV GV ( s) s V Hea Exchanger V (s) K H GH ( s) H (s) s H Sensor o Conroller delay s ( ) e (s) S Sensor K F GF ( s) s F Figure : Block diagram of closed loop emperaure conrol sysem wih addiive imevarying delays In he Fig. τ () porrays sensor o conroller delay (measuring delay) and τ () porrays conroller o acuaor delay (processing delay). Sysem descripion and problem saemen he general sae-space model of he addiive sae-delayed sysem is given by: x () = Ax() + A d x( τ () τ ()) x() = (), [ (τ + τ, )]. () where x() R n is he sae vecor, A R n n are sysem marices, () is he iniial condiion of he sysem for, [ (τ + τ, )]. Consider he sysem () wih he following sysem marices are: K P KV K I KV V V V K H H H A ; Ad K F F F he ime-varying addiive delays τ () and τ () are saisfying he following condiions: τ () τ, τ () τ, (4) τ () μ, τ () μ, (5) where τ and τ are he upper bound of he ime-varying addiive delays τ () and τ (); μ and μ are he derivaives of he delay respecively. o develop a robus sabiliy

5 Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem 5 crierion in LMI framework o ge o know he delay-dependen sabiliy of he sysem () subjec o he condiion saisfying (4) and (5) using LK funcional approach and reciprocal convex combinaion lemma [Ko and Park ()]. For driving he new sabiliy crierion, following lemmas are of he umos imporance. Lemma [Zhu and Yang ()]: For any posiive symmeric consan marix N R n n, scalars η and η saisfying η < η, a vecor valued funcion ω: [η, η ] R n such ha he inegraion concerned are well defined, hen following inequaliy holds: η ( ω(s)ds) η N η ( η ω(s)ds) η η ω (s)n ω(s)ds η (6) where η = η η Lemma : For a coninuous funcion f(x) and is derivaive f (x) are defined on [a, b], hen b f (x)dx = f(b) f(a). a (7) Lemma [Ko and Park ()]: For any vecors ε, ε, marices P >, Q and real numbers α >, α > wih α + α = saisfying P Q [ Q ], P () he following inequaliy condiion holds: ε α Pε ε α Pε [ ε ε ] [ P Q Q P ] [ε ε ] (9) Proof: he condiion () implies ha α ε α α ε P Q [ α Q P ] α, () ε [ α α ε ] [ α ] which gives α ε α Pε + α ε α Pε ε Qε + ε Qε () By using he relaionship α + α = on () and by rearranging he erms, we deduce he inequaliy condiion (9). 4 Main resuls In his secion, he new sabiliy crierion for he emperaure conrol sysem wih addiive ime-varying delay is presened in he following heorem. heorem: Given non-negaive scalars τ, τ, μ, μ he sysem () saisfying he condiions (4) and (5) is asympoically sable, if here exiss real, posiive, symmeric marices Z, R, Q i, i =,, ; free marices S, S, S of appropriae dimensions such ha following LMI condiions hold:

6 6 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, Z S S [ Z S ] () Z Z >, Q >, Q >, Q >, R > ; () Ξ <, (4) where Ξ = e Pe 5 + e 5 Pe + e (Q + Q + Q )e + e ( Q ( μ ))e + e ( Q ( (e e ) μ))e + e 4 ( Q )e 4 + e 5 Ue 5 [(e e ) ] (e e 4 ) wih e e e e e 4 5 I Ae U Z I where τ = τ + τ and μ = μ + μ. Z S S [ Z S ] (5) Z Proof: consider he LK funcional V() = i= V i () wih V () = x ()Px(), (6) V () = I A I d e,, τ () τ +θ, x (s)q x(s) ds + τ() x (s)q x(s) ds + τ x (s)q x(s) ds, (7) V () = τ x (s)zx (s) dsdθ, () where P, Q i, i =,,, and Z are real symmeric posiive definie marices, and τ() = (τ () + τ ()). Define ( ) x ( ) x ( ) x ( ) x. he ime-derivaive of he funcional V i (), i =,,, is compued along he rajecory of () as follows: he ime-derivaive of V () is given by V () = x ()Px() + x ()Px (),, V () = η ()(e Pe 5 + e 5 Pe )η() (9) he ime-derivaive of V () is given by V () = x ()(Q + Q + Q )x() ( τ ())x ( τ ())Q x( τ ()) ( τ ())x ( τ())q x( τ()) x ( τ )Q x( τ ) () Since in NCSs, τ () μ < and τ () μ <, using Eq. () can be expressed as an inequaliy as follows:

7 Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem 7 V () x ()(Q + Q + Q )x() ( μ )x ( τ ())Q x( τ ()) ( μ)x ( τ())q x( τ()) x ( τ )Q x( τ ). () In oher words, Eq. () can be revealed in erms of η() as follows: V () η ()(e (Q + Q + Q )e + e ( ( μ )Q )e + e ( ( μ)q )e + e 4 ( Q )e 4 )η() () he ime-derivaive V () is given by V () = x ()τ Zx () τ τ x (s)zx (s)ds. () he Eq. () can be rewrien by expanding he inegral par as follows: V () = η ()(e 5 Ue 5 )η() τ τ τ () x (s)zx (s)ds τ() τ x (s)zx (s)ds τ Which, for applying Lemma, is rewrien on Eq. (4), we ge V () η ()(e 5 Ue 5 )η() τ τ (τ() τ ()) [(τ() τ ()) τ (τ τ()) τ () τ() [(τ τ()) x τ() τ x (s)zx (s)ds] τ () τ() x (s)zx (s)ds (s)zx (s)ds] [(τ τ () ()) x (s)zx (s)ds]. (5) τ () Now, by applying Lemma on Eq. (5), we ge V () η ()(e 5 Ue 5 )η() δ ( δ ( δ ( τ () τ() τ () τ() τ where δ = (τ τ()), δ τ = (τ() τ ()), δ τ = (τ ()) τ Now, we can express Eq. (6) as follows: τ() τ x (s)ds) Z ( x (s)ds) τ () τ() x (s)ds) Z ( x (s)ds) τ () (4) x (s)ds) Z ( x (s)ds) (6) V () η ()(e 5 Ue 5 )η() δ η ()(e e 4 ) Z(e e 4 )η() δ η ()(e e ) Z(e e )η() δ η ()(e e ) Z(e e )η() (7) Since δ + δ + δ =,, he reciprocal convex combinaion echnique of Eq. (7) is deal using Lemma as follows: (e e ) V () η ()(e 5 Ue 5 )η() [(e e ) ] (e e 4 ) Z S S [ Z S Z (e e ) ] [(e e ) ] () (e e 4 ) Provided Z S S [ Z S ] (9) Z holds wih S, S and S being free marices of any dimensions.

8 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, By adding V (), V (), and V () are expressions from (9), () and () respecively, we obain he following quadraic bounding relaion. V () η ()Ξη() () If he condiion Ξ < and relaion (9) holds simulaneously, which he addiive imedelay sysem () wih saisfying he condiion (4) and (5) is asympoically sable. 5 Resuls and discussions In his discussion, he effeciveness of he sabiliy crierion is presened for emperaure conrol sysem wih addiive ime-varying delays is illusraed. he maximum allowable delay margin for addiive ime-varying delays τ and τ for sabiliy for a pick range of PI conroller gains is compued. he gains and ime consans of he emperaure conrol used in he analysis are as follows [Ko and Park ()]. able : Parameer values of emperaure conrol sysem Sysem Gain ime-consan Hea exchanger K H = 4 H = Valve K V =.5 V = Sensor K F =. F = For he aforesaid sysem parameers wih delay free condiion, he conroller capabiliy curve can be obained easily; i is illusraed in Fig.. From he Fig., i is clear ha for all values of K P and K I lying below he conroller capabiliy curve, he closed-loop sysem is asympoically sable. From he curve we choose K P = and K I =.5s, hen he closed loop sysem converges asympoically o equilibrium poin as illusraed in Fig. for a uni sep perurbaion in emperaure conrol sysem oupu variable (from is equilibrium value). Figure : Maximum value of K I for differen values of K P

