Dwell time-based stabilisation of switched delay systems using free-weighting matrices

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1 Internatonal Journal of Control ISSN: (Prnt) (Onlne) Journal homepage: Dwell tme-based stablsaton of swtched delay systems usng free-weghtng matrces Ahmet Taha Koru Akın Delbaşı & Htay Özbay To cte ths artcle: Ahmet Taha Koru Akın Delbaşı & Htay Özbay (2017): Dwell tme-based stablsaton of swtched delay systems usng free-weghtng matrces Internatonal Journal of Control DOI: / To lnk to ths artcle: Accepted author verson posted onlne: 01 Dec Publshed onlne: 20 Jan Submt your artcle to ths journal Artcle vews: 28 Vew related artcles Vew Crossmark data Full Terms & Condtons of access and use can be found at Download by: [Blkent Unversty] Date: 22 January 2017 At: 21:40

2 INTERNATIONAL JOURNAL OF CONTROL Dwell tme-based stablsaton of swtched delay systems usng free-weghtng matrces Ahmet Taha Koru a Akın Delbaşı b and Htay Özbay c a Department of Mechatroncs Engneerng Yıldız Techncal Unversty İstanbul Turkey; b Department of Control and Automaton Engneerng Yıldız Techncal Unversty İstanbul Turkey; c Department of Electrcal and Electroncs Engneerng Blkent Unversty Ankara Turkey ABSTRACT In ths paper we present a quas-convex optmsaton method to mnmse an upper bound of the dwell tme for stablty of swtched delay systems. Pecewse Lyapunov Krasovsk functonals are ntroduced and the upper bound for the dervatve of Lyapunov functonals s estmated by freeweghtng matrces method to nvestgate non-swtchng stablty of each canddate subsystems. Then a suffcent condton for the dwell tme s derved to guarantee the asymptotc stablty of the swtched delay system. Once these condtons are represented by a set of lnear matrx nequaltes dwell tme optmsaton problem can be formulated as a standard quas-convex optmsaton problem. Numercal examples are gven to llustrate the mprovements over prevously obtaned dwell tme bounds. Usng the results obtaned n the stablty case we present a nonlnear mnmsaton algorthm to synthesse the dwell tme mnmser controllers. The algorthm solves the problem wth successve lnearsaton of nonlnear condtons. ARTICLE HISTORY Receved 23 March 2016 Accepted 26 November 2016 KEYWORDS Tme-delay; dwell tme optmsaton; swtched delay systems; free-weghtng matrces 1. Introducton A swtched system s a dynamcal system that ncludes a set of subsystems and a dscrete swtchng event between them. General behavour of a swtched system s governed by followng dfferental equaton: ẋ(t) = f σ(t) (x(t)) t > t 0 where σ denotes the swtchng sgnal whch s a pecewse constant map from tme to an ndex set representng subsystems. See the survey of Ln and Antsakls (2009)fora revew of the recent results and further references. The stablty analyss encountered n swtched systems can be classfed nto three categores (Mahmoud 2010). The frst one s to fnd common Lyapunov functons so that the swtched systems are stable under any arbtrary swtchng sgnal (Fanshl Margalot & Chgansky 2009; Hou Fu & Duan 2013; Shorten Narendra & Mason 2003). The second one s to construct certan swtchng sgnals that make the swtched system asymptotcally stable (Lberzon & Morse 1999). The thrd category s the slow swtchng strateges such as dwell tme stabltyoraveragedwelltmestabltyforwhchthesystem s asymptotcally stable (Geromel & Colaner 2006; Hespanha 2004; Hespanha & Morse 1999; Mtra & Lberzon 2004; ZhangHanZhu&Huang2013). The class of swtchng sgnals can be restrcted to sgnals wth the property that the nterval between any consecutve swtchng tmes s not less than a value called the dwell tme. The swtched delay system s asymptotcally stable fallofthecanddatesubsystemsareasymptotcallystable and the dwell tme s large enough (Morse 1996). Most swtched systems do not share a common Lyapunov functon (Chen & Zheng 2010). Furthermore havng a common Lyapunov functon s a suffcent condton for the stablty under arbtrary