控制时滞系统输出反馈的能稳性.doc
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1、精品论文控制时滞系统输出反馈的能稳性蒋威安徽大学数学学院,合肥230039摘要: 本文研究了一个控制时滞系统的能稳性。 首先通过一个变换把含有控制时滞的系统转 变为没有控制时滞的系统,给出了这两种系统之间的关系。 然后讨论了没有控制时滞系统的输 出反馈的能稳性,给出了具有控制时滞系统能稳性的条件。 关键词:控制时滞系统;能稳性;输出反馈中图分类号: O175The stabilizability of control delay systems by output feedbackJIANG WeiSchool of Mathematical Sciences, Anhui Universit
2、y, Hefei 230039Abstract:In this paper, the stabilizability of a system with control delay will be discussed. Firstly a transformation is given, by which the discussed system with control delay is transformed to the system without control delay. The relationship of stabilizability of these two kind s
3、ystem is given. Then the problem of output feedback stabilizability of the system without control delay is discussed and the conditions for the stabilizability of the system with control delay are given.Key words: The system with control delay; Stabilizability; Output feedback.0 IntroductionIn many
4、practical systems, such as industry systems, economic systems, biological systems and so on, due to the signal transmission or the mechanical transmission need a length of time, the control variable must have time delay. For instance, in the space-flight industry systems, the order we give cannot be
5、 received by space-flight immediately. That is this order was gotten and executed by space-flight after a period of time. The period of this time is the time delay. This kind of system is the system with control delay. Usually, the mathematics model of such基金项目: National Natural Science Foundation o
6、f China ( No.11071001),the Doctoral Fund of Ministry of Education of China (No.20093401110001 ) and Ma jor Program of Educational Commission of Anhui Province of China (No. KJ2010ZD02).作者简介: Jiang Wei(1959-),male,professor,ma jor research direction:Functional differential equations andcontrol system
7、s. Correspondence author:Jiang Wei.system should be written as the form as x (t) = f (x(t), u(t), u(t h), t t0 ,x(t0 ) = x0 , u(t) = (t), t0 h t t0 ,(1)where x(t) Rn is the state vector, u(t) Rm the control vector, u() an admissible control ( that is, it is contained in the square integrable functio
8、ns L2 on every finite interval), h 0 is the control time delay, (t) is the initial control function and x0 is the initial state value. The linear form of the system (1) isx (t) =Ax(t) + Bu(t) + Cu(t h),t t0 ,x(t0 ) =x0 , u(t) = (t), t0 h t t0 ,(2)where A Rnn , and B, C Rnm are constant matrices.Gene
9、rally, the output equation of the system (2) can be written asy(t) = Ex(t).(3)here y(t) is the output vector, E Rln is a constant matrix.In 1, the function-controllability of nonlinear differential systems with state and control delay were investigated. In Ref.2, the controllability of a fractional
10、control system with control delay was discussed. In Refs. 3- 10, many time delay systems have been discussed. The most of the systems which were considered have as the form as( x (t) = f (x(t), x(t ), u(t), t t0 ,x(t) = (t),t0 t t0 ,(4)where 0 is the state time delay and (t) is the initial state fun
11、ction. For the controlproblems of these systems they obtained many consequences.For the stabilization, references 11 - 13 gave some results. But in 11, the authors just got the sufficient conditions for the stabilizability via state feedback of the system (2). In 12 and 13, the authors just consider
12、ed the output feedback stabilization of the control systems without delay.In this paper, we will discuss the stabilizability of the system (2) with control delay. Firstly we give a transformation, by which we transform the discussed system (2) with control delay into a system without control delay.
13、We obtain the relationship of stabilizability of these two kind systems. Then we discuss the stabilizability via output feedback of the system without control delay and give the conditions for the stabilizability of the system with control delay.1 An equivalent system of the system (2)Now we give an
14、 equivalent system for the system (2) with control delay. Let(t) = x(t) +then the system (2) can be written asZ ttheA(t+h) Cu()d,(5)Ah(t) = A(t) + (B + eC)u(t), t t0 ,0 0 t (t ) = x + R t00 h eA(t0 +h)C()d,(6) u(t) = (t), t0 h t t0 .Letthen we havez(t) =4 y(t) + EZ t theA(t+h) Cu()d,(7)z(t) = E(t).(
15、8)We can simply rewrite (2) (3) as( x (t) = Ax(t) + Bu(t) + Cu(t h), y(t) = Ex(t),(9)and (6) (8) as( (t) = A(t) + (B + eAh C)u(t), z(t) = E(t).(10)Theorem 1 If the system (9) is output feedback stabilizable, then the system (10) is stabilizable. On the other hand, if the system (10) is output feedba
16、ck stabilizable, then the system (9) is also stabilizable.Proof Firstly, if the system (9) is stabilizable via output feedback, there exist a matrixK such that the controlu(t) = Ky(t)(11)is the stabilizing control of system (9). Namely when t , x(t) 0.From (5), (9) and (11), we have|(t)| |x(t)| + h
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- 控制 系统 输出 反馈 能稳性
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