9 Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem 9 Figure : Evoluion of θ H () less han zero delay condiion when he nework delay τ is made zero, he sysem Eq. () becomes x () = (A + A d )x() () Now, i is observed ha he Eigenvalues of he sysem sae marix and inpu marix (A + A d ) depend on he conroller parameers K P and K I. ab. given he maximum values of K I (s ) for a fixed value of K P for which he emperaure conrol sysem is on he verge of insabiliy i.e. having one complex conjugae pair of eigenvalues on he jω axis. I is observed from he Fig. ha here is a endency for he curve o decrease gradually wih he increase of K I. able : Maximum inegral conroller gain for a given K P (s ) Eigen values of (A + A d ) ±.9j; -.4; ±.67j; -.5; ±.j; -.645; ±.67j; -.65; ±.5j; -.7; ±.9j; -.76; ±.7j; ; ±.9j; -.6; ±.45j; -.56; -.6 Now, when his PI conroller operaes aop a neworked environmen, he emperaure conrol sysem will be subjeced o addiive nework induced ime-varying delays τ () and τ (). Le he ime-derivaive of he addiive ime-varying delays be bounded as τ () =. and τ () =.. Le choose he conroller parameer K P = and K I =.5s, in he presence of nework induced delays, according o he sabiliy crierion are presened in heorem, he emperaure conrol sysem is sable up o τ = second τ =.5 seconds, and if nework induced delays is increased far beyond his value, he sysem oupu becomes unbounded evoluion. Hence, his analysis clearly brings o he impac of he nework induced delays on sabiliy and performance of he sysem. Le μ =., and μ =.. for differen values of K P and K I, from he capabiliy curve, he

10 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, maximum allowable delay bound τ for a given τ is given in abs. -7. Likewise, abs. - give he maximum allowable delay bound τ for a given τ. From he ables, i is clear ha as he various values of K P and K I, he maximum delay bound ha sysem can accommodae wihou losing sabiliy decreases. he corresponding sabiliy margin presened in Fig. 4 clearly indicaes lessening. able : Maximum allowable delay bound τ for a given τ able 4: Maximum allowable delay bound τ for a given τ able 5: Maximum allowable delay bound τ for a given τ able 6: Maximum allowable delay bound τ for a given τ able 7: Maximum allowable delay bound τ for a given τ able : Maximum allowable delay bound τ for a given τ

11 Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem able 9: Maximum allowable delay bound τ for a given τ able : Maximum allowable delay bound τ for a given τ - K P K I able : Maximum allowable delay bound τ for a given τ able : Maximum allowable delay bound τ for a given τ -5 K P K I Figure 4: Sabiliy margin for emperaure conrol of hea exchanger Simulaion sudy of he closed-loop emperaure conrol sysem was carried ou for zero nework delay is confined o he sable region shown in Fig. ; i is easy o infer he influence of nework delays on he sabiliy of he sysem. As nework delays increase, he response of he emperaure conrol sysem ges malformed and for higher delays, he closed-loop sysem becomes unsable.

12 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, 6 Conclusion In his research, he problem of delay-dependen sabiliy of emperaure conrol sysems wih addiive ime-varying delay has been addressed using Lyapunov-Krasovskii funcional echnique. In he sabiliy analysis, he emperaure conrol sysem is mahemaically modeled in erms of sae space model, and reciprocal convex combinaion echnique is employed in he sabiliy analysis o make i less number of decision variables. he proposed resuls are validaed benchmark emperaure conrol sysem wih ime-varying addiive delays. he deduced resuls of his paper impar more realisic operaing condiion in a real-ime emperaure conrol sysem. Acknowledgemen: One of he auhors (Dr. D. Prabhakaran) is graeful o All India Council for echnical Educaion (AICE), New Delhi, India for financial suppor (File No. /AICE/RFD/RPS (Policy-II) /-6). he auhors would like o express heir sincere hanks o Dr. S. R. K. Prasad (Corresponden, CI), Dr. R. Prabhakar (Secreary, CI), Dr. V. Selladurai (Principal, CI), Dr. M. hirumarimugan (Head of he Deparmen of Chemical Engineering, CI) for heir useful discussions and encouragemen during he course of work. Also, we like o acknowledgmen Dr.. Venkaachalam (Associae Professor of Physics, Coimbaore Insiue of echnology, Coimbaore, India) for heir valuable help in sudying dynamical sysem wih ime delays. References Baisha, A. P.; Joa, F. G. (4): Effecs of ime delay saisical parameers on he mos likely regions of sabiliy in an NCS. Conrol Engineering and Applied Informaics, vol. 6, pp. -. Chaibi, N.; isser, E. H.; Hmamed, A. (): Delay-dependen robus sabiliy of singular sysems wih addiive ime-varying delays. Inernaional Journal of Auomaion and Compuing, vol., no., pp Dey, R.; Ray, G.; Ghosh, S.; Rakshi, A. (): Sabiliy analysis for coninuous sysem wih addiive ime-varying delays: a less conservaive resul. Applied Mahemaics and Compuaion, vol. 5, no., pp Gahine, P.; Nemirovskii, A.; Laub, A. J.; Chilali, M. (995): Conrol oolbox for use wih Malab. Mahworks, Naick. Gao, H.; Chng,.; Lam, J. (): A new delay sysem approach o nework-based conrol. Auomaica, vol., no., pp Hamed, B. B.; Chaabane, M.; Kacem, W. (): Absolue sabiliy of nonlinear sysems wih wo addiive ime-varying delay componens. Inernaional Journal of Auomaion and Compuing, vol., no., pp Jiao, J. M. (): A sabiliy crierion for singular sysems wih wo addiive imevarying delay componens. Inernaional Journal of Auomaica, vol., no., pp Kakac, S.; Liu, H. (): Hea exchangers: selecion, raing and hermal design (nd ediion). CRC Press.

