Output Feedback Tracking Control for a Class of Switched Nonlinear Systems with Time-varying Delay

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1 Internatonal Journal of Automaton and Computng 11(6), December 014, DOI: /s Output Feedback Trackng Control for a Class of Swtched Nonlnear Systems wth Tme-varyng Delay Bn Wang Jun-Yong Zha Shu-Mn Fe Key Laboratory of Measurement and Control of CSE, Mnstry of Educaton, School of Automaton, Southeast Unversty, Nanjng, Jangsu 10096, Chna Abstract: Ths paper studes the problem of trackng control for a class of swtched nonlnear systems wth tme-varyng delay. Based on the average dwell-tme and pecewse Lyapunov functonal methods, a new exponental stablty crteron s obtaned for the swtched nonlnear systems. The desgned output feedback H controller can be obtaned by solvng a set of lnear matrx nequaltes (LMIs). Moreover, the proposed method does not need that a common Lyapunov functon exsts for the swtched systems, and the swtchng sgnal just depends on tme. A smulaton example s provded to demonstrate the effectveness of the proposed desgn scheme. Keywords: Trackng control, swtched nonlnear systems, exponental stablty, tme-varyng delay, H controller. 1 Introducton Swtched systems are a class of hybrd dynamcal systems whch consst of a famly of subsystems and a rule that orchestrates the swtchng among them. The local behavor s affected by the contnuous dynamcal subsystems and the dscrete dynamcal swtchng mechansms determne the global performance. In recent years, there are lots of sgnfcant achevements both n theory development and practcal applcatons of swtched systems 1 5. On the other hand, tme-delay systems are an mportant class of systems, whchareubqutousnrealworld,suchasnchemcalprocess, aerodynamcs, and communcaton networks systems. Sometmes, even a small delay may affect the system performance greatly. A stable system may become unstable or chaotc behavor may appear f delay s present n the system 6 8. Snce the swtched systems wth tme-delay have strong engneerng background, they have attracted consderable attenton, and some useful results have been obtaned There are several methods used n analyzng stablty of tme-delay swtched systems, such as the common Lyapunov functon, multple Lyapunov functons, pecewse Lyapunov functon and average dwell-tme. The work n 1 gves delay-dependent condtons for the exponental stablty of the swtched lnear systems wth tmevaryng delay by common Lyapunov functon. The work n 17 gves the exponental stablty crteron for a class of swtched lnear systems wth constant tme delay by combnng dwell-tme wth the pecewse Lyapunov functon. Trackng control for tme-delay swtched system s wdely used n robot control and guded mssle control. To the best of the authors knowledge,thessueoftrackngcontrolhas not been fully nvestgated for tme-delay swtched systems. Regular paper Specal Issue on Recent Advances on Complex Systems Control, Modellng and Predcton II Manuscrpt receved September 1, 013; accepted December 30, 013 Ths work was supported by Natonal Natural Scence Foundaton of Chna (Nos , , and ), Sx Talents Peaks Program of Jangsu Provnce (No. 014-DZXX-003) and the Fundamental Research Funds for the Central Unverstes (No. 4013R30006). Only a few results have been reported on trackng control for swtched systems The stablty of trackng control based on observer for tme-delay swtched lnear systems has been nvestgated n 1, but the proposed method needs that a common Lyapunov functon must exst for the swtched systems. By usng average dwell-tme, a new exponental stablty crteron of state feedback trackng control for swtched nonlnear systems wth tme-varyng delay has been obtaned n. In ths paper, we am to desgn an output feedback controller for a class of swtched nonlnear systems wth tme-varyng delay. Combnng average dwell-tme wth the pecewse Lyapunov functon, a new exponental stablty crteron for a class of swtched nonlnear systems wth tme-varyng delay s derved. Moreover, when there exsts a common Lyapunov functon for the system, t wll be exponentally stable under arbtrary swtchng law. Notatons. R n denotes n-dmensonal Eucldean space; R m n denotes the space of m n matrces wth real entres. L 0, ) s the space of square ntegrable functons on 0, ), and L1 loc (ϱ, ), R n ) s the space of locally Lebesgue ntegrable vector valued functons on ϱ, ), where ϱ s a scalar. For any gven τ > 0, let C n = C( τ,0, R n ) be the Banach space of contnuous mappng from ( τ,0, R n )tor n wth the topology of unform convergence. I represents dentty matrx wth approprate dmenson. P>0(,<, 0) denotes a postve defnte (postve sem-defned, negatve defnte, negatve sem-defnte) matrx. λ mn( ) andλ max( ) denote the mnmal and maxmal egenvalues of a square matrx. σ max( ) means the maxmal sngular value of a matrx. The superscrpt T stands for matrx transpose and the symmetrc terms n a symmetrc matrx are denoted by. Letx t C n be defned by x t(θ) =x(t + θ),θ τ,0. denotes the usual - norm, and x t cl =sup τ θ 0 { x(t + θ), ẋ(t + θ) }.