swtchng so t can be found conservatve (Ln & Antsakls 2009). In ths paper we present some results on the dwell tme stablty analyss and stablsaton of the swtched delay systems. A dwell tme s observed n many swtchng system applcatons. The tme ntervals between the change n the road condtons among dry wet and drt for a car on the road can be consdered as an example (Allerhand & Shaked 2011). Also the slow swtchng strateges wth dwell tme can avod chatterng problems whch can damage the physcal systems (Ish & Francs 2001). As a result the stablty analyss and stablsaton of swtched systems wth dwell tme are ncreasngly popular. The lterature s abounded wth varous approaches for the stablty analyss of tme-delay systems one can refer to Gu Khartonov and Chen (2003) forarevewonthe topc. Common methods to deal wth delay-dependent stabltyproblemsaremodeltransformatons.inths method pont wse delay system transferred nto a ds- CONTACT Ahmet Taha Koru ahtakoru@gmal.com 2017 Informa UK Lmted tradng as Taylor & Francs Group

3 2 A. T. KORU ET AL. trbuteddelaysystem.stabltyofthetransformedsystem s a suffcent condton for the stablty of the orgnal system. Hence stablty analyss wth model transformatons leads to a sort of conservatsm snce analyss operates on the transformed system nstead of the orgnal system (Gu et al. 2003). A less conservatve approach to stablty analyss s the free-weghtng matrces method whch does not nclude any model transformaton of the orgnal system (He Wang Xe & Ln 2007; Mahmoud2010; WuHe &She2010). In ths paper we present some results for swtched delay systems wth pontwse delays. There are recent results on dwell tme stablty of the swtched delay systems. In Sun Zhao and Hll (2006) and L Gao Agarwal and Kaynak (2013) stablty condtons are presented for a gven average dwell tme. In those papers the condtons nvolve exponental and blnear terms when the dwell tme s consdered as a free parameter. Hence the mnmsaton of the dwell tme and synthessng the dwell tme mnmser controllers wth those methods are not tractable. There are some optmsatonbased methods to mnmse the upper bound for the dwell tme (Çalışkan Özbay & Nculescu 2013; Yan& Özbay 2008). In Çalışkan et al. (2013) the calculaton of dwell tme s formulated as a sem-defnte programmng (SDP) n terms of lnear matrx nequaltes (LMIs). Pecewse Lyapunov Krasovsk functonals are derved by model transformaton methods. The upper bound of the dervatve of the Lyapunov functon s mnmsed whch ends up wth a sub-optmal soluton to the dwell tme mnmsaton problem. In Yan Özbay and Sansal (2011) parameter-varyng systems wth tme-delays are stablsed by swtchng control. The resultng dwell tme s mnmsed wth teratve search methods. The present paper proposes a quas-convex optmsaton approach to drectly mnmse the dwell tme and converges to global mnmum of represented upper bound of the dwell tme. To reduce conservatsm due to model transformatons we derve the stablty condtons by usng free-weghtng matrces. Thenotatontobeusednthepapersstandard:R (R + R + 0 ) stands for the set of real numbers (postve real numbers non-negatve real numbers) C s used to denote the set of dfferentable functons Z + symbolses the set of postve ntegers. The dentty matrces are denoted by I. WeuseX 0 ( 0) to denote a postve defnte (postve-semdefnte negatve defnte negatvesemdefnte) matrx. σ max [X] andσ mn [X] denotethe maxmum and mnmum sngular values of X respectvely. The astersk symbol ( ) denotes complex conjugate transpose of a matrx and x t denotes the translaton operator actng on the trajectory such as x t (θ) = x(t + θ) for some non-zero nterval θ [ τ 0]. The operator dag[x 1 X 2 X n ] denotes a block dagonal matrx whose elements on the man block dagonal are X 1 X 2 X n.thenorm s defned as the Eucledan norm for a vector n R n and the norm on C s defned as follows: { } f [ab] = max sup f (t) sup ḟ (t). t [ab] t [ab] Rest of the paper s organsed as follows. In Secton 2 prelmnares