13 Sabiliy Analysis of Nework Conrolled emperaure Conrol Sysem Ko, J. W.; Lee, W. I.; Park, P. G. (): Delayed sysem approach o he sabiliy of neworked conrol sysems. Inernaional Muli conference of Engineers and Compuer Scieniss, vol., pp. 6-. Ko, J. W.; Park, P. G. (): Reciprocally convex approach for he sabiliy of neworked conrol sysems. Inelligen Conrol and Innovaive Compuing, vol., pp. -9. Lam, J.; Goa, H.; Wang, C. (7): Sabiliy analysis for coninuous sysems wih wo addiive ime-varying delay componens. Sysem and Conrol Leers, vol. 56, no., pp Li,.; ian, J. (): Convex polyhedron mehod o sabiliy of coninuous sysems wih wo addiive ime-varying delay componens. Journal of Applied Mahemaics. Ramakrishnan, K.; Ray, G. (9): Delay-dependen sabiliy analysis of linear sysems wih addiive ime-varying delays. 5h Annual IEEE Conference on Auomaion Science and Engineering, Bangalore, India, pp. -6. Ramakrishnan, K.; Ray, G. (b): Sabiliy crieria wih less LMI variables for linear sysem wih addiive ime-delays. Journal of Applied Mahemaics and Compuing, vol. 6, no. -, pp Ramakrishnan, K.; Ray, G. (6): Sabiliy crierion for neworked conrol sysems wih addiive ime-varying sae-delays and bounded nonlineariy. Inernaional Journal of Sysems, Conrol and Communicaions, vol. 7, no., pp. 6-. Saunders, E. A. (9): Hea exchanges: selecion, design and consrucion. Longman Scienific and echnical, New York. Shao, H.; Han, Q. L. (): On sabilizaion for sysems wih wo addiive ime-varying inpu delays arising from neworked conrol sysems. Journal of he Franklin Insiue, vol. 49, no. 6, pp Shao, H.; Zhu, X.; Zhang, Z.; Zhong, G. (): Delay-dependen H conrol for sysems wih wo addiive ime-vary delays. Journal of Conrol Engineering and echnology, vol., no., pp Sharma, C.; Gupa, S.; Kumar, V. (): Modeling and simulaion of hea exchanger used in soda recovery. Proceedings of he World Congress on Engineering, vol.. Sephanopoulas, G. (): Chemical process conrol. Prenice-Hall of India Privae Limied, pp. 9-. Wu, H.; Liao, X.; Feng, W.; Guo, S.; Zhang, W. (9): Robus sabiliy analysis of uncerain sysems wih wo addiive ime-varying delay componens. Applied Mahemaical Modelling, vol., no., pp Xu, S.; Lam, J. (): A survey of linear marix inequaliy echniques in sabiliy analysis of delay sysems. Inernaional Journal of Sysems Science, vol. 9, no., pp Yeroglu, C. (5): Robus sabilizing PID conrollers for muliple ime delay sysems wih parameric uncerainy. Conrol Engineering and Applied Informaics, vol. 7, pp. -9. Zhao, Y.; Liu, G.; Rees, D. (): Packe-based dead-band conrol for inerne-based neworked conrol sysem. IEEE ransacions on Conrol Sysems echnology, vol., no. 5, pp

14 4 Copyrigh ech Science Press CMES, vol.4, no., pp.-4, Zhu, X. L.; Yang, G. H. (): Jensen inegral inequaliy approach o sabiliy analysis of coninuous-ime sysems wih ime-varying delay. IE Conrol heory and Applicaions, vol., no. 6, pp