Stability of linear systems with general sawtooth delay

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1 IMA Journal of Matematical Control an Information Page of 8 oi:0.093/imamci/nq03 Stability of linear systems wit general sawtoot elay KUN LIU, VLADIMIR SUPLIN AND EMILIA FRIDMAN Department of Electrical Engineering-Systems, Scool of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel liukun@eng.tau.ac.il suplin@eng.tau.ac.il Corresponing autor: emilia@eng.tau.ac.il Receive on 4 February 00; revise on June 00; accepte 3 August 00] It is well known tat in many particular systems, te upper boun on a certain time-varying elay tat preserves te stability may be iger tan te corresponing boun for te constant elay. Moreover, sometimes oscillating elays improve te performance (Miciels, W., Van Assce, V. & Niculescu, S. (005) Stabilization of time-elay systems wit a controlle time-varying elays an applications. IEEE Trans. Automat. Control, 50, ). Sawtoot elays τ wit τ = (almost everywere) can posses tis property (Louisell, J. (999) New examples of quencing in elay ifferential equations aving time-varying elay. Proceeigns of te 5t ECC, Karlsrue, Germany). In tis paper, we sow tat general sawtoot elay, were τ 0 is constant (almost everywere), also can posses tis property. By te existing Lyapunov-base metos, te stability analysis of suc systems can be performe in te framework of systems wit boune fast-varying elays. Our objective is to evelop qualitatively new metos tat can guarantee te stability for sawtoot elay wic may be not less tan te analytical upper boun on te constant elay tat preserves te stability. We suggest two metos. One meto evelops a novel input output approac via a Wirtinger-type inequality. By tis meto, we recover te result by Mirkin (007, Some remarks on te use of time-varying elay to moel sample-an-ol circuits. IEEE Trans. Automat. Control, 5, 09 ) for τ = an we sow tat for any integer τ, te same maximum boun tat preserves te stability is acieve. Anoter meto extens piecewise continuous (in time) Lyapunov functionals tat ave been recently suggeste for te case of τ = in Friman (00, A refine input elay approac to sample-ata control. Automatica, 46, 4 47) to te general sawtoot elay. Te time-epenent terms of te functionals improve te results for all values of τ, toug te most essential improvement correspons to τ =. Keywors: time-varying elay; Lyapunov-base metos; LMI.. Introuction Over te past ecaes, muc effort as been investe in te analysis an esign of uncertain systems wit time-varying elays (see, e.g. Kolmanovskii & Myskis, 999; Niculescu, 00; Karitonov & Niculescu, 00; Friman & Sake, 003; Ricar, 003; Gu et al., 003; He et al., 007; Park & Ko, 007). Te elay uner consieration as been eiter ifferentiable wit a known upper boun 0 τ < or piecewise continuous witout any constraints on te elay erivative (fast-varying elay) (Friman & Sake, 003). In te existing Lyapunov-base metos, te maximum elay boun tat preserves te stability correspons to = 0 an tis boun is usually a ecreasing function of. However, it is well known (see examples in Louisell, 999 an iscussions on quencing in Papacristooulou et al., 007 as well as Example below) tat in many particular systems, te upper boun on a certain time-varying elay tat preserves te stability may be iger tan te corresponing Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 c Te autor 00. Publise by Oxfor University Press on bealf of te Institute of Matematics an its Applications. All rigts reserve.

2 of 8 K. LIU ET AL. boun for te constant elay. Moreover, sometimes oscillating elays improve te performance (Miciels et al., 005). Recently, a iscontinuous Lyapunov function meto was introuce to sample-ata control systems (correspons to te sawtoot elay wit τ = (almost everywere)) in Nagstabrizi et al. (008). Tis meto improve te existing Lyapunov-base results an it inspire a piecewise continuous (in time) Lyapunov Krasovskii functional (LKF) approac to sample-ata systems (Friman, 00). Te LKFs in te latter paper are time epenent an tey o not grow after te sampling times. Te introuce novel iscontinuous terms of Lyapunov functionals in Friman (00) lea to qualitatively new results, allowing a superior performance uner te sampling, tan te one uner te constant elay. Te input elay approac to sample-ata control was also recently revise by using te scale small gain teorem an a tigter upper boun on te L -inuce norm of te uncertain term (Mirkin, 007). In te present paper, stability of systems wit boune sawtoot elay τ is analyse, were τ 0 is piecewise constant. By te existing Lyapunov-base metos, te stability analysis of suc systems can be performe in te framework of systems wit boune fast-varying elays. Our objective is to evelop qualitatively new metos tat can guarantee te stability for sawtoot elay wic may be not less tan te analytical upper boun on te constant elay tat preserves te stability. We suggest two metos. One meto evelops a novel input output (I O) approac via a Wirtinger-type inequality. By tis meto, we recover te result by Mirkin (007) for τ = an we sow tat for any integer τ, te same maximum boun tat preserves te stability is acieve. Anoter meto extens irect Lyapunov approac to systems wit a general form of sawtoot elay. By constructing appropriate iscontinuous LKFs, we obtain sufficient elay-epenent conitions tat guarantee te exponential stability of systems in terms of linear matrix inequalities (LMIs). Te time-epenent terms of LKFs improve te results for all values of τ, toug te most essential improvement correspons to τ =. A conference version of iscontinuous LKF approac was presente in Liu & Friman (009). Notations. Trougout te paper, te superscript stans for matrix transposition, R n enotes te n-imensional Eucliean space wit vector norm, R n m is te set of all n m real matrices an te notation P > 0, for P R n n means tat P is symmetric an positive efinite. Te symmetric elements of te symmetric matrix will be enote by. L is te space of square integrable functions v: 0, ) R n wit te norm v L = 0 v(t) t] /. Te space of functions φ: a, b] R n, wic are absolutely continuous on a, b), ave a finite lim θ b φ(θ) an ave square integrable firstorer erivatives is enote by W n a, b) wit te norm b ] φ Wn a,b) = max φ(θ) + φ(s) s. θ a,b] a We also enote W = W n, 0) an x t (θ) = x(t + θ)(θ, 0]). Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00. Problem formulation Consier te system { ẋ(t) = Ax(t) + A x(t τ(t)), x(t) = φ(t), t, 0], (.)

3 GENERAL SAWTOOTH DELAY 3 of 8 FIG.. Case A. FIG.. Case A. were x(t) R n is te state vector, A an A enote te constant matrices, φ(t) is te initial function, τ(t) 0, ] is te time-varying elay. It is assume tat te elay function as te form of sawtoot (see Figs an ), satisfying eiter A or A below: were > 0 an t k = k. A: τ(t) = (t t k ), t t k, t k+ ), k = 0,,..., (.) A: τ(t) = (t k+ t), t t k, t k+ ), k = 0,,..., (.3) It is clear tat uner A, we ave τ = > 0 an uner A, we ave τ = < 0. Bot cases can be analyse by using time-inepenent Lyapunov functionals corresponing to systems wit fast-varying elays. Our objective is to erive elay-epenent stability criteria for system (.) tat improve te recent results for fast-varying elays (see, e.g. Park & Ko, 007). 3. I O approac via Wirtinger-type inequality We recall te following Wirtinger-type inequality (Hary et al., 934): let z W a, b) be a scalar function wit z(a) = 0. Ten, b a z (ξ)ξ 4(b a) π b a ż (ξ)ξ. (3.) Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 Tis inequality is trivially extene to te vector case. LEMMA 3. Let z W n a, b). Assume tat z(a) = 0. Ten, for any n n-matrix R > 0, te following inequality ols: b a z (ξ)rz(ξ)ξ 4(b a) π b a ż (ξ)rż(ξ)ξ. (3.)

4 4 of 8 K. LIU ET AL. Proof. We ave b a z (ξ)rz(ξ)ξ = 4(b a) π 4(b a) π b a b a ξ ( ] z (ξ)rz(ξ) ξ ż (ξ)rz(ξ) z (ξ)rz(ξ) ) ξ 4(b a) π b a ż (ξ)rż(ξ)ξ. (3.3) System (.) can be rewritten as follows: ẋ(t) = (A + A )x(t) A We present te latter as te following forwar system: wit te feeback t τ(t) { ẋ = (A + A )x(t) + A u(t) y(t) = ẋ(t), Assume tat A + A is Hurwitz an y(t) = 0 for t 0. ẋ(s)s. (3.4) (3.5) u(t) = y(s)s. (3.6) t τ(t) LEMMA 3. Assume tat te time elay is given by (.), were N. Ten te following ols: Proof. Defining u L π y L. (3.7) k u s (t) = y(s)s, u st (t) = y(s)s, t t k, t k+ ), t k t τ we note tat u(t) = u s (t) + u st (t). We will prove next te following bouns: u s L π y L, (3.8) u st L ( ) π y L, (3.9) Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 wic imply (3.7) since u L u s L + u st L π y L.

