CONTROL OF SWITCHED LINEAR PARABOLIC SYSTEMS

Transcription

1 Electronc Journal of Dfferental Equatons, Vol (2014), No. 131, pp ISSN: URL: or ftp ejde.math.txstate.edu H CONTROL OF SWITCHED LINEAR PARABOLIC SYSTEMS LEPING BAO, SHUMIN FEI, LIN CHAI Abstract. The H control problem of a class of swtched lnear parabolc systems s consdered. By applyng the multple Lyapunov functon method and the average dwell tme scheme, suffcent condtons for exponental stablty and the H control synthess are establshed n terms of LMIs and a famly of swtchng sgnals. The advantage n ths work les n the fact that suffcent condtons completely depend on the system parameters and the system can be analyzed by usng numercal softwares. At the end of the paper, an example s gven to llustrate the obtaned result. 1. Introducton Durng the prevous decade, the study of swtched systems has attracted consderable attenton because of ts sgnfcance n both theoretcal research and practcal applcatons [1, 3, 6, 10, 11, 15, 18, 20, 22, 23]. A swtched system s a dynamcal system descrbed by a famly of contnuous-tme subsystems and a rule that governs the swtchng among them. In many realstc cases, swtched systems can be descrbed by partal dfferental equatons (PDE) or a combnaton of ordnary dfferental equatons (ODE) and PDE such as n chemcal ndustry process and bomedcal engneerng. We refer to these swtched systems as dstrbuted parameter swtched systems (DPSS) or nfnte dmensonal swtched systems [5, 13]. However, there are very few works concernng DPSS (see, eg. [1, 7, 9, 14, 16, 19, 21] and the references cted theren). For example, Sasane generalzed the results presented n [15] to nfnte dmensonal Hlbert spaces [21], and showed when all subsystems are stable and commutatve parwse, the swtched lnear system s stable under an arbtrary swtchng va the common Lyapunov functon. Mchel and Sun provded the stablty condtons for swtched nonlnear systems on Banach spaces under constraned swtchng [14]. Hante and Sgalott gave necessary and suffcent condtons n term of the exstence of common Lyapunov functons for DPSS [1, 7]. It seems that the majorty of works deal wth the stablty of DPSS. The H control s an nterestng research topc n the feld of swtched systems. Up to now, most of results n the lterature are regardng the H control of swtched 2000 Mathematcs Subject Classfcaton. 34K20, 35R15, 34D10. Key words and phrases. Swtched systems; parabolc systems; H control; lnear matrx nequaltes; average dwell tme. c 2014 Texas State Unversty - San Marcos. Submtted August 16, Publshed June 10,

