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Automatica 60 (2015) 189–192
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New results on stability analysis for systems with discrete distributed delay✩ Hong-Bing Zeng a , Yong He b , Min Wu b , Jinhua She b,c a
School of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, China
b
School of Automation, China University of Geosciences, Wuhan 430074, China
c
School of Engineering, Tokyo University of Technology, Tokyo 192-0982, Japan
article
info
Article history: Received 15 March 2015 Received in revised form 23 June 2015 Accepted 30 June 2015 Available online 30 July 2015 Keywords: Systems with time delay Integral inequality Stability Lyapunov–Krasovskii functional
abstract The integral inequality technique is widely used to derive delay-dependent conditions, and various integral inequalities have been developed to reduce the conservatism of the conditions derived. In this study, a new integral inequality was devised that is tighter than existing ones. It was used to investigate the stability of linear systems with a discrete distributed delay, and a new stability condition was established. The results can be applied to systems with a delay belonging to an interval, which may be unstable when the delay is small or nonexistent. Three numerical examples demonstrate the effectiveness and the smaller conservatism of the method. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since the time delay present in many systems is often a source of instability, the stability of time delay systems has been widely studied, and various methods of evaluating the stability of such systems have been developed (see Gu, Kharitonov, & Chen, 2003). Among them, an analytical method based on features of the characteristic equation often yields unconservative results. However, it is not suitable for uncertain systems or systems with a time-varying delay. In contrast, the Lyapunov–Krasovskii functional method can handle these systems and has thus attracted much attention in the field of control engineering (see Kim, 2011; Park & Ko, 2007; Sun, Liu, Chen, & Rees, 2010). When the Lyapunov–Krasovskii functional method is used to derive a delay-dependent condition, it is necessary to deal with t the integral t −h x˙ T (s)Rx˙ (s)ds in the derivative of the Lyapunov
✩ This work was supported in part by the National Natural Science Foundation of China (61125301, 61210011, 61304064, 61273157), and the Natural Science Foundation of Hunan Province (2015JJ3064, 2015JJ5021). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits. E-mail addresses: [email protected] (H.-B. Zeng), [email protected] (Y. He), [email protected] (M. Wu), [email protected] (J. She).
http://dx.doi.org/10.1016/j.automatica.2015.07.017 0005-1098/© 2015 Elsevier Ltd. All rights reserved.
–Krasovskii functional. Ways of doing that include a descriptor model transformation (Fridman & Shaked, 2002), the freeweighting matrix technique (Wu, He, She, & Liu, 2004), and integral inequality methods (Han, 2005; Zhang & Han, 2014). Recently, Seuret and Gouaisbaut (2013) presented a Wirtinger-based inequality that is less conservative than others. For systems with a time-varying delay, Zeng, He, Wu, and She (2015) proposed a free-matrix-based integral inequality that yields less conservative stability conditions. However, the integral inequality in Zeng et al. (2015) makes use of information about only a single integral of the system state, which leaves room for improvement. This paper concerns a new integral inequality that was developed using information about a double integral of the system state. It includes the Wirtinger-based inequality. A new delay-dependent stability criterion based on the new inequality is presented that can be applied to systems with a delay belonging to an interval. Finally, three numerical examples demonstrate the advantages of the method and its superiority over the others. Throughout this paper, the superscripts ‘−1’ and ‘T ’ stand for the inverse and transpose of a matrix, respectively; Rn denotes n-dimensional Euclidean space; Rn×m is the set of all n × m real matrices; P > 0 means that the matrix P is symmetric and positive definite; ‘∗’ denotes symmetric terms in a symmetric matrix; Sym{X } = X + X T ; I is the identity matrix; and 0 is a zero matrix. If the dimensions of a matrix are not explicitly stated, the matrix is assumed to have compatible dimensions.
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H.-B. Zeng et al. / Automatica 60 (2015) 189–192
2. Preliminaries
It is easy to see that
−2ζ T (s)N x˙ (s) ≤ ζ T (s)NR−1 N T ζ (s) + x˙ T (s)Rx˙ (s).
Consider a system with a discrete distributed delay:
x˙ (t ) = Ax(t ) + A1 x(t − h) + A2 x(t ) = φ(t ),
t
x(s)ds, t −h
t ∈ [−h, 0],
(1)
Integrating (3) from α to β yields
−2ϑ T N1 (e1 − e2 )ϑ − 2ϑ T N2 (e1 + e2 − 2e3 )ϑ
where x(t ) ∈ Rn is the state vector; A, A1 , A2 ∈ Rn×n are constant system matrices; the delay, h, is a constant satisfying h ∈ [hmin , hmax ]; and the initial condition, φ(t ), is a continuous vector-valued function in t ∈ [−h, 0]. a delay-dependent condition, the double integral 0 To t derive T ˙ x ( s ) R x˙ (s)dsdθ is usually used in the Lyapunov–Krasovskii −h t +θ
− 2ϑ T N3 (e1 − e2 − 6e3 + 6e4 )ϑ β −α T ≤ (β − α)ϑ T N1 R−1 N1T ϑ + ϑ N2 R−1 N2T ϑ 3 β (β − α) T x˙ T (s)Rx˙ (s)ds. ϑ N3 R−1 N3T ϑ + +
functional, which yields the integral t −h x˙ T (s)Rx˙ (s)ds. The key to reducing conservatism is how to deal with this integral. Many researchers have attempted to estimate its bounds, and various integral inequalities have been developed (see Gu et al., 2003; Han, 2005; Seuret & Gouaisbaut, 2013; Zhang & Han, 2014). The goal of this study was to develop a new integral inequality that yields less conservative conditions. The following nomenclature is used to simplify vector and matrix representations:
Rearranging (4) yields (2), and this completes the proof.
5
t
v1 ( t ) =
t
x(s)ds,
v2 (t ) =
t −h
t
t −h
T
T 1
Remark 2. In the proof of Lemma 1, ζ (s) contains two functions, f1 (s) and f2 (s). Integrating f1 (s)˙x(s) and f2 (s)˙x(s) from α to β yields
β
β s
two integrals, α x(s)ds and α α x(u)duds. Less conservative results are obtained by employing the free matrices N1 , N2 , and β N3 to deal with the relationships among x(α), x(β), α x(s)ds, and
α α x(u)duds. It is easy to prove that Corollary 5 in Seuret and Gouaisbaut (2013) is a special case of Lemma 1 by setting N1 =
x(u)duds, t −h
1
β−α
T 2
T −R R 0 0 , N2 =
T −R − R 2R 0 , and N3 = 0.
3
β−α
The following stability criterion is based on the new integral inequality. Theorem 3. Given a constant h ∈ [hmin , hmax ], the system (1) is asymptotically stable if there exist P (∈ R3n×3n ) > 0, Q (∈ Rn×n ) ≥ 0, R(∈ Rn×n ) > 0, and any matrices N1 , N2 , N3 ∈ R4n×n , such that the following LMI is feasible:
3. Main results First, a new integral inequality is given: Lemma 1. Let x be a differentiable function: [α, β ] → Rn . For symmetric matrices R(∈ Rn×n ) > 0, and N1 , N2 , N3 ∈ R4n×n , the following inequality holds:
√
Φ1 + Φ2 ∗ Φ= ∗ ∗
√
hN1 −R
√
hN2 0 −3R
∗ ∗
hN3 0 < 0, 0 −5R
∗
(5)
where
β
(4)
α
β s
s
T η1 (t ) = x (t ) v (t ) v (t ) , T 1 T 2 T T T ξ (t ) = x (t ) x (t − h) v1 (t ) 2 v2 (t ) , h h ei = 0n×(i−1)n In 0n×(4−i)n , i = 1, 2, . . . , 4, Γ = A A1 hA2 0 .
(3)
x˙ (s)Rx˙ (s)ds ≤ ϑ Ω ϑ, T
− α
T
(2)
where 1
1
N3 R−1 N3T ) 3 5 + Sym{N1 Π1 + N2 Π2 + N3 Π3 },
Ω = τ (N1 R−1 N1T +
N2 R−1 N2T +
Π1 = e1 − e2 , Π2 = e1 + e2 − 2e3 , Π3 = e1 − e2 − 6e3 + 6e4 , T β β s 1 2 T T T T ϑ = x (β) x (α) x (s)ds 2 x (u)duds , τ α τ α α τ = β − α. Proof. Define 2s − β − α
, β −α 6s2 − 6(β + α)s + β 2 + 4βα + α 2 f 2 ( s) = , (β − α)2 T N = N1T N2T N3T , T ζ (s) = ϑ T f1 (s)ϑ T f2 (s)ϑ T .
f 1 ( s) =
Φ1 = Sym{Π4T P Π5 } + eT1 Qe1 − eT2 Qe2 + hΓ T RΓ , Φ2 = Sym{N1 Π1 + N2 Π2 + N3 Π3 }, T h2 T T T Π4 = e1 he3 e4 , 2
Π5 = Γ T
eT1 − eT2
heT3 − heT2
T
,
and Πi , i = 1, 2, 3, are defined in Lemma 1. Proof. Choose the following Lyapunov–Krasovskii functional candidate: V (xt ) = η1T (t )P η1 (t ) +
t
xT (s)Qx(s)ds t −h
0
t
x˙ T (s)Rx˙ (s)dsdθ .
+ −h
(6)
t +θ
The derivative of V (xt ) is V˙ (xt ) = ξ T (t )Φ1 ξ (t ) −
t
x˙ T (s)Rx˙ (s)ds. t −h
(7)
H.-B. Zeng et al. / Automatica 60 (2015) 189–192 Table 1 Upper bound on h obtained for Example 6.
Applying Lemma 1, we get t
x˙ T (s)Rx˙ (s)ds
− t −h
h h ≤ ξ T (t ) hN1 R−1 N1T + N2 R−1 N2T + N3 R−1 N3T 3 5 + Sym{N1 Π1 + N2 Π2 + N3 Π3 } ξ (t ) = ξ T (t )(Φ2 + Φ3 )ξ (t )
(8)
where Φ3 = hN1 R−1 N1T + 3h N2 R−1 N2T + 5h N3 R−1 N3T . So, we have V˙ (xt ) ≤ ξ T (t )(Φ1 + Φ2 + Φ3 )ξ (t ).
Remark 4. Unlike the method in Zeng et al. (2015), Theorem 3 takes information on the double integral of the system state into consideration, which helps to reduce the conservatism. It should be pointed out that Lemma 1 can be easily applied to stability analysis for systems with time-varying delay by using a procedure similar to the one in Zeng et al. (2015). The following corollary reduces the computation complexity through the elimination of the free matrices N1 , N2 , N3 in Theorem 3. Corollary 5. Given a constant h ∈ [hmin , hmax ], the system (1) is asymptotically stable if there exist P (∈ R3n×3n ) > 0, Q (∈ Rn×n ) ≥ 0, R(∈ Rn×n ) > 0, such that Φ < 0, where Φ is defined in Theorem 3 with
N2 = N3 =
1
−R R 0 0
T
h 3
−R −R 2R 0
T
h
,
T −R R 6R −6R .
4. Numerical examples The three numerical examples in this section demonstrate the effectiveness of our method. Example 6. Consider the system (1) with
A=
−2 0
0 , −0.9
−1 A1 = −1
0 , −1
0 A2 = 0
0 . 0
This example often appears in the literature. Table 1 lists the maximum admissible upper bounds obtained by the method described in this paper, along with those obtained by other methods, and the corresponding number of variables (NVs). The table shows that, although Corollary 5 contains fewer variables than Theorem 3 does, it produces the same results. Below, we consider the stability of systems with a delay belonging to an interval with a non-zero lower bound, which may be unstable when the delay is small or nonexistent, as illustrated in the next example. Example 7. Consider the system (1) with
A=
0 −2
1 , 0.1
A1 =
hmax
NVs
Gu et al. (2003) (N = 1) Gu et al. (2003) (N = 2) Kao and Rantzer (2007) Park and Ko (2007) Seuret and Gouaisbaut (2013) Seuret and Gouaisbaut (2014) (N = 2) Seuret and Gouaisbaut (2014) (N = 4) Seuret and Gouaisbaut (2014) (N = 6) Theorem 3 Corollary 5 The analytical bounds
6.059 6.165 6.1107 4.472 6.059 3.21 5.28 6.12 6.1664 6.1664 6.1725
7.5n2 + 2.5n 9.5n2 + 3.5n 1.5n2 + 9n + 9 11.5n2 + 4.5n 3n2 + 2n 9n2 + 3n 19n2 + 4n 33n2 + 5n 17.5n2 + 2.5n 5.5n2 + 2.5n
Table 2 Delay interval in which stability of system in Example 7 is guaranteed. Methods
hmin
hmax
Gu et al. (2003) (N = 1) Gu et al. (2003) (N = 2) Seuret and Gouaisbaut (2013) Park et al. (2015) Theorem 3 Analytical bounds
0.1006 0.1003 0.1003 0.1002 0.100169 0.100169
1.4272 1.6921 1.5406 1.5954 1.7122 1.7178
Since Re(eig (A + A1 )) = 0.05 > 0, the system is unstable when h = 0. Few methods can analyze the stability of this system. Table 2 lists the results obtained from Theorem 3 and other methods reported in the literature. Example 8. Consider the system (1) with 0.2 0.2
A=
0 , 0.1
A1 =
0 0
0 , 0
A2 =
−1 −1
0 . −1
An eigenvalue analysis shows that the system is stable for any constant delay in the interval [0.2000, 2.04]. Chen and Zheng (2007), Park, Kwon, Park, Lee, and Cha (2015), and Seuret and Gouaisbaut (2013) reported that the stability was guaranteed for any delay in the intervals [0.2001, 1.6339], [0.2000, 1.877], and [0.2000, 1.9504], respectively. However, from Theorem 3 we find that the system is stable for any constant delay in the interval [0.2000, 2.0395]. Clearly, Theorem 3 yields the least conservative results.
,
h 5
Methods
(9)
If (Φ1 + Φ2 + Φ3 ) < 0, which is equivalent to LMI (5) (as can be seen by taking the Schur complement), then V˙ (xt ) < 0. So, the system (1) is asymptotically stable. This completes the proof.
N1 =
191
0 1
0 , 0
A2 =
0 0
0 . 0
5. Conclusion This paper concerns the stability of systems with a discrete distributed delay. A new integral inequality that yields less conservative stability criteria has been developed. Numerical examples demonstrate the advantages of the method. The method described here is easy to apply to uncertain systems and systems with a timevarying delay, which helps to reduce the conservatism of the conditions derived. References Chen, W. H., & Zheng, W. X. (2007). Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica, 43(1), 95–104. Fridman, E., & Shaked, U. (2002). An improved stabilization method for linear timedelay systems. IEEE Transactions on Automatic Control, 47(11), 1931–1937. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Han, Q. L. (2005). Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, 41(12), 2171–2176. Kao, C. Y., & Rantzer, A. (2007). Stability analysis of systems with uncertain timevarying delays. Automatica, 43(6), 959–970. Kim, J. H. (2011). Note on stability of linear systems with time-varying delay. Automatica, 47(9), 2118–2121. Park, P. G., & Ko, J. W. (2007). Stability and robust stability for systems with a timevarying delay. Automatica, 43(10), 1855–1858. Park, M. J., Kwon, O. M., Park, J. H., Lee, S. M., & Cha, E. J. (2015). Stability of time-delay systems via Wirtinger-based double integral inequality. Automatica, 55(5), 204–208.
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H.-B. Zeng et al. / Automatica 60 (2015) 189–192
Seuret, A., & Gouaisbaut, F. (2013). Wirtinger-based integral inequality: Application to time-delay systems. Automatica, 49(9), 2860–2866. Seuret, A., & Gouaisbaut, F. (2014). Complete Quadratic Lyapunov functionals using Bessel-Legendre inequality. In: 2014 European control conference, ECC, Strasbourg, France, June 24–27, 2014, pp. 448–453. Sun, J., Liu, G. P., Chen, J., & Rees, D. (2010). Improved delay-range-dependent stability criteria for linear systems with time-varying delays. Automatica, 46(2), 466–470.
Wu, M., He, Y., She, J. H., & Liu, G. P. (2004). Delay-dependent criteria for robust stability of time-varying delay systems. Automatica, 40(8), 1435–1439. Zeng, H. B., He, Y., Wu, M., & She, J. (2015). Free-matrix-based integral inequality for stability analysis of systems with time-varying delay. IEEE Transactions on Automatic Control, http://dx.doi.org/10.1109/TAC.2015.2404271. Zhang, X. M., & Han, Q. L. (2014). New stability criterion using a matrix-based quadratic convex approach and some novel integral inequalities. IET Control Theory & Applications, 8(12), 1054–1061.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
New results on stability analysis for systems with discrete distributed delay✩ Hong-Bing Zeng a , Yong He b , Min Wu b , Jinhua She b,c a
School of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, China
b
School of Automation, China University of Geosciences, Wuhan 430074, China
c
School of Engineering, Tokyo University of Technology, Tokyo 192-0982, Japan
article
info
Article history: Received 15 March 2015 Received in revised form 23 June 2015 Accepted 30 June 2015 Available online 30 July 2015 Keywords: Systems with time delay Integral inequality Stability Lyapunov–Krasovskii functional
abstract The integral inequality technique is widely used to derive delay-dependent conditions, and various integral inequalities have been developed to reduce the conservatism of the conditions derived. In this study, a new integral inequality was devised that is tighter than existing ones. It was used to investigate the stability of linear systems with a discrete distributed delay, and a new stability condition was established. The results can be applied to systems with a delay belonging to an interval, which may be unstable when the delay is small or nonexistent. Three numerical examples demonstrate the effectiveness and the smaller conservatism of the method. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since the time delay present in many systems is often a source of instability, the stability of time delay systems has been widely studied, and various methods of evaluating the stability of such systems have been developed (see Gu, Kharitonov, & Chen, 2003). Among them, an analytical method based on features of the characteristic equation often yields unconservative results. However, it is not suitable for uncertain systems or systems with a time-varying delay. In contrast, the Lyapunov–Krasovskii functional method can handle these systems and has thus attracted much attention in the field of control engineering (see Kim, 2011; Park & Ko, 2007; Sun, Liu, Chen, & Rees, 2010). When the Lyapunov–Krasovskii functional method is used to derive a delay-dependent condition, it is necessary to deal with t the integral t −h x˙ T (s)Rx˙ (s)ds in the derivative of the Lyapunov
✩ This work was supported in part by the National Natural Science Foundation of China (61125301, 61210011, 61304064, 61273157), and the Natural Science Foundation of Hunan Province (2015JJ3064, 2015JJ5021). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits. E-mail addresses: [email protected] (H.-B. Zeng), [email protected] (Y. He), [email protected] (M. Wu), [email protected] (J. She).
http://dx.doi.org/10.1016/j.automatica.2015.07.017 0005-1098/© 2015 Elsevier Ltd. All rights reserved.
–Krasovskii functional. Ways of doing that include a descriptor model transformation (Fridman & Shaked, 2002), the freeweighting matrix technique (Wu, He, She, & Liu, 2004), and integral inequality methods (Han, 2005; Zhang & Han, 2014). Recently, Seuret and Gouaisbaut (2013) presented a Wirtinger-based inequality that is less conservative than others. For systems with a time-varying delay, Zeng, He, Wu, and She (2015) proposed a free-matrix-based integral inequality that yields less conservative stability conditions. However, the integral inequality in Zeng et al. (2015) makes use of information about only a single integral of the system state, which leaves room for improvement. This paper concerns a new integral inequality that was developed using information about a double integral of the system state. It includes the Wirtinger-based inequality. A new delay-dependent stability criterion based on the new inequality is presented that can be applied to systems with a delay belonging to an interval. Finally, three numerical examples demonstrate the advantages of the method and its superiority over the others. Throughout this paper, the superscripts ‘−1’ and ‘T ’ stand for the inverse and transpose of a matrix, respectively; Rn denotes n-dimensional Euclidean space; Rn×m is the set of all n × m real matrices; P > 0 means that the matrix P is symmetric and positive definite; ‘∗’ denotes symmetric terms in a symmetric matrix; Sym{X } = X + X T ; I is the identity matrix; and 0 is a zero matrix. If the dimensions of a matrix are not explicitly stated, the matrix is assumed to have compatible dimensions.
190
H.-B. Zeng et al. / Automatica 60 (2015) 189–192
2. Preliminaries
It is easy to see that
−2ζ T (s)N x˙ (s) ≤ ζ T (s)NR−1 N T ζ (s) + x˙ T (s)Rx˙ (s).
Consider a system with a discrete distributed delay:
x˙ (t ) = Ax(t ) + A1 x(t − h) + A2 x(t ) = φ(t ),
t
x(s)ds, t −h
t ∈ [−h, 0],
(1)
Integrating (3) from α to β yields
−2ϑ T N1 (e1 − e2 )ϑ − 2ϑ T N2 (e1 + e2 − 2e3 )ϑ
where x(t ) ∈ Rn is the state vector; A, A1 , A2 ∈ Rn×n are constant system matrices; the delay, h, is a constant satisfying h ∈ [hmin , hmax ]; and the initial condition, φ(t ), is a continuous vector-valued function in t ∈ [−h, 0]. a delay-dependent condition, the double integral 0 To t derive T ˙ x ( s ) R x˙ (s)dsdθ is usually used in the Lyapunov–Krasovskii −h t +θ
− 2ϑ T N3 (e1 − e2 − 6e3 + 6e4 )ϑ β −α T ≤ (β − α)ϑ T N1 R−1 N1T ϑ + ϑ N2 R−1 N2T ϑ 3 β (β − α) T x˙ T (s)Rx˙ (s)ds. ϑ N3 R−1 N3T ϑ + +
functional, which yields the integral t −h x˙ T (s)Rx˙ (s)ds. The key to reducing conservatism is how to deal with this integral. Many researchers have attempted to estimate its bounds, and various integral inequalities have been developed (see Gu et al., 2003; Han, 2005; Seuret & Gouaisbaut, 2013; Zhang & Han, 2014). The goal of this study was to develop a new integral inequality that yields less conservative conditions. The following nomenclature is used to simplify vector and matrix representations:
Rearranging (4) yields (2), and this completes the proof.
5
t
v1 ( t ) =
t
x(s)ds,
v2 (t ) =
t −h
t
t −h
T
T 1
Remark 2. In the proof of Lemma 1, ζ (s) contains two functions, f1 (s) and f2 (s). Integrating f1 (s)˙x(s) and f2 (s)˙x(s) from α to β yields
β
β s
two integrals, α x(s)ds and α α x(u)duds. Less conservative results are obtained by employing the free matrices N1 , N2 , and β N3 to deal with the relationships among x(α), x(β), α x(s)ds, and
α α x(u)duds. It is easy to prove that Corollary 5 in Seuret and Gouaisbaut (2013) is a special case of Lemma 1 by setting N1 =
x(u)duds, t −h
1
β−α
T 2
T −R R 0 0 , N2 =
T −R − R 2R 0 , and N3 = 0.
3
β−α
The following stability criterion is based on the new integral inequality. Theorem 3. Given a constant h ∈ [hmin , hmax ], the system (1) is asymptotically stable if there exist P (∈ R3n×3n ) > 0, Q (∈ Rn×n ) ≥ 0, R(∈ Rn×n ) > 0, and any matrices N1 , N2 , N3 ∈ R4n×n , such that the following LMI is feasible:
3. Main results First, a new integral inequality is given: Lemma 1. Let x be a differentiable function: [α, β ] → Rn . For symmetric matrices R(∈ Rn×n ) > 0, and N1 , N2 , N3 ∈ R4n×n , the following inequality holds:
√
Φ1 + Φ2 ∗ Φ= ∗ ∗
√
hN1 −R
√
hN2 0 −3R
∗ ∗
hN3 0 < 0, 0 −5R
∗
(5)
where
β
(4)
α
β s
s
T η1 (t ) = x (t ) v (t ) v (t ) , T 1 T 2 T T T ξ (t ) = x (t ) x (t − h) v1 (t ) 2 v2 (t ) , h h ei = 0n×(i−1)n In 0n×(4−i)n , i = 1, 2, . . . , 4, Γ = A A1 hA2 0 .
(3)
x˙ (s)Rx˙ (s)ds ≤ ϑ Ω ϑ, T
− α
T
(2)
where 1
1
N3 R−1 N3T ) 3 5 + Sym{N1 Π1 + N2 Π2 + N3 Π3 },
Ω = τ (N1 R−1 N1T +
N2 R−1 N2T +
Π1 = e1 − e2 , Π2 = e1 + e2 − 2e3 , Π3 = e1 − e2 − 6e3 + 6e4 , T β β s 1 2 T T T T ϑ = x (β) x (α) x (s)ds 2 x (u)duds , τ α τ α α τ = β − α. Proof. Define 2s − β − α
, β −α 6s2 − 6(β + α)s + β 2 + 4βα + α 2 f 2 ( s) = , (β − α)2 T N = N1T N2T N3T , T ζ (s) = ϑ T f1 (s)ϑ T f2 (s)ϑ T .
f 1 ( s) =
Φ1 = Sym{Π4T P Π5 } + eT1 Qe1 − eT2 Qe2 + hΓ T RΓ , Φ2 = Sym{N1 Π1 + N2 Π2 + N3 Π3 }, T h2 T T T Π4 = e1 he3 e4 , 2
Π5 = Γ T
eT1 − eT2
heT3 − heT2
T
,
and Πi , i = 1, 2, 3, are defined in Lemma 1. Proof. Choose the following Lyapunov–Krasovskii functional candidate: V (xt ) = η1T (t )P η1 (t ) +
t
xT (s)Qx(s)ds t −h
0
t
x˙ T (s)Rx˙ (s)dsdθ .
+ −h
(6)
t +θ
The derivative of V (xt ) is V˙ (xt ) = ξ T (t )Φ1 ξ (t ) −
t
x˙ T (s)Rx˙ (s)ds. t −h
(7)
H.-B. Zeng et al. / Automatica 60 (2015) 189–192 Table 1 Upper bound on h obtained for Example 6.
Applying Lemma 1, we get t
x˙ T (s)Rx˙ (s)ds
− t −h
h h ≤ ξ T (t ) hN1 R−1 N1T + N2 R−1 N2T + N3 R−1 N3T 3 5 + Sym{N1 Π1 + N2 Π2 + N3 Π3 } ξ (t ) = ξ T (t )(Φ2 + Φ3 )ξ (t )
(8)
where Φ3 = hN1 R−1 N1T + 3h N2 R−1 N2T + 5h N3 R−1 N3T . So, we have V˙ (xt ) ≤ ξ T (t )(Φ1 + Φ2 + Φ3 )ξ (t ).
Remark 4. Unlike the method in Zeng et al. (2015), Theorem 3 takes information on the double integral of the system state into consideration, which helps to reduce the conservatism. It should be pointed out that Lemma 1 can be easily applied to stability analysis for systems with time-varying delay by using a procedure similar to the one in Zeng et al. (2015). The following corollary reduces the computation complexity through the elimination of the free matrices N1 , N2 , N3 in Theorem 3. Corollary 5. Given a constant h ∈ [hmin , hmax ], the system (1) is asymptotically stable if there exist P (∈ R3n×3n ) > 0, Q (∈ Rn×n ) ≥ 0, R(∈ Rn×n ) > 0, such that Φ < 0, where Φ is defined in Theorem 3 with
N2 = N3 =
1
−R R 0 0
T
h 3
−R −R 2R 0
T
h
,
T −R R 6R −6R .
4. Numerical examples The three numerical examples in this section demonstrate the effectiveness of our method. Example 6. Consider the system (1) with
A=
−2 0
0 , −0.9
−1 A1 = −1
0 , −1
0 A2 = 0
0 . 0
This example often appears in the literature. Table 1 lists the maximum admissible upper bounds obtained by the method described in this paper, along with those obtained by other methods, and the corresponding number of variables (NVs). The table shows that, although Corollary 5 contains fewer variables than Theorem 3 does, it produces the same results. Below, we consider the stability of systems with a delay belonging to an interval with a non-zero lower bound, which may be unstable when the delay is small or nonexistent, as illustrated in the next example. Example 7. Consider the system (1) with
A=
0 −2
1 , 0.1
A1 =
hmax
NVs
Gu et al. (2003) (N = 1) Gu et al. (2003) (N = 2) Kao and Rantzer (2007) Park and Ko (2007) Seuret and Gouaisbaut (2013) Seuret and Gouaisbaut (2014) (N = 2) Seuret and Gouaisbaut (2014) (N = 4) Seuret and Gouaisbaut (2014) (N = 6) Theorem 3 Corollary 5 The analytical bounds
6.059 6.165 6.1107 4.472 6.059 3.21 5.28 6.12 6.1664 6.1664 6.1725
7.5n2 + 2.5n 9.5n2 + 3.5n 1.5n2 + 9n + 9 11.5n2 + 4.5n 3n2 + 2n 9n2 + 3n 19n2 + 4n 33n2 + 5n 17.5n2 + 2.5n 5.5n2 + 2.5n
Table 2 Delay interval in which stability of system in Example 7 is guaranteed. Methods
hmin
hmax
Gu et al. (2003) (N = 1) Gu et al. (2003) (N = 2) Seuret and Gouaisbaut (2013) Park et al. (2015) Theorem 3 Analytical bounds
0.1006 0.1003 0.1003 0.1002 0.100169 0.100169
1.4272 1.6921 1.5406 1.5954 1.7122 1.7178
Since Re(eig (A + A1 )) = 0.05 > 0, the system is unstable when h = 0. Few methods can analyze the stability of this system. Table 2 lists the results obtained from Theorem 3 and other methods reported in the literature. Example 8. Consider the system (1) with 0.2 0.2
A=
0 , 0.1
A1 =
0 0
0 , 0
A2 =
−1 −1
0 . −1
An eigenvalue analysis shows that the system is stable for any constant delay in the interval [0.2000, 2.04]. Chen and Zheng (2007), Park, Kwon, Park, Lee, and Cha (2015), and Seuret and Gouaisbaut (2013) reported that the stability was guaranteed for any delay in the intervals [0.2001, 1.6339], [0.2000, 1.877], and [0.2000, 1.9504], respectively. However, from Theorem 3 we find that the system is stable for any constant delay in the interval [0.2000, 2.0395]. Clearly, Theorem 3 yields the least conservative results.
,
h 5
Methods
(9)
If (Φ1 + Φ2 + Φ3 ) < 0, which is equivalent to LMI (5) (as can be seen by taking the Schur complement), then V˙ (xt ) < 0. So, the system (1) is asymptotically stable. This completes the proof.
N1 =
191
0 1
0 , 0
A2 =
0 0
0 . 0
5. Conclusion This paper concerns the stability of systems with a discrete distributed delay. A new integral inequality that yields less conservative stability criteria has been developed. Numerical examples demonstrate the advantages of the method. The method described here is easy to apply to uncertain systems and systems with a timevarying delay, which helps to reduce the conservatism of the conditions derived. References Chen, W. H., & Zheng, W. X. (2007). Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica, 43(1), 95–104. Fridman, E., & Shaked, U. (2002). An improved stabilization method for linear timedelay systems. IEEE Transactions on Automatic Control, 47(11), 1931–1937. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Han, Q. L. (2005). Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, 41(12), 2171–2176. Kao, C. Y., & Rantzer, A. (2007). Stability analysis of systems with uncertain timevarying delays. Automatica, 43(6), 959–970. Kim, J. H. (2011). Note on stability of linear systems with time-varying delay. Automatica, 47(9), 2118–2121. Park, P. G., & Ko, J. W. (2007). Stability and robust stability for systems with a timevarying delay. Automatica, 43(10), 1855–1858. Park, M. J., Kwon, O. M., Park, J. H., Lee, S. M., & Cha, E. J. (2015). Stability of time-delay systems via Wirtinger-based double integral inequality. Automatica, 55(5), 204–208.
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