5 GENERAL SAWTOOTH DELAY 5 of 8 By using (3.3), we obtain u s L = k=0 k+ t k (x(t k ) x(t)) (x(t k ) x(t))t x(t)) t (x(t k) x(t))t 4 π k=0 4(t k+ t k ) k=0 k+ t k π k+ t k t (x(t k) ẋ T (t)ẋ(t)t = 4 π ẋ L. (3.0) Similarly, we ave u st L = k=0 k+ t k (x(t (t t k )) x(t k )) (x(t (t t k )) x(t k ))t ( ) 4(t k+ t k ) k=0 Using te following cange of variables we arrive to u st L ( ) π k+ t k ẋ T (t (t t k ))ẋ(t (t t k ))t. s = t (t t k ) s = ( )t, =, 3,... t = tk t = t k+, s = t k s = ( )t k+ + t k 4(t k+ t k ) k=0 ( ) 4 π k=0 π k k (k+) ( )t k+ +t k ẋ (s)ẋ(s)s ẋ T (s)ẋ(s)s = ( ) 4 π ẋ L. REMARK 3. For =, te boun of Lemma 3. coincies wit te boun of Mirkin (007), were sample-ata control wit variable sampling t k+ t k was analyse by using te lifting tecnique. We note tat te bouns in (3.0) are vali also for t k+ t k, i.e. for =, we recover result of Mirkin (007). Moreover, Lemma 3. strengtens te result of Mirkin (007), sowing tat te same boun ols for any N if t k+ t k =. It follows tat stability of (3.5) can be verifie by using te small gain teorem (Gu et al., 003). Namely, (3.5) is stable if Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 tere exists non-singular M R n n suc tat MG M <, (3.) were G(s) = ss I (A + A )] π A. (3.)

6 6 of 8 K. LIU ET AL. We can verify te conition (3.) via te following LMI: P(A + A ) + (A + A ) P π P A (A + A ) R R π A R < 0 (3.3) R for P > 0, R > 0. THEOREM 3. For N, (3.5) (3.6) is I O stable (an, tus, (.) (.) is asymptotically stable) if one of te following conitions is satisfie: Conition (3.) ols, were G is given by (3.). Tere exist positive n n-matrices P, R suc tat LMI (3.3) is feasible. 4. Lyapunov Krasovskii approac 4. Lyapunov-base exponential stability DEFINITION 4. Te system (.) is sai to be exponentially stable if tere exists constants μ > 0 an δ > 0 suc tat x(t) μ e δ(t t0) φ W for t t 0. LEMMA 4. (Friman, 00). Let tere exist positive numbers β, δ an a functional V : R W L, 0] R suc tat β φ(0) V (t, φ, φ) δ φ W. (4.) Let te function ˉV (t) = V (t, x t, ẋ t ) is continuous from te rigt for x(t) satisfying (.) absolutely continuous for t t k an satisfies Given α > 0, if along (.) lim t t k ten (.) is exponentially stable wit te ecay rate α. 4. Exponential stability: Case A ˉV (t) ˉV (t k ). (4.) ˉV (t) + α ˉV (t) 0, almost for all t, (4.3) We start wit te case, were τ satisfies τ = > 0. We consier separately 0 < an >. (i) Wen 0 <, we are looking for te functional of te form Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 were V 0 (x t, ẋ t ) = x (t)px(t) + V (t, x t, ẋ t ) = ˉV (t) = V 0 (x t, ẋ t ) + V i (t, x t, ẋ t ) + V 3 (x t ), (4.4) i= 0 t e α(s t) x (s)sx(s)s + t+θ e α(s t) ẋ (s)rẋ(s)s θ, (4.5)

7 GENERAL SAWTOOTH DELAY 7 of 8 V (t, x t, ẋ t ) = τ X+X ξ (t) X + X X X + X+X ] ξ (t), (4.6) V (t, x t, ẋ t ) = τ t τ e α(s t) ẋ (s)u ẋ(s)s, (4.7) V 3 (x t ) = t τ e α(s t) x (s)qx(s)s, (4.8) wit ξ (t) = col{x(t), x(t τ)}, α 0, P > 0, S > 0, R > 0, U > 0, Q > 0. Te above functional coincies wit te one introuce in Friman (00) for S = R = Q = 0, were τ = was consiere. Te positive terms, epening on S, R, Q, guarantee tat te results will be not worse tan for te case of time-varying elays, were te above functional wit U = X = X can be applie. We note tat te term V 3 is non-negative before te jumps at t = t k an it becomes zero just after te jumps (because t t=tk = (t τ) t=tk ). Te time-epenent terms V an V vanis before te jumps (because τ = ) an after te jumps (because τ = 0 an tus x(t τ) = x(t)). Tus, ˉV oes not increase after te jumps an te conition lim t t ˉV (t) ˉV (t k ) ols. k To guarantee tat V > 0 in te sense tat V satisfies (4.), we assume tat ] P + X+X X X X X + > 0. (4.9) X+X Differentiating ˉV, we fin along (.) ˉV (t) + α ˉV (t) x (t)pẋ(t) + ẋ (t) R + τ ] U ẋ(t) + αx (t)px(t)] e α t ẋ (s)rẋ(s)s e α t τ ẋ (s)u ẋ(s)s + x (t)s + Q]x(t) x (t )e α Sx(t ) ( )ẋ (t τ) τ U e α ẋ(t τ) ( )x (t τ)e α Qx(t τ) X+X α( τ) ξ (t) X + X ξ (t) X X + X+X + τ ẋ (t)(x + X )x(t) + ẋ (t)( X + X )x(t τ) +( )x (t)( X + X )ẋ(t τ) +( )ẋ (t τ) ( X X + X + X ) ] x(t τ). Following He et al. (007), we employ te representation t ẋ (s)rẋ(s)s = τ t ẋ (s)rẋ(s)s t τ ẋ (s)rẋ(s)s. (4.0) Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00