2 2 L. BAO, S. FEI, L. CHAI EJDE-2014/131 systems whch are descrbed by ODEs [6, 12, 23]. For example, the stablty, the L 2 - gan analyss and the H control for swtched systems va the multple Lyapunovlke functon methods s consdered n [23]. The dynamc output feedback n H control desgn of swtched lnear systems are studed n terms of lnear matrx nequaltes (LMIs) n [6]. To the best of our knowledge, the H control has not been nvestgated for DPSS. Motvated by the above consderaton, we study the H control synthess for swtched lnear parabolc systems n ths paper. The man contrbutons of the present paper can be summarzed as follows: Frstly, the concept of H control s extended to DPSS. Secondly, suffcent condtons for the exponental stablzaton and the H control synthess of DPSS are developed n terms of LMIs and a class of swtchng sgnals. Compared wth the work n [7], our suffcent condtons completely depend on the system parameters. In ths artcle, L 2 (, R n ) denotes the Hlbert space of square ntegrable n dmensonal vector-valued functons ν(x), x wth the norm ν L2 = νt νdx. L 2 [, ; L 2 (, R n )) s the Hlbert space of square ntegrable functons ν(t, ) L 2 [, ) wth values ν(, x) L 2 (, R n ). H 2 (, R n ) and H0 2 (, R n ) denote the classcal Sobolev spaces defned by H 2 (, R n ) = {ν L 2 (, R n ) : 2 ν x L 2 2 (, R n )} and H0 1 (, R n ) = {ν L 2 (, R n ν ) : x L 2(, R n ), ν(, t) = 0} respectvely. γ M (P )(γ m (P )) denotes the largest (smallest) egenvalue of P. The symmetrc elements of the matrx wll be denoted by T. 2. Problem formulaton and prelmnares Consder the swtched lnear parabolc systems ν(x, t) t = D σ(t) ν(x, t) + A σ(t) ν(x, t) + B σ(t) u(x, t) + C σ(t) ω(x, t) y(x, t) = E σ(t) ν(x, t) + F σ(t) ω(x, t) ν(x, t) = 0, wth the statc state feedback control ν( ) = ν 0 (x, t) [, + ) (2.1) u(x, t) = K σ(t) ν(x, t) (2.2) where ν(x, t) L 2 [, ; L 2 (, R n )) s a vector-valued functon representng the state of the process, u(x, t) L 2 [, ; L 2 (, R s )) denotes the manpulated nput, ω(x, t) L 2 [, ; L 2 (, R p )) s the dsturbance, and y(x, t) L 2 [, ; L 2 (, R q )) denotes the measured output wth (x, t) [, + ). = [0, 2] [0, 2] R 2 s a bounded doman wth the smooth boundary. denotes the Laplace operator;.e., = 2 k=1 2 x 2 k. D = dag(d 1, d 2,..., d n ) represent postve dagonal matrces, and A, B, C, E, F ( = 1, 2,..., m) represent constant matrces of compatble dmensons. σ(t) : [, ) Θ s the swtchng sgnal mappng tme to some fnte ndex set Θ = {1, 2,..., m}, and the swtchng sgnal σ(t) s a pecewse contnuous (from the rght) functon dependng on tme or state or both. The dscontnutes of σ(t) are called swtchng tmes or swtchng nstants. The nteger m s the number of models (called subsystems) of the swtched system. The objectve of ths artcle s to establsh suffcent condtons of the H control for the system (2.1) (2.2). That s, we look for controller gan matrces K ( Θ) and a class of swtchng sgnals σ(t), such that

3 EJDE-2014/131 CONTROL OF SWITCHED SYSTEMS 3 1. When ω = 0, the system (2.1) s exponentally stablzed by the state feedback control (2.2). 2. The system (2.1) s exponentally stablzed by (2.2) wth the H dsturbance level γ > 0,.e., for a prescrbed scalar γ > 0, the performance ndex s J(ω) = [y T (x, s)y(x, s) γ 2 ω T (x, s)ω(x, s)] dx ds 0 for all non-zero ω(x, t) L 2 [, ; L 2 (, R p )) under the zero ntal condton. The followng s the defnton of average dwell tme (ADT) [10]. Defnton 2.1. Gven some famly of swtchng sgnals σ(t) Θ, for each σ(t) and each t > s, let N σ (s, t) denote the number of swtchng of σ(t) n the open nterval (s, t). If N σ (s, t) N 0 + t s τ a holds for τ a > 0 and N 0 > 0, then the postve constant τ a s called the ADT and N 0 s the chatter bound. Lemma 2.2 (Poncare s nequalty [4]). Let the scalar functon u H0 1 (, R) wth 1, then we have n u 2 dx γ 2 ( u ) 2 dx = γ 2 u 2 dx (2.3) x =1 where 1 : 0 x δ( = 1, 2,..., n), γ = δ n, and = ( x 1, x 2,..., 3. Stablzaton analyss of swtched parabolc systems x n ). In ths secton, we consder the exponental stablzaton problem of the swtched lnear parabolc system ν(x, t) t = D σ(t) ν(x, t) + A σ(t) ν(x, t) + B σ(t) u(x, t) ν(x, t) = 0, ν( ) = ν 0 (x, t) [, + ) (3.1) wth the statc state feedback control (2.2). We assume that the state of system does not jump at swtchng nstants;.e., the state trajectory s contnuous and the swtchng sgnal σ(t) has the fnte swtchng number n any fnte tme nterval [10]. We start wth the well-posedness problem of the closed-loop system (2.2)-(3.1). Defne the state functon z(t) as z(t) = ν(, t) on the Hlbert space H = L 2 (, R n ) wth the norm L2, then the equaton of closed-loop (2.2) and (3.1) can be rewrtten as ż(t) = Ãσ(t)z(t) + f σ(t) (t), t (3.2) n H, where the nfntesmal operators à (Ãx = D x) have the dense doman W = D(Ã) = {ν H 2 (, R n ) H 1 0 (, R n ) : ν(x) = 0, x }, f σ(t) (t) = [A σ(t) + B σ(t) K σ(t) ]z(t). As we know, the nfntesmal operators à generate analytcal semgroups T (t) [17]. Because the state of system (2.2)-(3.1) does not jump at swtchng nstants, for every ntal value ν 0 W, there exsts a unque soluton for system (2.2)-(3.1). Thus, the ntal problem (2.2)-(3.1) turns out to be well-posed on the tme nterval [, ).

