The presented method gives analytical formulas for the coefficients of the Lyapunov functional
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(2) J. Duda. 328 ceeded with delay-depended stability analysis for linear neutral systems, constructed the Lyapunov functional and derived sufficient delay-dependent conditions in terms of linear matrix inequalities. Han (2004a) obtained a delay-dependent stability criterion for neutral systems with a time-varying discrete delay. This criterion was expressed in the form of an LMI and was obtained using the Lyapunov direct method. Han (2004b) investigated robust stability of uncertain neutral systems with discrete and distributed delays, which was based on descriptor model transformation and the decomposition technique, and formulated stability criteria in the form of LMIs. Han (2005a) considered the stability for linear neutral systems with norm-bounded uncertainties in all system matrices and derived a new delay-dependent stability criterion. Neither model transformation, nor the bounding technique for cross terms is involved in derivation of the stability criterion. Han (2005b) developed the discretized Lyapunov functional approach to investigation of the stability of linear neutral systems with mixed neutral and discrete delays. Stability criteria, which are applicable to linear neutral systems with both small and nonsmall discrete delays, are formulated in the form of LMIs. Han (2009a) studied the problem of stability of linear time delay systems, both retarded and neutral types, using the discrete delay N -decomposition approach to derive some more general new discrete delay dependent stability criteria. Han (2009b) employed the delay-decomposition approach to derive some improved stability criteria for linear neutral systems and to deduce some sufficient conditions for the existence of the Lyapunov functional for a system with k-non-commensurate neutral time delays of a delayed state feedback controller, which ensure asymptotic stability and a prescribed H1 performance level of the corresponding closed-loop system. Gu and Liu (2009) investigated the stability of coupled differentialfunctional equations using the discretized Lyapunov functional method and set forth the stability condition in the form of LMIs, suitable for numerical computation. This paper presents a method of determining the Lyapunov functional for a linear time-invariant system with an interval time-varying delay. The functional is constructed for a system with a time-varying delay with a given time derivative which is calculated on the system trajectory. We assume that a time derivative of the Lyapunov functional is a quadratic form. This assumption enables calculation of the value of the integral quadratic performance index of parametric optimization of a system with a time delay. The presented method gives analytical formulas for the coefficients of the Lyapunov functional. The novelty of the result lies in the extension of the Repin method to the system with an interval time-varying delay. To the best of the author’s knowledge, such extension has not been reported in the literature. An example illustrating that method is also presented.. 2. Problem formulation Let us consider a linear system with a time-varying delay, whose dynamics are described with the equation dx(t) = Ax(t) + Bxt (−τ (t)), dt x(t0 ) = x0 ∈ Rn ,. (1). xt0 = Φ ∈ W 1,2 ([−r, 0), Rn ), where we have A, B ∈ Rn×n ; x(t) ∈ Rn ; xt ∈ W 1,2 ([−r, 0), Rn ), xt (θ) = x(t + θ); t ≥ t0 ; θ ∈ [−r, 0); τ (t) is a time-varying delay satisfying the condition 0 ≤ τ (t) ≤ r; dτ (t)/dt = 1; t ≥ t0 , where r is a positive constant. W 1,2 ([−r, 0), Rn ) is the space of all absolutely continuous functions [−r, 0) → Rn with derivatives in L2 ([−r, 0), Rn ), the space of Lebesgue square integrable functions on the interval [−r, 0) with values in Rn . The state of the system (1) is the vector x(t) for t ≥ t0 . S(t) = xt. (2). The state space is defined by the formula X = Rn × W 1,2 ([−r, 0), Rn ).. (3). We define a Lyapunov functional, positive definite and differentiable, whose derivative computed on the trajectory of the system (1) is negative definite, V (x(t), xt , t) = xT (t)α(t)x(t) ˆ0 +. xT (t)β (θ + τ (t)) xt (θ) dθ. −τ (t). ˆ0 ˆ0 +. xTt (θ)δ θ + τ (t), σ + τ (t). −τ (t) θ. · xt (σ) dσ dθ,. (4). for t ≥ t0 , where α ∈ C 1 ([t0 , ∞), Rn×n ) , α(t) is positive definite, β ∈ C 1 ([0, τ (t)], Rn×n ), δ ∈ C 1 (Ω, Rn×n ), Ω = {(θ, σ) : θ ∈ [0, τ (t)], σ ∈ [θ, 0]}; 0 ≤ τ (t) ≤ r. C 1 is the space of continuously differentiable functions. The Lyapunov functional for a time-delay system with interval time-varying delay defined by (4) is more general than the ones found in the existing literature. To the best of the author’s knowledge, such a functional has not been reported in the literature yet..
