Stabilization of Linear Systems with Distributed Input Delay

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1 2 Ameicn Contol Confeence Miott Wtefont, Bltimoe, MD, USA June 3-July 2, 2 FB9.5 Stbiliztion of Line Systems with Distibuted nput Dely Gego Goebel, Ulich Münz, Fnk Allgöwe Abstct his ppe pesents line mtix inequlity LM conditions to design stbilizing stte-feedbck contolles fo line systems with distibuted input delys. A Lypunov- Ksovskii functionl ppoch is used nd the obtined pmetic mtix inequlities e efomulted s LMs using the full-block S-pocedue nd convex hull elxtion. hese LM conditions cn be used, fo exmple, to design stbilizing contolle fo netwoked contol systems NCS. n this setup, the kenel of the distibuted input dely descibes the pobbility density of the pcket-dely in the communiction chnnel. his contolle design fo NCS is illustted in n exmple. ndex ems ime dely systems, distibuted input dely, stbiliztion, full-block S-pocedue, convex hull elxtion, netwoked contol systems.. NRODUCON ime dely systems DS hve expeienced get inteest in ecent yes s cn be seen in [3] nd efeences theein. n the field of engineeing, DS e often used to model tnspottion o communiction phenomen, e.g. [7], [22]. A ough clssifiction of dely systems is obtined by distinguishing between discete nd distibuted dely systems nd gin by distinguishing between delys tht ppe in the sttes nd those ppeing in the input of the system. Most publictions on DS conside nlysis nd design of DS with discete o distibuted stte delys, e.g. [6], [3], [4], [8], [23], [29], [32] nd efeences theein. Yet, the contolle design fo systems with input dely is much moe chllenging becuse no diect contol ction cn be pplied to the system but only fte some dely. Contolle design lgoithms fo DS with delyed nd undelyed input e pesented fo instnce in [2], [28], [3]. he moe chllenging poblem of DS with delyed but without undelyed input hs been consideed fo exmple in [3] [5], [], [3] fo discete input delys. Howeve, thee e only vey few esults fo distibuted input delys, e.g. [], [], which lso ssume undelyed inputs, nd [24], which ssumes γ-distibuted dely kenel. Distibuted input delys ppe, e.g., in netwoked contol systems NCS whee the digitl communiction chnnel with stochstic pcket dely nd loss is modeled using distibuted delys, cf. [24]. n this ppe, we pesent design conditions fo stte feedbck contolles fo line systems with distibuted input dely. We ssume the dely kenel of the input is given in G. Goebel, U. Münz, nd F. Allgöwe e with the nstitute fo Systems heoy nd Automtic Contol, Univesity of Stuttgt, Pfffenwlding 9, 755 Stuttgt, Gemny. gego goebel@web.de, {muenz,llgowe}@ist.uni-stuttgt.de. U. Münz nd F. Allgöwe cknowledge finncil suppot by he Mth- Woks Foundtion nd by the Pioity Pogmme 35 Contol heoy of Digitlly Netwoked Dynmicl Systems of the Gemn Resech Foundtion DFG. line fctionl fom. As will be explined lte on, this ssumption is not estictive. We choose suitble Lypunov- Ksovskii functionl nd efomulte the esulting pmetic mtix inequlity using the full-block S-pocedue nd convex-hull elxtion in tems of LM. Simil guments hve been used in ou pevious publictions [7], [2], [2] tht conside distibuted delys in the stte. Hee, we extend these concepts to systems with distibuted input delys. Ou fist min esult is still quite estictive on the dely kenel becuse the kenel hs to be nonzeo in the nge of the distibuted dely integl. his estiction is emoved in ou second min esult, whee we intoduce dditionl fee mtices. he ltte condition cn be used to design stte-feedbck contolles fo NCS, which is illustted in n exmple. he ppe is stuctued s follows: We stt by giving ou poblem fomultion nd some mthemticl peliminies in Section. n Section, ou min esults e pesented, the new contolle design conditions. n Section V, we show the esults of n exemply ppliction of ou theoy befoe the wok is concluded in Section V. fo A B B C nd AB fo Nottion: We wite A B C B AB. By V we men the time deivtive long tjectoies of the consideed DS. Fo the signl xτ, τ, <, we denote by x t the section xθ, θ [t,t] fo < t. Futhemoe,,,, denote positive nd negtive semi-definiteness fo mtices, espectively.. PROBLEM SAEMEN AND PRELMNARES A. Poblem Sttement We conside the following clss of system: ẋt Axt xt φt, t [,] Fθut θdθ whee xt R n is the system stte, ut R m is the input, φt is the initil condition, A R n n is dynmic mtix, possibly with eigenvlues in the ight hlf plne, nd Fθ R n m, F : [,] R n m, is the so-clled dely kenel. he dely kenel cn be consideed s weighting function to the input ove the pst time. he only ssumption we mke is tht the dely kenel cn be expessed s line fctionl epesenttion LFR Fθ D F C F θ θa F B F, 2 fo suitble mtices A F R n F n F s well s B F,C F, nd D F of ppopite dimensions, i.e. F is mtix of tionl //$26. 2 AACC 58

2 functions in θ. We use LFR fo the kenel F becuse it llows to tnsfom the pmetic mtix inequlity obtined in the poof into set of LMs. Since on the one hnd ny function cn be ppoximted bitily pecise by tionl function of sufficiently high ode nd on the othe hnd ny tionl function cn be witten in fom of LFR, this ssumption is not estictive. he im of this wok is to deive LM conditions to design stte feedbck contolle of the fom ut Kxt, t [,, 3 tht symptoticlly stbilizes system fo given A,A F,B F,C F,D F. B. Peliminies We use the following esults on mtix inequlities in the poofs of ou min theoems. Full-Block S-pocedue: he full-block S-pocedue is poweful tool tht cn be used to simplify the pmete dependence of mtix inequlities. t is fomulted in the following lemm: Lemm [9], [26]: Suppose Gδ : [δ,δ 2 ] R n n 2,δ < δ 2, nd Gδ cn be witten s Gδ D G C G δ ng δa G B G. hen the following holds: n2 QP S P n2 Gδ SP, δ [δ R P Gδ,δ 2 ], if nd only if thee exist mtices Q Q, R R nd S stisfying δ Q S δ S, δ [δ R,δ 2 ], 4 nd Q S A G B G S R A G B G Qp S p C G D G S. 5 p R p C G D G Fo futhe detils on the full-block S-pocedue see fo exmple [9], [26]. 2 Convex Hull elxtion: he convex hull elxtion is tool tht cn be used to efomulte pmete dependent mtix inequlities like 4 in LM: Lemm 2 [25]: Suppose Q. hen δ if nd only if Q S S R δi δ Q S S R δi, δ [δ,δ 2 ], 6, i,2. 7 Futhe detils on the convex hull elxtion cn be found in [25]. 3 Jensen s nequlity: Fo the poof of ou min esult, we lso use Jensen s inequlity given in the next lemm: Lemm 3 [8]: Fo ny constnt, symmetic, positive definite mtix M R n n, scl γ >, vecto function ω : [,γ] R n such tht the following integls e well defined, we hve γ γ ω smωsds γ ωsds M Fo poof of lemm 3, see fo exmple [8].. MAN RESULS γ ωsds. n this section, we pesent the min esults of this ppe. heoem gives fist esult to the poblem stted in Section -A. As we e going to show, the conditions imply some estictions concening the dely kenel. We pesent n extension of ou fist esult in heoem 2 tht elxes this estiction. A. Stbiliztion fo F he following theoem povides LM condition to design stte feedbck contolles 3. As will be discussed below, it equies F. his estiction is emoved in heoem 2. heoem : Conside system with kenel F given in 2. he system is symptoticlly stbilizble by stte feedbck contolle ut Kxt,K R m n, if thee exist positive definite, symmetic mtices P, P 2, nd P 3, s well s mtices K, R, Q, S of ppopite dimensions with R R nd Q Q such tht 2 QSS R nd 8 on top of the next pge hold. hen, stbilizing contolle is K K P 3. Poof: Conside the Lypunov-Ksovskii functionl cndidte Vx t V x t V 2 x t with V x t x tp xt θ V 2 x t θ θ sẋ t sp 2 ẋt sdsdθ, with P nd P 2 being positive definite mtices of ppopite dimensions. Fo positive definite, symmetic mtix P 3 nd with the definition yt : ẋt, the time deivtive of V x t long the solutions of is V x t 2x tp yt2x tp 3 ẋt 2x tp 3 yt }{{} 2y tp 3 ẋ t 2y tp 3 yt }{{} xt P P 3 xt yt P P 3 2P 3 yt 2x tp 3 Axt2x tp 3 FθKxt θdθ 2y tp 3 Axt2y tp 3 FθKxt θdθ 58

3 QSA F A FS A F RA F SCF A FRCF B F K SCF A FRC F A P 3 P 3 A P 2 C F RCF P 2 D F K P P 3 P 3 A C F RCF P 2 K D 8 F 2 P P 2 C F RCF with xt xt θ yt M θ xt xt θ dθ yt M θ P 3 AA P 3 P 3 FθK P P 3 A P 3 K F θp 3. 2 P 3 n ode to compute the time deivtive of V 2, note tht the deivtive of θ θ sẋ t spẋt sds is θẋ tpẋt t t θ ẋ spẋsds. his cn be shown substituting the integl vible nd splitting up the integl. hus, we bound the deivtive of V 2 using Lemm 3 s t V 2 x t θ 2 ẋ tp 2 ẋt θ θ 2 ẋ tp 2 ẋt 3 3 y tp 2 yt t θ t t θ ẋ sp 2 ẋsdsdθ ẋ sds P 2 t t θ ẋsdsdθ P 2 [xt xt θ]dθ 3 2 y tp 2 yt x tp 2 xt 2x t θp 2 xt x t θp 2 xt θ hus, we get dθ P 2 P 2 P 2 P 2 xt xt θ dθ. 3 2 P 2 yt Vx t xt xt θ yt Mθ xt xt θ dθ 9 yt with Mθ ψ P 3FθK P 2 P P 3 A P 3 P 2 K F θp 3 2 P P 2 nd ψ P 3AA P 3 P 2. Stbility of with feedbck 3 is gunteed if Vx t <, see fo exmple [7]. his is chieved, e.g., if Mθ is negtive definite fo ll θ [,]. n ode to lineize Mθ in the fee mtices P,P 2,P 3 nd K, we define P 3 digp3,p3,p 3 nd compute Mθ P 3 MθP 3 s ψ Fθ K P 2 Mθ P P 3 P 3 A P 2 K F θ 2 P P 2 with ψ A P 3 P 3 A P 2 nd P P3 P P3, P 2 P3 P 2P3, P 3 P3, K KP3. Finlly, Mθ is line in the fee mtices nd we cn pply the full-block S-pocedue nd the convex hull elxtion to emove the pmete dependence on θ. Note tht Mθ ψ P 2 P P 3 P 3 A P 2 K 2 P P 2 Gθ with Gθ F θ F θ. Fo the coesponding LFR Gθ D G C G θ θa G B G, we hve A G A F, B G CF,,C F, C G B F, D G D F,,D F. Applying Lemm povides Mθ if nd only if thee exist mtices Q Q, R R nd S stisfying 5 with Q p,s p,r p given in nd θ Q S θ S, θ [,]. 2 R Applying the convex hull elxtion in Lemm 2 to the ltte inequlity leds to R, Q nd 2 QSS R. Multiplying out 5 leds to 8. he LM 8 evels the estiction this solution bes: n unstble system cn only be stbilized with this LM if D F. his cn lso be seen fom in the poof. D F implies F nd this in tun implies tht K cnnot ende Mθ fo ll θ [,] if A hs eigenvlues in the ight hlf plne. n ode to emove this estiction, we pesent n extension to ou fist theoem in the next section. B. Stbiliztion fo F he following theoem gives solution to the contolle design poblem in Section -A tht does not equie D F fo unstble open loop systems. heoem 2: Conside system with kenel F given in 2. he system is symptoticlly stbilizble by stte feedbck contolle ut Kxt, K R m n, if fo some, thee exist mtices K, S, nd symmetic mtices 582