4 4 L. BAO, S. FEI, L. CHAI EJDE-2014/131 Lemma 3.1. For the gven scalar λ > 0, f there exst postve constants α, β, a constant µ 1, and contnuous functons V C(H [, + ), R + ) such that the functons V (t) = V (ν, t) are absolutely contnuous along the solutons ν of system (2.2)-(3.1) and satsfy α ν(t) L2 V (t) β ν(t) L2 (3.3) V (t) + λv (t) 0. (3.4) Furthermore, the Lyapunov functon of the system satsfes V (t) µv j (t),, j Θ. (3.5) Then the closed-loop system (2.2)-(3.1) s exponentally stable for the arbtrary swtchng sgnal σ(t) wth the ADT τ a > lnµ λ. Proof. For t [t k, t k+1 )(k = 0, 1... ), from (3.4) t follows that Ths, together wth (3.5) gves It s easy to show that V σ(t) (t) e λ(t t k) V σ(tk )(t k ). V σ(t) (t) µv σ(t k )(t k )e λ(t t k). V σ(t) (t) µv σ(t k )(t k )e λ(t t k) µv σ(tk 1 )(t k 1 )e λ(t t k) e λ(t k t k 1 ) µ 2 V σ(t k 1 )(t k 1 )e λ(t t k 1)... µ k V σ(t0 )( )e λ(t t0) for all t and a constant µ 1. Note that when kτ a t, we have V σ(t) (t) e λ(t t0) e klnµ lnµ (λ V σ(t0 )( ) e τa )(t t0) V σ(t0 )( ). (3.6) Combng (3.3) and (3.6), we obtan Thus we have ν(t) L2 ν(t) L2 V σ(t)(t) α 1 lnµ e (λ τa )(t t0) V σ(t0 )( ). α 1 lnµ e (λ τa )(t t0) γ ν 0 L2 γ lnµ e (λ τa )(t t0) ν 0 L2 α α where γ = max Θ {β } and α = mn Θ {α }. Let h = λ lnµ τ a > 0, and we have τ a > lnµ λ. It s obvous that the system (2.2)-(3.1) s exponentally stable for the arbtrary swtchng sgnal σ(t) wth the ADT τ a > lnµ Theorem 3.2. For the gven scalar λ > 0, f there exst dagonal matrces X > 0, and matrces Y > 0 such that Π = 2D X + A X + X T A T λ. + B Y + Y T B T + λx < 0. (3.7) Then system (3.1) can be exponentally stablzed by the state feedback control (2.2) wth K = Y X 1 for the arbtrary swtchng sgnal σ(t) wth the ADT τ a > ln(µ)/λ, where µ s determned by {γ M (X k )} µ = max. (3.8) k,l Θ γ m (X l )

5 EJDE-2014/131 CONTROL OF SWITCHED SYSTEMS 5 Proof. Choose the multple Lyapunov functon for the system (2.2)-(3.1) V (t) = V σ(t) (t) = ν T (x, t)p σ(t) ν(x, t) dx (3.9) wth constant dagonal matrces P > 0. It s not dffcult to see that there exst postve numbers α, β and a constant µ 1 such that nequaltes (3.3) and (3.5) hold. For nequaltes (3.5), we can choose µ = max k,l Θ { γ M (P k ) γ m(p l ) } [22]. Dfferentatng V (t) wth respect to t along the trajectory of the closed-loop system (2.2)-(3.1), we have V (t) + λv (t) = [ ν(x, t)] T D P ν(x, t)dx + ν T (x, t)p D ν(x, t)dx (3.10) + ν T (x, t)[p A + P B K ]ν(x, t)dx + ν T (x, t)[a T P + K T B T P ]ν(x, t)dx + ν T (x, t)λp ν(x, t)dx. Because D and P are postve dagonal matrces, we fnd that P D = D P. Thus we have [ ν(x, t)] T D P ν(x, t)dx + ν T (x, t)p D ν(x, t)dx 2λ max (P D ) ν T (x, t)iν(x, t)dx 2λ max (P D ) [ ν 1 (x, t),..., ν n (x, t)] ν 1(x, t)... dx ν n (x, t) 2λ max (P D ) [ν 1 (x, t) ν 1 (x, t) + + ν n (x, t) ν n (x, t)]dx. Accordng to Gauss s dvergence theorem, Poncare s nequalty (2.3) and takng nto account the boundary condton n (3.1), we obtan [ ν(x, t)] T D P ν(x, t)dx + ν T (x, t)p D ν(x, t)dx 2λ max (P D ) [ν1(x, 2 t) + + ν n ( 2 x, t)]dx (3.11) 2λ max (P D ) ν T (x, t)iν(x, t)dx ν T (x, t)( 2P D )ν(x, t)dx < 0. Substtutng (3.11) nto (3.10) yelds where When V (t) + λv (t) ν T (x, t)γ ν(x, t) dx Γ = 2P D + P A + A T P + P B K + K T B T P + λp. Γ = 2P D + P A + A T P + P B K + K T B T P + λp < 0, (3.12)