8 8 of 8 K. LIU ET AL. We apply te Jensen s inequality (Gu et al., 003) t τ ẋ (s)r + U]ẋ(s)s τ τ t Here, for τ = 0, we unerstan by For τ = 0, te vector ẋ (s)rẋ(s)s τ ẋ (s)sr + U] t τ τ t ẋ (s)s R t τ τ t ẋ(s)s, ẋ(s)s. (4.) ẋ(s)s = lim ẋ(s)s = ẋ(t). τ t τ τ 0 τ t τ t τ τ t ẋ(s)s is efine similarly as ẋ(t ). Ten, enoting v = ẋ(s)s, v = τ ẋ(s)s, τ t τ τ t we obtain ˉV (t) + α ˉV (t) x (t)pẋ(t) + ẋ (t) R + τ ] U ẋ(t) + αx (t)px(t)] e α τ v R + U]v e α τ v Rv + x (t)s + Q]x(t) x (t )e α Sx(t ) ( )ẋ (t τ) τ U e α ẋ(t τ) ( )x (t τ)e α Qx(t τ) α( τ) + τ X+X ξ (t) X + X X X + X+X ] ξ (t) ẋ (t)(x + X )x(t) + ẋ (t)( X + X )x(t τ) +( )x (t)( X + X )ẋ(t τ) +( )ẋ (t τ) ( X X + X + X ) ] x(t τ). (4.) Following He et al. (004), we insert free-weigting n n-matrices by aing te following expressions to ˉV : Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 0 = x (t)y + ẋ (t)y + x (t τ)t ] x(t) + x(t τ) + τv ], 0 = x (t)z + ẋ (t)z ] x(t τ) + x(t ) + ( τ)v ]. (4.3) We use furter te escriptor meto (Friman, 00), were te rigt-an sie (RHS) of te expression 0 = x (t)p + ẋ (t)p 3 ]Ax(t) + A x(t τ) ẋ(t)], (4.4) wit some n n-matrices, P, P 3 is ae into te RHS of (4.).

9 GENERAL SAWTOOTH DELAY 9 of 8 Setting we obtain tat η (t) = col{x(t), ẋ(t), x(t ), v, v, x(t τ), ẋ(t τ)}, ˉV (t) + α ˉV (t) η (t)ψ η (t) < 0, (4.5) if te following inequality Ψ = ols, were Φ α( τ) (X + X ) Φ + τ (X + X ) Z τy ( τ)z Φ 6 Φ 7 Φ + τ U Z τy ( τ)z Φ 6 0 S e α τ R + U] e α 0 τ T 0 τ R e α 0 0 Φ 66 Φ 67 Φ 77 Φ = A P + P A + α P + S + Q Y Y, Φ = P P + A P 3 Y, Φ 6 = Y Z + P A α( τ) T + (X X ), Φ 7 = ( ) τ ( X + X ), Φ = P 3 P3 + R, Φ 6 = Y Z + P 3 A τ (X X ), Φ 66 = ( )Q e α + T + T α( τ)] X + X X X, Φ 67 = ( ) τ (X + X X X ), Φ 77 = ( ) τ U e α. Te latter inequality for τ 0 an τ leas to te following LMIs: Φ α (X + X ) Φ + X+X Z Z Φ + U Z Z Ψ = Φ 6 τ=0 Φ 7 τ=0 Φ 6 τ=0 0 S e α R e α 0 0 Φ 66 τ=0 Φ 67 τ=0 < 0 (4.6) (4.7) < 0, (4.8) Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 Ψ = Φ 77 τ=0 Φ (X + X ) Φ Z Y Φ 6 τ= Φ Z Y Φ 6 τ= S e α 0 0 R + U] e α T Φ 66 τ= < 0. (4.9)