2 606 Internatonal Journal of Automaton and Computng 11(6), December 014 Problem formulaton and prelmnares Consder the swtched nonlnear system wth tmevaryng delay ẋ(t) =f σ(x(t)) + A σx(t)+d σx(t d(t)) + B σu(t)+ω(t) y(t) =C σx(t) x(t) =ϕ(t), τ t 0 (1) where x R n s the system state vector, u(t) R p s the control nput, y(t) R q s the output, ω(t) R n s the bounded exogenous dsturbance whch belongs to L 0, ) and L1 loc (ϱ, ), R n ). f σ( ) :R n R n s a known nonlnear functon; σ(t) :, ) M {1,,,m 0} s the swtchng sgnal. d(t) denotes the tme-varyng delay satsfyng 0 <d(t) τ; ϕ(t) s a contnuous vector-valued ntal functon. Moreover, σ(t) = means that the -th subsystem s actvated, A,B,C,D are known constant matrces wth approprate dmensons. Wthout loss of generalty, we state that the followng assumptons hold. Assumpton 1. U R n n, M are known constant matrces, the functon f (x(t)) s Lpschtz for all x(t) R n and ˆx(t) R n, and satsfes f (x(t)) f (ˆx(t)) U (x(t) ˆx(t)). () Assumpton. For M, the subsystem (A,C,D ) are detectable 1 and B are full row rank. Defnton System (1) s sad to be exponentally stablzable under control law u(t) and swtchng law σ(t), f the soluton x(t) of system (1) through (,ϕ) R + C n satsfes x(t) κ x t0 cl e λ(t ), t (3) for some constants κ 0andλ>0. Suppose that the state observer s of the form ˆx(t) =f σ(ˆx(t)) + A σ ˆx(t)+D σ ˆx(t d(t))+ B σu(t)+l σ(y(t) ŷ(t)) ŷ(t) =C σ ˆx(t) (4) where y(t) s the measurable output of system (1), and L σ s the observer gan matrx to be determned later. The reference model s gven as ẋ r(t) =A rx r(t)+r(t) (5) where x r(t) R n s the reference state, A r s a Hurwtz matrx, r(t) s the bounded reference nput whch belongs to L 0, ) andl1 loc (ϱ, ), R n ), respectvely. Now, defne trackng error e r(t) =x(t) x r(t), and consder the H trackng performance as e α(t ) e T r (t)e r(t)dt γ ω T (t) ω(t)dt, t (6) where ω(t) =ω T (t),r T (t) T,α,γ are postve constants. Defne the dfference between the real state and the observer state, the observer state and the reference state as e(t) =x(t) ˆx(t), ê r(t) =ˆx(t) x r(t). Desgn the output feedback controller u(t) =K σ ˆx(t)+F σx r(t) Bσ T (B σbσ T ) 1 f σ(ˆx(t)) (7) where K σ, F σ are the output feedback gans. Combnng (1), (4), (5) and (7), we can obtan the augmented systems ė(t) =(A σ L σc σ)e(t)+d σe(t d(t)) + f σ(x(t)) f σ(ˆx(t)) + ω(t) (8) ˆx(t) =(A σ + B σk σ)ˆx(t)+b σf σx r(t)+d σ ˆx(t d(t))+ L σc σe(t) ẋ r(t) =A rx r(t)+r(t). (9) Let ˆx(t) A σ + B σk σ B σf σ x(t) =, Ā σ =, x r(t) 0 A r L σc σe(t) D σ 0 g σ(t) =, Dσ =. r(t) 0 0 Then, system (9) can be rewrtten as x(t) =Āσ x(t)+ D σ x(t d(t)) + g σ(t). (10) Defne the swtchng sequences of system (8) and (9) Σ {( 0,), ( 1,t 1),, ( k, ), k M} (11) whch means the k -th subsystem s actvated at tme. Defnton 0, 1. For system (1), f there exst control nput u(t) and swtchng law σ(t), such that: 1) the closedloop (8) and (9) are exponentally stable when ω(t) 0; ) performance ndex (6) s satsfed when ω(t) 0 under zero ntal condtons, that s, x(t) =0,x r(0) = 0, ˆx(t) =0,t τ,0. Then system (1) s sad to have observer-based H model reference trackng performance. Defnton 3 1. For the swtched sgnal σ(t) andany t τ 0, N σ(t, τ) denotes the system swtchng tmes n the open nterval (τ, t). If N σ(t, τ) N 0 + t τ (1) τ a holds for τ a > 0andN 0 0, then τ a s called average dwell-tme. Wthout loss of generalty, as commonly used n the lterature, we assume N 0 =0. To conclude ths secton, we recall the followng lemmas. Lemma 1 9. Let U, V be real matrces of approprate dmensons. Then, for any matrx Q>0ofapproprate dmenson and scalar ɛ>0, t holds that UV + V T U T ɛ 1 UQ 1 U T + ɛv T QV. Lemma 13. For any constant matrx N > 0, scalar τ>0, any t 0, + ), vector functon y :t τ,t R n, such that the ntegratons n the followng are well defned, then ( t τ T y(s)ds) N y(s)ds τ y(s) T Ny(s)ds. t τ t τ Consder the lnear tme-varyng delay system (10) wthout swtchng and ts homogeneous system as x(t) =Ā x(t)+ D x(t d(t)) + g(t) (13) x(t) =Ā x(t)+ D x(t d(t)). (14)

3 B. Wang et al. / Output Feedback Trackng Control for a Class of Swtched Nonlnear Systems wth 607 Itcanberewrttennoperatorform 1 x(t) =L(t, x t)+g(t), t ϱ x(t) =L(t, x t), x ϱ = φ, t ϱ where the operator L(t, φ) s lnear n φ, and has the form L(t, φ) = Aφ(0) +Dφ( d(t)), n whch φ(θ) = x(t + θ),θ τ,0. Suppose there s an m L1 loc (ϱ, ), R +)such that L(t, φ) m(t) φ (15) for all t (, ),φ C n. Lemma 3 (Varaton-of-constants). Let x(ϱ, φ, g)(t) denote the soluton of system (13), and x(ϱ, φ, 0)(t) denote the soluton of the correspondng homogeneous system (14). Denote { x(ϱ, φ, 0)(t + θ) by x t(ϱ, φ, 0)(θ), τ θ 0, 0, τ θ<0 X 0(θ) =.If x t(ϱ, φ, 0) T (t, ϱ)φ, then I, θ =0 T (t, ϱ) s a contnuous lnear operator. And f (15) s satsfed and g(t) L1 loc (ϱ, ), R n ), then x t(ϱ, φ, g) =T (t, ϱ)φ + ϱ T (t, s)x 0g(s)ds, t ϱ. (16) Proof. The proof follows the same lnes as n 1, 7. 