4 M 2 A P 3 P 3 A P 2 P 4 P 2 P P 3 P 3 A P 2 K 2 P P 2 P 5 Φ P 3 P P 3 P 3 A P 2 K Φ 2 3 Q, R, P, P 2, P 3, P 4, P 5 such tht η 2 QηSS R fo η : mx{, }, nd Q S S M R A G B 2 G C G D G 4 whee A A G F à F B C G F B F C, B G F CF C F C F D F D F, D G D F D F,, with à F A F A F, B F A F B F, C F C F C F A F A F nd D F D F C F A F B F nd M 2 s given in 3 on top of this pge with Φ A P 3 P 3 A P 2 P 4 nd Φ 2 2 P P 2 P 5. hen, stbilizing contolle is K K P 3. Poof: We use the sme Lypunov-Ksovskii functionl cndidte s in the poof of heoem nd conside 9. n the sequel of the poof of heoem, we deived conditions such tht Mθ is negtive definite fo ll θ [,]. his equies M which cn only be chieved fo unstble open loop systems if D F. he ide of this theoem is to dmit Mθ fo some θ [,] s long s thee e othe intevls in [,] tht compenste fo those θ whee Mθ. heefoe, we split up the whole integl into two pts nd shift the constnt fctos x tp 4 xt nd y tp 5 yt fo symmetic mtices P 4,P 5 fom one pt to the othe one Vx t Mθ xt xt θ dθ yt P 4 xt xt θ dθ P 5 yt P 4 xt xt θ dθ P 5 yt M θ xt xt θ dθ yt M 2 θ xt xt θ dθ yt fo some, nd M θ Mθ P 4 nd P 5 P 4 M 2 θ Mθ. P 5 o show negtive definiteness of Vx t, it is sufficient to poof M θ, θ, nd M 2 θ, θ,. hese conditions e stisfied if M θ M 3 θ θ,mx{, }. M 2 θ he tnsition fom Mθ to Mθ in M 2 θ cn be done by eplcing Fθ by Fθ in. Hence, we intoduce Fθ : Fθ nd clculte the pmetes fo the coesponding LFR Fθ D F C F θ θã F B F s given in heoem 2. We futhemoe use P 3 defined in the poof of heoem to lineize M 3 θ nd intoduce the mtices P 4 P3 P 4 P3 nd P 5 P3 P 5P3. Using these esults we find suitble decomposition to be ble to pply Lemm : M 3 θ digp 3,P 3 M 3 θdigp 3,P 3 M 2 Gθ whee Gθ F θ F θ F θ F θ nd M 2 given in 3. he LFR of Gθ is Gθ D G C G θ A G B G with A G,B G,C G,D G given in the heoem. Using Lemm, we get M 3 θ fo ll θ [,mx{, }] if nd only if thee exist mtices Q Q, R R nd S stisfying 4 nd θ Q S θ S, θ [,mx{, }]. R 5 Applying the convex hull elxtion in Lemm 2 to the ltte inequlity leds to Q,R, nd η 2 QηSS R fo η : mx{, }. 583

5 ut FθKxt θdθ Dynmicl System ẋt Axt But xt 5 nomlized histogm LFR ppoximtion Contolle nsmission Chnnel Fθxt θdθ K Fθ θ Fθ 5 Fig.. chnnel. Netwoked contol system with distibuted dely tnsmission Note tht 4 is in fct LM fo given,, i.e. it is line in the fee mtices K,S,Q,R, P, P 2, P 3, P 4, P 5. heefoe, stte feedbck contolle cn be designed efficiently bsed on heoem 2 using, e.g., SeDuMi [27]. his is illustted in n exmple in the next section. Following the poof of heoem 2, it is lso stightfowd to see how to del with dely kenels F tht stisfy Fθ fo ll θ [,], i.e. F is zeo in some intevl in [,]: We split up the intevl [,] into intevls nd [,] \ nd tet both pts s in the fome poof. V. EXAMPLE: CONROLLER DESGN FOR NEWORKED CONROL SYSEMS n this section, we illustte the theoeticl findings in this ppe on netwoked contol system NCS. We conside n unstble line dynmicl system tht is to be stbilized vi digitl netwok by stte feedbck contolle. he digitl netwok intoduces ndomly distibuted, fst vying delys between sensos, contolle, nd ctutos. he communiction delys e often modeled using time-vying delys, see [4], [5], [3] fo exmple. Yet, fst time-vying pcket-delys cn lso be modeled in simplified fom using distibuted delys whee the dely kenel descibes the pobbility density of the pcket-dely, see [6], [9], [24]. n this exmple, we follow the ltte ide nd conside NCS of the fom ẋt AxtBut, ut FθKxt θdθ, 6 whee A R n n, B R n m, xt R n is the system stte, ut is the contol input, nd F : R R descibes the pobbility density of the pcket delys, see lso Figue. he im of this exmple is to design the contolle gin K such tht the closed loop system is symptoticlly stble. Note tht 6 is identicl to, 3 if BFθ : R R n m in 6 equls Fθ : R R n m in. Fo the simultions, we implemented the bove system in MALAB/SMULNK nd modeled the digitl communiction netwok using simevents [5]. SimEvents povides SMULNK blocks fo geneting, queueing, nd outing pckets. Using these blocks, we cn simulted elistic communiction chnnels. Fo suitble choice of the netwok pmetes, we obtined the pcket dely histogm in Figue 2 with n vege pcket dely of bout 2ms. he histogm is nomlized such tht it esembles pobbility θ [s] Fig. 2. Nomlized histogm of pcket dely mesuements in simevents netwok nd tionl ppoximtion of the pobbility density of the dely. density. We ppoximted this pobbility density function with simple tionl function Fθ with LFR mtices 2 35 A F, B 35 F, C F, D 3542 F. Bsed on Figue 2, we tke.45. Note tht D F, i.e. heoem cnnot povide solution. nsted, we use heoem 2. Consideing gin Figue 2, we see tht.26 is esonble choice such tht Fθ is the lge fo θ,] nd Fθ the smll fo θ [,. Fo ou simultions, we conside n unstble scl system with A.36 nd B. We solve the LM condition in heoem 2 with SeDuMi [27] vi the ylmip [2] intefce nd obtin the stbilizing feedbck contolle K Simultion esults fo the closed loop system with the pcket-bsed communiction chnnel modeled with simevents blocks e pesented in Figue 3. Clely, the system is stbilized by the deived contolle. Fo compison, the esult of simultion of the sme system with the sme contolle but without delys in the input, i.e. with n idel communiction link, is depicted in the Figue 3 s well. We see tht the obtined contolle chieves simil pefomnce in tems of convegence time s the closed loop with idel link. V. CONCLUSONS We pesented two LM conditions fo stbilizing stte feedbck contolles fo line systems with distibuted input dely. Menwhile the fist one hs some estictions concening the dely kenel, the second one cn be pplied to bod viety of systems of the given clss. Futhemoe, we pesented n exemply ppliction to ou LM conditions: A stbilizing contolle fo system tht is contolled vi ndomly delyed communiction chnnel ws designed. REFERENCES [] M. Aiol nd A. Pionti. H optiml teminl contol fo line systems with delyed sttes nd contols. Automtic, 44: , 28. [2] W.-H. Chen nd W. X. Zheng. On impoved obust stbiliztion of uncetin systems with unknown input dely. Automtic, 426:67 72,

6 x with digitl communiction netwok with idel link t [s] Fig. 3. Exemply simultion of closed loop system with stbilizing contolle K with nd without digitl communiction netwok. [3] M. Di Loeto, J.-J. Loiseu, nd J.-F. Lfy. Distubnce ttenution by dynmic output feedbck fo input-dely systems. Automtic, 58: , 28. [4] E. Fidmn. Robust smpled-dt H contol of line singully petubed systems. EEE ns. Autom. Contol, 53:47 475, 26. [5] E. Fidmn, A. Seuet, nd J.