and problem defnton are ntroduced. In Secton 3 dwell tme stablty condton s gven. In Secton 4 quas-convex optmsaton of the upper bound of the dwell tme and some numercal examples are proposed. In Secton 5 dwell tme mnmsng controller synthess s presented wth some numercal examples to llustrate effectveness of the proposed algorthm. Conclusons are summarsed n Secton Prelmnares and problem defnton Consder a class of swtched delay system gven by ẋ(t) = A σ(t) x(t) σ(t) : + Ā σ(t) x(t r σ(t) (t)) t 0 x(θ ) = ϕ(θ) θ [ τ max 0] (1) where x(t) R n s the pseudo-state and σ (t)sthepecewse swtchng sgnal such that σ(t) : R + P P := { m} s an ndex set m Z + s the number of subsystems and ntal condton belongs to Banach space of contnuous functons such that ϕ( ) C. Tme-delay r σ (t) (t) s a tme-varyng dfferentable functon that satsfes 0 r σ(t) (t) τ σ(t) (2) ṙ σ(t) (t) d σ(t) (3) where τ σ (t) d σ (t) > 0arepecewseconstants.Wentroduce the quadropule := ( A τ d ) R n n R n n R R to descrbe the th canddate subsystem of Equaton (1) and τ max = max P τ. Defnton 2.1: Aswtcheddelaysystemsstable f there exsts a functon β of class K such that x(t) β( x [t0 τ max t 0])

4 INTERNATIONAL JOURNAL OF CONTROL 3 along every soluton of Equaton (1). Furthermore a swtched delay system s asymptotcally stable when t s stable and lm x(t) = 0. t + Lemma 2.1: (See Gu et al. 2003). Consderthenonswtched lnear subsystem of the system (1) for an P. Supposeu v w : R + 0 R+ 0 are contnuous nondecreasng functons satsfyng u (0) = v (0) = 0 w (s) > 0 for s > 0. If there exsts a contnuous functonal V such that u ( x(t) ) V (t x t ) v ( x [t τ t]) t t 0 (4) V (t x t ) w ( x(t) ) t t 0 (5) then the soluton x = 0 of the subsystem s unformly asymptotcally stable. Let us construct the followng pecewse Lyapunov functon: t V (t x t ) := x T (t)p x(t) τ t t+θ t r (t) x T (s)q x(s)ds ẋ T (s)z ẋ(s)dsdθ P (6) Lemma 2.2: (See Wu He & She 2010).Consderthenonswtched lnear subsystem foran P of the swtched system (1) wth varyng delays r (t). Gvenscalarτ > 0 and d > 0 for whch both Equatons (2) and (3) hold the th subsystem s asymptotcally stable f there exst symmetrc matrces P 0Q 0Z 0and X11 X X := 12 0 (7) X 22 and any approprately dmensoned matrces N 1 and N 2 such that the followng LMIs hold: where φ 11 φ 12 τ A T Z φ := φ 22 τ Ā T Z 0 (8) τ Z X 11 X 12 N 1 ψ := X 22 N 2 0 (9) Z φ 11 = P A + A T P + N 1 + N1 T + Q + τ X 11 φ 12 = P Ā N 1 + N2 T + τ X 12 φ 22 = N 2 N2 T (1 d )Q + τ X Man results The followng proposton s a modfed verson of a result obtaned n Çalışkan et al. (2013). In the correspondng proposton the tme T scalculatedasthetmenstant after whch norm of the states does not exceed the predefned parameter ρ for the non-swtched case. Furthermore after the dwell tme T + τ max thenormofthestate functonal does not exceed ρ. Notethatthenormofthe statefunctonalscomputedas x [t τmax t].thetmet s related to the upper bounds defned n Lemma 2.1. Proposton 3.1: For any non-swtchng lnear subsystem satsfyng Lemma 2.1 wth lm u (s) assume s there exsts a non-decreasng functon u d such that u d ( ẋ(t) ) V (t x t ). For an arbtrary ρ 0 <ρ<δ 2 x [t0 τ t 0] δ 1 mples x [t τmax t] ρ t > t 0 + τ max + T (δ 1 ρ) where v and w are defned as n Lemma 2.1u(δ 2 ) = v(δ 1 ) and T (δ 1 ρ)= v (δ 1 ) w (ρ). Proof: Let T > 0andlet x(t 1 ) >ρfor a tme nstant t 1 > t 0 + T.Functonw s non-decreasng by defnton as a result nf ρ<s<δ2 w (s) = w (ρ). Sncethesubsystem s stable and V s a Lyapunov Krasovsk functonal from Lemma 2.1wehavethefollowng: Ths mples V (t x t ) w (ρ) t 0 t t 1. V (t x t ) V (t 0 x 0 ) (t t 0 )w (ρ) v (δ 1 ) (t t 0 )w (ρ). Let T > v (δ 1 )/w (ρ). Then for every t > t 0 + T we have V (t x t ) 0. However we assume that there s a tme nstant t 1 > t 0 + T such that x(t 1 ) >ρ. Ths mples that V (t 1 x t1 ) u ( x(t 1 ) ) >u (ρ) > 0 Ths s a contradcton. Therefore tme nstant t 1 cannot exst and ths mples x(t) ρ t > t 0 + v (δ 1 ) w (ρ).

