韓彥武,湯紅吉
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一類具有2個加性變時滯的系統(tǒng)的指數(shù)穩(wěn)定性分析
韓彥武,湯紅吉
(南通大學理學院,江蘇南通 226019)
考慮了一類具有2個加性變時滯的系統(tǒng)的指數(shù)穩(wěn)定性問題.通過把時滯區(qū)間分別分成2個小區(qū)間,構造一個適當?shù)腖yapunov-Krasovskii泛函(LKF),該LKF整體正定,不要求每一部分正定.運用積分不等式和倒凸組合的方法,得出了系統(tǒng)指數(shù)穩(wěn)定的充分條件,并以線性矩陣不等式的形式表示.數(shù)值實例表明了該方法的有效性.
加性變時滯;時滯分解;指數(shù)穩(wěn)定;倒凸組合
時滯廣泛存在于各類系統(tǒng)中,如生物系統(tǒng)、神經(jīng)網(wǎng)絡和網(wǎng)絡化控制系統(tǒng)等.時滯的存在可能會引發(fā)系統(tǒng)振蕩甚至使系統(tǒng)失穩(wěn),因此時滯系統(tǒng)的穩(wěn)定性分析成為系統(tǒng)理論領域的熱點問題之一[1-14].文獻[1-2]構造了包含三重積分的增廣LKF,得出了較好的結(jié)果;文獻[3-4]在LKF求導時,利用Newton-Leibniz公式,引入了自由權矩陣;文獻[5-7]利用積分不等式、凸組合或倒凸組合得出了時滯系統(tǒng)穩(wěn)定的充分條件;文獻[8-9]利用時滯分解的方法,分析了系統(tǒng)的穩(wěn)定性.
本文針對一類加性變時滯系統(tǒng),研究其指數(shù)穩(wěn)定性問題.運用時滯分解的方法,把時滯區(qū)間進行分解(可以是平均分解,也可以是不平均分解),構造一個適當?shù)腖KF,利用積分不等式和倒凸組合的方法,得出系統(tǒng)指數(shù)漸近穩(wěn)定的充分條件,并以線性矩陣不等式的形式表示.
考慮具有2個加性變時滯的系統(tǒng)
證明構造Lyapunov-Krasovskii泛函(LKF)
注1通常,LKF表示為若干正定二次型和的形式,這樣可以保證LKF的正定性.但在定理中,不需要是正定矩陣,由式(4)保證了LKF(6)的正定性.
注3由于本文考慮的是指數(shù)穩(wěn)定性問題,所以在定理中,為便于估計,需要()是正定矩陣,若只考慮漸近穩(wěn)定問題,則只需要是對稱矩陣[14]756.
表1 對于給定的,的最大值
表1 對于給定的,的最大值
方法來源 1 1.2 1.5方法來源 1 1.2 1.5 文獻[10]0.4150.3760.248文獻[13]0.8730.6730.373 文獻[11]0.5120.4060.283文獻[14]0.9880.8360.563 文獻[12]0.5830.5190.421定理1.1260.9440.652
表2 對于給定的和,的最大值
表2 對于給定的和,的最大值
k 1 1.2 1.5 0.050.8730.6770.366 0.10.6770.4700.148
由表1可以看出,與文獻[10-14]相比較,利用本文時滯分解方法可以得出較好的結(jié)果.
本文主要研究了具有2個加性變時滯系統(tǒng)的指數(shù)穩(wěn)定性問題.綜合利用時滯分解、積分不等式和倒凸組合技巧,得出系統(tǒng)指數(shù)穩(wěn)定的充分條件,并用LIMs表示.數(shù)值實例說明了本文時滯分解方法的有效性.
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Exponential stability analysis for a class of system with two-additive time-varying delays
HAN Yan-wu,TANG Hong-ji
(School of Science,Nantong University,Nantong 226019,China)
Deals with the exponential stability analysis of dynamic systems with two additive time-varying delay.By decomposing one delay interval into two subintervals which may be unequal,an appropriate Lyapunov-Krasovskii functional(LKF)is constructed whose each term is not positive definite while the the sum of each term is positive definite.The integral inequality method and the reciprocally convex technique are utilized to deal with the derivative of the LKF.The delay-dependent exponential stability criterion obtained from this method is expressed in terms of the linear matrix inequalities(LMIs).Anumerical example is used to show the effectiveness of this method.
additive time-varying delay;delay decomposing;exponential stability;reciprocally convex technique
1007-9831(2016)11-0001-05
O231
A
10.3969/j.issn.1007-9831.2016.11.001
2016-09-05
國家自然科學基金資助項目(61273013,61374061)
韓彥武(1977-),男,黑龍江依蘭人,講師,碩士,從事微分方程理論與應用研究.E-mail:ntuhyw@163.com