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Stabilization of neutral time-delay systems with actuator saturation via auxiliary time-delay feedback
Automatica 52 (2015) 242–247
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Brief paper
Stabilization of neutral time-delay systems with actuator saturation via auxiliary time-delay feedback✩ Yonggang Chen a , Shumin Fei b , Yongmin Li c a
School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, PR China
b
Key Laboratory of Measurement and Control of CSE (Ministry of Education), School of Automation, Southeast University, Nanjing 210096, PR China
c
School of Science, Huzhou Teachers College, Huzhou 313000, PR China
article
info
Article history: Received 9 December 2013 Received in revised form 17 September 2014 Accepted 9 November 2014
Keywords: Stabilization Neutral systems Actuator saturation Auxiliary time-delay feedback
abstract This paper investigates the stabilization problem for neutral time-delay systems with actuator saturation. Different from the existing techniques, the auxiliary time-delay feedback is introduced for the first time in this paper. Based on such a technique, the saturation nonlinearity is represented as the convex combination of state feedback and auxiliary time-delay feedback. By employing free-weighting matrix technique and Jensen integral inequalities, and performing the accurate estimation of the lower bounds of L–K functionals, the improved delay-dependent local stabilization conditions are proposed in terms of linear matrix inequalities (LMIs). Numerical examples illustrate the reduced conservatism of the proposed conditions in this paper. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Time-delays are frequently encountered in various practical systems, such as chemical engineering systems, biological systems and manufacturing processes (Gu, Kharitonov, & Chen, 2003). There are two types of time-delay systems, i.e., retarded type and neutral type. The retarded type contains delays only in its states, while the neutral type contains delays in both its states and its derivatives of the states. On the other hand, it is well recognized that LMI-based approaches are more convenient for solving corresponding synthesis problems, and delay-dependent results are generally less conservative than delay-independent ones especially when the size of delay is small (Xu & Jam, 2008). During the past two decades, several important techniques have been proposed to obtain LMI-based delay-dependent analysis and synthesis conditions for time-delay systems, see, e.g., Chen and Zheng (2007), Fridman (2001), Han (2009), He, Wang, Lin, and Wu (2005);
✩ This work was supported by the National Natural Science Foundations of China under Grants 61304061, 61273119, 61174076 and 61374086. This work was also partly supported by the Natural Science Foundation of Henan Institute of Science and Technology (201319). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fen Wu under the direction of Editor Roberto Tempo. E-mail addresses: [email protected] (Y. Chen), [email protected] (S. Fei), [email protected] (Y. Li).
http://dx.doi.org/10.1016/j.automatica.2014.11.015 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
He, Wang, Xie, and Lin (2007); He, Wu, She, and Liu (2004), Li, Jing, and Karimi (2014), Qian, Liu, and Fei (2012) and Sun, Liu, and Chen (2009). In many practical control applications, actuator saturation is often inevitable, and its existence may deteriorate the performance of a control system and even cause the instability of closedloop system. Therefore, considerable attention has been devoted to linear systems subject to saturating controllers during the past decades, see e.g., Alamo, Cepeda, and Limon (2005), Gomes da Silva and Tarbouriech (2005); Hu and Lin (2001), Hu, Lin, and Chen (2002), Lin (1998), Tarbouriech, Garcia, Gomes da Silva, and Queinnec (2011), Zhou (2013) and Zhou, Lin, and Duan (2008). Generally speaking, the current research can be classified into two categories according to whether the open-loop poles are located on the closed left-half plane, i.e., global/semi-global stabilization, and local stabilization and anti-windup design. For the local stabilization and anti-windup design, two dominant approaches are proposed to deal with the saturation nonlinearity, one is the polytopic models (Alamo et al., 2005; Hu & Lin, 2001; Hu et al., 2002; Zhou, 2013) and the other is the generalized sector condition (Gomes da Silva & Tarbouriech, 2005). In particular, it is worth mentioning that the saturation representation proposed in Alamo et al. (2005) and Tarbouriech et al. (2011) with the compact notation proposed in Zhou (2013) contains more slack variables, and thus is less conservative than that in Hu and Lin (2001) and Hu et al. (2002) for the multiple input systems.
Y. Chen et al. / Automatica 52 (2015) 242–247
For the systems with both time-delay and actuator saturation, global/semi-global stabilization were well investigated in Lin and Fang (2007), Yakoubi and Chitour (2007), Zhou, Lin, and Duan (2010) and Zhou, Lin, and Duan (2012) under the assumption that the open-loop poles are located on the closed left-half plane. Removing such a restriction on open-loop poles, the problem of local stabilization has been widely studied in Cao, Lin, and Hu (2002), Fridman, Pila, and Shaked (2003), Gomes da Silva, Seuret, Fridman, and Richard (2005, 2011), Tarbouriech and Gomes da Silva (2000) and Zhang, Boukas, and Haidar (2008) by incorporating the techniques of analyzing the stability of time-delay systems. However, it should be pointed out that the techniques of representing saturation nonlinearity in the above references are the same as the cases without delay. It is clear that the time-delay information is completely neglected when dealing with the saturation nonlinearity, which may result in some conservative results. In this paper, we resist the stabilization problem for neutral systems with time-varying delay and actuator saturation. Different from the existing techniques, the auxiliary time-delay feedback is proposed in this paper. Based on the polytopic approach proposed in Alamo et al. (2005), Tarbouriech et al. (2011) and Zhou (2013), the saturation nonlinearly is firstly represented by the convex combination of state feedback and auxiliary time-delay feedback. By incorporating Lyapunov–Krasovskii (L–K) functional theory, free-weighting matrix technique and integral inequalities, and performing the accurate estimation of the lower bounds of L–K functionals, then the improved stabilization conditions are obtained in terms of LMIs. Compared with the existing results, the main novelty of this paper is that the auxiliary time-delay feedback is introduced for the first time when representing the saturation nonlinearity, and the accurate estimation of the lower bounds of L–K functionals is performed to obtain LMI-based conditions. Finally, the reduced conservatism of the proposed conditions in this paper is shown by numerical examples. Notation. λM (P ) denotes the maximum eigenvalue of matrix P. A real symmetric matrix P > 0(≥ 0) denotes P being a positive definite (positive semi-definite) matrix. I denotes an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The space of the continuously differentiable vector functions φ over [−h, 0] is denoted by C 1 [−h, 0]. ∥ · ∥ and ∥ · ∥∞ denote the 2-norm and ∞-norm, respectively, and maxt ∈[−h,0] ∥φ(t )∥ is denoted by ∥φ∥c . I[1, π] denotes the set of integers whose elements are 1, 2, . . . , π . Dm denotes the set of m × m diagonal matrices with diagonal elements either 1 or 0. em,k ∈ R1×m denotes a row vector whose k-th element is 1 and the others are zero, and ⊗ denotes the Kronecker product. 2. Problem formulation Consider the following neutral time-delay system with actuator saturation x˙ (t ) − C x˙ (t − h(t )) = Ax(t ) + Ad x(t − h(t )) + Bsat (u(t )),
(1)
x(t ) = φ(t ),
(2)
∀t ∈ [−h, 0],
where x(t ) ∈ Rn is the state vector, u(t ) ∈ Rm is the input vector, A, Ad , B and C are known real constant matrices with appropriate dimensions, h(t ) denotes time-varying delay that satisfies 0 ≤ h(t ) ≤ h and h˙ (t ) ≤ µ < 1. sat (u) : Rm → Rm is the vector valued standard saturation function described by sat (u) = [sat (u1 ) sat (u2 ) · · · sat (um )]T , where sat (uj ) = sgn(uj ) min{1, |uj |}, j ∈ I[1, m]. The controller used in this paper is the
243
following state feedback u(t ) = Kx(t ),
(3)
where K ∈ Rm×n is the gain matrix to be designed. In this paper, it is assumed that φ(t ) is continuously differentiable over [−h, 0], and one of our interests is to estimate the domain of attraction of the following form
˙ c ≤ ρ2 , Xρ = φ(t ) ∈ C 1 [−h, 0] : ∥φ∥c ≤ ρ1 , ∥φ∥
(4)
where ρ1 and ρ2 are some scalars to be maximized. ↔
Lemma 1 (Zhou (2013)). Let m ≥ 1 be a given integer, and v ∈ R m ↔
be such that ∥v∥∞ ≤ 1, where m = m2m−1 . Let the elements in Dm be labeled as Di , i ∈ I[1, 2m ], and the function fm be defined as fm (0) = 0 and fm (i) =
fm (i − 1) + 1, fm (j),
Di + Dj ̸= Im , Di + Dj = Im ,
∀j ∈ I[1, i] ∃j ∈ I[1, i].