5 4 A. T. KORU ET AL. Smlarly assumng there s a tme nstant t 1 > t 0 + T such that ẋ(t 1 ) ρ V (t 1 x t1 ) u d ( ẋ(t 1 ) ) >u d (ρ) > 0 whch s also a contradcton. Hence x(t) <ρ ẋ(t) <ρ t > t 0 + T If we wat for a perod of maxmum tme-delay such that t > t 0 + T + τ max thenequalty x(t) [t τmax t] ρ holds whch concludes the proof. Now some specfc upper and lower bounds for the Lyapunov functon (6)canbegvenasv (s) μ s 2 wth and μ := σ max [P ] + τ 2 σ max [Q ] τ 2 σ max [Z ] (10) u (s) = σ mn [P ] s 2 (11) respectvely. Another lower bound of the Lyapunov functon wth respect to norm of ẋ(t) can be defned as u d (s) := 1 2 τ 2 σ mn [Z ] s 2 where u d ( ẋ(t) ) V (t x t ). The upper bounds v (s) can be calculated va LMI condtons defned n Lemma 2.2 due to Equaton (10). In order to formulate the upper bounds of the dervatve of the Lyapunov functons w (s) asanlmifeasbltyproblem we need a new result stated as Proposton 3.2. Remark 3.1: In the proof of Proposton 3.2annequalty from the proof of Lemma 2.2 n Wu et al. (2010) wllbe used specfcally: V (t x t ) η T 1 (t) η 1 (t) t t r (t) where ψ s defned n Equaton (9)and η T 2 (t s)ψ η 2 (t s)ds η 2 (t s) = [ x T (t) x T (t r (t)) ẋ T (s) ] T [ φ11 + τ = A T Z A φ 12 + τ A T Z ] Ā φ 22 + τ Ā T Z. Ā (12) Note that Equaton (8) s the Schur complement of.for more nformaton about the proof we refer to Wu et al. (2010). Proposton 3.2: Consder the system (1) wth each satsfyng Lemma 2.2 fthereexstmatrcesw T = W 0 such that followng LMIs hold: φ 11 + W φ 12 τ A T Z φ := φ 22 τ Ā T Z 0 P τ Z (13) then V (t x t ) x T (t)w x(t) for all P. Proof: Consder the nequalty (12). Snce ψ 0 we know that V (t x t ) η1 T (t) η 1 (t). Boundng ths nequalty η T 1 (t) η 1 (t) x T (t)w x(t) yelds η T 1 (t)d η 1 (t) 0where [ φ11 + W D := + τ A T Z A φ 12 + τ A T Z ] Ā φ 22 + τ Ā T Z. Ā Snce φ s the Schur complement of D fequaton (13)holdsthen V (t x t ) x T (t)w x(t). Afterdefnnganewvarable λ := σ mn [W ] (14) we can select the upper bound functon for the dervatve of the Lyapunov functon as w (s) = λ s 2. Assume that the Lemma 2.1 s satsfed for the system (1). There exsts a δ 2 >δ 1 > 0suchthatu(δ 2 ) = v(δ 1 ). For such δ 2 Lemma2.1 mples that x(t) δ 2 for all t > t 0 f x [t0 τ t 0] δ 1.Henceforu(s) andv(s) defned n Equatons (10)and(11) followng nequalty holds: where x(t) β x [t0 τ t 0] P (15) μ β = max P σ mn [P ]. Consder the kth swtchng nstant t k.thedwelltme τ D s defned as the tme nstant after whch the norm of the state functonals for any t k > t k 1 + τ D does notexceedthenormofthestatefunctonalattmet k 1. Hence ρ n Proposton 3.1 s defned as a fracton of the norm of the state functonal at the swtchng nstant t k 1. As a result the dwell tme defned n ths paper should bestrctlygreaterthanthemaxmumofallthepossble delays. The fracton s a pre-defned number α (0 1). Theorem 3.1: Consder the swtched delay system descrbed n Equaton (1). Assume all of the canddate subsystems satsfy Lemma 2.2. Then the swtched delay system s asymptotcally stable for all swtchng sgnals