3 Man results 3.1 Observer gan matrx desgn Consder the smplfed system of (8) when ω(t) = 0 ė(t) = (A σ L σc σ)e(t)+d σe(t d(t))+ f σ(x(t)) f σ(ˆx(t)). (17) Theorem 1. For system (17), f there exst scalar α> 0,τ > 0, matrces H > 0,G > 0, L, M and any nvertble matrx Y wth approprate dmensons, such that the followng LMIs hold Θ = ϕ 11 ϕ 1 Y τy D ϕ Y τy D I 0 τe ατ G < 0 (18) and the average dwell-tme satsfes ln μ1 τ a1 > (19) α where ϕ 11 = Y (A +D )+(A +D ) T Y T L C C T L T + αh +U T U,ϕ 1 = H Y +(A +D ) T Y T C T L T,ϕ = τg Y Y T and μ 1 1satsfes H μ 1H j, G μ 1G j,, j M, j. (0) Then, system (17) s exponentally stable and the observer gan matrx s gven by L = Y 1 L. Proof. Choose the pecewse Lyapunov functonal canddate as V (e(t)) = V σ(e(t)) = V 1σ(e(t)) + V σ(e(t)) V 1σ(e(t)) = e T (t)h σe(t) V σ(e(t)) = 0 τ t+θ e α(t s) ė T (s)g σė(s)dsdθ. (1) Durng any nterval,+1 ), we let σ(t) = k =, then V (e(t)) = V (e(t)). Along the trajectores of system (17), the tme dervatve of V (e(t)) s gven as V (e(t)) = e T (t)h ė(t) αv (e(t)) + τė T (t)g ė(t) t τ From Assumpton 1, we can get e α(t s) ė T (s)g ė(s)ds. () f (x(t)) f (ˆx(t)) T f (x(t)) f (ˆx(t)) e T (t)u T U e(t). (3) Substtutng (3) nto (), one has V (e(t)) + αv (e(t)) e T (t)(αh + U T U )e(t)+ e T (t)h ė(t)+τė T (t)g ė(t) e ατ ė T (s)g ė(s)ds f (x(t)) f (ˆx(t)) T f (x(t)) f (ˆx(t)). (4) For the free weghtng matrx Y and ė(s)ds = e(t) e(t d(t)), t holds that e T (t), ė T Y (t) (A + D L C )e(t) ė(t) Y D ė(s)ds + f (x(t)) f (ˆx(t)) = 0. (5) Let ξ(t, s) =e T (t), ė T (t),f T (x(t)) f T (ˆx(t)), ė T (s) T. Substtutng (5) nto (4), t yelds V (e(t)) + αv (e(t)) 1 d(t) ξ T (t, s) Θ ξ(t, s)ds (6) ϕ 11 ϕ 1 Y d(t)y D where Θ ϕ = Y d(t)y D < 0. By I 0 d(t)e ατ G proper transformaton to (18), we can get Θ < 0, then V (e(t)) + αv (e(t)) 0. (7) Let t k be the left lmt of, whch s before the swtchng at. Usng (0) and (1), one has V (e( )) μ 1V j(e(t k )),, j M, j. (8) Let any t,+1 ), then N σ(t, )=k. Combnng (11), (7) and (8), we can obtan V k (e(t)) V k (e( ))e α(t tk) μ 1V k 1 (e( 1 ))e α(t tk 1) μ k 1V 0 (e())e α(t t0) = e α(t )+k ln μ 1 V 0 (e()). (9) From Defnton 3, we can ge ln μ 1 t τ a1 ln μ 1. Let λ = α ln μ 1 τ a1 > 0, then, a e(t) e λ(t ) V 0 (e()) b e λ(t ) e t0 cl (30)

4 608 Internatonal Journal of Automaton and Computng 11(6), December 014 where a = mn M{λ mn(h )},b = max M{λ max(h )+ τ λmax(g)}. From (30), we can obtan b e(t) e λ(t t0) e t0 cl. (31) a By Defnton 1, we know that system (17) s exponentally stable. 3. Stablty and H performance Theorem. For system (10), f there exst scalars α> 0,τ > 0,γ > 0, and matrces P > 0,S > 0, M, such that the followng nequaltes hold Φ = φ 11 P D P τ(ā + D ) T φ 0 τ D T 1 γ I τi S 1 and the average dwell-tme satsfes < 0 (3) ln μ τ a =max{τ a1,τ a }, τ a > (33) α where φ 11 = P (Ā + D )+(Ā + D ) T P + αp + Q, φ = e ατ S, Q I I =,andμ 1satsfes I I P μ P j, S μ S j,, j M, j. (34) Then system (10) s exponentally stable, and the H model reference trackng performance n (1) s guaranteed. Proof. From (3), usng Schur complement, t yelds Λ = φ 11 P D P φ 0 1 γ I + τ Ā + D D I T S Ā + D D I < Q (35) where φ 11 = φ 11 Q. Frst, consder the nomnal system of system (10): x(t) =Āσ x(t)+ D σ x(t d(t)). (36) Defne the pecewse Lyapunov functonal canddate as V ( x(t)) = V σ( x(t)) = V 1σ( x(t)) + V σ( x(t)) V 1σ( x(t)) = x T (t)p σ x(t) V σ( x(t)) = τ 0 τ t+θ e α(t s) x T (s)s σ x(s)dsdθ. Let x(s)ds = z(t) = x(t d(t)) x(t), then x(t) = (Ā + D ) x(t) + D z(t). Then, along the trajectores of system (36), the tme dervatve of V ( x(t)) s gven by V ( x(t)) x T (t)p (Ā + D )+(Ā + D ) T P x(t)+ x T (t)p Dz(t) αv ( x(t)) + τ x T (t)s x(t) τe ατ x T (s)s x(s)ds. (37) By Lemma, t can be obtaned that τ x T (s)s x(s)ds z T (t)s z(t). (38) Let η(t) = x T (t),z T (t) T. Substtutng (38) nto (37), t yelds where Φ = V ( x(t)) + αv ( x(t)) η T (t) Φ η(t) (39) P(Ā + D )+(Ā + D ) T P + αp P D + e ατ S τ Ā + D T D S Ā + D D. (40) From (35), (39) and (40), one can obtan V ( x(t)) + αv ( x(t)) 0. (41) By usng the same means as the proof of Theorem 1, we can get x(t) b1 a 1 e λ 1(t ) x t0 cl. (4) where λ 1 = α ln μ τ a > 0, a 1 =mn M{λ mn(p )},b 1 = max M{λ max(p )+ τ 3 λmax(s)}. Then, system (36) s exponentally stable. Next, the stablty of system (10) wll be analysed wth LC g (t) 0when ω(t) =0,g (t) = e(t). Snce g 0 σ(t) L1 loc (ϱ, ), R n ), for any t t j,t j+1), by Lemma 3, we get the soluton of (10) wth the ntal condton (,φ 0 ): x t(,φ 0,g σ)=t j (t, t j)φ j + T (t, s)x 0g j (s)ds = t j T j (t, t j)t j 1 (t j,t j 1)φ j 1 + T j (t, t j) j t j 1 T (t j,s)x 0g j 1 (s)ds + T (t, )φ 0 + t j T (t, s)x 0g j (s)ds = = T (t, s)x 0g σ(s)ds (43) where T (t, )=T j (t, t j) T j 1 (t j,t j 1) T 0 (t 1,)sa contnuous pecewse lnear operator. Recallng the above analyss, system (36) s exponentally stable. There exst η 1 > 0,κ 1 > 0 such that T (t, ) κ 1e η 1(t ) T (t, s)x 0 κ 1e η 1(t s), t s. (44) If (18), (19) and (0) hold, system (8) s exponentally stable b by Theorem 1. From (31), there exsts a scalar B 0 = a e t0 cl max M{σ max(l C )}, such that g σ(t) L σc σ e(t) B 0e λ(t ), t. (45) Substtutng (44) and (45) nto (43), and choosng η 1 λ,

5 B. Wang et al. / Output Feedback Trackng Control for a Class of Swtched Nonlnear Systems wth 609 t yelds x t(,φ 0,g σ) κ 1e η 1(t ) φ 0 + κ 1e η 1(t s) B 0e λ(s ) ds κ 1e η 1(t ) φ 0 + κ1b0 η 1 λ (e λ(t ) e η 1(t ) ) κ 1( φ 0 + B0 η 1 λ )e β(t ) (46) where β =mn{λ, η 1}. We know that system (10) s exponentally stable when ω(t) = 0. Second, we wll prove that performance ndex (6) s satsfed under zero ntal condton x()=0,e()=0wth ω(t) 0. Letς(t) = x T (t),z T (t),g T (t) T. Usng the same method as the above analyss, t mples V ( x(t)) + αv ( x(t)) ς T (t)λ 1 ς(t)+ γ g T (t)g (t) Q 0 0 ς T (t) ς(t)+ 1 γ g T (t)g (t) = ê T r (t)ê r(t)+ 1 γ g T (t)g (t). (47) Lettng Q = 1 I Rn n, by Lemma 1, we can get ê T r (t)ê r(t) = e(t) e r(t) T e(t) e r(t) = e T r (t)e r(t) e T (t)e(t)+e T r (t)e(t) e T r (t)e r(t) e T (t)e(t)+e T r (t)qe r(t)+ e T (t)q 1 e(t) = 1 et r (t)e r(t)+ e(t). (48) Smlar to the reasonng n the above proof, let e t(,ψ 0,ω) denote the soluton of system (8) wth the ntal condton (,ψ 0 ), ψ 0 = e()=0,t 1(t, ) denotes a contnuous pecewse lnear operator. As the nomnal system of (8) s exponentally stable, by Lemma 3, there exst κ > 0,η > 0 such that e t(,ψ 0,ω)=T 1(t, )ψ 0 + T 1(t, s)x 0ω(s)ds T 1(t, ) κ e η (t ) T 1(t, s)x 0 κ e η (t s), t s. (49) Accordng to Cauchy-Schwartz Inequalty, one can get from (49): e(t) κ e η (t s) ds e η (t s) ω(s) ds κ e η (t s) ω(s) ds. (50) η Let λ =max M{σ max(l C )}, thas g T (t)g (t) =e T (t)c T L T L C e(t)+r T (t)r(t) λ e(t) +r T (t)r(t). (51) Combnng (47), (48), (50) and (51), t holds that V ( x(t)) + αv ( x(t)) 1 et r (t)e 1 r(t)+(1+ γ λ ) κ e η (t s) ω(s) ds + 1 η γ r T (t)r(t), t,+1 ). (5) Let Γ(s) = 1 et r (s)e r(s) + (1 + 1 γ λ ) κ s η e η(s θ) ω(θ) dθ + 1 γ r T (s)r(s), V k ( x(t)) = V k ( x( )) + V k ( x( )) α V k ( x(s))ds + μ V k 1 ( x( 1 )) μ α μ k 1 Γ(s)ds α μ k V 0 ( x() μ k α 1 μ k 1 Γ(s)ds α μ Nσ(t,) V 0 ( x() α k Vk ( x(s))ds Γ(s)ds 1 V k 1 ( x(s))ds+ V k ( x(s))ds + V 0 ( x(s))ds+ V k ( x(s))ds + V k ( x(s))ds+ Γ(s)ds Γ(s)ds μ Nσ(t,s) Γ(s)ds. (53) Under zero ntal condton, V ( x()) = 0, and V k ( x(t)) 0, we can get μ Nσ(t,s) Γ(s)ds 0. (54) When t, pre- and post-multplyng (54) by e Nσ(t,)lnμ κ and lettng γ, t can be η κ λ concluded that 1 e T r (s)e r(s)e N σ(s,)lnμ ds s (1 + 1 γ λ ) κ e η (s θ) ω(θ) dθds+ η t 0 1 γ r T (s)r(s)ds = (1 + 1 γ λ ) κ η ω(s) ds+ 1 γ r T (s)r(s)ds 1 γ ω T (s) ω(s)ds. (55) By Defnton 3 and (33), N σ(s, )lnμ s τ a ln μ α(s ), t follows from (55) that 1 e α(t ) e T r (t)e r(t)dt 1 γ ω T (t) ω(t)dt. Remark 1. The average dwell-tme τ a1 s desgned to guarantee that system (17) s exponentally stable, and τ a

6 610 Internatonal Journal of Automaton and Computng 11(6), December 014 can guarantee system (36) s exponentally stable. The average dwell-tme τ a of the whole system s chosen as the maxmum value between τ a1 and τ a, whch can guarantee that system (8) and (9) are exponentally stable wth H performance ndex. In 1, a condton s requred to hold that (8) has the common Lyapunov functon and the desgned swtchng law reles on the states x(t), x(t) and x(s) whch contans the dervatve term. The proposed method just needs that (8) satsfes the average dwell-tme (19) lowerng the conservatsm, whch s one of the man contrbutons of ths paper. The asymptotc stablty of system (10) s obtaned n 1, whle the proposed method can guarantee that system (10) s exponentally stable wth g (t) H controller desgn Frst, let ψ 11 B F D S 1 0 Ω 11 =, Ω 1 = ψ 0 0 ψ 17 0 X 1 Ω 14 = F T B T X A T, Ω 15 = r X where ψ 11 = X 1(A +D ) T +(A +D )X 1 +αx 1 +B K+ K T B T,ψ = A rx + X A T r + αx,ψ 17 = X 1(A + D ) T + K T B T,andΩ 4 =Ω T 1. Theorem 3. Consder the augmented system (10), and suppose that Assumptons 1 and and (18) (0) hold. For the gven scalars τ>0,α > 0,γ > 0, there exst matrces X 1 > 0,X > 0,S 1 > 0,S > 0, K, F, M, such that the followng LMIs hold Ω = Ω 11 Ω 1 I τω 14 Ω 15 e ατ S 0 τω γ I τi 0 S 0 I and average dwell-tme satsfes (33), μ 1satsfes < 0 (56) X 1 μ X 1j,X μ X j, S 1 μ S 1j,S μ S j,, j M, j. (57) Then, the system (10) s exponentally stable, and (1) has the H model reference trackng performance. The desgned controller gan matrces are gven by K = K X 1 1,F = F X 1. Proof. Choose the form of postve defnte matrces as P =,S P1 0 Ŝ1 0 P =. Preand post-multplyng (3) Ŝ by dag{p 1,S 1,I,I}, denote X = P 1 X1 0 =, S X = S 1 = S1 0, K S = K X 1, F = F X, and X QX = X1 X1 X. By usng Schur complement for X (3) n Theorem, we can easly get (56). Remark. The work n proposed an exponental stablty crteron for a class of swtched nonlnear systems wth tme-varyng delay. However, the desgned controller reles on the system state x(t) and reference state x r(t). In ths paper, we desgn an observer to deal wth the unavalable state x(t) and construct an output feedback controller to track the reference sgnal x r(t). 4 Numercal example Consder system (1) and reference system (5) wth A 1 =,B 1 = C 1 = ,D 1 = A =,B = C = ,D = cos(0.01x 1) A r =,f 1(x(t)) = cos(0.01x ) 0.cos(0.01x 1) f (x(t)) =. 0.cos(0.01x ) We adopt the parameters below: d(t) = sn t, τ = 0.3,α= 1,γ = 0.5, and the Lpschtz matrces are gven by U 1 =, U = Frst, let μ 1 = 7, by Theorem 1, we obtan H 1 =,L = H =,L = G 1 =,G = and the average dwell-tme satsfes τ a1 > s. However, by usng the proposed method n 1, we cannot obtan the feasble solutons, so we cannot get the correspondng common Lyapunov functon. Then, consderng the system (9), and usng Theorem 3, we can get the common Lyapunov functon and a set of solutons X 1 =,X = S 1 =,S = K 1 =,K = F 1 =,F = where = 1,, thus, we can get the average dwell-tme τ a > 0.