6 6 L. BAO, S. FEI, L. CHAI EJDE-2014/131 we have V (t) + λv (t) < 0 (for all ν(x, t) 0). If the swtchng sgnal σ(t) satsfes the ADT τ a > ln(µ)/λ, all condtons n Lemma 3.1 hold. Hence, the closed-loop system (2.2)-(3.1) s exponentally stable. Left- and rght- multplyng (3.12) by P 1 and lettng X = P 1 and Y = K P 1, t s not dffcult to see that equalty (3.12) s equvalent to (3.7) and µ = max k,l Θ { γ M (P k ) γ m(p l ) } leads to (3.8) mmedately. Consequently, the proof s completed. 4. H control synthess In the secton, we consder the H control problem for system (2.1) (2.2). Lemma 4.1. For gven scalars λ > 0 and γ > 0, f there exst dagonal matrces P > 0 such that [ Γ + E T E P C + E T F ] C T P + F T E γ 2 I + F T F < 0, (4.1) then for any t [t k, t k+1 ), along the trajectory of system (2.1)-(2.2), we have t V (t) e λ(t tk) V (t k ) e λ(t s) Υ(x, s) dx ds (4.2) where Γ = 2P D + P A + A T P + P B K + K T B T P + λp, Υ(x, s) = y T (x, s)y(x, s) γ 2 ω T (x, s)ω(x, s). Proof. Dfferentatng V (t) wth respect to t along the trajectory of the closed-loop system (2.1) (2.2) and usng the smlar argument descrbed n the prevous secton, we have V (t) + λv (t) η T (x, t) t k [ ] Γ P C C T P η(x, t)dx, (4.3) 0 where η T (x, t) = [ν(x, t), ω(x, t)] T. It follows from (4.1) and (4.3) that [ V (t) + λv (t) < η T E T (x, t) E E T F ] F T E γ 2 I + F T F η(x, t)dx = Υ(x, s)dx. By calculaton, we have Integratng leads to (4.2). d dt (eλt V (t)) < e λt Υ(x, s)dx. (4.4) Theorem 4.2. For gven scalars λ > 0 and γ > 0, f there exst dagonal matrces P > 0 such that [ Γ + E T E P C + E T F ] C T P + F T E γ 2 I + F T F < 0. (4.5) Then, the system (2.1) can be exponentally stablzed by the state feedback control (2.2) wth the H dsturbance level γ > 0 for the arbtrary swtchng sgnal σ(t) wth the ADT τ a > lnµ λ, where µ s determned by µ = max k,l Θ{ γ M (P k ) γ m(p l ) }.