6 INTERNATIONAL JOURNAL OF CONTROL 5 satsfyng dwell tme requrement τ D τ D = 1 α 2 max P μ λ + τ max for any α (0 1) (16) wth μ and λ beng defned n Equatons (10) and (14) respectvely. Proof: Let us choose ρ = αδ k 1 where δ k denotes norm of the state at the kth swtchng nstant such that δ k = x [tk τ max t k ]. Let us restrct ourselves to swtchng sgnals to sgnals for whch the tme nterval between two consecutve swtchng nstants s larger than dwell tme such that t k t k 1 >τ D.Introducngthsdwelltmeas τ D = max P T (δ k 1 αδ k 1 ) + τ max leadsustoannequaltyfromproposton3.1 as x [tk τ max t k ] α x [tk 1 τ max t k 1 ] t k > t k 1 + τ D (17) where T (δ k 1 αδ k 1 ) = v (δ k 1 ) w (αδ k 1 ) = From Equatons (15)and(17) x(t) β x [tk τ max t k ] βα x [tk 1 τ max t k 1 ] μ α 2 λ.. βα k x [t0 τ max t 0] βα x [t0 τ max t 0] α (0 1) whch s satsfyng the stablty condton descrbed n Defnton 2.1. Remark 3.2: The parameter α s the rato of the norms of the state functonals at the consecutve swtchng nstants as n Equaton (17). Hence t can be regarded as a measure of the decay rate. Ths parameter quantfes a tradeoffbetweenthedwelltmeandthedecayrate.e.the larger α the smaller dwell tme but the slower decay rate (tshouldbestrctlylessthan1forstabltyoftheswtched system). 4. Mnmum dwell tme va quas-convex optmsaton In order to mnmse the dwell tme n Equaton (16) we can defne the optmsaton problem wth a cost functon f (μ λ ) := max P μ /λ for a gven α. Thssaquasconvex functon snce t s the composton of a convex functon wth a nondecreasng functon (Bullo & Lberzon 2006). It s known that an optmsaton problem wth a quas-convex cost functon and convex constrants can be solved by teratve methods such as bsecton algorthm (Boyd & Vandenberghe 2004). We defne a new free varable t to bound the cost functon: μ λ t P (18) The parameters μ and λ are related wth the egenvalues of P Q Z and W as n Equatons (10) and(14). We defne p q and z to defne maxmum egenvalues of P Q and Z respectvely.sothenequalty(18) canberewrtten as p + τ q τ 2z tλ < 0. Wth respect to the free parameters P Q Z W X 11 X 12 X 22 N 1 N 2 p q z w t u theupperboundofthe dwell tme s mnmsed va followng optmsaton problem: mnmse t (19) subject to dag [P Q Z W X ] 0 P dag [P Q Z W ] dag [ p I q I z I λ I ] P ψ 0 φ 0 P p + τ q τ 2 z tλ < 0 P where X ψ and φ are defned n Equatons (7) (9) and (13)respectvely.Thenthedwelltmecanbechosen as τ D = αt + τ max for any α (0 1). However the optmsaton problem nvolves a blnear matrx nequalty when t s consdered as a free parameter. Searchng for mnmum t wth bsecton algorthm generates a sequence of lnear SDP feasblty problems whch can easly be solved by SeDuM Sturm (1999). In ths secton the examples are taken from publshed papers for comparson purposes. Examples can be foundnçalışkan et al. (2013) Yan and Özbay (2008)and Chen and Zheng (2010) respectvely. Example 4.1: Let 1 and 2 be A 1 = = A 2 = τ 1 = 0.3s d 1 = = τ 2 = 0.6s d 2 = 0.

7 6 A. T. KORU ET AL. 150 Dwell Tme τ D Tme Delay τ Fgure 1. Dwell tme results for dfferent delay values of the swtched delay system descrbed n the Example 4.1 where delays are fxed such that d 1 = d 2 = 0 wth upper bounds τ 1 = τ and τ 2 = 2τ. Table 1. Dwell tme for dfferent τ and d values of Example 4.1 α = τ 1 τ 2 d 1 d 2 τ D 0.15 s 0.3 s 0 s 0 s 0.69 s 0.15 s 0.3 s 0.15 s 0.3 s 0.69 s 0.3 s 0.6 s 0 s 0 s 1.11 s 0.3 s 0.6 s 0.3 s 0.3 s 1.11 s 0.3 s 0.6 s 0.6 s 0.6 s 1.11 s 0.6s 1.2s 0s 0s 2.54s 0.6 s 1.2 s 0.3 s 0.3 s 2.76 s 0.6 s 1.2 s 0.6 s 0.6 s 3.51 s Table 2. Dwell tme for dfferent τ and d values of Example 4.2 α = τ 1 τ 2 d 1 d 2 τ D 0.08 s 0.1 s 0 s 0 s 0.46 s 0.16 s 0.2 s 0.15 s 0.15 s 0.58 s 0.3 s 0.4 s 0.2 s 0.2 s 0.84 s 0.6 s 0.8 s 0 s 0 s 1.38 s 0.9 s 1.2 s 0 s 0 s 2.39 s 0.9 s 1.2 s 0.3 s 0.3 s 2.58 s 0.9 s 1.2 s 0.6 s 0.6 s 3.15 s 0.9 s 1.2 s 0.9 s 0.9 s s Dwell tme results for dfferent delay values for ths example can be seen n Fgure 1. Correspondng mnmum dwell tmes for dfferent τ and d values are llustratedntable 1. Example 4.2: Let 1 and 2 be A 1 = A 2 = = τ 1 = 0.155s d 1 = = τ 2 = 0.2s d 2 = 0. Correspondng mnmum dwell tmes for dfferent τ and d values are llustrated n Table 2.Notethatsecond subsystem of Example 4.2 s unstable for d 2 > As d 2 Table 3. Dwell tme for α = Example Chen and Zheng (2010) 43.99s Yan and Özbay (2008) 6.51s Çalışkan et al. (2013) 3.4 s 0.72 s Present work 1.11 s 0.58 s s comng closer to the stablty lmts dwell tme ncreases dramatcally that sts n shaded row of Table 2. Example 4.3: Ths example s the Case 3 of Example 4.1 from the paper (Chen & Zheng 2010). Let 1 be A 1 = = A 2 = τ 1 = 1.82 d 1 = = τ 2 = 1.82 d 2 = 0 Comparson of present paper wth prevous works for Examples canbeseennTable 3. Example 1.1: Ths example s a slghlty modfed verson of Example 4.1 of Sun and Ge (2011)wtha = 50 where a s a parameter used n correspondng example. In the example system s not guaranteed stable under arbtrary swtchng. Let the subsystems be: A 1 = = τ 1 = 0.01 d 1 = A 2 = = 1 1 τ 2 = 0.05 d 2 = 0.1. Resultng dwell tme s τ D = wth T =

8 INTERNATIONAL JOURNAL OF CONTROL 7 5. Dwell tme mnmsng controller synthess Consder a class of swtched delay systems gven by σ(t) : ẋ(t) = A σ(t) x(t) + Ā σ(t) x(t r σ(t) (t)) + B σ(t) u(t) t 0 x(θ ) = ϕ(θ) θ [ τ max 0] (20) We ntroduce the quntet := ( A B τ d ) R n n R n n R n m R R to descrbe the th canddate subsystem of Equaton (20) and τ max = max P τ. Lemma 5.1: (See Wu et al. 2010). Consderanynonswtchng lnear subsystem of the swtched delay system (20) wth a delay r (t). Forgvenscalarτ and d whch both Equatons (2) and (3) hold f there exst matrces L > 0T 0R > 0and Y11 Y Y := 12 0 Y 22 and any approprately dmensoned matrces M 1 M 2 and V such that the followng matrx nequaltes hold: where τ (L A T + V T B T ) = 22 τ L Ā T 0 (21) τ R Y 11 Y 12 M 1 = Y 22 M 2 L R 1 L 0 (22) 11 = L A T + A L + B V + V T B T + M 1 + M1 T + T + τ Y = Ā L M 1 + M T 2 + τ Y = M 2 M T 2 (1 d )T + τ Y 22 then the subsystem can be stablsed by control law u(t) = K x(t) and the controller gan s K = V L 1. Proof: After applyng memoryless state-feedback controller to closed-loop system ẋ(t) = (A + B K ) x(t) + Ā x(t r (t)) let us replace the A wth A + B K pre-andpostmultply (8) by dag P 1 P 1 Z 1 pre- and postmultply (9) bydag P 1 P 1 P 1 andmakethefollowng change of varables: L := P 1 T := P 1 Q P 1 R := Z 1 (23) M 1 := P 1 N 1 P 1 M 2 := P 1 N 2 P 1 V = K P 1 Y := dag [ P 1 P 1 ] X dag [ P 1 P 1 ] These operatons end up wth Equatons (21) and(22) whch complete the proof. Due to the term L R 1 L condton (22)nLemma5.1 s not an LMI. In order to handle ths term let us defne a new varable S forwhch and replace Equaton (22)wth L R 1 L S (24) Y 11 Y 12 M 1 := Y 22 M 2 0. (25) S Inequalty (24) s equvalent to L 1 RL 1 S 1 whchthe Schur complement allows us to wrte as [ S 1 L 1 R 1 We ntroduce new varables ] 0. (26) J = L 1 U = S 1 H = R 1 (27) so that we can re-wrte the condton (26)as U J 0. (28) H Ths lftng provdes us to use LMIs (28)and(25)nstead of (22) n order to make the condton LMI. Proposton 5.1: Consder the system (20) wth each satsfyng Lemma 5.1 f there exsts a matrx W = W T 0 such that followng LMIs hold: 11 + L W L 12 τ (L A T + V T B T ) 22 τ L Ā T 0 τ R P (29) then V (t x t ) x T (t)w x(t) for all P. Proof: Let us pre- and post-multply the LMI (13) n Proposton 3.2 by dag P 1 P 1 P 1 and make the change of varables defned n Equaton (23).

9 8 A. T. KORU ET AL. SmlartothecondtonnLemma5.1 Equaton(29) s also not an LMI and we handle ths term wth the same procedure. By defnng the new varables C and O where C L W L 0 whose Schur complement s C L 0 (30) O and assumng O = W 1 thenwecanreplacethenonconvex representaton n Equaton (29)wthC as a convex one 11 + C 12 τ (L A T + V T B T ) := 22 τ L Ā T 0. τ R (31) Now we defne lower and upper bounds for the Lyapunov functons: [ μ := σ max L 1 λ := σ mn [W ]. ] + τ σ max [ L 1 T L 1 ] τ 2 σ max R 1 subject to dag [L T R Y W ] 0 P dag [J [ F H W ] ] dag j I f I h I λ I P 0 0 P U J C L 0 0 H O F J 0 P E [ ] L I S I 0 0 J U R I 0 P H [ ] T I O I 0 0 P E W j + τ f τ 2h t λ < 0 P. (33) The cost functon n Equaton (33) s mnmsed to satsfy the nequalty constrants (22) and(29). Durng the mnmsaton procedure J U H O E converge to L 1 S 1 R 1 W 1 T 1 respectvely. We overcome the nonlnearty of the cost functon of Equaton (33) by usng lnearsaton method provded n Ghaou Oustry and AtRam (1997). The lnearsaton of the cost functon s RepeatngthesameprocedurewthnewvarablesofF and E (F L 1 T L 1 E := T 1 ) and re-wrtng μ F J 0 E μ = σ max [J ] + τ σ max [F ] τ 2 σ max [H ]. (32) we obtan LMI condtons. Consder the upper bound of the term μ / λ t. For aconstanttfeasbltyofthedwelltmeτ D = t + τ max s the followng nonlnear SDP mnmsaton problem: [ mn trace (L J + S U P + R H + T E + O W ) ] f = constant + trace [ P ( L J 0 + L 0 J + S U 0 + S 0 U + R H 0 + R 0 H + T E 0 + T 0 E + O W 0 + O 0 W ) ]. The lnearsed cost functon f s mnmsed teratvely. The ( cost functon s re-lnearsed around new pont L k J k Sk U k Rk Hk T k Ek Ok W ) k n each step. The nonlnear condtons (22) and(29) arecheckedn each teraton up to a pre-defned number of maxmum teratons. If the condtons (22) (29) and(33) aresats- fed t s a proper dwell tme. The mnmsaton of the dwell tme problem s a nested optmsaton problem where the outer loop (Steps 1 and 4) s a bsecton algorthm wth the cost functon f := t and the nner loop (Steps 2 4) s the optmsaton problemdefnednequaton(33) wththecostfuncton f. If the nner loop s concluded successvely t s halved n ts bsecton nterval otherwse doubled (Step 4). Step 1. Choose a suffcently large ntal t u > 0suchthat thereexstsasoluton.sett l = 0. Step 2. Set the teraton ndex k to 0 and t = (t u + t l )/2. Fnd a feasble set for the free parameters

10 INTERNATIONAL JOURNAL OF CONTROL 9 (L J S U R H T E O W Y M 1 M 2 V j h f w ) subject to condtons n Equaton (33). Set L 0 = L J 0 = J S 0 = S U 0 = U = T E 0 = E O 0 = O and W 0 = W. Step 3. Solve the followng convex optmsaton problem for the same free parameters n Step 2: R 0 = R H 0 = H T 0 mn trace [ P ( L J k + L k J + S U k + R H k + R k H + T E k + T k E + O W k + O k W ) ] s.t. condtons n Equaton (33). + S k U Set L k+1 = L J k+1 = J S k+1 = S U k+1 R k+1 = R H k+1 = H T k+1 = T E k+1 = U = E O k+1 = O and W k+1 = W. Step 4. If specfed tolerance such that t u t l < tolssatsfed then set K = V L 1 for all P and ext. The dwell tme s τ D = t + τ max. Else f Equatons (22) and(29) aresatsfedthen set t u = tandreturntostep2. Otherwse set k = k + 1andgotoStep3. If there s no feasble solutons after specfed number of teratons then set t l = t and return to Step 2. Example 5.1: Ths example s from Yan Özbay and Şansal (2014). In the correspondng paper stablsaton of a lnear tme-varyng system guaranteed wth a swtchng controller. In order to acheve that lnear parameter varyng (LPV) system s represented as a swtchng delay system wth two nomnal subsystems and uncertanty bounds are determned. Then controllers are desgned wth robust stablty condtons. Synthessed controllers are K 1 = [ ] K 2 = [ ] andtheresultngdwelltmesfoundtobe0.92seconds. In ths paper we only consdered the nomnal subsystems of the swtchng delay system representaton. The two nomnal subsystems are defned as [ 3 1 A 1 = [ 1 B 1 = 1 [ A 2 = 1 2 ] = ] τ 1 = 0.2 d 1 = 0.01 ] = B 2 = 1 τ 1 2 = d 2 = 0.01 Resultngcontrollersofouralgorthmare K 1 = [ ] K 2 = [ ] andthedwelltmesτ d = 0.49 seconds. Example 5.2: Ths example s a slghtly modfed verson of the example of Yuan and Wu (2015) where the swtched system n queston s a non-delayed system whch does not admt a common Lyapunov functon. In the correspondng example swtched lnear plant s n the form: ẋ = A 0σ (t) x + B 0σ (t) w + B 1σ (t) u where w s the dsturbance A 01 = B 11 = A 02 = B 12 = By usng A 0 and B 1 we generated our example. Let 1 and 2 be A 1 = (1 λ) A 01 1 = λ A 01 B 1 = B 11 τ 1 = τ d 1 = 0.01 A 2 = (1 λ) A 02 2 = λ A 02 B 2 = B 12 τ 2 = 1.6 τ d 2 = For λ = 0.9 and τ = 0.05 resultng dwell tme s seconds and controllers are K 1 = [ ] K 2 = [ ] In Fgure 2 mnmum upper bounds for the dwell tmes can be seen for varous τ and λ values. Dwell tme growslnearlyforlargedelayvalueswhereasanexponentalgrowthsobservednfgure 1. The key dfference between two examples s that the subsystems n Example 5.2 can be stablsed ndependent of delays but

11 10 A. T. KORU ET AL. 30 τ 1 = 0.05 τ 2 = λ =0.1 Dwell Tme (sec) λ τ Fgure 2. Dwell tme results for dfferent τ and λ values of Example 5.2. thestablty ofthesubsystems n Example4.1 depends on delays. 6. Conclusons Weperformedthecalculatonofanupperboundofdwell tme by quas-convex optmsaton methods to ensure stablty of lnear swtched delay system. LMI condtons of free-weghtng matrces method are used to fnd approprate Lyapunov Krasovsk functonals for nonswtchng subsytems. By combnng these condtons wth a cost functon whch represents the upper bound of dwell tme the upper bound s optmsed usng a bsecton algorthm where each step s a lnear SDP feasbltyproblem.bythenumercalexamplestsshownthat the results obtaned n Çalışkan et al. (2013) and Yan and Özbay (2008) can be mproved usng the method proposed n the present paper. In addton to ths a dwell tme mnmsng controller synthess algorthm s also developed n ths work. Although the condtons are nonlnear and the correspondng set s non-convex ths algorthm successvely lnearse the condtons and turn the problem nto a lnear SDP. The numercal examples are gventollustratetheeffcencyoftheproposedmethod. Less conservatve condtons for the stablty of the delayed swtchng systems can be found n papers presentng average dwell tme methods (see Sun et al. 2006; Ches Colaner Geromel Mddleton & Shorten 2012). However the average dwell tme condtons are nonconvex due to exponental and blnear terms when the dwell tme s consdered as a free parameter n optmsaton.representatonofthedwelltmenthepresentpaper s more conservatve but the dwell tme mnmser controller synthessaton problem s tractable due to convex nature of the condtons. A typcal applcaton of the swtched control scheme s the network congeston control systems (Zhao Zhang Sh & Lu 2012). Due to the tme-delay nature of the network systems the method presented n ths paper can contrbute to the research on applcaton of the network congeston control systems. Dsclosure statement No potental conflct of nterest was reported by the authors. ORCID Ahmet Taha Koru References Allerhand L. & Shaked U. (2011). Robust stablty and stablzaton of lnear swtched systems wth dwell tme. IEEE Transactons on Automatc Control 56(2) Boyd S. & Vandenberghe L. (2004). Convex optmzaton. Cambrdge: Cambrdge Unversty Press. BulloF.&LberzonD.(2006). Quantzed control va locatonal optmzaton. IEEE Transactons on Automatc Control 51(1) Çalışkan S.Y. Özbay H. & Nculescu S.I. (2013). Dwell-tme computaton for stablty of swtched systems wth tme delays. IET Control Theory & Applcatons 7(10) Chen W.H. & Zheng W.X. (2010). Delay-ndependent mnmum dwell tme for exponental stablty of uncertan swtched delay systems. IEEE Transactons on Automatc Control 55(10) Ches G. Colaner P. Geromel J.C. Mddleton R. & Shorten R. (2012). A nonconservatve LMI Condton for stablty of swtched systems wth guaranteed dwell tme. IEEE Transactons on Automatc Control 57(5) Fanshl L. Margalot M. & Chgansky P. (2009). On the stablty of postve lnear swtched systems under arbtrary

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