10 0 of 8 K. LIU ET AL. Denoting: η i (t) = col{x(t), ẋ(t), x(t ), v i, x(t τ), ẋ(t τ)}(i =, ), te latter two LMIs imply (4.6) because τ η (t)ψ η (t) + τ η (t)ψ η (t) = η (t)ψ η (t) < 0, an Ψ is tus convex in τ 0, ]. (ii) For >, we consier te following LKF: were V 0 is given by (4.5) an V (t, x t, ẋ t ) = ˉV (t) = V 0 (x t, ẋ t ) + V i (t, x t, ẋ t ), (4.0) V (t, x t, ẋ t ) = τ V (t, x t, ẋ t ) = τ i= X+X ] ξ (t) X + X X X + ξ X+X (t), (4.) t τ e α(s t) ẋ (s)u ẋ(s)s, (4.) wit ξ (t) = col { x(t), x ( t τ )}, α 0, U > 0. To guarantee tat V > 0, we assume (4.9). Since t t=tk = ( t τ ), we see tat ˉV oes not grow after te jumps. t=t k Differentiating ˉV, we ave along (.) ˉV (t) + α ˉV (t) x (t)pẋ(t) + ẋ (t) R + τ ] U ẋ(t) + αx (t)px(t)] e α ẋ (s)rẋ(s)s α t e ẋ (s)u ẋ(s)s t + x (t)sx(t) x (t )e α Sx(t ) α( τ) + τ We employ te representation t τ X+X ξ (t) X + X X X + X+X ẋ (t)(x + X )x(t) + ẋ (t)( X + X )x ] ξ (t) ( t τ )]. Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 τ τ ẋ (s)rẋ(s)s = ẋ t (s)rẋ(s)s ẋ (s)rẋ(s)s ẋ (s)rẋ(s)s. t t t τ t τ Similar to (4.), applying te Jensen s inequality an ten enoting v = τ t τ ẋ(s)s, v = τ ẋ(s)s, v 3 = τ t t τ ẋ(s)s, ( ) t τ

11 GENERAL SAWTOOTH DELAY of 8 we obtain ˉV (t) + α ˉV (t) x (t)pẋ(t) + ẋ (t) R + τ τ R v e α + U e α ] v e ] U ẋ(t) + αx (t)px(t)] α τ v Rv e α v 3 Rv 3 +x (t)sx(t) x (t )e α Sx(t ) ] α( τ) X+X ξ (t) X + X X X + X+X ξ (t) + τ ẋ (t)(x + X )x(t) + ẋ (t)( X + X )x ( t τ )]. Similar to (4.3), we insert free-weigting n n-matrices by aing te following expressions to ˉV : 0 = x (t)y + ẋ (t)y + x ( t τ ) ] ( T x(t) + x t τ ) + τ ] v, 0 = x (t)z + ẋ (t)z + x (t τ)t ] x(t τ) + x(t ) + ( τ)v ], ( 0 = x (t)m + ẋ (t)m ] x t τ ) ( ) + x(t τ) + v 3 ]. Similar to (4.4), te same expression is ae into te RHS of (4.). Setting η (t) = col { x(t), ẋ(t), x(t ), v, v, v 3, x ( t τ ), x(t τ) }, we obtain tat if te inequality Ψ = ˉV (t) + α ˉV (t) η (t)ψ η (t) < 0, (4.3) Ω α( τ) (X + X ) Φ + τ (X + X ) Z τ Y ( τ)z ( ) M Ω 7 Ω 8 Φ + τ U Z τ Y ( τ)z ( ) M Ω 7 Ω 8 S e α T Ω τ T 0 τ R e α 0 0 ( τ)t R e α 0 0 Ω 77 0 T + T ols, were Ω = A P + P A + α P + S Y Y, Ω 7 = Y M T α( τ) + (X X ), Ω 8 = M Z + P A, Ω 7 = Y M τ (X X ), Ω 8 = M Z + P 3 A, Ω 44 = τ ] R e α + U e α, Ω 77 = T + T α( τ)] X + X X X. < 0 (4.4) (4.5) Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00

12 of 8 K. LIU ET AL. Te latter inequality for τ 0 an τ leas to te following LMIs: Ω α (X + X ) Φ + X+X Z Z ( ) M Φ + U Z Z ( ) M Ψ = Ω 7 τ=0 Ω 8 Ω 7 τ=0 Ω 8 S e α T R e α 0 0 T R e α 0 0 Ω 77 τ=0 0 < 0 an Ψ = T + T Ω (X + X ) Φ Z Φ Z Y Y ( ) M ( ) M Ω 7 τ= Ω 8 Ω 7 τ= Ω 8 S e α T Ω 44 τ= 0 T 0 R e α 0 0 Ω 77 τ= 0 T + T Similarly, we can obtain tat Ψ is also convex in τ 0, ]. We summarize te results in te following teorem. (4.6) < 0. (4.7) THEOREM 4. (i) Given α > 0, 0 <, let tere exist n n-matrices P > 0, R > 0, U > 0, S > 0, Q > 0, X, X, T, P, P 3, Y i an Z i (i =, ) suc tat te LMIs (4.9), (4.8), (4.9) wit notations given in (4.7) are feasible. Ten, (.) is exponentially stable wit te ecay rate α for all elays 0 τ satisfying A. (ii) Given α > 0, >, let tere exist n n-matrices P > 0, R > 0, U > 0, S > 0, X, X, P, P 3, T i, Y i, Z i an M i (i =, ) suc tat te LMIs (4.9), (4.6), (4.7) wit notations given in (4.5) are feasible. Ten, (.) is exponentially stable wit te ecay rate α for all elays 0 τ satisfying A. REMARK 4. LMIs of Teorem 4. wit X = X = U = 0 give sufficient conitions for exponential stability of (.) wit τ(t) 0, ]: (i) for all slowly varying elays an (ii) for fast-varying elays. In te numerical examples, tese conitions lea to te same results as te results of Park & Ko (007), owever, tey posses a fewer number of slack matrices. 4.3 Exponential stability: Case A For τ = < 0, we also iffer between 0 < an >. (i) Wen 0 <, we are looking for te functional of te form: Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 V (t, x t, ẋ t ) = ˉV (t) = V 0 (x t, ẋ t ) + V i (t, x t, ẋ t ) + V 3 (x t ), (4.8) were V 0 is given by (4.5) an V (t, x t, ẋ t ) = τ X+X ] ξ 3 (t) X + X X X + ξ X+X 3 (t), i=