Then for any u ∈ Rm , there holds sat (u) ∈ co Di u + Di− v : i ∈ I[1, 2m ] ,
↔
where ‘‘co’’ denotes the convex hull, and Di− ∈ Rm× m is defined as − Di− = e2m−1 ,fm (i) ⊗ D− i with Di = I − Di . ↔
↔
Assume that there exist matrices U ∈ R m ×n , V ∈ R m ×n and ↔ m ×n
W ∈R such that the restrictive condition ∥v(t )∥∞ = ∥Ux(t )+ Vx(t − h(t )) + Wx(t − h)∥∞ ≤ 1 holds for t ≥ 0, then it follows from (1) and (3), and Lemma 1 that the closed-loop system can be written as x˙ (t ) =
2m
λi (t ) A + B(Di K + Di− U ) x(t )
i=1
+ (Ad + BDi− V )x(t − h(t )) + BDi− Wx(t − h) + C x˙ (t − h(t )) , χ (t ), where λ1 (t ) ≥ 0, . . . , λ2m (t ) ≥ 0 and
2m
i =1
(5)
λi (t ) = 1.
Remark 1. In Cao et al. (2002), Fridman et al. (2003) and Zhang et al. (2008), the auxiliary feedback of the from v(t ) = Hx(t ) was introduced under the assumption that |hl x(t )| ≤ u¯ l , l ∈ [1, m]. Different from the techniques in Cao et al. (2002), Fridman et al. (2003) and Zhang et al. (2008), the auxiliary time-delay feedback v(t ) = Ux(t )+ Vx(t − h(t ))+ Wx(t − h) is introduced in this paper. Compared with some existing results, our proposed stabilization conditions will be more slack due to the introduction of timedelay feedback matrices V and W , and thus the larger estimates of the domain of attraction can be obtained by the conditions in this paper. 3. Main results In this section, we will establish the improved local stabilization conditions in terms of LMIs. Theorem 1. For given scalars h, µ < 1 and δ ̸= 0, if there Q¯ Q¯ 12 exist symmetric matrices P¯ > 0, Q¯ 1 = ¯ 11 > 0, Q¯ 2 = T ¯
¯
Q21 T Q¯ 22
Q¯ 22 Q¯ 23
Q12
Q13
¯ k , N¯ k , k > 0, Z¯ > 0, and any matrices X , Y , G, H , L, M
244
Y. Chen et al. / Automatica 52 (2015) 242–247
↔
∈ I[1, 6], such that for ∀i ∈ I[1, 2m ], ∀l ∈ I[1, m], the following LMIs hold
¯T ¯ i hM Ω < 0, ∗ −hZ¯ 1 Gl Hl ¯ 1 −Q¯ 13 /h ∗ Θ ∗ ∗ Q¯ 13 /h ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯i Ω ∗
2[xT (t )T1 + x˙ T (t )T2 ][χ (t ) − x˙ (t )] = 0.
hN¯ T < 0, −hZ¯
(6)
Ll
0
0
−Q¯ 23 /h
¯ / −¯ /
¯2 Θ
T Q12 h T Q12 h
0 Q¯ 23 /h
0
∗ ∗
Q¯ 11 /h
0 ≥ 0, T ¯ −Q22 /h 0
V˙ (t ) ≤
i ¯ 12 Ω ¯ 22 Ω
i ¯ 13 Ω ¯ 23 Ω
i ¯ 14 Ω i ¯ 24 Ω
¯ 15 Ω ¯ 25 Ω
∗
¯ 33 Ω
−N¯ 5T
∗ ∗
∗ ∗
i ¯ 34 Ω ¯ 44 Ω
∗ ¯ = M ¯ 1T M
∗
N¯ = N¯ 1T
¯ 3T M
N¯ 2T
N¯ 3T
∗ ¯ 4T M
¯ 5T M
N¯ 4T
N¯ 5T
∗ ¯ 6T , M N¯ 6T ,
λi (t )ξ T (t ) Ωi + h(t )M T Z −1 M
i =1
=
2m
λi (t )ξ T (t ) θ1 (t )(Ωi + hM T Z −1 M )
i =1
+ θ2 (t )(Ωi + hN T Z −1 N ) ξ (t ),
, 0 0 −Q¯ 23
(12)
where
¯ 26 Ω ¯ 36 Ω
δ CX T ¯ 55 Ω
∗
∗
¯ 2T M
¯ 6T M
2m
+ (h − h(t ))N T Z −1 N ξ (t )
(7)
¯3 Θ
∗
(11)
Adding the left side of (11) to the time derivative of V (t ), and combining with (9)–(10), then one can obtain that
where Gl , Hl and Ll are the l-th rows of the matrices G, H and L, respectively, and
¯i Ω11 ∗ ∗ ¯i = Ω ∗ ∗
the following equation holds
i Ω11
i Ω12
i Ω13
i Ω14
Ω15
M6T
∗
Ω22
Ω23
i Ω24
Ω25
Ω33 ∗ ∗ ∗
i Ω34
N5T
Ω26 Ω36 , 0 0
Ωi =
∗ ∗ ∗ ∗
M1T
M =
N =
N1T
∗ ∗ ∗ ∗ M2T N2T
M3T
Ω44 ∗ ∗
M4T
N3T
−
N4T
T2 C
Ω55 ∗
M5T N5T
−Q23
M6T N6T
,
,
i ¯ 11 and Ω = AX T + B(Di Y + Di− G)+[AX T + B(Di Y + Di− G)]T + Q¯ 11 + i i ¯ ¯ ¯ 1T , Ω ¯ 2T − M ¯ 1 + N¯ 1 , Ω ¯ 12 ¯ 13 Q21 + M1 + M = Ad X T + BDi− H + M = − i T T T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ BDi L + M3 − N1 , Ω14 = P + Q12 + Q22 − X + M4 +δ[AX + B(Di Y + ¯ 5T , Ω ¯ 2 −M ¯ 2T + N¯ 2 + ¯ 15 = CX T + M ¯ 22 = −(1 − µ)Q¯ 11 − M Di− G)]T , Ω − i T ¯ T T T T ¯ ¯ ¯ ¯ ¯ 4T + ¯ 24 = δ(Ad X + BDi H ) − M N2 , Ω23 = −M3 + N3 − N2 , Ω T ¯ T T ¯ T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ N4 , Ω25 = −(1 − µ)Q12 − M5 + N5 , Ω26 = −M6 + N6 , Ω33 = i ¯ 34 ¯ 36 = −Q¯ 22 − N¯ 6T , Ω ¯ 44 = −Q¯ 21 − N¯ 3 − N¯ 3T , Ω = δ(BDi− L)T − N¯ 4T , Ω T ¯ 55 = −(1 −µ)Q¯ 13 , Θ ¯ 1 = P¯ + Q¯ 13 /h + hZ¯ + Q¯ 13 + Q¯ 23 −δ(X + X ), Ω T ¯ ,Θ ¯ 2 = Q¯ 22 Q¯ 23 /h + 2Z /h − 2Z¯ /h, Θ3 = Q¯ 21 /h + 2Z¯ /h2 , then for any initial function φ(t ) satisfying V (0) ≤ 1, where V (t ) is defined in (8),
i and Ω11 = T1 [A + B(Di K + Di U )] + [A + B(Di K + Di− U )]T T1T + i Q11 + Q21 + M1 + M1T , Ω12 = T1 (Ad + BDi− V ) + M2T − M1 + − i i T N1 , Ω13 = T1 BDi W + M3 − N1 , Ω14 = P + Q12 + Q22 − − T T T T1 + M4 + [A + B(Di K + Di U )] T2 , Ω15 = T1 C + M5T , Ω22 = i −(1 − µ)Q11 − M2 − M2T + N2 + N2T , Ω23 = −M3T + N3T − N2 , Ω24 = − T T T T T T (Ad + BDi V ) T2 − M4 + N4 , Ω25 = −(1 −µ)Q12 − M5 + N5 , Ω26 = i −M6T + N6T , Ω33 = −Q21 − N3 − N3T , Ω34 = (BDi− W )T T2T − T T N4 , Ω36 = −Q22 − N6 , Ω44 = hZ + Q13 + Q23 − T2 − T2T , Ω55 = −(1 − µ)Q13 , θ1 (t ) = h(t )/h, θ2 (t ) = (h − h(t ))/h. It is seen from (12) that if the following inequalities hold
the neutral time-delay system (1) can be asymptotically stabilized by the controller (3) with K = YX −T .
Ωi + hM T Z −1 M < 0,
Proof. Choose the following augmented L–K functional
then one can obtain that V˙ (t ) < 0, which gives that
V (t ) = x (t )Px(t ) + T
V (t ) < V (0),
t t −h(t )
ηT (s)Q1 η(s)ds
ηT (s)Q2 η(s)ds 0
t
x˙ T (s)Z x˙ (s)dsdθ ,
+ −h
where η(s) =
(8)
t +θ
x(s) x˙ (s)
, Qi =
Qi1
Qi2
T Qi2
Qi3
> 0, P > 0, i = 1, 2,
and Z > 0. For any matrices M and N, it follows from the fact ±2aT b ≤ aT Za + bT Z −1 b that
t
x˙ (s)Z x˙ (s)ds ≤ 2ξ (t )M T
− t −h(t )
T
T
t
x˙ (s)ds
t −h(t ) T −1
+ h(t )ξ (t )M Z M ξ (t ), t −h(t ) t −h(t ) − x˙ T (s)Z x˙ (s)ds ≤ 2ξ T (t )N T x˙ (s)ds T
t −h
t −h T
+ (h − h(t ))ξ (t )N Z T
−1
(9)
T
T
T
(14)
T
T
Θ
1
∗ where Θ = ∗ ∗ ∗
−Q13 /h Q13 /h ∗ ∗ ∗
−Q23 /h 0 Q23 /h
(15) T Q12 /h T −Q12 /h
Θ2 0
T −Q22 /h, and
0
Q11 /h
∗ ∗
ζ T (t )
0
Θ3 t ∗ t = xT (t ) xT (t − h(t )) xT (t − h) t −h(t ) xT (s)ds t −h xT (s)ds , T Θ1 = P + Q13 /h + Q23 /h + 2Z , Θ2 = Q22 /h − 2Z /h, Θ3 = 2 Q21 /h + 2Z /h . Assume that
FlT Fl ≤ Θ ,
↔
l ∈ I[1, m], Vl
Wl
(16) 0
0], then it can be seen that
|Ul x(t ) + Vl x(t − h(t )) + Wl x(t − h)|2
N ξ (t ),
(10)
T
x˙ T (t − h) . Also, for any matrices T1 and T2 , it is clear from (5) that
t ≥ 0.
where Fl = [Ul
where ξ (t ) = x (t ) x (t − h(t )) x (t − h) x˙ (t ) x˙ (t − h(t ))
(13)
V (t ) ≥ ζ T (t )Θ ζ (t ) ≥ xT (t )Px(t ),
t −h
Ωi + hN T Z −1 N < 0,
For the functional V (t ) defined in (8), using the Jensen integral inequalities, see Sun et al. (2009), one can obtain that
t
+
−
= ζ T (t )FlT Fl ζ (t ) ≤ ζ T (t )Θ ζ (t ),
↔
l ∈ I[1, m].
(17)
For any initial function φ(t ) satisfying V (0) ≤ 1, it is obvious from (14)–(15) and (17) that the assumption ∥Ux(t ) + Vx(t − h(t ))
Y. Chen et al. / Automatica 52 (2015) 242–247
+Wx(t −h)∥∞ ≤ 1 can be guaranteed. From the above discussions, it can be concluded that the closed-loop system (5) is locally asymptotically stable at the origin if the inequalities (13) and (16) hold. To obtain LMI-based conditions, one set T1 = T , T2 = δ T , δ ̸= 0, it is seen that the matrix T is invertible if the matrix inequalities in (13) are feasible. Therefore, it is reasonable to introduce the following new variables X = T −1 ,
P¯ = XPX T ,
Z¯ = XZX ,
Q¯ ij = XQij X T ,
¯ k = XMk X , M
T
Y = KX T ,
G = UX T ,
(18)
N¯ k = XNk X ,
T
T
H = VX T ,
(19)
L = WX T .
(20)
Performing some congruence transformations for (13) and (16) as in Zhang et al. (2008), and noting (18)–(20), then one can obtain the LMIs (6)–(7). This completes the proof. Remark 2. It can be seen that the accurate estimation of the lower bound of L–K functional (8) is performed and the inequality (17) is introduced to satisfy ∥v(t )∥∞ ≤ 1. It should be pointed out that the condition (17) is less conservative than the condition ε(P , 1) ⊆ L (U ) widely used in the existing references even for V = W = 0, where ε(P , 1) = {x ∈ Rn : xT Px ≤ 1} and L (U ) = {x ∈ Rn : ∥Ux∥∞ ≤ 1}. In fact, it is clear from (15) that the condition xT UlT Ul x ≤ xT Px is sufficient but not necessary for the condition xT (t )UlT Ul x(t ) ≤ ζ T (t )Θ ζ (t ), and meanwhile, it is easy to prove ↔
that ε(P , 1) ⊆ L (U ) is equivalent to xT UlT Ul x ≤ xT Px, l ∈ I[1, m]. For the case of constant delay, i.e., h(t ) = h, one can choose the following simple L–K functional V (t ) = xT (t )Px(t ) +
245
defined in (21), the neutral system (1) can be asymptotically stabilized by the controller (3) with K = YX −T . Remark 3. Compared with the functionals employed in Gomes da t Silva et al. (2005, 2011), the additional terms 2 t −h(t ) xT (s)Q12 x˙ (s)ds
t
t
xT (s)Q22 x˙ (s)ds and 2 t −h xT (s)Q12 x˙ (s)ds are introduced in (8) and (21), respectively. It should be pointed out that such terms are also effective in reducing the conservatism, which will be shown in Example 1.
+2
t −h
In the following part, we will discuss the estimation and optimization of the domain of attraction Xρ . Noting (18)–(20) and functional (8), it can be seen that the domain of attraction Xρ in Theorems 1–2 can be respectively bounded by the following inequalities
¯ −T )ρ12 + 0.5h2 λM (X −1 Z¯ X −T )ρ22 V (0) ≤ λM (X −1 PX + hΣi2=1 Σj2=1 λM (X −1 Λij X −T )ρj2 ≤ 1,
(24)
¯ −T )ρ12 + 0.5h3 λM (X −1 Z¯ X −T )ρ22 V (0) ≤ λM (X −1 PX + hΣj2=1 λM (X −1 Λ1j X −T )ρj2 ≤ 1,
(25)
where Q¯ i ≤ diag{Λi1 , Λi2 }, i = 1, 2. Now, we introduce the inequality X −1 X −T = (X T X )−1 ≤ rI. By the fact (X − α I )T (X − α I ) ≥ 0 and Schur complement, where α is a scalar, it can be deduced that X −1 X −T ≤ rI holds if the following LMI holds
rI I
I α(X + X T ) − α 2 I
≥ 0.
(26)
t
ηT (s)Q1 η(s)ds
Furthermore, we define the following matrix inequalities
t −h
0
t
x˙ T (s)Z x˙ (s)dsdθ .
+h −h
(21)
t +θ
Setting t W = 0 in (5), and using the well-known inequality −h t −h x˙ T (s)Z x˙ (s)ds ≤ −[xT (t ) − xT (t − h)]Z [x(t ) − x(t − h)], and along the proof of Theorem 1, then one can obtain the following simple stabilization condition. Theorem 2. For given scalars h and ̸ 0, if there exist symmetric ¯ δ = Q11 Q¯ 12 ¯ ¯ matrices P > 0, Q1 = ¯ T > 0, Z¯ > 0, and any matrices ¯ Q12
Q13
↔
X , Y , G, H, such that for ∀i ∈ I[1, 2 ], ∀l ∈ I[1, m], the following LMIs hold m
¯i Ξ11 ∗ ∗
i ¯ 12 Ξ −Q¯ 11 − Z¯
i ¯ 13 Ξ i ¯ 23 Ξ
∗
¯ 33 Ξ
∗
∗
∗ 1
Gl
Hl
∗ ∗ ∗
Ψ¯ 1
−Q¯ 13 /h
∗ ∗
Q¯ 13 /h
∗
CX T
0
−Q¯ 12 < 0, T δ CX −Q¯ 13
Ψ¯ 2 ≥ 0, T −Q¯ 12 /h
P¯ ≤ pI ,
Z¯ ≤ zI ,
Q¯ i ≤ diag{Λi1 , Λi2 }, Q¯ 1 ≤ diag{Λ11 , Λ12 },
(27)
Λij ≤ qij I , Λ1j ≤ q1j I ,
i, j = 1, 2,
(28)
j = 1, 2.
(29)
Similar to the techniques proposed in Tarbouriech and Gomes da Silva (2000) and Zhang et al. (2008), it can be seen that the maximization of the estimates of the domain of attraction Xρ in Theorems 1–2 can be respectively formulated as follows Pb.1.
min
¯ k ,N¯ k ,Λij P¯ ,Q¯ i ,Z¯ ,X ,Y ,G,H ,L,M
β,
s.t., ∀s ∈ I[1, 2m ],
∀l ∈ I[1, m], LMIs (6)–(7) and (26)–(28) hold, Pb.2. min γ , s.t., ∀s ∈ I[1, 2m ], P¯ ,Q¯ 1 ,Z¯ ,X ,Y ,G,H ,Λ1j
(22)
for all l ∈ I[1, m], LMIs (22)–(23), (26)–(27) and (29) hold, where β = e ∗ r + p + hΣi2=1 Σj2=1 qij + 0.5h2 z , γ = e ∗ r + p +
hΣj2=1 q1j + 0.5h3 z, and e denotes weighting parameter. (23)
Ψ¯ 3
i ¯ 11 where Ξ = [AX T + B(Di Y + Di− G)] + [AX T + B(Di Y + Di− G)]T + i i ¯ ¯ ¯ 12 Q11 − Z , Ξ = Ad X T + BDi− H + Z¯ , Ξ¯ 13 = P¯ + Q¯ 12 − X T + − T T ¯ i T δ[AX + B(Di Y + Di G)] , Ξ23 = δ(Ad X + BDi− H )T , Ξ¯ 33 = h2 Z¯ + T Q¯ 13 − δ(X + X T ), Ψ¯ 1 = P¯ + Q¯ 13 /h + 2hZ¯ , Ψ¯ 2 = Q¯ 12 /h − 2Z¯ , Ψ¯ 3 = ¯ ¯ Q11 /h + 2Z /h, then for any φ(t ) satisfying V (0) ≤ 1, where V (t ) is
Remark 4. It is seen that the Theorems 1–2 are based on the unity saturation level. For the case that non-unity saturation level occurs in system (1) with sat (uj ) = sgn(uj ) min{|uj |, u¯ j }, u¯ j > 0, j ∈ I[1, m], the corresponding conditions can also be applicable by substituting B˜ = [¯u1 b1 u¯ 2 b2 · · · u¯ m bm ]and Y˜ = [yT1 /¯u1 yT2 /¯u2 · · · yTm /¯um ]T for the matrices B and Y in Theorems 1–2, respectively, where bj is the jth column of B, and yj is the jth row of Y .
246
Y. Chen et al. / Automatica 52 (2015) 242–247
Fig. 1. The domain of attraction Xρ in Example 1.
Fig. 2. The domain of attraction Xρ in Example 2.
4. Numerical examples
paper can provide a larger estimate of the domain of attraction Xρ , which is also shown in Fig. 2.
Example 1 (Gomes da Silva et al., 2011). Consider the neutral timedelay system (1)–(2), where u¯ 1 = 15, h = 1, and
1 0.3
A=
c 0
C =
1.5 0 , Ad = −2 0 0 10 , B= .
c
−1 0
,
1
For this example, by solving Pb.1 with c = 0.2, µ = 0.1, δ = 1, α = 0.5 and e = 109 , then one can obtain 11.028ρ12 + 6.1828ρ22 ≤ 105 and controller gain K = [−0.2359 −0.0453]. In Gomes da Silva et al. (2005, 2011), the proposed initial conditions for guaranteeing the stability are respectively bounded by 51ρ12 + 9.34ρ22 ≤ 104 and 11.4ρ12 + 8.57ρ22 ≤ 105 . For the case that ρ1 = ρ2 , ρ , it is seen that the estimated domains of attraction Xρ by Pb.1 in this paper, and the results in Gomes da Silva et al. (2005, 2011) are bounded by ρ ≤ 76.2262, ρ ≤ 12.88 and ρ ≤ 70.74, respectively, which are also shown in Fig. 1. When the delay bound is set as h = 3.53, the proposed domain of attraction Xρ in Gomes da Silva et al. (2011) is bounded by ρ ≤ 62.8 × 10−3 , while by Pb.1 in this paper, one can obtain a larger bound ρ = 38.6276. By solving Pb.2 with c = µ = 0, δ = 1, α = 0.4 and e = 109 , the obtained upper bound of Xρ is ρ1 = ρ2 = ρ , 84.6074 with K = [−0.2223 −0.0246], which is larger than ρ = 67.0618 proposed in Cao et al. (2002) and ρ = 79.43 obtained in Fridman et al. (2003). By solving Pb.2 with c = 0.2, µ = 0, δ = 1, α = 0.5, e = 109 , one can obtain the upper bound ρ = 80.6453 of Xρ . While by setting H = 0 and Q¯ 12 = 0 in LMIs (23)–(24), respectively, the smaller bounds ρ = 68.9012 and ρ = 74.5137 are obtained, respectively, which implies that the auxiliary time-delay feedback matrix V and the additional t term 2 t −h xT (s)Q12 x˙ (s)ds are important in reducing the possible conservatism. Example 2 (Fridman et al., 2003). Consider the neutral time-delay system (1)–(2), where C = 0, u¯ 1 = 5, h = 1.854, µ = 0, and
A=
0.5 0.5
−1 , −0.5
Ad =
0.6 0
0.4 , −0.5
B=
1 . 1
For this example, by solving Pb.2 with α = 0.5, δ = 4.8 and e = 8000, one can obtain the bound ρ1 = ρ2 = ρ , 0.6348 of the domain of attraction Xρ , and gain matrix K = [−2.2346 0.0580]. Compared with ρ = 0.091 and ρ = 0.4521 proposed in Fridman et al. (2003) and Zhang et al. (2008), respectively, it is clear that this
5. Conclusion In this paper, by introducing the auxiliary time-delay feedback, and performing the accurate estimation of the lower bounds of L–K functionals, we have proposed the improved delaydependent stabilization conditions for neutral time-delay systems with actuator saturation in terms of LMIs. Different from the existing techniques, the time-delay information has been utilized sufficiently in this paper, which can result in the larger estimates of the domain of attraction. The proposed conditions of this paper have been well illustrated by numerical examples. References Alamo, T., Cepeda, A., & Limon, D. (2005). Improved computation of ellipsoidal invariant sets for saturated control systems. In Proceedings of the 44th IEEE conference on decision and control, Seville, Spain (pp. 6216–6221). Cao, Y.-Y., Lin, Z., & Hu, T. (2002). Stability analysis of linear time-delay systems subject to input saturation. IEEE Transactions on Circuits and Systems I, 49(2), 233–240. 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. (2001). New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. Systems & Control Letters, 43(4), 309–319. Fridman, E., Pila, A., & Shaked, U. (2003). Regional stabilization and H∞ control of time-delay systems with saturating actuators. International Journal of Robust and Nonlinear Control, 13(9), 885–907. Gomes da Silva, J. M. Jr., Seuret, A., Fridman, E., & Richard, J. P. (2005). Stabilization of neutral systems with saturating inputs. In Proceedings of the 16th IFAC world congress, Praga, Czech Republic (pp. 448–453). Gomes da Silva, J. M., Jr., Seuret, A., Fridman, E., & Richard, J. P. (2011). Stabilisation of neutral systems with saturating control inputs. International Journal of Systems Science, 42(7), 1093–1103. Gomes da Silva, J. M., Jr., & Tarbouriech, S. (2005). Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Transactions on Automatic Control, 50(1), 106–111. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Han, Q.-L. (2009). A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica, 45(2), 517–524. He, Y., Wang, Q.-G., Lin, C., & Wu, M. (2005). Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems. International Journal of Robust and Nonlinear Control, 15(18), 923–933. He, Y., Wang, Q.-G., Xie, L., & Lin, C. (2007). Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 52(2), 293–299. He, Y., Wu, M., She, J. H., & Liu, G. P. (2004). Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Systems & Control Letters, 51(1), 57–65. Hu, T., & Lin, Z. (2001). Control systems with actuator saturation: analysis and design. Boston: Birkhäuser. Hu, T., Lin, Z., & Chen, B. M. (2002). An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 38(2), 351–359.
Y. Chen et al. / Automatica 52 (2015) 242–247 Li, H., Jing, X., & Karimi, H. R. (2014). Output-feedback based H∞ control for vehicle suspension systems with control delay. IEEE Transactions on Industrial Electronics, 61(1), 436–446. Lin, Z. (1998). Low gain feedback. London: Springer-Verlag. Lin, Z., & Fang, H. (2007). On asymptotic stabilizability of linear systems with delayed input. IEEE Transactions on Automatic Control, 52(6), 998–1013. Qian, W., Liu, J., & Fei, S. (2012). New augmented Lyapunov functional method for stability of uncertain neutral systems with equivalent delays. Mathematics and Computers in Simulation, 84, 42–50. Sun, J., Liu, G. P., & Chen, J. (2009). Delay-dependent stability and stabilization of neutral time-delay systems. International Journal of Robust and Nonlinear Control, 19(1), 1364–1375. Tarbouriech, S., Garcia, G., Gomes da Silva, J. M., Jr., & Queinnec, I. (2011). Stability and stabilization of linear systems with saturating actuators. London: SpringerVerlag. Tarbouriech, S., & Gomes da Silva, J. M., Jr. (2000). Synthesis of controllers for continuous-time delay systems with saturating controls via LMI’s. IEEE Transactions on Automatic Control, 45(1), 105–111. Xu, S., & Jam, L. (2008). A survey of linear matrix inequality techniques in stability analysis of delay systems. International Journal of Systems Science, 39(12), 1095–1113. Yakoubi, K., & Chitour, Y. (2007). Linear systems subject to input saturation and time delay: global asymptotic stabilization. IEEE Transactions on Automatic Control, 52(5), 874–879. Zhang, L., Boukas, E.-K., & Haidar, A. (2008). Delay-range-dependent control synthesis for time-delay systems with actuator saturation. Automatica, 44(10), 2691–2695. Zhou, B. (2013). Analysis and design of discrete-time linear systems with nested actuator saturations. Systems & Control Letters, 62(10), 871–879. Zhou, B., Lin, Z., & Duan, G. (2008). A parametric Lyapunov equation approach to the design of low gain feedback. IEEE Transactions on Automatic Control, 53(6), 1548–1554. Zhou, B., Lin, Z., & Duan, G. (2010). Stabilization of linear systems with input delay and saturation—a parametric Lyapunov equation approach. International Journal of Robust and Nonlinear Control, 20(13), 1502–1519. Zhou, B., Lin, Z., & Duan, G. (2012). Truncated predictor feedback for linear systems with long time-varying input delays. Automatica, 48(10), 2387–2399.
247 Yonggang Chen received his B.Sc. and M.Sc. degrees in Mathematics from Henan Normal University in 2003 and 2006, respectively. He received his Ph.D. degree in Control Theory and Control Engineering from Southeast University, in 2013. He is currently a lecture of the School of Mathematical Sciences in Henan Institute of Science and Technology. His research interests include time-delay systems, constrained control and switched system control.
Shumin Fei received his M.S. degree in Mathematics from Anhui University, Ph.D. degree in Engineering from Beijing University of Aeronautics and Astronautics, in 1985 and 1995, respectively. From 1995 to 1997, he was a postdoctoral research fellow of the Research Institute in Southeast University. Presently, he is a professor of the College of Automation in Southeast University. His research interests include nonlinear system control, switched control, time-delay system and so on.
Yongmin Li received his B.S. in Mathematics from Shaanxi Normal University, M.S. in Operational Research and Cybernetics from Guizhou University and Ph.D. in Control Theory and Control Engineering from Nanjing University of Science and Technology, in 1992, 2002 and 2008 respectively. He is currently an associate professor of the School of Science, Huzhou Teachers College, Huzhou, China. His research interests include robust control, antiwindup design and time-delay systems.
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Brief paper
Stabilization of neutral time-delay systems with actuator saturation via auxiliary time-delay feedback✩ Yonggang Chen a , Shumin Fei b , Yongmin Li c a
School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, PR China
b
Key Laboratory of Measurement and Control of CSE (Ministry of Education), School of Automation, Southeast University, Nanjing 210096, PR China
c
School of Science, Huzhou Teachers College, Huzhou 313000, PR China
article
info
Article history: Received 9 December 2013 Received in revised form 17 September 2014 Accepted 9 November 2014
Keywords: Stabilization Neutral systems Actuator saturation Auxiliary time-delay feedback
abstract This paper investigates the stabilization problem for neutral time-delay systems with actuator saturation. Different from the existing techniques, the auxiliary time-delay feedback is introduced for the first time in this paper. Based on such a technique, the saturation nonlinearity is represented as the convex combination of state feedback and auxiliary time-delay feedback. By employing free-weighting matrix technique and Jensen integral inequalities, and performing the accurate estimation of the lower bounds of L–K functionals, the improved delay-dependent local stabilization conditions are proposed in terms of linear matrix inequalities (LMIs). Numerical examples illustrate the reduced conservatism of the proposed conditions in this paper. © 2014 Elsevier Ltd. All rights reserved.
1. Introduction Time-delays are frequently encountered in various practical systems, such as chemical engineering systems, biological systems and manufacturing processes (Gu, Kharitonov, & Chen, 2003). There are two types of time-delay systems, i.e., retarded type and neutral type. The retarded type contains delays only in its states, while the neutral type contains delays in both its states and its derivatives of the states. On the other hand, it is well recognized that LMI-based approaches are more convenient for solving corresponding synthesis problems, and delay-dependent results are generally less conservative than delay-independent ones especially when the size of delay is small (Xu & Jam, 2008). During the past two decades, several important techniques have been proposed to obtain LMI-based delay-dependent analysis and synthesis conditions for time-delay systems, see, e.g., Chen and Zheng (2007), Fridman (2001), Han (2009), He, Wang, Lin, and Wu (2005);
✩ This work was supported by the National Natural Science Foundations of China under Grants 61304061, 61273119, 61174076 and 61374086. This work was also partly supported by the Natural Science Foundation of Henan Institute of Science and Technology (201319). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Fen Wu under the direction of Editor Roberto Tempo. E-mail addresses: [email protected] (Y. Chen), [email protected] (S. Fei), [email protected] (Y. Li).
http://dx.doi.org/10.1016/j.automatica.2014.11.015 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
He, Wang, Xie, and Lin (2007); He, Wu, She, and Liu (2004), Li, Jing, and Karimi (2014), Qian, Liu, and Fei (2012) and Sun, Liu, and Chen (2009). In many practical control applications, actuator saturation is often inevitable, and its existence may deteriorate the performance of a control system and even cause the instability of closedloop system. Therefore, considerable attention has been devoted to linear systems subject to saturating controllers during the past decades, see e.g., Alamo, Cepeda, and Limon (2005), Gomes da Silva and Tarbouriech (2005); Hu and Lin (2001), Hu, Lin, and Chen (2002), Lin (1998), Tarbouriech, Garcia, Gomes da Silva, and Queinnec (2011), Zhou (2013) and Zhou, Lin, and Duan (2008). Generally speaking, the current research can be classified into two categories according to whether the open-loop poles are located on the closed left-half plane, i.e., global/semi-global stabilization, and local stabilization and anti-windup design. For the local stabilization and anti-windup design, two dominant approaches are proposed to deal with the saturation nonlinearity, one is the polytopic models (Alamo et al., 2005; Hu & Lin, 2001; Hu et al., 2002; Zhou, 2013) and the other is the generalized sector condition (Gomes da Silva & Tarbouriech, 2005). In particular, it is worth mentioning that the saturation representation proposed in Alamo et al. (2005) and Tarbouriech et al. (2011) with the compact notation proposed in Zhou (2013) contains more slack variables, and thus is less conservative than that in Hu and Lin (2001) and Hu et al. (2002) for the multiple input systems.
Y. Chen et al. / Automatica 52 (2015) 242–247
For the systems with both time-delay and actuator saturation, global/semi-global stabilization were well investigated in Lin and Fang (2007), Yakoubi and Chitour (2007), Zhou, Lin, and Duan (2010) and Zhou, Lin, and Duan (2012) under the assumption that the open-loop poles are located on the closed left-half plane. Removing such a restriction on open-loop poles, the problem of local stabilization has been widely studied in Cao, Lin, and Hu (2002), Fridman, Pila, and Shaked (2003), Gomes da Silva, Seuret, Fridman, and Richard (2005, 2011), Tarbouriech and Gomes da Silva (2000) and Zhang, Boukas, and Haidar (2008) by incorporating the techniques of analyzing the stability of time-delay systems. However, it should be pointed out that the techniques of representing saturation nonlinearity in the above references are the same as the cases without delay. It is clear that the time-delay information is completely neglected when dealing with the saturation nonlinearity, which may result in some conservative results. In this paper, we resist the stabilization problem for neutral systems with time-varying delay and actuator saturation. Different from the existing techniques, the auxiliary time-delay feedback is proposed in this paper. Based on the polytopic approach proposed in Alamo et al. (2005), Tarbouriech et al. (2011) and Zhou (2013), the saturation nonlinearly is firstly represented by the convex combination of state feedback and auxiliary time-delay feedback. By incorporating Lyapunov–Krasovskii (L–K) functional theory, free-weighting matrix technique and integral inequalities, and performing the accurate estimation of the lower bounds of L–K functionals, then the improved stabilization conditions are obtained in terms of LMIs. Compared with the existing results, the main novelty of this paper is that the auxiliary time-delay feedback is introduced for the first time when representing the saturation nonlinearity, and the accurate estimation of the lower bounds of L–K functionals is performed to obtain LMI-based conditions. Finally, the reduced conservatism of the proposed conditions in this paper is shown by numerical examples. Notation. λM (P ) denotes the maximum eigenvalue of matrix P. A real symmetric matrix P > 0(≥ 0) denotes P being a positive definite (positive semi-definite) matrix. I denotes an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The space of the continuously differentiable vector functions φ over [−h, 0] is denoted by C 1 [−h, 0]. ∥ · ∥ and ∥ · ∥∞ denote the 2-norm and ∞-norm, respectively, and maxt ∈[−h,0] ∥φ(t )∥ is denoted by ∥φ∥c . I[1, π] denotes the set of integers whose elements are 1, 2, . . . , π . Dm denotes the set of m × m diagonal matrices with diagonal elements either 1 or 0. em,k ∈ R1×m denotes a row vector whose k-th element is 1 and the others are zero, and ⊗ denotes the Kronecker product. 2. Problem formulation Consider the following neutral time-delay system with actuator saturation x˙ (t ) − C x˙ (t − h(t )) = Ax(t ) + Ad x(t − h(t )) + Bsat (u(t )),
(1)
x(t ) = φ(t ),
(2)
∀t ∈ [−h, 0],
where x(t ) ∈ Rn is the state vector, u(t ) ∈ Rm is the input vector, A, Ad , B and C are known real constant matrices with appropriate dimensions, h(t ) denotes time-varying delay that satisfies 0 ≤ h(t ) ≤ h and h˙ (t ) ≤ µ < 1. sat (u) : Rm → Rm is the vector valued standard saturation function described by sat (u) = [sat (u1 ) sat (u2 ) · · · sat (um )]T , where sat (uj ) = sgn(uj ) min{1, |uj |}, j ∈ I[1, m]. The controller used in this paper is the
243
following state feedback u(t ) = Kx(t ),
(3)
where K ∈ Rm×n is the gain matrix to be designed. In this paper, it is assumed that φ(t ) is continuously differentiable over [−h, 0], and one of our interests is to estimate the domain of attraction of the following form
˙ c ≤ ρ2 , Xρ = φ(t ) ∈ C 1 [−h, 0] : ∥φ∥c ≤ ρ1 , ∥φ∥
(4)
where ρ1 and ρ2 are some scalars to be maximized. ↔
Lemma 1 (Zhou (2013)). Let m ≥ 1 be a given integer, and v ∈ R m ↔
be such that ∥v∥∞ ≤ 1, where m = m2m−1 . Let the elements in Dm be labeled as Di , i ∈ I[1, 2m ], and the function fm be defined as fm (0) = 0 and fm (i) =
fm (i − 1) + 1, fm (j),
Di + Dj ̸= Im , Di + Dj = Im ,
∀j ∈ I[1, i] ∃j ∈ I[1, i].
Then for any u ∈ Rm , there holds sat (u) ∈ co Di u + Di− v : i ∈ I[1, 2m ] ,
↔
where ‘‘co’’ denotes the convex hull, and Di− ∈ Rm× m is defined as − Di− = e2m−1 ,fm (i) ⊗ D− i with Di = I − Di . ↔
↔
Assume that there exist matrices U ∈ R m ×n , V ∈ R m ×n and ↔ m ×n
W ∈R such that the restrictive condition ∥v(t )∥∞ = ∥Ux(t )+ Vx(t − h(t )) + Wx(t − h)∥∞ ≤ 1 holds for t ≥ 0, then it follows from (1) and (3), and Lemma 1 that the closed-loop system can be written as x˙ (t ) =
2m
λi (t ) A + B(Di K + Di− U ) x(t )
i=1
+ (Ad + BDi− V )x(t − h(t )) + BDi− Wx(t − h) + C x˙ (t − h(t )) , χ (t ), where λ1 (t ) ≥ 0, . . . , λ2m (t ) ≥ 0 and
2m
i =1
(5)
λi (t ) = 1.
Remark 1. In Cao et al. (2002), Fridman et al. (2003) and Zhang et al. (2008), the auxiliary feedback of the from v(t ) = Hx(t ) was introduced under the assumption that |hl x(t )| ≤ u¯ l , l ∈ [1, m]. Different from the techniques in Cao et al. (2002), Fridman et al. (2003) and Zhang et al. (2008), the auxiliary time-delay feedback v(t ) = Ux(t )+ Vx(t − h(t ))+ Wx(t − h) is introduced in this paper. Compared with some existing results, our proposed stabilization conditions will be more slack due to the introduction of timedelay feedback matrices V and W , and thus the larger estimates of the domain of attraction can be obtained by the conditions in this paper. 3. Main results In this section, we will establish the improved local stabilization conditions in terms of LMIs. Theorem 1. For given scalars h, µ < 1 and δ ̸= 0, if there Q¯ Q¯ 12 exist symmetric matrices P¯ > 0, Q¯ 1 = ¯ 11 > 0, Q¯ 2 = T ¯
¯
Q21 T Q¯ 22
Q¯ 22 Q¯ 23
Q12
Q13
¯ k , N¯ k , k > 0, Z¯ > 0, and any matrices X , Y , G, H , L, M
244
Y. Chen et al. / Automatica 52 (2015) 242–247
↔
∈ I[1, 6], such that for ∀i ∈ I[1, 2m ], ∀l ∈ I[1, m], the following LMIs hold
¯T ¯ i hM Ω < 0, ∗ −hZ¯ 1 Gl Hl ¯ 1 −Q¯ 13 /h ∗ Θ ∗ ∗ Q¯ 13 /h ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
¯i Ω ∗
2[xT (t )T1 + x˙ T (t )T2 ][χ (t ) − x˙ (t )] = 0.
hN¯ T < 0, −hZ¯
(6)
Ll
0
0
−Q¯ 23 /h
¯ / −¯ /
¯2 Θ
T Q12 h T Q12 h
0 Q¯ 23 /h
0
∗ ∗
Q¯ 11 /h
0 ≥ 0, T ¯ −Q22 /h 0
V˙ (t ) ≤
i ¯ 12 Ω ¯ 22 Ω
i ¯ 13 Ω ¯ 23 Ω
i ¯ 14 Ω i ¯ 24 Ω
¯ 15 Ω ¯ 25 Ω
∗
¯ 33 Ω
−N¯ 5T
∗ ∗
∗ ∗
i ¯ 34 Ω ¯ 44 Ω
∗ ¯ = M ¯ 1T M
∗
N¯ = N¯ 1T
¯ 3T M
N¯ 2T
N¯ 3T
∗ ¯ 4T M
¯ 5T M
N¯ 4T
N¯ 5T
∗ ¯ 6T , M N¯ 6T ,
λi (t )ξ T (t ) Ωi + h(t )M T Z −1 M
i =1
=
2m
λi (t )ξ T (t ) θ1 (t )(Ωi + hM T Z −1 M )
i =1
+ θ2 (t )(Ωi + hN T Z −1 N ) ξ (t ),
, 0 0 −Q¯ 23
(12)
where
¯ 26 Ω ¯ 36 Ω
δ CX T ¯ 55 Ω
∗
∗
¯ 2T M
¯ 6T M
2m
+ (h − h(t ))N T Z −1 N ξ (t )
(7)
¯3 Θ
∗
(11)
Adding the left side of (11) to the time derivative of V (t ), and combining with (9)–(10), then one can obtain that
where Gl , Hl and Ll are the l-th rows of the matrices G, H and L, respectively, and
¯i Ω11 ∗ ∗ ¯i = Ω ∗ ∗
the following equation holds
i Ω11
i Ω12
i Ω13
i Ω14
Ω15
M6T
∗
Ω22
Ω23
i Ω24
Ω25
Ω33 ∗ ∗ ∗
i Ω34
N5T
Ω26 Ω36 , 0 0
Ωi =
∗ ∗ ∗ ∗
M1T
M =
N =
N1T
∗ ∗ ∗ ∗ M2T N2T
M3T
Ω44 ∗ ∗
M4T
N3T
−
N4T
T2 C
Ω55 ∗
M5T N5T
−Q23
M6T N6T
,
,
i ¯ 11 and Ω = AX T + B(Di Y + Di− G)+[AX T + B(Di Y + Di− G)]T + Q¯ 11 + i i ¯ ¯ ¯ 1T , Ω ¯ 2T − M ¯ 1 + N¯ 1 , Ω ¯ 12 ¯ 13 Q21 + M1 + M = Ad X T + BDi− H + M = − i T T T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ BDi L + M3 − N1 , Ω14 = P + Q12 + Q22 − X + M4 +δ[AX + B(Di Y + ¯ 5T , Ω ¯ 2 −M ¯ 2T + N¯ 2 + ¯ 15 = CX T + M ¯ 22 = −(1 − µ)Q¯ 11 − M Di− G)]T , Ω − i T ¯ T T T T ¯ ¯ ¯ ¯ ¯ 4T + ¯ 24 = δ(Ad X + BDi H ) − M N2 , Ω23 = −M3 + N3 − N2 , Ω T ¯ T T ¯ T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ N4 , Ω25 = −(1 − µ)Q12 − M5 + N5 , Ω26 = −M6 + N6 , Ω33 = i ¯ 34 ¯ 36 = −Q¯ 22 − N¯ 6T , Ω ¯ 44 = −Q¯ 21 − N¯ 3 − N¯ 3T , Ω = δ(BDi− L)T − N¯ 4T , Ω T ¯ 55 = −(1 −µ)Q¯ 13 , Θ ¯ 1 = P¯ + Q¯ 13 /h + hZ¯ + Q¯ 13 + Q¯ 23 −δ(X + X ), Ω T ¯ ,Θ ¯ 2 = Q¯ 22 Q¯ 23 /h + 2Z /h − 2Z¯ /h, Θ3 = Q¯ 21 /h + 2Z¯ /h2 , then for any initial function φ(t ) satisfying V (0) ≤ 1, where V (t ) is defined in (8),
i and Ω11 = T1 [A + B(Di K + Di U )] + [A + B(Di K + Di− U )]T T1T + i Q11 + Q21 + M1 + M1T , Ω12 = T1 (Ad + BDi− V ) + M2T − M1 + − i i T N1 , Ω13 = T1 BDi W + M3 − N1 , Ω14 = P + Q12 + Q22 − − T T T T1 + M4 + [A + B(Di K + Di U )] T2 , Ω15 = T1 C + M5T , Ω22 = i −(1 − µ)Q11 − M2 − M2T + N2 + N2T , Ω23 = −M3T + N3T − N2 , Ω24 = − T T T T T T (Ad + BDi V ) T2 − M4 + N4 , Ω25 = −(1 −µ)Q12 − M5 + N5 , Ω26 = i −M6T + N6T , Ω33 = −Q21 − N3 − N3T , Ω34 = (BDi− W )T T2T − T T N4 , Ω36 = −Q22 − N6 , Ω44 = hZ + Q13 + Q23 − T2 − T2T , Ω55 = −(1 − µ)Q13 , θ1 (t ) = h(t )/h, θ2 (t ) = (h − h(t ))/h. It is seen from (12) that if the following inequalities hold
the neutral time-delay system (1) can be asymptotically stabilized by the controller (3) with K = YX −T .