(3) A Lyapunov functional for a system with a time-varying delay. 3. Determination of the coefficients of the Lyapunov functional. 329 Lyapunov functional, we can easily determine the value of the square performance index since. We compute the derivative of the functional (4) on the trajectory of the system (1),. ˆ∞ J=. xT (t)W x(t)dt = V (x0 , Φ, t0 ).. (7). t0. dV (x(t), xt , t) dt dα(t) T = x (t) AT α(t) + α(t)A + dt T + β(τ (t)) x(t) + xt (−τ (t)) B T α(t) + αT (t) + β T (0) ˆ0 + −τ (t). dτ (t) −1 dt. x(t). dβ (θ + τ (t)) xT (t) AT β (θ + τ (t)) + dt. + δ (0, θ + τ (t)) ˆ0 ˆ0 +. xTt (θ). −τ (t) θ. dτ (t) −1 dt. dβ (θ + τ (t)) dt dβ (θ + τ (t)) − + δ T (θ + τ (t), τ (t)) = 0, dθ. B T β (θ + τ (t)). (8) (9). dτ (t) −1 dt. (10). = 0, (11). xt (θ) dθ dδ (θ + τ (t), σ + τ (t)) ∂δ (θ + τ (t), σ + τ (t)) − dt ∂θ ∂δ (θ + τ (t), σ + τ (t)) − = 0, (12) ∂σ. ∂δ (θ + τ (t), σ + τ (t)) ∂θ. ∂δ (θ + τ (t), σ + τ (t)) xt (σ) dσd θ − ∂σ. for t ≥ t0 , θ ∈ [−τ (t), 0], σ ∈ [θ, 0], where 0 ≤ τ (t) ≤ r.. −. We introduce new variables, (5). for t ≥ t0 , where α ∈ C 1 [t0 , ∞), Rn×n ;. (13) (14). dδ (θ + τ (t), σ + τ (t)) dδ(ξ, η) = dt dt ∂δ(ξ, η) d τ (t) ∂δ(ξ, η) dτ (t) = + , ∂ξ dt ∂η dt. Ω = {(θ, σ) : θ ∈ [0, τ (t)], σ ∈ [θ, 0]} , 0 ≤ τ (t) ≤ r. We identify the coefficients of the Lyapunov functional (4) assuming that the derivative (5) satisfies the relationship for t ≥ t0 ,. ξ = θ + τ (t), η = σ + τ (t), and calculate the derivatives,. δ ∈ C 1 (Ω, Rn×n ),. dV (x(t), xt , t) = −xT (t)W x(t) dt. + δ (0, θ + τ (t)). dδ (θ + τ (t), σ + τ (t)) dt. β ∈ C 1 ([0, τ (t)], Rn×n ),. dα(t) + β(τ (t)) = −W, dt. dτ (t) T T T B α(t) + α (t) + β (0) − 1 = 0, dt AT α(t) + α(t)A +. AT β (θ + τ (t)) +. dβ (θ + τ (t)) + δ T (θ + τ (t), τ (t)) xt (θ) dθ − dθ ˆ0 T xt (−τ (t)) B T β (θ + τ (t)) + −τ (t). From Eqns. (5) and (6) we obtain the set of equations. (6). where W ∈ Rn×n is a positive definite matrix. When the relationship (6) holds and we know the. (15). ∂δ (θ + τ (t), σ + τ (t)) ∂δ(ξ, η) ∂δ(ξ, η) = = , (16) ∂θ ∂θ ∂ξ ∂δ (θ + τ (t), σ + τ (t)) ∂δ(ξ, η) ∂δ(ξ, η) = = , (17) ∂σ ∂σ ∂η dβ (θ + τ (t)) dβ(ξ) ∂ξ dβ(ξ) dτ (t) = = , dt dξ ∂t dξ dt dβ (θ + τ (t)) dβ(ξ) ∂ξ dβ(ξ) = = . dθ dξ ∂θ dξ. (18) (19).
(4) J. Duda. 330 The formula (12) takes the form ∂δ(ξ, η) ∂δ(ξ, η) + =0 ∂ξ ∂η. with respect to ξ using the relationship (26). We get dβ (−ξ + τ (t)) dξ. −2. dτ (t) −1 β T (ξ)B =− dt. −1 dτ (t) + −1 AT β (−ξ + τ (t)) dt. (20). for t ≥ t0 , θ ∈ [−τ (t), 0], σ ∈ [θ, 0], ξ ∈ [0, τ (t)] , η ∈ [ξ, τ (t)] , where 0 ≤ τ (t) ≤ r. The solution of Eqn. (12) is given by δ (θ + τ (t), σ + τ (t)) = δ (ξ, η) = f (ξ − η) = f (θ − σ). (21). for t ≥ t0 , θ ∈ [−τ (t), 0], σ ∈ [θ, 0], 0 ≤ τ (t) ≤ r, where f ∈ C 1 ([−r, r], Rn×n ) . Taking (21) into account, from Eqn. (11) we get the relationship δ (0, θ + τ (t)) = f (−τ (t) − θ). −1 d τ (t) = 1− B T β (θ + τ (t)) . dt (22) Hence. −1 dτ (t) f (ξ) = 1 − B T β (−ξ) dt. (23). for ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r. From (21) we obtain δ T (θ + τ (t), τ (t)) = f T (θ). −1 dτ (t) = 1− β T (−θ) B. dt (24) We substitute the term (24) into (10) and obtain dβ (θ + τ (t)) dt. −1 dτ (t) dβ (θ + τ (t)) + 1− − β T (−θ) B = 0. dθ dt (25). for ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r. In such a way, we get the set of differential equations, ⎧. −1 dβ (ξ) dτ (t) ⎪ ⎪ =− −1 AT β (ξ) ⎪ ⎪ ⎪ dξ dt ⎪ ⎪. −2. ⎪ ⎪ dτ (t) ⎪ ⎪ − 1 + β T (−ξ + τ (t)) B, ⎪ ⎨ dt. −2 ⎪ dβ (−ξ + τ (t)) ⎪ dτ (t) ⎪ ⎪ = − − 1 β T (ξ)B ⎪ ⎪ dξ dt ⎪ ⎪. −1. ⎪ ⎪ ⎪ dτ (t) ⎪ ⎩ −1 + AT β (−ξ + τ (t)) dt (28) for each fixed t ≥ t0 , ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r. with the initial conditions β (0)and β (τ (t)). There is a relationship between β(ξ) and β(−ξ + τ (t)) such that (29) β (ξ) τ (t) = β (−ξ + τ (t)) τ (t) . ξ=. ξ=. 2. −1 dτ (t) −1 AT β (ξ) dt. −2 dτ (t) + −1 β T (−ξ + τ (t)) B (26) dt. dβ (ξ) =− dξ. for ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r. We calculate the derivative of the term β(−ξ + τ (t)). 2. We calculate the derivative of Eqn. (9) with respect to t, BT. dα(t) dαT (t) + dt dt. AT β (θ + τ (t)) +. Taking (13), (18) and (19) into account from Eqn. (25) we get the relationship. (27). dβ T (0) dτ (t) −1 dt dt 2 d τ (t) + β T (0) = 0. (30) d t2. +. From Eqn. (26) it follows that d β T (0) d τ (t) =− dt dt +. d τ (t) dt. d τ (t) −1 dt. −1. d τ (t) −1 dt. −2. β T (0)A B T β(τ (t)). (31). From Eqn. (8) we obtain d α(t) = −AT α(t) − α(t)A − β(τ (t)) − W. dt. (32). We substitute the terms (31) and (32) into Eqn. (30),.
(5) A Lyapunov functional for a system with a time-varying delay. t ≥ t0 ; xt , Φ ∈ W 1,2 ([−r, 0), R); x (t) ∈ R; a, b ∈ R; τ (t) is a time-varying delay satisfying the condition 0 ≤ τ (t) ≤ r; dτ (t)/dt = 1; t ≥ t0 , where r is positive constant.. and, after calculations, we get B T AT α(t) + αT (t) + α(t) + αT (t) A. d τ (t) d 2 τ (t) A− I + β (0) dt d t2. −1 d τ (t) − −1 B T β(τ (t)) + B T β T (τ (t)) dt = −B T W + W T . T. V (x (t) , xt , t) ˆ0 2. = α(t)x (t) +. β (ξ) . ξ= τ (t) 2. 2. Having the solution of the set of differential equations (28) and taking (13), (21) and (23) into account, we obtain the matrices β (θ + τ (t)) = β (ξ) , (37) ξ=θ+τ (t). δ (θ + τ (t), σ + τ (t)). −1 dτ (t) = 1− B T β (σ − θ) dt. +. δ (θ + τ (t), σ + τ (t)) xt (θ) xt (σ) dσ dθ. −τ (t) θ. (40). (34). ξ=. ˆ0 ˆ0. We obtain the coefficients of the functional as below.. dτ (t) − 1 = 0, (35) dt = β (−ξ + τ (t)) τ (t) . (36). B α(t) + αT (t) + β T (0) T. β (θ + τ (t)) x (t) xt (θ) dθ −τ (t). d2 τ (t) dτ (t) A− I dt dt2. −1. dτ (t) −1 − B T β(τ (t)) + B T β T (τ (t)) dt = −B T W + W T , + β T (0). The Lyapunov functional is given by. (33). The matrix α(t) and the initial conditions of the system (28) are obtained by solving Eqns. (33), (9) and (29): B T AT α(t) + αT (t) + α(t) + αT (t) A. 331. Equation (28) takes the form ⎡ ⎤ dβ(ξ) ⎢ ⎥ dθ ⎣ ⎦ dβ (−ξ + τ (t)) dθ ⎡. a. . ⎢ 1 − dτdt(t) ⎢ ⎢ =⎢ b ⎢ − 2 ⎣ dτ (t) 1 − dt ×. ⎤. b 1−. −. 2 ⎥ ⎥ ⎥ ⎥ a ⎥ ⎦ dτ (t). dτ (t) dt. 1−. dt. β(ξ). (41). β (−ξ + τ (t)). for t ≥ t0 , ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r. (38). for t ≥ t0 , θ ∈ [−τ (t), 0], σ ∈ [θ, 0] where 0 ≤ τ (t) ≤ r. In this way, we have obtained all the parameters of the Lyapunov functional (4).. 4. Example. The fundamental matrix of the differential equation (41) is given by Q. ⎡. ⎢ ⎢ =⎢ ⎢ ⎣. chλξ +. a. . λ 1−. − λ 1−. dτ (t) dt. b dτ (t) dt. shλξ. 2 shλξ (42). Let us consider the system described by dx (t) = ax (t) + bxt (−τ (t)) , dt x (t0 ) = x0 ∈ R, xt0 = Φ,. (39). λ 1− chλξ −. b dτ (t) dt. ⎤ 2 shλξ. a λ 1 − dτdt(t). ⎥ ⎥ ⎥, ⎥ shλξ ⎦.
(6) J. Duda. 332 . where. a2 λ=. Hence. 2 dτ (t) − b2 1− dt .. 2 dτ (t) 1− dt. 0.5. α(t). 0.45. (43). 0.4 0.35 0.3 0.25. . β(ξ) β(−ξ + τ (t)). . = Q(ξ). β(0) β(τ (t)). . 0.2. (44). 0.15 0.1. for t ≥ t0 , ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r. We need the initial conditions of the set of differential equations (41) to obtain , (45) β(θ + τ (t)) = β (ξ) . 0.05 0. 0. 2. 4 6 time in sec. 8. 10. Fig. 1. Parameter α(t). 1. β(ξ). ξ=θ+τ (t). −1 dτ (t) δ (θ + τ (t), σ + τ (t)) = 1 − bβ (σ − θ) . dt (46) for t ≥ t0 , θ ∈ [−τ (t), 0], σ ∈ [θ, 0],. 0.8. 0.6. 0.4. 0.2. 0. where 0 ≤ τ (t) ≤ r.. −0.2. The initial conditions of the differential equation (41) and the coefficient α(t) are obtained by solving the set of equations (34)–(36), which take the following form:. dτ (t) d2 τ (t) − β(0) 4abα(t) + a dt dt2 1 bβ(τ (t)) = −2bw, (47) + 1 − dτ (t) dt − 1. dτ (t) − 1 β(0) = 0, (48) 2bα(t) + dt (49). p1 β(0) + p2 β(τ (t)) = 0, where a λτ (t) + 2 λ 1 − dτdt(t) λτ (t) b , + 2 sh 2 λ 1 − dτdt(t). 0. 0.1. 0.2 0.3 0 ≤ ξ ≤ τ(t). 0.4. 0.5. Fig. 2. Parameter β(ξ).. and β(ξ), obtained with a Matlab code, for the given values of parameters of the system (39). a = −1, b = −0.5, w = 1,. t τ (t) = r 1 − exp − , r = 0.5, T. T = 1.. The parameter α(t) is positive for r ≤ 1.25, so we have obtained a region of stability for the system (39). Let us consider the system described by equation. p1 = ch. λτ (t) p2 = −ch ⎛ 2. −0.4. (50). ⎞. b a ⎟ λτ (t) ⎜ + . +⎝ 2 ⎠ sh dτ (t) 2 λ 1 − dt λ 1 − dτdt(t) (51) Figures 1 and 2 show graphs of the functions α(t). ⎧⎡ ⎤ dx1 (t) . ⎪ ⎪ ⎪ x a a (t) ⎢ ⎥ 11 12 1 ⎪ dt ⎪ ⎪ ⎣ ⎦= ⎪ dx2 (t) ⎪ x2 (t) a21 a22 ⎪ ⎪ ⎪ ⎪ dt. . ⎪ ⎪ ⎪ ⎪ xt1 (−τ (t)) b11 b12 ⎪ ⎨ + , xt2 (−τ (t)) b21 b22. ⎪ ⎪ ⎪ ⎪ x10 x1 (t0 ) ⎪ ⎪ , = ⎪ ⎪ x20 x2 (t0 ) ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ x (θ) (θ) Φ t 1 ⎪ 1 ⎪ = . ⎩ xt2 (θ) Φ2 (θ) (52).
(7) A Lyapunov functional for a system with a time-varying delay The Lyapunov functional is defined by. We introduce ⎡ p11 ⎢ p21 ⎢ ⎢ p31 ⎢ ⎢ p41 Qτ (t)/2 =⎢ e ⎢ p51 ⎢ ⎢ p61 ⎢ ⎣ p71 p81. V (x(t), xt , t) # α11 (t) α12 (t) " x1 (t) = x1 (t) x2 (t) α21 (t) α22 (t) x2 (t) ˆ0. ". +. #. x1 (t) x2 (t). −τ (t). ×. β11 (θ + τ (t)). β12 (θ + τ (t)). β21 (θ + τ (t)). β22 (θ + τ (t)). ˆ0 ˆ0 +. ". xt1 (θ). xt2 (θ). . xt1 (θ) xt2 (θ). dθ. #. ⎡ ⎢ ⎢ =⎢ ⎢ ⎣. δ22 (θ + τ (t), σ + τ (t)) . xt1 (σ) xt2 (σ). ×. (53). dσdθ. for t ≥ t0 ,. θ ∈ [−τ (t), 0],. p14 p24 p34 p44 p54 p64 p74 p84. p15 p25 p35 p45 p55 p65 p75 p85. p16 p26 p36 p46 p56 p66 p76 p86. p17 p27 p37 p47 p57 p67 p77 p87. p18 p28 p38 p48 p58 p68 p78 p88. ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎦. Now we give the formulas for determination of the set of the initial conditions of the equation (54) and the matrix α: ⎤ ⎡ col α(t) Z ⎣ col β(0) ⎦ col β(τ (t)). δ21 (θ + τ (t), σ + τ (t)) δ12 (θ + τ (t), σ + τ (t)). p13 p23 p33 p43 p53 p63 p73 p83. (59). δ11 (θ + τ (t), σ + τ (t)). ×. p12 p22 p32 p42 p52 p62 p72 p82. −τ (t) θ. . 333. σ ∈ [θ, 0],. where 0 ≤ τ (t) ≤ r.. where. −2b11 w11 − b21 w12 − b21 w21 −b11 w12 − b11 w21 − 2b21 w22 −2b12 w11 − b22 w12 − b22 w21 −b12 w12 − b12 w21 − 2b22 w22 0(8,1). ⎤ ⎥ ⎥ ⎥ , (60) ⎥ ⎦. ⎡. ⎤ D Z12 Z13 Z = ⎣ Z21 Z22 0(4,4) ⎦ , 0(4,4) Z32 Z33 ⎤ ⎡ d11 d12 d13 d14 ⎢ d21 d22 d23 d24 ⎥ ⎥ D=⎢ ⎣ d31 d31 d33 d34 ⎦ , d41 d42 d43 d44. (61). (62). The set of equations (28) becomes d dξ. . col β(ξ) col β(−ξ + τ (t)). . =Q. . col β(ξ) col β(−ξ + τ (t)). (54). for ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r, Q = [−Q1 Q2 ] ,. (55). and Q1 and Q2 are given by (56) and (57), respectively, while col β(0) col β(ξ) Qξ (58) =e col β(τ (t)) col β(−ξ + τ (t)) for ξ ∈ [0, τ (t)] , where 0 ≤ τ (t) ≤ r.. d11 = 4a11 b11 + 2a12 b21 ,. (63). d12 = d13 = 2a21 b11 + a22 b21 + a11 b21 ,. (64). d14 = 2a21 b21 ,. (65). d21 = 2a12 b11 ,. (66). d22 = d23 = a11 b11 + a22 b11 + 2a12 b21 ,. (67). d24 = 2a21 b11 + 4a22 b21 ,. (68). d31 = 4a11 b12 + 2a12 b22 ,. (69). d32 = d33 = 2a21 b12 + a22 b22 + a11 b22 ,. (70). d34 = 2a21 b22 ,. (71). d41 = 2a12 b12 ,. (72). d42 = d43 = a11 b12 + a22 b12 + 2a12 b22 ,. (73). d44 = 2a21 b12 + 4a22 b22 , " 1 2# Z12 = Z12 Z12 ,. (74) (75).
(8) J. Duda. 334 ⎡. a11 dτ (t) −1 dt a12 dτ (t) −1 dt. ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ b11 ⎢ Q1 = ⎢ . 2 ⎢ dτ (t) ⎢ −1 ⎢ dt ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ b12 ⎢ ⎢ . 2 ⎢ dτ (t) ⎢ − 1 ⎢ dt ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎣ ⎡. b11. 2 ⎢ dτ (t) ⎢ −1 ⎢ dt ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ b12 ⎢ ⎢ . 2 ⎢ dτ (t) ⎢ − 1 ⎢ dt ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ a11 Q2 = ⎢ ⎢ dτ (t) ⎢ −1 ⎢ dt ⎢ ⎢ ⎢ ⎢ ⎢ a12 ⎢ ⎢ dτ (t) ⎢ −1 ⎢ dt ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ ⎢ 0 ⎣. a21 dτ (t) −1 dt a22 dτ (t) −1 dt. 0 0. b21 dτ (t) −1 dt. 2. 0. b22 dτ (t) −1 dt. 2. b21 dτ (t) −1 dt. b22 dτ (t) −1 dt. 2. 2. a21 dτ (t) −1 dt a22 dτ (t) −1 dt. 0. 0. a11 dτ (t) −1 dta 12. a21 dτ (t) −1 dta. dτ (t) −1 dt 0. dτ (t) −1 dt 0. b11 dτ (t) −1 dt. 2. 22. b12 dτ (t) −1 dt. b21 dτ (t) −1 dt. 2. 0. 2. 0. 0. 0. 0. b22 dτ (t) −1 dt. 2. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. (56). ⎤. 0. 0. 0. 0. ⎤. 0. b11 dτ (t) −1 dt. 0. 2. 0. b12 dτ (t) −1 dt 0. b21 dτ (t) −1 dt. 2. 0. 2. b22 dτ (t) −1 dt 0. 2. 0. 0. a11 dτ (t) −1 dta 12. a21 dτ (t) −1 dta. dτ (t) −1 dt. dτ (t) −1 dt. 22. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. (57).
(9) A Lyapunov functional for a system with a time-varying delay. ⎡. 1 Z12. ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣. dτ (t) d2 τ (t) a11 − dt dt2 dτ (t) a12 dt 0 0. ⎤. dτ (t) a21 dt dτ (t) d2 τ (t) a22 − dt dt2 0 0. ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦. 335. Z22. ⎡. ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎢ ⎣. (76) ⎡ 2 Z12. 0 0. ⎢ ⎢ 2 ⎢ = ⎢ dτ (t) a11 − d τ (t) ⎢ dt dt2 ⎣ dτ (t) a12 dt. Z13. ⎡. 1 Z13. 0 0 dτ (t) a21 dt dτ (t) d2 τ (t) a22 − dt dt2. ⎤ b21 dτ (t) −1 dt b11. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ b22 ⎥ − ⎥ dτ (t) ⎥ −1 ⎥ ⎦ dt −. b12 (79). ⎡. 2 Z13. b21. ⎢ ⎢ b11 ⎢ − ⎢ dτ (t) ⎢ −1 ⎢ ⎢ dt ⎢ =⎢ b22 ⎢ ⎢ ⎢ ⎢ ⎢ b12 ⎢ ⎣ − dτ (t) −1 dt ⎡. Z21. 2b11 ⎢ 0 =⎢ ⎣ 2b12 0. ⎛ ⎜ b21 ⎝1 −. 0. b21 b11 b22 b12. b21 b11 b22 b12. p11 − p51 ⎢ p21 − p61 =⎢ ⎣ p31 − p71 p41 − p81. ⎤. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎞ ⎥ ⎥ ⎥ ⎟ ⎥ ⎠ ⎦. ⎡. p15 − p55 ⎢ p25 − p65 =⎢ ⎣ p35 − p75 p45− p85. ⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦ (82). p12 − p52 p22 − p62 p32 − p72 p42 − p82. p13 − p53 p23 − p63 p33 − p73 p43 − p83. ⎤ p14 − p54 p24 − p64 ⎥ ⎥, p34 − p74 ⎦ p44 − p84 (83). p16 − p56 p26 − p66 p36 − p76 p46− p86. p17 − p57 p27 − p67 p37 − p77 p47 − p87. ⎤ p18 − p58 p28 − p68 ⎥ ⎥. p38 − p78 ⎦ p48 − p88 (84). Now we obtain the matrix δ(θ + τ (t), σ + τ (t)), δ11 (θ + τ (t), σ + τ (t)) b21 b11 β (θ − σ) + β (θ − σ), = dτ (t) 11 dτ (t) 21 1− 1− dt dt (85) δ12 (θ + τ (t), σ + τ (t)) b21 b11 β (θ − σ) + β (θ − σ), = dτ (t) 12 dτ (t) 22 1− 1− dt dt (86). 1 ⎟ ⎠ dτ (t) −1 dt 0. δ21 (θ + τ (t), σ + τ (t)) b22 b12 β11 (θ − σ) + β (θ − σ), = dτ (t) dτ (t) 21 1− 1− dt dt (87). 1 dτ (t) −1 dt. δ22 (θ + τ (t), σ + τ (t)) b22 b12 β12 (θ − σ) + β22 (θ − σ) = dτ (t) dτ (t) 1− 1− dt dt. ⎛ ⎜ b22 ⎝1 −. ⎞. ⎡. (77) (78). ⎞. 1 ⎟ ⎢ b ⎜ ⎠ ⎢ 11 ⎝1 − dτ (t) ⎢ −1 ⎢ ⎢ dt ⎢ ⎢ 0 ⎢ =⎢ ⎢ ⎛ ⎞ ⎢ ⎢ 1 ⎜ ⎟ ⎢ ⎢ b12 ⎝1 − ⎠ dτ (t) ⎢ ⎢ −1 ⎣ dt 0. ⎥ ⎥ ⎥ ⎥, ⎥ ⎦. Z32. Z33. " 1 2# = Z13 Z13 ,. ⎛. ⎤. dτ (t) −1 0 0 0 dt dτ (t) −1 0 0 0 dt dτ (t) −1 0 0 0 dt dτ (t) 0 0 0 −1 dt. ⎤ 0 2b21 ⎥ ⎥, 0 ⎦ 2b22. (80). (88). for t ≥ t0 , θ ∈ [−τ (t), 0], σ ∈ [θ, 0], where 0 ≤ τ (t) ≤ r. (81). Figures 3–7 show graphs of the functions α(t), β(ξ), obtained with a Matlab code, for the following matrices.
(10) J. Duda. 336 A, B, W of the system (52): −1 0.3 −1 0.4 A= , B= , 0.5 −2 0.1 −1 1 1 W = , (89) 0 1. 15. 9. x 10. α21(t). 8. 7. and a time delay given by the function. t τ (t) = r 1 − exp − , T. 6. 5. where r = 0.5, T = 1.. 4 0.5. α11(t). 0.45 3 0.4. 0. 2. 4. 6. 8. 10. t. 0.35. Fig. 5. Parameter α21 (t).. 0.3 0.25 0.3. α22(t). 0.2 0.28 0.15 0.1. 0.26 0.24 0. 2. 4. 6. 8. 10. t. 0.22. Fig. 3. Parameter α11 (t).. 0.2 0.18 0.16 0.14. 15. −3. x 10. 0.12 α12(t) 0.1. 0. 2. 4. 6. 8. 10. t −4. Fig. 6. Parameter α22 (t).. −5 1.5. β11(ξ) β21(ξ) β12(ξ) β22(ξ). 1 −6. 0.5 0. −7. −0.5 −1. −8. −1.5 −2. −9. −2.5 0. 2. 4. 6. 8. 10. t. Fig. 4. Parameter α12 (t).. The matrix α(t) is positive definite for r ≤ 0.86, so we have obtained a stability region for the system (52).. −3. 0. 0.1. 0.2 0.3 0 ≤ ξ ≤ τ(t). 0.4. 0.5. Fig. 7. Elements of the matrix β(ξ)..
(11) A Lyapunov functional for a system with a time-varying delay. 5. Conclusions The paper presented the procedure of determining the coefficients of the Lyapunov functional given by the formula (4) for a linear system with an interval time-varying delay, described by Eqn. (1). This article extends the method presented by Repin to the system with an interval timevarying delay. The presented method allows achieving analytical formulas on the coefficients of the Lyapunov functional which can be used to examine the stability of time delay systems with an interval time-varying delay and in the process of parametric optimization for calculation of the square performance index given by the formula (7).. Acknowledgment The author wishes to thank the editors and the reviewers for their suggestions, which have improved the quality of the paper.. References Duda, J. (1986). Parametric Optimization Problem for Systems with Time Delay, Ph.D. thesis, AGH University of Science and Technology, Cracow. Duda, J. (1988). Parametric optimization of neutral linear system with respect to the general quadratic performance index, Archiwum Automatyki i Telemechaniki 33(3): 448–456. Duda, J. (2010a). Lyapunov functional for a linear system with two delays, Control & Cybernetics 39(3): 797–809. Duda, J. (2010b). Lyapunov functional for a linear system with two delays both retarded and neutral type, Archives of Control Sciences 20(LVI): 89–98. Fridman, E. (2001). New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems, Systems & Control Letters 43(4): 309–319. Górecki, H., Fuksa, S., Grabowski, P., Korytowski, A. (1989). Analysis and Synthesis of Time Delay Systems, John Wiley & Sons, Chichester/New York, NY/Brisbane/Toronto/Singapore. Gu, K. (1997). Discretized LMI set in the stability problem of linear time delay systems, International Journal of Control 68(4): 923–934. Gu, K. and Liu, Y. (2009). Lyapunov–Krasovskii functional for uniform stability of coupled differential-functional equations, Automatica 45(3): 798–804. Han, Q.L. (2004). On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty, Automatica 40(6): 1087–1092. Han, Q.L. (2004). A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays, Automatica 40(10): 1791–1796. Han, Q.L. (2005). On stability of linear neutral systems with mixed time delays: A discretised Lyapunov functional approach, Automatica 41(7): 1209–1218.. 337 Han, Q.L. (2009). A discrete delay decomposition approach to stability of linear retarded and neutral systems, Automatica 45(2): 517–524. Infante, E.F. and Castelan, W.B. (1978). A Lyapunov functional for a matrix difference-differential equation, Journal of Differential Equations 29: 439–451. Ivanescu, D., Niculescu, S.I., Dugard, L., Dion, J.M. and Verriest, E.I. (2003). On delay-dependent stability for linear neutral systems, Automatica 39(2): 255–261. Kharitonov, V.L. (2005). Lyapunov functionals and Lyapunov matrices for neutral type time delay systems: A single delay case, International Journal of Control 78(11): 783– 800. Kharitonov, V.L. (2008). Lyapunov matrices for a class of neutral type time delay systems, International Journal of Control 81(6): 883–893. Kharitonov, V.L. and Hinrichsen, D. (2004). Exponential estimates for time delay systems, Systems & Control Letters 53(5): 395–405. Kharitonov, V.L. and Plischke, E. (2006). Lyapunov matrices for time-delay systems, Systems & Control Letters 55(9): 697– 706. Kharitonov, V.L., Zhabko, A.P. (2003). Lyapunov–Krasovskii approach to the robust stability analysis of time-delay systems, Automatica 39(1): 15–20. Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer Academic Publishers, Dordrecht. Repin, Yu. M. (1965). Quadratic Lyapunov functionals for systems with delay, Prikladnaja Matiematika i Miechanika 29: 564–566. Respondek, J.S. (2008). Approximate controllability of the n-th order infinite dimensional systems with controls delayed by the control devices, International Journal of Systems Science 39(8): 765–782. Richard, J.P. (2003). Time-delay systems: An overview of some recent advances and open problems, Automatica 39(10): 1667–1694. Wang, D., Wang, W. and Shi, P. (2009). Exponential H-infinity filtering for switched linear systems with interval timevarying delay, International Journal of Robust and Nonlinear Control 19(5): 532–551. Józef Duda graduated from the Faculty of Electrical Engineering, Automatics and Electronics, AGH University of Science and Technology in Cracow, Poland. He received an M.Sc. in electronics/automatic control in 1981. After completing doctoral studies, he received his Ph.D. degree in automatic control from the AGH University in 1986. His main research concerns time delay systems and application of Lyapunov functionals to stability analysis of time delay systems and in the process of parametric optimization.. Received: 27 January 2011 Revised: 26 May 2011 Re-revised: 1 August 2011.
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