13 GENERAL SAWTOOTH DELAY 3 of 8 V (t, x t, ẋ t ) = τ t ( τ) e α(s t) ẋ (s)u ẋ(s)s, V 3 (x t ) = e α(s t) x (s)qx(s)s, t ( τ) wit ξ 3 (t) = col{x(t), x(t ( τ))}, α 0, U > 0, Q > 0 an (4.9) also satisfie to guarantee tat V > 0. We note tat V 3 is non-negative before te jump at t = t k an it becomes zero just after te jump (because t t=tk = (t ( τ)) t=tk ). Te time-epenent terms V an V vanis before te jumps (because τ = 0) an after te jumps (because τ = an tus x(t ( τ)) = x(t)). Hence, te conition lim t t ˉV (t) ˉV (t k ) ols. k (ii) For >, te iscontinuous Lyapunov functional is moifie as follows: were V 0 is also given by (4.5) an V (t, x t, ẋ t ) = ˉV (t) = V 0 (x t, ẋ t ) + V i (t, x t, ẋ t ), (4.9) i= V (t, x t, ẋ t ) = τ X+X ξ 4 (t) X + X X X + X+X V (t, x t, ẋ t ) = τ t τ e α(s t) ẋ (s)u ẋ(s)s, ] ξ 4 (t), wit ξ 4 (t) = col { x(t), x ( t τ )}, α 0, U > 0 an (4.9) also satisfie to guarantee tat V > 0. Also in tis case, ˉV oes not grow after te jumps. Similar to Teorem 4., we obtain te following teorem. THEOREM 4.3 (i) Given α > 0, 0 <, let tere exist n n-matrices P > 0, R > 0, U > 0, S > 0, Q > 0, X, X, P, P 3, P, P 3, Y i j, M i j, Z i j an T i j (i, j =, ) suc tat (4.9) an te following four LMIs: (4.30), for τ 0 an τ, an (4.3), for τ an τ, Θ Θ + τ (X + X ) M Θ + τ U M τy τy ( τ)z ( τ)z τ M τ M Θ 7 Θ 8 + Z M Θ 7 Θ 8 + Z M S e α τ R + U]e α 0 0 τ T 0 0 Θ ( τ)t 0 τ R e α T + T Θ 9 0 T 0 Θ 88 + T + T Θ 89 < 0, Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 Θ 99 (4.30)

14 4 of 8 K. LIU ET AL. Ξ were Ξ + τ ) Y ( τ)y (τ )Z ( τ)m Ξ 7 Θ 8 + M Z Θ 9 Ξ + τ U Y ( τ)y (τ )Z ( τ)m Ξ 7 Θ 8 + M Z 0 S e α T 0 0 τ R e α 0 0 ( τ)t 0 0 τ R e α 0 0 (τ )T 0 Ξ T T T 0 Θ 88 T T Θ 99 Θ = A P + P A + α P + S + Q Y Y ατ (X + X ), Θ = P P + A P 3 Y, Θ 7 = Y Z T + P A, Θ 8 = ατ (X X ), Θ 9 = ( ) τ ( X + X ), Θ = P 3 P 3 + R, Θ 7 = Y Z + P 3 A, Θ 8 = τ (X X ), Θ 55 = τ R + U]e α, Θ 88 = ( )Q e α ( ατ) X + X X X, Θ 89 = ( ) τ (X + X X X ), Θ 99 = ( ) τ U e α, Ξ = A P + P A + α P + S + Q M M ατ (X + X ), Ξ = P P + A P 3 M, Ξ 7 = Z Y + P A, Ξ = P 3 P 3 + R, Ξ 7 = Z Y + P 3 A, Ξ 66 = τ R + U]e α, Θ 89 < 0, (4.3) are feasible. Ten (.) is exponentially stable wit te ecay rate α for all elays 0 τ satisfying A. (ii) Given α > 0, >, let tere exist n n-matrices P > 0, R > 0, U > 0, S > 0, X, X, P, P 3, P, P 3, Y i j, M i j, Z i j an T i j (i, j =, ) suc tat (4.9) an te following four LMIs: (4.3), for τ 0 an τ +, an (4.33), for τ + an τ, Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00

15 GENERAL SAWTOOTH DELAY 5 of 8 Σ Θ + τ ) M τy (+)τ Z ( τ )M Θ 7 Θ 8 + Z M Θ + τ U M τy (+)τ Z ( τ )M Θ 7 Θ 8 + Z M S e α Σ τ T 0 Σ (+)τ T Σ T + T T < 0, (4.3) were Σ 88 +T +T Υ Ξ + τ ) Y ( τ)y (+)τ Z τ M Ξ 7 Θ 8 +M Z Ξ + τ U Y ( τ)y (+)τ Z τ M Ξ 7 Θ 8 +M Z S e α T 0 τ R e α 0 0 ( τ)t 0 (+)τ R e α (+)τ 0 0 Υ T T Σ 88 T T Σ = A P + P A + α P + S Y Y ατ (X + X ), Σ 44 = τ R e α + Ue α ], Σ 55 = ( + )τ R e α + Ue α ], Σ 66 = ( τ ) R e α, Σ 88 = ( ατ) X + X X X, Υ = A P + P A + α P + S M M ατ (X + X ), Υ 66 = τ R e α + U e α ], T T < 0, (4.33) are feasible. Ten (.) is exponentially stable wit te ecay rate α for all elays 0 τ satisfying A. 5. Examples EXAMPLE 5. Consier te system from Yue et al. (005) ] ] 0 0 ẋ(t) = x(t) + K x(t τ(t)), K = ]. Te stability of tis system was stuie by many autors (see Nagstabrizi et al., 008 an te references terein). For te constant sampling, it was foun in Nagstabrizi et al. (008) tat te system Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00

16 6 of 8 K. LIU ET AL. remains stable for all constant samplings <.7 an becomes unstable for samplings >.7. Moreover, te above system wit constant elay τ is asymptotically stable for τ.6 an becomes unstable for τ >.7. Te latter means tat all te existing metos, tat are base on time-inepenent Lyapunov functionals, corresponing to stability analysis of systems wit fast-varying elays, cannot guarantee te stability for te samplings wit te upper boun >.7. In Nagstabrizi et al. (008), te upper boun on te constant sampling interval tat preserves te stability is foun to be =.377, improving all te existing LMI-base results. For ifferent τ, by applying Lyapunov Krasovskii(L K) approac wit α = 0 an I O approac via Wirtinger-type inequality, we obtain te maximum value of given in Table. Our results for X = X =U = 0 coincie wit te ones of Park & Ko (007). We see tat iscontinuous terms of LKFs improve te performance. Moreover, te I O approac via Wirtinger-type inequality improves te result for >, N compare to time-epenent L K approac. EXAMPLE 5. We consier te following simple an muc-stuie problem (Papacristooulou et al., 007; Friman, 00): ẋ(t) = x(t τ(t)). (5.) It is well known tat te equation ẋ(t) = x(t τ) wit constant elay τ is asymptotically stable for τ < π/ an unstable for τ > π/, wereas for te fast-varying elay, it is stable for τ <.5 an tere exists a estabilizing elay wit an upper boun >.5. Te latter means tat all te existing metos, tat are base on time-inepenent Lyapunov functionals, corresponing to stability analysis of systems wit fast-varying elays, cannot guarantee te stability for te samplings, wic may be >.5. It is easy to ceck, tat te system remains stable for all constant samplings < an becomes unstable for samplings >. Conitions of Nagstabrizi et al. (008) an of Mirkin (007) guarantee asymptotic stability for all variable samplings up to.8 an.57, respectively. For ifferent τ, by applying our metos, we obtain te maximum value of given in Table.We can also see tat iscontinuous terms of LKFs improve te performance an te I O approac via Wirtinger-type inequality improves te result for >, N compare to time-epenent L K approac. EXAMPLE 5.3 Consier te system (De Souza & Li, 999) ] ] 0 0 ẋ(t) = x(t) + x(t τ(t)) TABLE Example : maximum value of for ifferent τ \ τ \ τ τ N L K approac I O approac.3659 X = X = U = Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 TABLE Example : maximum value of for ifferent τ \ τ \ τ τ N L K approac I O approac.57 X = X = U =

17 TABLE 3 Example 3: maximum value of for ifferent τ GENERAL SAWTOOTH DELAY 7 of 8 \ τ \ τ τ N L K approac I O approac.57 X = X = U = TABLE 4 Example 4: maximum value of for ifferent τ \ τ \ τ τ N Our meto I O approac X = X = U = It is well known tat tis system is stable for constant elay τ 6.7, wereas in Park & Ko (007), it was foun tat te system is stable for all fast-varying elays τ.86. For ifferent τ, by applying our metos, we obtain te maximum value of given in Table 3. Our results for X = X = U = 0 coincie wit te ones of Park & Ko (007). Also in tis Example, te time-epenent terms of LKFs improve te performance. However, te I O approac via Wirtinger-type inequality as not improve te results for >, N an te result is worse tan te result for te fast-varying elays. EXAMPLE 5.4 Consier te system (Karitonov & Niculescu, 00): ] ] ẋ(t) = x(t) + x(t τ(t)). For ifferent τ, by applying our metos, we obtain te maximum value of given in Table 4. REMARK 5. Simulations in all te examples above sow tat our results are conservative (at least for ) an tat te value of maximum tat preserves te stability grows for growing >. 6. Conclusions In tis paper, two metos ave been introuce to investigate elay-epenent stability problem for systems wit sawtoot elay wit constant τ 0. One meto evelops a novel I O approac via a Wirtinger-type inequality. Te result by Mirkin (007) is recovere for τ = an for any integer τ, te same maximum boun tat preserves te stability is acieve. Anoter meto improves stability criteria by constructing piecewise continuous (in time) Lyapunov functionals. Toug te most essential improvement correspons to τ =, te time-epenent terms improve te results for all values of τ. Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00 Funing Israel Science Founation (754/0); Cina Scolarsip Council. REFERENCES DE SOUZA, C. E. & LI, X. (999) Delay-epenent robust H control of uncertain linear state-elaye systems. Automatica, 35, FRIDMAN, E. (00) New Lyapunov-Krasovskii functionals for stability of linear retare an neutral type systems. Syst. Control Lett., 43,

18 8 of 8 K. LIU ET AL. FRIDMAN, E. (00) A refine input elay approac to sample-ata control. Automatica, 46, FRIDMAN, E. & SHAKED, U. (003) Delay-epenent stability an H control: constant an time-varying elays. Int. J. Control, 76, GU, K., KHARITONOV, V. & CHEN, J. (003) Stability of Time-Delay Systems. Boston, MA: Birkauser. HARDY, G., LITTLEWOOD, J. E. & POLYA, G. (934) Inequalities. Cambrige, UK: Cambrige University Press. HE, Y., WANG, Q. G., LIN, C. & WU, M. (007) Delay-range-epenent stability for systems wit time-varying elay. Automatica, 43, HE, Y., WU, M., SHE, J. & LIU, G. (004) Delay-epenent robust stability criteria for uncertain neutral systems wit mixe elays. Syst. Control Lett., 5, KHARITONOV, V. & NICULESCU, S. (00) On te stability of linear systems wit uncertain elay. IEEE Trans. Autom. Control, 48, 7-3. KOLMANOVSKII, V. & MYSHKIS, A. (999) Introuction to te Teory an Applications of Functional Dierential Equations. Dorrect, Neterlans: Kluwer Acaemic. LIU, K. & FRIDMAN, E. (009) Discontinuous Lyapunov functionals for linear systems wit sawtoot elays. Proceeings of te 8t IFAC Worksop on Time-Delay Systems, Sinaya, Romania. LOUISELL, J. (999) New examples of quencing in elay ifferential equations aving time-varying elay. Proceeings of te 5t European Control Conference. Karlsrue, Germany. MICHIELS, W., VAN ASSCHE, V. & NICULESCU, S. (005) Stabilization of time-elay systems wit a controlle time-varying elays an applications. IEEE Trans. Automat. Control, 50, MIRKIN, L. (007) Some remarks on te use of time-varying elay to moel sample-an-ol circuits. IEEE Trans. Automat. Control, 5, 09. NAGHSHTABRIZI, P., HESPANHA, J. & TEEL, A. (008) Exponential stability of impulsive systems wit application to uncertain sample-ata systems. Syst. Control Lett., 57, NICULESCU, S. (00) Delay Effects on Stability: A Robust Control Approac. Lecture Notes in Control an Information Sciences, vol. 69. Lonon: Springer. PAPACHRISTODOULOU, A., PEET, M. & NICULESCU, S. (007) Stability analysis of linear systems wit timevarying elays: elay uncertainty an quencing. Proceeings of te 46t IEEE CDC, New Orleans, USA. PARK, P. G. & KO, J. W. (007) Stability an robust stability for systems wit a time-varying elay. Automatica, 43, RICHARD, J. P. (003) Time-elay systems: an overview of some recent avances an open problems. Automatica, 39, YUE, D., HAN, Q. L. & LAM, J. (005) Networke-base robust H control of systems wit uncertainty. Automatica, 4, Downloae from imamci.oxforjournals.org at TEL AVIV UNIVERSITY on November 4, 00