Ωi + hM T Z −1 M < 0,
Proof. Choose the following augmented L–K functional
then one can obtain that V˙ (t ) < 0, which gives that
V (t ) = x (t )Px(t ) + T
V (t ) < V (0),
t t −h(t )
ηT (s)Q1 η(s)ds
ηT (s)Q2 η(s)ds 0
t
x˙ T (s)Z x˙ (s)dsdθ ,
+ −h
where η(s) =
(8)
t +θ
x(s) x˙ (s)
, Qi =
Qi1
Qi2
T Qi2
Qi3
> 0, P > 0, i = 1, 2,
and Z > 0. For any matrices M and N, it follows from the fact ±2aT b ≤ aT Za + bT Z −1 b that
t
x˙ (s)Z x˙ (s)ds ≤ 2ξ (t )M T
− t −h(t )
T
T
t
x˙ (s)ds
t −h(t ) T −1
+ h(t )ξ (t )M Z M ξ (t ), t −h(t ) t −h(t ) − x˙ T (s)Z x˙ (s)ds ≤ 2ξ T (t )N T x˙ (s)ds T
t −h
t −h T
+ (h − h(t ))ξ (t )N Z T
−1
(9)
T
T
T
(14)
T
T
Θ
1
∗ where Θ = ∗ ∗ ∗
−Q13 /h Q13 /h ∗ ∗ ∗
−Q23 /h 0 Q23 /h
(15) T Q12 /h T −Q12 /h
Θ2 0
T −Q22 /h, and
0
Q11 /h
∗ ∗
ζ T (t )
0
Θ3 t ∗ t = xT (t ) xT (t − h(t )) xT (t − h) t −h(t ) xT (s)ds t −h xT (s)ds , T Θ1 = P + Q13 /h + Q23 /h + 2Z , Θ2 = Q22 /h − 2Z /h, Θ3 = 2 Q21 /h + 2Z /h . Assume that
FlT Fl ≤ Θ ,
↔
l ∈ I[1, m], Vl
Wl
(16) 0
0], then it can be seen that
|Ul x(t ) + Vl x(t − h(t )) + Wl x(t − h)|2
N ξ (t ),
(10)
T
x˙ T (t − h) . Also, for any matrices T1 and T2 , it is clear from (5) that
t ≥ 0.
where Fl = [Ul
where ξ (t ) = x (t ) x (t − h(t )) x (t − h) x˙ (t ) x˙ (t − h(t ))
(13)
V (t ) ≥ ζ T (t )Θ ζ (t ) ≥ xT (t )Px(t ),
t −h
Ωi + hN T Z −1 N < 0,
For the functional V (t ) defined in (8), using the Jensen integral inequalities, see Sun et al. (2009), one can obtain that
t
+
−
= ζ T (t )FlT Fl ζ (t ) ≤ ζ T (t )Θ ζ (t ),
↔
l ∈ I[1, m].
(17)
For any initial function φ(t ) satisfying V (0) ≤ 1, it is obvious from (14)–(15) and (17) that the assumption ∥Ux(t ) + Vx(t − h(t ))
Y. Chen et al. / Automatica 52 (2015) 242–247
+Wx(t −h)∥∞ ≤ 1 can be guaranteed. From the above discussions, it can be concluded that the closed-loop system (5) is locally asymptotically stable at the origin if the inequalities (13) and (16) hold. To obtain LMI-based conditions, one set T1 = T , T2 = δ T , δ ̸= 0, it is seen that the matrix T is invertible if the matrix inequalities in (13) are feasible. Therefore, it is reasonable to introduce the following new variables X = T −1 ,
P¯ = XPX T ,
Z¯ = XZX ,
Q¯ ij = XQij X T ,
¯ k = XMk X , M
T
Y = KX T ,
G = UX T ,
(18)
N¯ k = XNk X ,
T
T
H = VX T ,
(19)
L = WX T .
(20)
Performing some congruence transformations for (13) and (16) as in Zhang et al. (2008), and noting (18)–(20), then one can obtain the LMIs (6)–(7). This completes the proof. Remark 2. It can be seen that the accurate estimation of the lower bound of L–K functional (8) is performed and the inequality (17) is introduced to satisfy ∥v(t )∥∞ ≤ 1. It should be pointed out that the condition (17) is less conservative than the condition ε(P , 1) ⊆ L (U ) widely used in the existing references even for V = W = 0, where ε(P , 1) = {x ∈ Rn : xT Px ≤ 1} and L (U ) = {x ∈ Rn : ∥Ux∥∞ ≤ 1}. In fact, it is clear from (15) that the condition xT UlT Ul x ≤ xT Px is sufficient but not necessary for the condition xT (t )UlT Ul x(t ) ≤ ζ T (t )Θ ζ (t ), and meanwhile, it is easy to prove ↔
that ε(P , 1) ⊆ L (U ) is equivalent to xT UlT Ul x ≤ xT Px, l ∈ I[1, m]. For the case of constant delay, i.e., h(t ) = h, one can choose the following simple L–K functional V (t ) = xT (t )Px(t ) +
245
defined in (21), the neutral system (1) can be asymptotically stabilized by the controller (3) with K = YX −T . Remark 3. Compared with the functionals employed in Gomes da t Silva et al. (2005, 2011), the additional terms 2 t −h(t ) xT (s)Q12 x˙ (s)ds
t
t
xT (s)Q22 x˙ (s)ds and 2 t −h xT (s)Q12 x˙ (s)ds are introduced in (8) and (21), respectively. It should be pointed out that such terms are also effective in reducing the conservatism, which will be shown in Example 1.
+2
t −h
In the following part, we will discuss the estimation and optimization of the domain of attraction Xρ . Noting (18)–(20) and functional (8), it can be seen that the domain of attraction Xρ in Theorems 1–2 can be respectively bounded by the following inequalities
¯ −T )ρ12 + 0.5h2 λM (X −1 Z¯ X −T )ρ22 V (0) ≤ λM (X −1 PX + hΣi2=1 Σj2=1 λM (X −1 Λij X −T )ρj2 ≤ 1,
(24)
¯ −T )ρ12 + 0.5h3 λM (X −1 Z¯ X −T )ρ22 V (0) ≤ λM (X −1 PX + hΣj2=1 λM (X −1 Λ1j X −T )ρj2 ≤ 1,
(25)
where Q¯ i ≤ diag{Λi1 , Λi2 }, i = 1, 2. Now, we introduce the inequality X −1 X −T = (X T X )−1 ≤ rI. By the fact (X − α I )T (X − α I ) ≥ 0 and Schur complement, where α is a scalar, it can be deduced that X −1 X −T ≤ rI holds if the following LMI holds
rI I
I α(X + X T ) − α 2 I
≥ 0.
(26)
t
ηT (s)Q1 η(s)ds
Furthermore, we define the following matrix inequalities
t −h
0
t
x˙ T (s)Z x˙ (s)dsdθ .
+h −h
(21)
t +θ
Setting t W = 0 in (5), and using the well-known inequality −h t −h x˙ T (s)Z x˙ (s)ds ≤ −[xT (t ) − xT (t − h)]Z [x(t ) − x(t − h)], and along the proof of Theorem 1, then one can obtain the following simple stabilization condition. Theorem 2. For given scalars h and ̸ 0, if there exist symmetric ¯ δ = Q11 Q¯ 12 ¯ ¯ matrices P > 0, Q1 = ¯ T > 0, Z¯ > 0, and any matrices ¯ Q12
Q13
↔
X , Y , G, H, such that for ∀i ∈ I[1, 2 ], ∀l ∈ I[1, m], the following LMIs hold m
¯i Ξ11 ∗ ∗
i ¯ 12 Ξ −Q¯ 11 − Z¯
i ¯ 13 Ξ i ¯ 23 Ξ
∗
¯ 33 Ξ
∗
∗
∗ 1
Gl
Hl
∗ ∗ ∗
Ψ¯ 1
−Q¯ 13 /h
∗ ∗
Q¯ 13 /h
∗
CX T
0
−Q¯ 12 < 0, T δ CX −Q¯ 13
Ψ¯ 2 ≥ 0, T −Q¯ 12 /h
P¯ ≤ pI ,
Z¯ ≤ zI ,
Q¯ i ≤ diag{Λi1 , Λi2 }, Q¯ 1 ≤ diag{Λ11 , Λ12 },
(27)
Λij ≤ qij I , Λ1j ≤ q1j I ,
i, j = 1, 2,
(28)
j = 1, 2.
(29)
Similar to the techniques proposed in Tarbouriech and Gomes da Silva (2000) and Zhang et al. (2008), it can be seen that the maximization of the estimates of the domain of attraction Xρ in Theorems 1–2 can be respectively formulated as follows Pb.1.
min
¯ k ,N¯ k ,Λij P¯ ,Q¯ i ,Z¯ ,X ,Y ,G,H ,L,M
β,
s.t., ∀s ∈ I[1, 2m ],
∀l ∈ I[1, m], LMIs (6)–(7) and (26)–(28) hold, Pb.2. min γ , s.t., ∀s ∈ I[1, 2m ], P¯ ,Q¯ 1 ,Z¯ ,X ,Y ,G,H ,Λ1j
(22)
for all l ∈ I[1, m], LMIs (22)–(23), (26)–(27) and (29) hold, where β = e ∗ r + p + hΣi2=1 Σj2=1 qij + 0.5h2 z , γ = e ∗ r + p +
hΣj2=1 q1j + 0.5h3 z, and e denotes weighting parameter. (23)
Ψ¯ 3
i ¯ 11 where Ξ = [AX T + B(Di Y + Di− G)] + [AX T + B(Di Y + Di− G)]T + i i ¯ ¯ ¯ 12 Q11 − Z , Ξ = Ad X T + BDi− H + Z¯ , Ξ¯ 13 = P¯ + Q¯ 12 − X T + − T T ¯ i T δ[AX + B(Di Y + Di G)] , Ξ23 = δ(Ad X + BDi− H )T , Ξ¯ 33 = h2 Z¯ + T Q¯ 13 − δ(X + X T ), Ψ¯ 1 = P¯ + Q¯ 13 /h + 2hZ¯ , Ψ¯ 2 = Q¯ 12 /h − 2Z¯ , Ψ¯ 3 = ¯ ¯ Q11 /h + 2Z /h, then for any φ(t ) satisfying V (0) ≤ 1, where V (t ) is
Remark 4. It is seen that the Theorems 1–2 are based on the unity saturation level. For the case that non-unity saturation level occurs in system (1) with sat (uj ) = sgn(uj ) min{|uj |, u¯ j }, u¯ j > 0, j ∈ I[1, m], the corresponding conditions can also be applicable by substituting B˜ = [¯u1 b1 u¯ 2 b2 · · · u¯ m bm ]and Y˜ = [yT1 /¯u1 yT2 /¯u2 · · · yTm /¯um ]T for the matrices B and Y in Theorems 1–2, respectively, where bj is the jth column of B, and yj is the jth row of Y .
246
Y. Chen et al. / Automatica 52 (2015) 242–247
Fig. 1. The domain of attraction Xρ in Example 1.
Fig. 2. The domain of attraction Xρ in Example 2.
4. Numerical examples
paper can provide a larger estimate of the domain of attraction Xρ , which is also shown in Fig. 2.
Example 1 (Gomes da Silva et al., 2011). Consider the neutral timedelay system (1)–(2), where u¯ 1 = 15, h = 1, and
1 0.3
A=
c 0
C =
1.5 0 , Ad = −2 0 0 10 , B= .
c
−1 0
,
1
For this example, by solving Pb.1 with c = 0.2, µ = 0.1, δ = 1, α = 0.5 and e = 109 , then one can obtain 11.028ρ12 + 6.1828ρ22 ≤ 105 and controller gain K = [−0.2359 −0.0453]. In Gomes da Silva et al. (2005, 2011), the proposed initial conditions for guaranteeing the stability are respectively bounded by 51ρ12 + 9.34ρ22 ≤ 104 and 11.4ρ12 + 8.57ρ22 ≤ 105 . For the case that ρ1 = ρ2 , ρ , it is seen that the estimated domains of attraction Xρ by Pb.1 in this paper, and the results in Gomes da Silva et al. (2005, 2011) are bounded by ρ ≤ 76.2262, ρ ≤ 12.88 and ρ ≤ 70.74, respectively, which are also shown in Fig. 1. When the delay bound is set as h = 3.53, the proposed domain of attraction Xρ in Gomes da Silva et al. (2011) is bounded by ρ ≤ 62.8 × 10−3 , while by Pb.1 in this paper, one can obtain a larger bound ρ = 38.6276. By solving Pb.2 with c = µ = 0, δ = 1, α = 0.4 and e = 109 , the obtained upper bound of Xρ is ρ1 = ρ2 = ρ , 84.6074 with K = [−0.2223 −0.0246], which is larger than ρ = 67.0618 proposed in Cao et al. (2002) and ρ = 79.43 obtained in Fridman et al. (2003). By solving Pb.2 with c = 0.2, µ = 0, δ = 1, α = 0.5, e = 109 , one can obtain the upper bound ρ = 80.6453 of Xρ . While by setting H = 0 and Q¯ 12 = 0 in LMIs (23)–(24), respectively, the smaller bounds ρ = 68.9012 and ρ = 74.5137 are obtained, respectively, which implies that the auxiliary time-delay feedback matrix V and the additional t term 2 t −h xT (s)Q12 x˙ (s)ds are important in reducing the possible conservatism. Example 2 (Fridman et al., 2003). Consider the neutral time-delay system (1)–(2), where C = 0, u¯ 1 = 5, h = 1.854, µ = 0, and
A=
0.5 0.5
−1 , −0.5
Ad =
0.6 0
0.4 , −0.5
B=
1 . 1
For this example, by solving Pb.2 with α = 0.5, δ = 4.8 and e = 8000, one can obtain the bound ρ1 = ρ2 = ρ , 0.6348 of the domain of attraction Xρ , and gain matrix K = [−2.2346 0.0580]. Compared with ρ = 0.091 and ρ = 0.4521 proposed in Fridman et al. (2003) and Zhang et al. (2008), respectively, it is clear that this
5. Conclusion In this paper, by introducing the auxiliary time-delay feedback, and performing the accurate estimation of the lower bounds of L–K functionals, we have proposed the improved delaydependent stabilization conditions for neutral time-delay systems with actuator saturation in terms of LMIs. Different from the existing techniques, the time-delay information has been utilized sufficiently in this paper, which can result in the larger estimates of the domain of attraction. The proposed conditions of this paper have been well illustrated by numerical examples. References Alamo, T., Cepeda, A., & Limon, D. (2005). Improved computation of ellipsoidal invariant sets for saturated control systems. In Proceedings of the 44th IEEE conference on decision and control, Seville, Spain (pp. 6216–6221). Cao, Y.-Y., Lin, Z., & Hu, T. (2002). Stability analysis of linear time-delay systems subject to input saturation. IEEE Transactions on Circuits and Systems I, 49(2), 233–240. 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. (2001). New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. Systems & Control Letters, 43(4), 309–319. Fridman, E., Pila, A., & Shaked, U. (2003). Regional stabilization and H∞ control of time-delay systems with saturating actuators. International Journal of Robust and Nonlinear Control, 13(9), 885–907. Gomes da Silva, J. M. Jr., Seuret, A., Fridman, E., & Richard, J. P. (2005). Stabilization of neutral systems with saturating inputs. In Proceedings of the 16th IFAC world congress, Praga, Czech Republic (pp. 448–453). Gomes da Silva, J. M., Jr., Seuret, A., Fridman, E., & Richard, J. P. (2011). Stabilisation of neutral systems with saturating control inputs. International Journal of Systems Science, 42(7), 1093–1103. Gomes da Silva, J. M., Jr., & Tarbouriech, S. (2005). Antiwindup design with guaranteed regions of stability: an LMI-based approach. IEEE Transactions on Automatic Control, 50(1), 106–111. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Han, Q.-L. (2009). A discrete delay decomposition approach to stability of linear retarded and neutral systems. Automatica, 45(2), 517–524. He, Y., Wang, Q.-G., Lin, C., & Wu, M. (2005). Augmented Lyapunov functional and delay-dependent stability criteria for neutral systems. International Journal of Robust and Nonlinear Control, 15(18), 923–933. He, Y., Wang, Q.-G., Xie, L., & Lin, C. (2007). Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Transactions on Automatic Control, 52(2), 293–299. He, Y., Wu, M., She, J. H., & Liu, G. P. (2004). Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Systems & Control Letters, 51(1), 57–65. Hu, T., & Lin, Z. (2001). Control systems with actuator saturation: analysis and design. Boston: Birkhäuser. Hu, T., Lin, Z., & Chen, B. M. (2002). An analysis and design method for linear systems subject to actuator saturation and disturbance. Automatica, 38(2), 351–359.
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247 Yonggang Chen received his B.Sc. and M.Sc. degrees in Mathematics from Henan Normal University in 2003 and 2006, respectively. He received his Ph.D. degree in Control Theory and Control Engineering from Southeast University, in 2013. He is currently a lecture of the School of Mathematical Sciences in Henan Institute of Science and Technology. His research interests include time-delay systems, constrained control and switched system control.
Shumin Fei received his M.S. degree in Mathematics from Anhui University, Ph.D. degree in Engineering from Beijing University of Aeronautics and Astronautics, in 1985 and 1995, respectively. From 1995 to 1997, he was a postdoctoral research fellow of the Research Institute in Southeast University. Presently, he is a professor of the College of Automation in Southeast University. His research interests include nonlinear system control, switched control, time-delay system and so on.
Yongmin Li received his B.S. in Mathematics from Shaanxi Normal University, M.S. in Operational Research and Cybernetics from Guizhou University and Ph.D. in Control Theory and Control Engineering from Nanjing University of Science and Technology, in 1992, 2002 and 2008 respectively. He is currently an associate professor of the School of Science, Huzhou Teachers College, Huzhou, China. His research interests include robust control, antiwindup design and time-delay systems.