7 B. Wang et al. / Output Feedback Trackng Control for a Class of Swtched Nonlnear Systems wth 611 Accordng to the swtchng law (33), we choose the average dwell-tme τ a = s. Let w(t) =,r(t) = w1(t) w (t) r1(t) sn t, 5s t 0 s, ω r (t) =r (t) = t, =1,, (t) 0, others and choose the ntal condtons ϕ(t) = 0,t 0.3, 0),x(0) = 0.4, 0.3 T, ˆx(0) = 0.3, 0. T,x r(t) = 0.6, 0.5 T. Then, the smulaton results are shown n Fgs It can be seen that the state error s exponentally stable after s when ω(t) = 0, and the real state tracks the reference state perfectly. When the system s subjected to the bounded exogenous dsturbance, we can see that t has H trackng performance under the desgned controller and swtchng law. Fg. 1 The desgned swtchng sgnal σ(t) Fg. 4 5 Conclusons The state x and the reference state x r In ths paper, output feedback trackng control for swtched nonlnear system wth tme-varyng delay has been studed. A new swtchng law s desgned and exponental stablty crteron for the system s derved. Average dwelltme and pecewse Lyapunov functon have been used to obtan the stablty of the augmented system and H trackng performance when the system has the bounded exogenous dsturbance. Besdes, the desgned observer and controller can be obtaned by solvng a set of LMIs, and the proposed method does not need that a common Lyapunov functon exsts for the swtched nonlnear systems. A numercal example s gven to show the effectveness of the proposed method. If the nonlnear functon f σ( ) s unknown, how to deal wth ths problem wll be our further study ssue. References Fg. 3 Fg. State error e(t) The state x 1 and the reference state x r1 1 J. P. Hespanha, A. S. Morse. Stablty of swtched systems wth average dwell-tme. In Proceedngs of the 38th Conference on Decson & Control, IEEE, Phoenx, Arzona, USA, pp , G. S. Zha, B. Hu, K. Yasuda, A. N. Mchel. Stablty analyss of swtched systems wth stable and unstable subsystems: An average dwell tme approach. In Proceedngs of the Amercan Control Conference, IEEE, Chcago, USA, pp , H. Ln, P. J. Antsakls. Stablty and stablzablty of swtched lnear systems: A survey of recent results. IEEE Transactons on Automatc Control, vol. 54, no., pp , W. M. Xang, J. Xao, M. N. Iqbal. Robust observer desgn for nonlnear uncertan swtched systems under asynchronous swtchng. Nonlnear Analyss: Hybrd Systems, vol. 6, no. 1, pp , D. W. Dng, G. H. Yang. H statc output feedback control for dscrete-tme swtched lnear systems wth average dwell tme. IET Control Theory and Applcatons, vol. 4, no. 3, pp , J. Y. Zha, W. T. Zha. Global adaptve output feedback control for a class of nonlnear tme-delay systems. ISA Transactons, vol. 53, no. 1, pp. 9, W. T. Zha, J. Y. Zha, S. M. Fe. Output feedback control for a class of stochastc hgh-order nonlnear systems wth tme-varyng delays. Internatonal Journal of Robust and Nonlnear Control, vol. 4, no. 16, pp , T. Wang, M. X. Xue, C. Zhang, S. M. Fe. Improved stablty crtera on dscrete-tme systems wth tme-varyng and dstrbuted delays. Internatonal Journal of Automaton and Computng, vol. 10, no. 3, pp , 013.

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