7 EJDE-2014/131 CONTROL OF SWITCHED SYSTEMS 7 Proof. It follows from (4.5) that [ Γ + E T E P C + E T F ] [ ] [ Γ P C T P + F T E γ 2 I + F T F = C E T γ 2 + E E T F ] I F T E F T F < 0. Snce [ E T E F T E E T F ] [ ] E T [E F T F = ] F 0, we have [ ] Γ P C γ 2 < 0. I Accordng to the Schur complement [2], we have Γ < 0. By vrtue of the proof of Theorem 3.1, the closed-loop system (2.1) (2.2) s exponentally stable when ω = 0. On the other hand, combnng (3.5) and (4.2), we obtan t V (t) µe λ(t tk) V (t k ) e λ(t s) Υ(x, s) dx ds. Followng [22] and repeatng the above procedure, we obtan V (t) t1 µ k e λt V (ν 0 ) µ k t2 µ k 1 t 1 = µ k e λt V (ν 0 ) F T t k e λ(t s) Υ(x, s) dx ds e λ(t s) Υ(x, s) dx ds t t e λ(t s)+nσ(s,t)lnµ Υ(x, s) dx ds. t k e λ(t s) Υ(x, s) dx ds (4.6) Because V (t) > 0, the zero ntal condton mples V (ν 0 ) = 0. Usng (4.6) yelds t e λ(t s)+nσ(s,t)lnµ Υ(x, s) dx ds 0. It follows from e Nσ(s,t)lnµ 1(µ 1, N σ (s, t) > 0) that t e λ(t s) [y T (x, s)y(x, s) γ 2 ω T (x, s)ω(x, s)] dx ds 0. (4.7) Notce that snce N σ (, s) N 0 + s t0 τ a, N 0 > 0, and τ a > lnµ λ, we derve that N σ (, s)lnµ N 0 lnµ + λ(s ). Multplyng both sdes of (4.7) by e [N0lnµ+λ(s t0)] gves t e λ(t s) [N0lnµ+λ(s t0)] y T (x, s)y(x, s) dx ds t Thus we obtan t e λt y T (x, s)y(x, s) dx ds e λ(t s) [N0lnµ+λ(s t0)] γ 2 ω T (x, s)ω(x, s) dx ds. t (4.8) e λt γ 2 ω T (x, s)ω(x, s) dx ds. (4.9)

8 8 L. BAO, S. FEI, L. CHAI EJDE-2014/131 Integratng both sdes from t = to gves y T (x, s)y(x, s) dx ds γ 2 ω T (x, s)ω(x, s) dx ds (4.10).e., J(ω) 0. Ths completes the proof. The matrx nequaltes (4.5) are not LMIs. Left- and rght- multplyng (4.5) by dag{p 1, I}. Let X = P 1, Y = K P 1. It follows from the Schur complement that (4.5) s equvalent to the LMIs Π C + X E T F X 0 γ 2 I + F T F 0 0 ( E T E ) 1 0 < 0, (4.11) I where Π = 2D X + A X + X T AT + B Y + Y T B T + λx. Next, we show a result whch can be obtaned usng matlab software. Theorem 4.3. For gven scalars λ > 0 and γ > 0, f there exst dagonal matrces X > 0 and matrces Y > 0, such that the LMIs (4.11) are feasble. Then the system (2.1) can be exponentally stablzed by the state feedback control (2.2) wth K = Y X 1 wth the H dsturbance level γ > 0 for the arbtrary swtchng sgnal σ(t) wth the ADT τ a > lnµ λ, where µ s determned by (3.8). Example 4.4. Consder the swtched parabolc equatons (2.1) under the state feedback (2.2). Suppose there are two subsystems wth parameters D 1 = [1 0; 0 2], D 2 = [2 0; 0 1], A 1 = [1 3; 2 3], A 2 = [1 2; 3 1], B 1 = [6 7; 5 1], B 2 = [5 2; 3 1], C 1 = [3 5; 4 2], C 2 = [5 2; 3 2], E 1 = [1 0; 0 1], E 2 = [1 0; 0 1], F 1 = [1 2; 2 1], F 2 = [2 1; 1 2]. Set λ = 0.6, γ = 0.8, usng Theorem 4.3, by resolvng LMIs (4.11), we obtan X 1 = [ ; ], X 2 = [ ; ]. The state feedback matrces are K 1 = [ ], K 2 = [ ] From (3.8) we obtan that µ = and τ a > lnµ λ = So system (2.1) can be exponentally stablzed. Concluson. By usng multple Lyapunov functon and ADT method, we establsh some new crtera for the exponental stablzaton and the H control synthess of swtched lnear parabolc systems va the state feedback. All the results are gven n terms of LMIs and a class of sgnals whch can be easly tested by matlab software. Acknowledgments. Ths work s supported by the Natonal Scence Foundaton of Chna (No , ), and by the Natural Scence Foundaton of Jangsu Provnce (BK ).

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10 10 L. BAO, S. FEI, L. CHAI EJDE-2014/131 Ln Cha School of Automaton, Southeast Unversty, Nanjng, Jangsu , Chna E-mal address: