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On stability of a class of switched nonlinear systems
Automatica 49 (2013) 305–307
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
On stability of a class of switched nonlinear systems✩ Yuangong Sun a,1 , Long Wang b a
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
b
Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
article
info
Article history: Received 29 February 2012 Received in revised form 9 July 2012 Accepted 28 September 2012 Available online 24 November 2012 Keywords: Stability Switched system Sector nonlinearity Disturbance input Delay
abstract This note is focused on the stability analysis for a class of switched nonlinear systems with disturbance input and delay. Sufficient conditions in terms of linear inequalities are presented such that the switched system is asymptotically stable for arbitrary switching, any admissible sector nonlinearities and disturbances, and any constant delay. We not only drop a condition of one result given in Aleksandrov, Chen, Platonov, and Zhang (2011), but also extend the main result to a more general switched nonlinear system with disturbance input and delay. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Recently, in Aleksandrov et al. (2011), the authors considered the following switched nonlinear system
The stability theory of switched systems has historically assumed a position of great importance in system theory and has been studied extensively in recent years (Decarlo, Branicky, Pettersson, & Lennartson, 2000; Liberzon & Morse, 1999; Lin & Antsaklis, 2009; Sun & Ge, 2011). For the stability analysis of switched linear systems under arbitrary switching, an usual method is to find a common quadratic Lyapunov function. Finding a common quadratic Lyapunov function for a family of subsystems is still an open problem, even though some progress has been made (Agrachev & Liberzon, 2001; Cheng, 2004; Cheng, Guo, & Huang, 2003). If the switched linear system is positive, a common linear copositive Lyapunov function is usually applied to the stability for any switching (Bundfuss & Dür, 2009; Fornasini & Valcher, 2010; Knorn, Mason, & Shorten, 2009; Liu & Dang, 2011; Sun, 2012). Compared with switched linear systems, the stability problem of switched positive nonlinear systems receives less attention (Zhao & Dimirovski, 2004).
x˙ = Aσ f (x),
✩ This work was supported by the National Natural Science Foundation of China
under grant nos. 60704039, 10972002 and 61174217, and the Natural Science Foundation of Shandong Province under grant nos. ZR2010AL002 and JQ201119. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nuno C. Martins under the direction of Editor André L. Tits. E-mail addresses: [email protected] (Y. Sun), [email protected] (L. Wang). 1 Tel.: +86 531 82769115; fax: +86 531 82769115. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.10.011
(1)
where x = (x1 , x2 , . . . , xn )T is the state vector, the piecewise constant function σ : [0, ∞) → N = {1, 2, . . . , N } is the switching (k) signal, N is a positive integer, Ak = [aij ], k ∈ N , are n × n-
dimensional constant matrices, f (x) = (f1 (x1 ), f2 (x2 ), . . . , fn (xn ))T , and the scalar functions fi (i = 1, 2, . . . , n), which are called admissible sector nonlinearities, are continuous for |xi | ≤ H, 0 < H ≤ +∞, and satisfy xi f (xi ) > 0 for xi ̸= 0. By using the Lyapunov direct approach, the absolute stability of the individual system shown in (1) was studied in Barbashin (1970), Kazkurewicz and Bhaya (1999) and Persidskij (1969). (k) Under the assumption that aij ≥ 0 for i ̸= j and k ∈ N , i.e., Ak is a Metzler matrix, it was proved that the kth subsystem of (1) is absolutely stable if and only if Ak is Hurwitz. In the general case (k) without assuming that aij ≥ 0 for i ̸= j and k ∈ N , it is sufficient to find a common Lyapunov function to guarantee the stability of the switched system (1) for arbitrary switching (Liberzon & Morse, 1999). In fact, based on the method given in Barbashin (1970), one can choose the following Lyapunov function V (x) =
n i=1
λi
xi
fi (s)ds,
(2)
0
where λi > 0 are constants for i = 1, 2, . . . , n. The existence of such a common Lyapunov function of form (2) can be determined by a set of linear matrix inequalities: ATk Λ + ΛAk < 0,
k ∈ N,
where Λ = diag{λ1 , . . . , λn }.
306
Y. Sun, L. Wang / Automatica 49 (2013) 305–307
In a recent paper (Aleksandrov et al., 2011), the common Lyapunov function of form (2) has been extended to the following general form: V (x) =
n
λi
xi
γ
fi (s)ds,
(3)
0
i=1
where λ1 , . . . , λn are defined as above, γ > 0 is a given rational with odd numerator and denominator. It was shown that in many cases, the Lyapunov function (3) may be more effective than the Lyapunov function (2) (Aleksandrov et al., 2011). Throughout this note, say A ≻ 0 (≺ 0) if all elements of matrix A are positive (negative). In Aleksandrov et al. (2011), it was proved that the existence of a common Lyapunov function of form (3) can be determined by two auxiliary systems of inequalities A¯ k θ ≺ 0,
ξ A¯ k ≺ 0, T
k ∈ N, k ∈ N,
n
≤
x˙ = Aσ f (x) + Bσ f (x(t − τ )) + Dσ w,
(6) (k)
where f , σ and Aσ are defined as in (1), Bk = [bij ] and Dk = [d(ijk) ] are respectively n × n-dimensional and n × m-dimensional constant matrices for k ∈ N , τ > 0 is a constant delay, the m-dimensional vector w = [w1 , w2 , . . . , wm ]T is the disturbance input ∞ and satisfies w ∈ {w(t ) : w is bounded on [0, ∞), and ∥w(t )∥dt < ∞}. When we do not take the time delay into 0 account, system (6) takes the form: x˙ = Aσ f (x) + Dσ w.
(7)
For convenience, we say system (6) is absolutely stable if each solution of system (6) satisfies limt →∞ ∥x(t )∥ = 0 for arbitrary switching, any admissible sector nonlinearities and disturbance input, and any constant delay. Say system (7) is absolutely stable if each solution of system (7) satisfies limt →∞ ∥x(t )∥ = 0 for arbitrary switching and any admissible sector nonlinearities and disturbance input. In this note, by using some different analytical techniques, we first show that condition (5) is sufficient to ensure that system (1) is absolutely stable for arbitrary switching and any admissible sector nonlinearities. That is, another condition (4) for absolute stability of system (1) is redundant. We also extend the main result to the more general switched time-delay system (6).
(k)
a¯ ij |fj (xj (t ))| +
j =1
m
|d(ijk) ||wj (t )|.
j =1
(ii) When xi (t ) = 0, noting that fi (xi (t )) = 0, we have D+ |xi (t )| = |D+ xi (t )|
n m (k) (k) = aij fj (xj (t )) + dij wj (t ) j =1 j=1 n
≤
(5)
system (1) is asymptotically stable for any switching and any admissible sector nonlinearities. Generally, it does not follow that the positive solution also exists for the the inequality (5) even when the inequality (4) exists a positive solution. We here further consider the following switched nonlinear system with disturbance input and delay
(k)
a¯ ij |fj (xj (t ))| +
j =1
m
|d(ijk) ||wj (t )|.
j =1
This completes the proof of Lemma 1.
Theorem 1. If there exists an n-dimensional vector ξ = (ξ1 , . . . , ξn )T ≻ 0 such that (5) holds, then system (7) is absolutely stable. Proof. Define a common Lyapunov function as follows: V (t ) =
n
ξi |xi (t )|.
(9)
i=1
For any t ≥ 0, assume that σ (t ) = k ∈ N . Calculating the right derivative V (t + h) − V (t )
D+ V = lim
h
h→0+
along solutions of the system (7) under arbitrary switching and any admissible sector nonlinearities and disturbance input, we get from (15) that D+ V (t ) ≤
n n
ξi a¯ (ijk) |fj (xj (t ))| +
n m
ξi |d(ijk) ||wj (t )|
i =1 j =1
i =1 j =1
= ξ A¯ k f¯ (x) + m(t ), T
(10)
where f¯ (x) = (|f1 (x1 )|, . . . , |fn (xn )|)T , m(t ) =
n m
ξi |d(ijk) ||wj (t )|.
i =1 j =1
(k)
(k)
Set η(k) = −ξ T A¯ k = (η1 , . . . , ηn )T and η = (η1 , . . . , ηn )T with (k)
ηi = min{ηi : k ∈ N } for i = 1, . . . , n. It is obvious that η ≻ 0 due to (5). Then, we can get from (10) that D+ V (t ) ≤ −ηT f¯ (x) + m(t ),
t ≥ 0.
(11)
If follows by an integration of (11) that
2. Main results We first establish a sufficient condition for the absolute stability of system (7). For the particular case when w ≡ 0, (7) reduces to (1). The following Lemma is required. Lemma 1. Assume that x(t ) is a solution of system (7). Then we have n
D+ |xi (t )| = D+ xi (t ) sign xi (t )
(4)
(k) (k) where the n-dimensional vectors θ ≻ 0, ξ ≻ 0, A¯ k = [¯aij ], a¯ ii = (k) (k) (k) aii and a¯ ij = |aij | for i ̸= j, i, j = 1, 2, . . . , n. Consequently,
D+ |xi (t )| ≤
Proof. Noting that |xi | is not differentiable at 0, we divide the proof into the following two cases. (i) When xi (t ) ̸= 0, by the fact that fi (xi ) sign xi = |fi (xi )| and fj (xj ) sign xi ≤ |fj (xj )| for i ̸= j, we have
(k)
a¯ ij |fj (xj (t ))| +
j =1
where D+ |xi (t )| = limh→0+
m
|dij(k) ||wj (t )|,
(8)
V (t ) +
h
η f¯ (x(s))ds ≤ V (0) + T
0
t
m(s)ds.
(12)
0
By the assumption on the disturbance input w , we have that m(t ) ∞ is bounded on [0, ∞) and 0 m(s)ds < ∞. Consequently, we get from (9) and (12) that each solution of system (7) is bounded on [0, ∞), and ∞
ηT f¯ (x(s))ds < ∞.
j =1
|xi (t +h)|−|xi (t )|
t
(13)
0
denotes the right deriva(k)
tive of |xi (t )| for i = 1, 2, . . . , n, σ (t ) = k ∈ N , and a¯ ij is defined as above.
On the other hand, the boundedness of the function m(t ) and solutions of system (7) imply that D+ xi is bounded on [0, ∞). Thus, xi is uniformly continuous on [0, ∞) for i = 1, 2, . . . , n. By the
Y. Sun, L. Wang / Automatica 49 (2013) 305–307
continuity of the function f¯ , it is not difficult to further conclude that ηT f¯ (x(t )) is uniformly continuous on [0, ∞). Such a uniform continuity together with (13) will imply that ηT f¯ (x(t )) → 0 as t → +∞. Consequently, xi (t ) → 0 for i = 1, . . . , n as t → +∞. This completes the proof of Theorem 1. Remark 1. For the case when w ≡ 0, we drop condition (4) in Theorem 1 that is required in Aleksandrov et al. (2011). Next, we extend Theorem 1 to the switched delay system (6). (k) (k) (k) (k) Denote A˜ k = [˜aij ], where a˜ ij = a¯ ij + b˜ ij , a¯ ij is defined as above, (k)
and b˜ ij = max{|bij | : k ∈ N } for i, j = 1, . . . , n. Similar to the proof of Lemma 1, the following lemma can be obtained immediately. Lemma 2. Assume that x(t ) is a solution of system (6). Then we have D+ |xi | ≤
n [¯a(ijk) |fj (xj )| + |b(ijk) ∥fj (xj (t − τ ))|]
+
|d(ijk) ∥wj |,
i = 1, 2, . . . , n,
j =1
where σ (t ) = k ∈ N for t ∈ [0, ∞). Theorem 2. If there exists an n-dimensional vector ξ = (ξ1 , . . . , ξn )T ≻ 0 such that
ξ T A˜ k ≺ 0,
k ∈ N,
(14)
then system (6) is absolutely stable. Proof. Choose a common Lyapunov function of the following form: V (t ) =
n
ξi |xi | +
n
i=1
j =1
b˜ ij |fj (xj (s))|ds . t −τ
Based on the definition of b˜ ij and by Lemma 2, along the solution of system (6) we get D+ V (t ) ≤
n
ξi a¯ (ijk) |fj (xj )| + |b(ijk) ||fj (xj (t − τ ))|
i,j=1
+
n m
ξi |d(ijk) | |wj (t )|
i=1 j=1
+
n
ξi b˜ ij [|fj (xj )| − |fj (xj (t − τ ))|]
i,j=1
≤ ξ T A˜ k f¯ (x) + m(t ) = −ζ T f¯ (x) + m(t ),
(15) (k)
where ζ = (ζ1 , ζ2 , . . . , ζn ) with ζi = min{ζi T
: k ∈ N }, ζi(k)
is the ith entry of ξ T A˜ k , and m(t ) is defined as in Theorem 1. By integrating on both sides of (15) and by the definition of V (t ), we have that n i =1
ξi |xi | +
t
ζ T f¯ (x(s))ds ≤ V (0) + 0
t
Based on the above analysis, Theorem 2 can also be extended to the following time-varying nonlinear system x˙ = A(t )f (x) + B(t )f (x(t − τ )) + D(t )w,
(16)
where A(t ) = [aij (t )], B(t ) = [bij (t )] and D(t ) = [dij (t )] are piecewise continuous matrix functions of appropriate dimensions on [0, ∞), f , τ and w are defined as above. In fact, if we assume that aii (t ) ≤ aˆ ii , |aij (t )| ≤ aˆ ij for i ̸= j,
|bij (t )| ≤ bˆ ij , dij (t ) ≤ dˆ ij for i, j = 1, 2, . . . , n, where aˆ ij , bˆ ij and dˆ ij are constants, then Lemma 2 is also valid. Therefore, similar to the proof of Theorem 2, we can conclude that system (16) is absolutely stable if there exists an n-dimensional vector ξ ≻ 0 such that ξ T (Aˆ + Bˆ ) ≺ 0, where Aˆ = [ˆaij ] and Bˆ = [bˆ ij ].
In this note, we further study the stability of a class of switched nonlinear systems with disturbance input and delay. By choosing an appropriate common Lyapunov function, sufficient conditions in terms of linear inequalities are established to guarantee that each solution of the switched system converges to zero for any switching signal, any admissible sector nonlinearities and disturbance input, and any constant delay. Even when the switched system reduces to the particular case, one of the conditions given in a recent paper (Aleksandrov et al., 2011) is dropped. The method used in this note can also be applied to a class of time-varying nonlinear system with disturbance input and delay. References
t
(5) holds if and only if all possible matrices A¯ π := [col1 (A¯ π (1) ), . . . , coln (A¯ π (n) )] are Hurwitz for every map π : {1, . . . , n} → N .
3. Conclusions
j =1 m
307
m(s)ds. 0
The remaining details of proof are similar to that of Theorem 1 and we will omit the details. Remark 2. Generally speaking, linear inequality conditions (5) and (14) in Theorems 1 and 2 can be tested by some results given in Fornasini and Valcher (2010) and Knorn et al. (2009). For example, there exists an n-dimensional vector ξ ≻ 0 such that
Agrachev, A. A., & Liberzon, D. (2001). Lie-algebraic stability criteria for switched systems. SIAM Journal on Control and Optimization, 40(1), 253–269. Aleksandrov, A. Yu., Chen, Y., Platonov, A. V., & Zhang, L. (2011). Stability analysis for a class of switched nonlinear systems. Automatica, 47(10), 2286–2291. Barbashin, E. A. (1970). Introduction to the theory of stability. Groningen: WoltersNoordhoff Publishing. Bundfuss, S., & Dür, M. (2009). Copositive Lyapunov functions for switched systems over cones. Systems & Control Letters, 58(5), 342–345. Cheng, D. (2004). Stabilization of planar switched systems. Systems & Control Letters, 51(2), 79–88. Cheng, D., Guo, L., & Huang, J. (2003). On quadratic functions. IEEE Transactions on Automatic Control, 48(5), 885–890. Decarlo, R. A., Branicky, M. S., Pettersson, S., & Lennartson, B. (2000). Perspectives and results on the stability and stabilizability of hybrid systems. Hybrid Systems, Proceedings of the IEEE, 1069–1082 (special issue). Fornasini, E., & Valcher, M. E. (2010). Linear copositive Lyapunov functions for continuous-time positive switched systems. IEEE Transactions on Automatic Control, 55(8), 1933–1937. Kazkurewicz, E., & Bhaya, A. (1999). Matrix diagonal stability in systems and computation. Boston, Basel, Berlin: Birkhauser. Knorn, F., Mason, O., & Shorten, R. (2009). On linear co-positive Lyapunov functions for sets of linear positive systems. Automatica, 45(8), 1943–1947. Liberzon, D., & Morse, A. S. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19(5), 59–70. Lin, H., & Antsaklis, P. J. (2009). Stability and stabilizability of switched linear systems: a survey of recent results. IEEE Transactions on Automatic Control, 54(2), 308–322. Liu, X., & Dang, C. (2011). Stability analysis of positive switched linear systems with delays. IEEE Transactions on Automatic Control, 56(7), 1684–1689. Persidskij, S. K. (1969). Problem of absolute stability. Avtomatika i Telemekhanika, (12), 5–11 (in Russian); English translation: (1970). Automation and Remote Control. 1889–1895. Sun, Y. (2012). Delay-independent stability of switched linear systems with unbounded time-varying delays. Abstract and Applied Analysis, 11. http://dx.doi.org/10.1155/2012/560897. Article ID 560897. Sun, Z., & Ge, S. S. (2011). Stability theory of switched dynamical systems. London: Springer-Verlag. Zhao, J., & Dimirovski, G. M. (2004). Quadratic stability of a class of switched nonlinear systems. IEEE Transactions on Automatic Control, 49(4), 574–578.
Contents lists available at SciVerse ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
On stability of a class of switched nonlinear systems✩ Yuangong Sun a,1 , Long Wang b a
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
b
Center for Systems and Control, College of Engineering, Peking University, Beijing 100871, China
article
info
Article history: Received 29 February 2012 Received in revised form 9 July 2012 Accepted 28 September 2012 Available online 24 November 2012 Keywords: Stability Switched system Sector nonlinearity Disturbance input Delay
abstract This note is focused on the stability analysis for a class of switched nonlinear systems with disturbance input and delay. Sufficient conditions in terms of linear inequalities are presented such that the switched system is asymptotically stable for arbitrary switching, any admissible sector nonlinearities and disturbances, and any constant delay. We not only drop a condition of one result given in Aleksandrov, Chen, Platonov, and Zhang (2011), but also extend the main result to a more general switched nonlinear system with disturbance input and delay. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Recently, in Aleksandrov et al. (2011), the authors considered the following switched nonlinear system
The stability theory of switched systems has historically assumed a position of great importance in system theory and has been studied extensively in recent years (Decarlo, Branicky, Pettersson, & Lennartson, 2000; Liberzon & Morse, 1999; Lin & Antsaklis, 2009; Sun & Ge, 2011). For the stability analysis of switched linear systems under arbitrary switching, an usual method is to find a common quadratic Lyapunov function. Finding a common quadratic Lyapunov function for a family of subsystems is still an open problem, even though some progress has been made (Agrachev & Liberzon, 2001; Cheng, 2004; Cheng, Guo, & Huang, 2003). If the switched linear system is positive, a common linear copositive Lyapunov function is usually applied to the stability for any switching (Bundfuss & Dür, 2009; Fornasini & Valcher, 2010; Knorn, Mason, & Shorten, 2009; Liu & Dang, 2011; Sun, 2012). Compared with switched linear systems, the stability problem of switched positive nonlinear systems receives less attention (Zhao & Dimirovski, 2004).
x˙ = Aσ f (x),
✩ This work was supported by the National Natural Science Foundation of China
under grant nos. 60704039, 10972002 and 61174217, and the Natural Science Foundation of Shandong Province under grant nos. ZR2010AL002 and JQ201119. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nuno C. Martins under the direction of Editor André L. Tits. E-mail addresses: [email protected] (Y. Sun), [email protected] (L. Wang). 1 Tel.: +86 531 82769115; fax: +86 531 82769115. 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.10.011
(1)
where x = (x1 , x2 , . . . , xn )T is the state vector, the piecewise constant function σ : [0, ∞) → N = {1, 2, . . . , N } is the switching (k) signal, N is a positive integer, Ak = [aij ], k ∈ N , are n × n-
dimensional constant matrices, f (x) = (f1 (x1 ), f2 (x2 ), . . . , fn (xn ))T , and the scalar functions fi (i = 1, 2, . . . , n), which are called admissible sector nonlinearities, are continuous for |xi | ≤ H, 0 < H ≤ +∞, and satisfy xi f (xi ) > 0 for xi ̸= 0. By using the Lyapunov direct approach, the absolute stability of the individual system shown in (1) was studied in Barbashin (1970), Kazkurewicz and Bhaya (1999) and Persidskij (1969). (k) Under the assumption that aij ≥ 0 for i ̸= j and k ∈ N , i.e., Ak is a Metzler matrix, it was proved that the kth subsystem of (1) is absolutely stable if and only if Ak is Hurwitz. In the general case (k) without assuming that aij ≥ 0 for i ̸= j and k ∈ N , it is sufficient to find a common Lyapunov function to guarantee the stability of the switched system (1) for arbitrary switching (Liberzon & Morse, 1999). In fact, based on the method given in Barbashin (1970), one can choose the following Lyapunov function V (x) =
n i=1
λi
xi
fi (s)ds,
(2)
0
where λi > 0 are constants for i = 1, 2, . . . , n. The existence of such a common Lyapunov function of form (2) can be determined by a set of linear matrix inequalities: ATk Λ + ΛAk < 0,
k ∈ N,
where Λ = diag{λ1 , . . . , λn }.
306
Y. Sun, L. Wang / Automatica 49 (2013) 305–307
In a recent paper (Aleksandrov et al., 2011), the common Lyapunov function of form (2) has been extended to the following general form: V (x) =
n
λi
xi
γ
fi (s)ds,
(3)
0
i=1
where λ1 , . . . , λn are defined as above, γ > 0 is a given rational with odd numerator and denominator. It was shown that in many cases, the Lyapunov function (3) may be more effective than the Lyapunov function (2) (Aleksandrov et al., 2011). Throughout this note, say A ≻ 0 (≺ 0) if all elements of matrix A are positive (negative). In Aleksandrov et al. (2011), it was proved that the existence of a common Lyapunov function of form (3) can be determined by two auxiliary systems of inequalities A¯ k θ ≺ 0,
ξ A¯ k ≺ 0, T
k ∈ N, k ∈ N,
n
≤
x˙ = Aσ f (x) + Bσ f (x(t − τ )) + Dσ w,
(6) (k)
where f , σ and Aσ are defined as in (1), Bk = [bij ] and Dk = [d(ijk) ] are respectively n × n-dimensional and n × m-dimensional constant matrices for k ∈ N , τ > 0 is a constant delay, the m-dimensional vector w = [w1 , w2 , . . . , wm ]T is the disturbance input ∞ and satisfies w ∈ {w(t ) : w is bounded on [0, ∞), and ∥w(t )∥dt < ∞}. When we do not take the time delay into 0 account, system (6) takes the form: x˙ = Aσ f (x) + Dσ w.
(7)
For convenience, we say system (6) is absolutely stable if each solution of system (6) satisfies limt →∞ ∥x(t )∥ = 0 for arbitrary switching, any admissible sector nonlinearities and disturbance input, and any constant delay. Say system (7) is absolutely stable if each solution of system (7) satisfies limt →∞ ∥x(t )∥ = 0 for arbitrary switching and any admissible sector nonlinearities and disturbance input. In this note, by using some different analytical techniques, we first show that condition (5) is sufficient to ensure that system (1) is absolutely stable for arbitrary switching and any admissible sector nonlinearities. That is, another condition (4) for absolute stability of system (1) is redundant. We also extend the main result to the more general switched time-delay system (6).
(k)
a¯ ij |fj (xj (t ))| +
j =1
m
|d(ijk) ||wj (t )|.
j =1
(ii) When xi (t ) = 0, noting that fi (xi (t )) = 0, we have D+ |xi (t )| = |D+ xi (t )|
n m (k) (k) = aij fj (xj (t )) + dij wj (t ) j =1 j=1 n
≤
(5)
system (1) is asymptotically stable for any switching and any admissible sector nonlinearities. Generally, it does not follow that the positive solution also exists for the the inequality (5) even when the inequality (4) exists a positive solution. We here further consider the following switched nonlinear system with disturbance input and delay
(k)
a¯ ij |fj (xj (t ))| +
j =1
m
|d(ijk) ||wj (t )|.
j =1
This completes the proof of Lemma 1.
Theorem 1. If there exists an n-dimensional vector ξ = (ξ1 , . . . , ξn )T ≻ 0 such that (5) holds, then system (7) is absolutely stable. Proof. Define a common Lyapunov function as follows: V (t ) =
n
ξi |xi (t )|.
(9)
i=1
For any t ≥ 0, assume that σ (t ) = k ∈ N . Calculating the right derivative V (t + h) − V (t )
D+ V = lim
h
h→0+
along solutions of the system (7) under arbitrary switching and any admissible sector nonlinearities and disturbance input, we get from (15) that D+ V (t ) ≤
n n
ξi a¯ (ijk) |fj (xj (t ))| +
n m
ξi |d(ijk) ||wj (t )|
i =1 j =1
i =1 j =1
= ξ A¯ k f¯ (x) + m(t ), T
(10)
where f¯ (x) = (|f1 (x1 )|, . . . , |fn (xn )|)T , m(t ) =
n m
ξi |d(ijk) ||wj (t )|.
i =1 j =1
(k)
(k)
Set η(k) = −ξ T A¯ k = (η1 , . . . , ηn )T and η = (η1 , . . . , ηn )T with (k)
ηi = min{ηi : k ∈ N } for i = 1, . . . , n. It is obvious that η ≻ 0 due to (5). Then, we can get from (10) that D+ V (t ) ≤ −ηT f¯ (x) + m(t ),
t ≥ 0.
(11)
If follows by an integration of (11) that
2. Main results We first establish a sufficient condition for the absolute stability of system (7). For the particular case when w ≡ 0, (7) reduces to (1). The following Lemma is required. Lemma 1. Assume that x(t ) is a solution of system (7). Then we have n
D+ |xi (t )| = D+ xi (t ) sign xi (t )
(4)
(k) (k) where the n-dimensional vectors θ ≻ 0, ξ ≻ 0, A¯ k = [¯aij ], a¯ ii = (k) (k) (k) aii and a¯ ij = |aij | for i ̸= j, i, j = 1, 2, . . . , n. Consequently,
D+ |xi (t )| ≤
Proof. Noting that |xi | is not differentiable at 0, we divide the proof into the following two cases. (i) When xi (t ) ̸= 0, by the fact that fi (xi ) sign xi = |fi (xi )| and fj (xj ) sign xi ≤ |fj (xj )| for i ̸= j, we have
(k)
a¯ ij |fj (xj (t ))| +
j =1
where D+ |xi (t )| = limh→0+
m
|dij(k) ||wj (t )|,
(8)
V (t ) +
h
η f¯ (x(s))ds ≤ V (0) + T
0
t
m(s)ds.
(12)
0
By the assumption on the disturbance input w , we have that m(t ) ∞ is bounded on [0, ∞) and 0 m(s)ds < ∞. Consequently, we get from (9) and (12) that each solution of system (7) is bounded on [0, ∞), and ∞
ηT f¯ (x(s))ds < ∞.
j =1
|xi (t +h)|−|xi (t )|
t
(13)
0
denotes the right deriva(k)
tive of |xi (t )| for i = 1, 2, . . . , n, σ (t ) = k ∈ N , and a¯ ij is defined as above.
On the other hand, the boundedness of the function m(t ) and solutions of system (7) imply that D+ xi is bounded on [0, ∞). Thus, xi is uniformly continuous on [0, ∞) for i = 1, 2, . . . , n. By the
Y. Sun, L. Wang / Automatica 49 (2013) 305–307
continuity of the function f¯ , it is not difficult to further conclude that ηT f¯ (x(t )) is uniformly continuous on [0, ∞). Such a uniform continuity together with (13) will imply that ηT f¯ (x(t )) → 0 as t → +∞. Consequently, xi (t ) → 0 for i = 1, . . . , n as t → +∞. This completes the proof of Theorem 1. Remark 1. For the case when w ≡ 0, we drop condition (4) in Theorem 1 that is required in Aleksandrov et al. (2011). Next, we extend Theorem 1 to the switched delay system (6). (k) (k) (k) (k) Denote A˜ k = [˜aij ], where a˜ ij = a¯ ij + b˜ ij , a¯ ij is defined as above, (k)
and b˜ ij = max{|bij | : k ∈ N } for i, j = 1, . . . , n. Similar to the proof of Lemma 1, the following lemma can be obtained immediately. Lemma 2. Assume that x(t ) is a solution of system (6). Then we have D+ |xi | ≤
n [¯a(ijk) |fj (xj )| + |b(ijk) ∥fj (xj (t − τ ))|]
+
|d(ijk) ∥wj |,
i = 1, 2, . . . , n,
j =1
where σ (t ) = k ∈ N for t ∈ [0, ∞). Theorem 2. If there exists an n-dimensional vector ξ = (ξ1 , . . . , ξn )T ≻ 0 such that
ξ T A˜ k ≺ 0,
k ∈ N,
(14)
then system (6) is absolutely stable. Proof. Choose a common Lyapunov function of the following form: V (t ) =
n
ξi |xi | +
n
i=1
j =1
b˜ ij |fj (xj (s))|ds . t −τ
Based on the definition of b˜ ij and by Lemma 2, along the solution of system (6) we get D+ V (t ) ≤
n
ξi a¯ (ijk) |fj (xj )| + |b(ijk) ||fj (xj (t − τ ))|
i,j=1
+
n m
ξi |d(ijk) | |wj (t )|
i=1 j=1
+
n
ξi b˜ ij [|fj (xj )| − |fj (xj (t − τ ))|]
i,j=1
≤ ξ T A˜ k f¯ (x) + m(t ) = −ζ T f¯ (x) + m(t ),
(15) (k)
where ζ = (ζ1 , ζ2 , . . . , ζn ) with ζi = min{ζi T
: k ∈ N }, ζi(k)
is the ith entry of ξ T A˜ k , and m(t ) is defined as in Theorem 1. By integrating on both sides of (15) and by the definition of V (t ), we have that n i =1
ξi |xi | +
t
ζ T f¯ (x(s))ds ≤ V (0) + 0
t
Based on the above analysis, Theorem 2 can also be extended to the following time-varying nonlinear system x˙ = A(t )f (x) + B(t )f (x(t − τ )) + D(t )w,
(16)
where A(t ) = [aij (t )], B(t ) = [bij (t )] and D(t ) = [dij (t )] are piecewise continuous matrix functions of appropriate dimensions on [0, ∞), f , τ and w are defined as above. In fact, if we assume that aii (t ) ≤ aˆ ii , |aij (t )| ≤ aˆ ij for i ̸= j,
|bij (t )| ≤ bˆ ij , dij (t ) ≤ dˆ ij for i, j = 1, 2, . . . , n, where aˆ ij , bˆ ij and dˆ ij are constants, then Lemma 2 is also valid. Therefore, similar to the proof of Theorem 2, we can conclude that system (16) is absolutely stable if there exists an n-dimensional vector ξ ≻ 0 such that ξ T (Aˆ + Bˆ ) ≺ 0, where Aˆ = [ˆaij ] and Bˆ = [bˆ ij ].
In this note, we further study the stability of a class of switched nonlinear systems with disturbance input and delay. By choosing an appropriate common Lyapunov function, sufficient conditions in terms of linear inequalities are established to guarantee that each solution of the switched system converges to zero for any switching signal, any admissible sector nonlinearities and disturbance input, and any constant delay. Even when the switched system reduces to the particular case, one of the conditions given in a recent paper (Aleksandrov et al., 2011) is dropped. The method used in this note can also be applied to a class of time-varying nonlinear system with disturbance input and delay. References
t
(5) holds if and only if all possible matrices A¯ π := [col1 (A¯ π (1) ), . . . , coln (A¯ π (n) )] are Hurwitz for every map π : {1, . . . , n} → N .
3. Conclusions
j =1 m
307
m(s)ds. 0
The remaining details of proof are similar to that of Theorem 1 and we will omit the details. Remark 2. Generally speaking, linear inequality conditions (5) and (14) in Theorems 1 and 2 can be tested by some results given in Fornasini and Valcher (2010) and Knorn et al. (2009). For example, there exists an n-dimensional vector ξ ≻ 0 such that
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