-P. Richd. Robust smpled-dt stbiliztion of line systems: An input dely ppoch. Automtic, 48:44 446, 24. [6] K. Gu. An impoved stbility citeion fo systems with distibuted delys. nt. J. Robust nd Nonline Contol, 39:89 83, 23. [7] K. Gu, V. L. Khitonov, nd J. Chen. Stbility of ime-dely Systems. Bikhäuse, Boston, 23. [8] Keqin Gu. An integl inequlity in the stbility poblem of time-dely systems. n Poc. 39th EEE Confeence on Decision nd Contol, 2. [9]. wski nd G. Shibt. LPV system nlysis vi qudtic septo fo uncetin implicit systems. EEE ns. Autom. Contol, 468:95 28, 2. [] M. Kstic. Compenstion of infinite-dimensionl ctuto nd senso dynmics. EEE Cont. Syst. Mgzine, 3:22 4, 2. [] S. Liu, L. Xie, nd H. Zhn. Line qudtic egultion fo line continuous-time systems with distibuted input dely. n Poc. Chin. Cont. Conf., pges 95 99, Kunming, Chin, 28. [2] J. Löfbeg. YALMP, 26. Avilble fom joloef/ylmip.php. [3] J. J. Loiseu, W. Michiels, S.-. Niculescu, nd R. Siphi, editos. opics in ime Dely Systems: Anlysis, Algoithms, nd Contol, volume 388 of LNCS. Spinge, Belin, 29. [4] C. Mie,. Hg, U. Münz, nd F. Allgöwe. Constuction of qudtic Lypunov-Ksovskii functionls fo line time-dely systems with multiple uncetin delys. n S. Sivsundm, edito, Mthemticl Anlysis nd Applictions in Engineeing Aeospce nd Sciences, volume 5 of Mthemticl Poblems in Engineeing Aeospce nd Sciences. Cmbidge, UK, 29. [5] he MthWoks. SimEvents.2 toolbox, 26. Avilble fom [6] W. Michiels, V. vn Assche, nd S.-. Niculescu. Stbiliztion of timedely systems with contolled time-vying dely nd pplictions. EEE ns. Autom. Contol, 5:493 54, 25. [7] U. Münz nd F. Allgöwe. L 2 -gin bsed contolle design fo line systems with distibuted delys nd tionl dely kenels. n Poc. FAC Wokshop on ime-dely Systems, Nntes, Fnce, 27. [8] U. Münz, C. Ebenbue,. Hg, nd F. Allgöwe. Stbility nlysis of time-dely systems in the fequency domin using positive polynomils. EEE ns. Autom. Contol, 545:9 24, 29. [9] U. Münz, A. Ppchistodoulou, nd F. Allgöwe. Consensus eching in multi-gent pcket-switched netwoks. nt. J. Contol, 825: , 29. [2] U. Münz, J. M. Riebe, nd F. Allgöwe. Robust stbility of distibuted dely systems. n Poc. FAC Wold Congess, pges , Seoul, South Koe, 28. [2] U. Münz, J. M. Riebe, nd F. Allgöwe. Robust stbiliztion nd H-infinity contol of uncetin distibuted dely systems. n J. J. Loiseu, W. Michiels, S.-. Niculescu, nd R. Siphi, editos, opics in ime Dely Systems: Anlysis, Algoithms, nd Contol, volume 388 of LNCS, pges Spinge, 29. [22] S.-. Niculescu. Dely Effects on Stbility: A Robust Contol Appoch. Spinge, London, 2. [23] J.-P. Richd. ime-dely systems: An oveview of some ecent dvnces nd open poblems. Automtic, 39: , 23. [24] O. Roesch, H. Roth, nd S.-. Niculescu. Remote contol of mechtonic systems ove communiction netwoks. n Poc. nt. Conf. Mechtonics nd Automtion, pges , Nig Flls, Cnd, 25. [25] C. W. Schee. Robust mixed contol nd line pmete-vying contol with full block multiplies. n L. EL Ghoui nd S.-. Niculescu, editos, Recent Advnces on LM methods in Contol, pges SAM, 2. [26] C. W. Schee. LPV contol nd full block multiplies. Automtic, 373:36 375, 2. [27] J. F. Stum. SeDuMi, 26. Avilble fom [28] F. Wu nd K. M. Gigoidis. LPV systems with pmete-vying time-delys: Anlysis nd contol. Automtic, 372:22 229, 2. [29] L. Xie, E. Fidmn, nd U. Shked. Robust H contol of distibuted dely systems with ppliction to combustion contol. EEE ns. Autom. Contol, 462:93 935, 2. [3] D. Yue. Robust stbiliztion of uncetin systems with unknown input dely. Automtic, 42:33 336, 24. [3] D. Yue, E. in, Z. Wng, nd J. Lm. Stbiliztion of systems with pobbilistic intevl input delys nd its pplictions to netwoked contol systems. EEE ns. Systems, Mn, nd Cybenetics Pt A: Systems nd Humns, 394: , 29. [32] F. Zheng nd P. Fnk. Robust contol of uncetin distibuted dely systems with ppliction to the stbiliztion of combustion in ocket moto chmbes. Automtic, 382: ,