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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
17、maxh0|eA | |C| |KE| |xt |. (12)There xt = x(t + ), h, 0, and|xt | = maxh0|x(t + )|.(13)From (12) (13), we have (t) 0 as x(t) 0. Thus the system (10) is stabilizable.On the contrary, if the system (10) is stabilizable via output feedback, there exists a matrixK such that the controlu(t) = K (t)(14)is
18、 the stabilizing control of system (10). Namely, when t , (t) 0.From (5) and (10) (14), we have|x(t)| |(t)| + h maxh0|eA | |C| |K E| |t |. (15)There t = (t + ), h, 0, and|t | = maxh0|(t + )|.(16)From (15) (16), we have x(t) 0 as (t) 0. That is the system (9) is stabilizable. Thiscompletes the proof.
19、Remark For general control systems, the immediate controller u(t) can relatively easy come true. That is why many systems havent or need not have the term u(t). We can say that no matter B = 0 or B = 0, all the results of this section are true.2 Stabilizability of the system with control delayFor th
20、e system (9) with B = 0, that is the system( x (t) = Ax(t) + Cu(t h), y(t) = Ex(t),(17)we consider linear quadratic cost with cross termZ W =(x0 (t)Qx(t) + 2x0 (t)Su(t h) + u0 (t)Ru(t)dt, (18)t0here Q 0, R 0, and S are constant weighting matrices of compatible dimensions satisfying inequalityFor the
21、 system (17), we have thatQ S0 R1 S 0.(19)Z t+Thenx(t + ) = eA x(t) +teA(t) Cu( h)d. (20)tx(t + h) = eAh x(t) + R t+h eA(t) Cu( h)d= eAh x(t) + R ttheA(t+h) Cu()d(21)= eAh (t),where (t) satisfyWe can divide (18) asW = R t0 +h 0( (t) = A(t) + eAh Cu(t), z(t) = E(t).0(22)t0 (x (t)Qx(t) + 2x (t)Su(t h)
22、dt(23)+ R 000t0 (x (t + h)Qx(t + h) + 2x (t + h)Su(t) + u (t)Ru(t)dt.But from (20) we know that when 0, hZ t0 +x(t0 + ) = eA x0 +t0eA(t0 ) C( h)d.That is u(t), t t0 cannot control x(t), t t0 , t0 + h.LetR t0 +hW =1 t0(x0 (t)Qx(t) + 2x0 (t)Su(t h)dt,(24)We have thatW2 = R (x0 (t + h)Qx(t + h) + 2x0 (
23、t + h)Su(t) + u0 (t)Ru(t)dt.t0From (21) and (24) we haveZ minu(t),tt00W = W1 + minu(t),tt00W2 .W2 =(0 (t)eA h QeAh (t) + 20 (t)eA h Su(t) + u0 (t)Ru(t)dt. (25)t0From 13, we have the following lemma.Lemma 1 Consider the system( x (t) = Ax(t) + Bu(t), y(t) = Cx(t).(26)Let Ei = C+ C; Ek = I Ei . If the
24、re exist matrix L of compatible dimensions and matricesQ 0, R 0, S 0 inZ V =(x0 (t)Qx(t) + 2x0 (t)Su(t) + u0 (t)Ru(t)dt (27)t0such that the constrained algebraic matrix equationA0 P + P A Ei (P B + L0 )R1 (B0 P + L)Ei + Q = 0, Q S0 R1 S 0,S = LEi B0 P Ekhas a unique solution P 0, then the system (26
25、) is stabilizable via output feedback.(28)Theorem 2 For the system (17), let Ei = E+ E;Ek = I Ei . If there exist matrix Lof compatible dimensions and matrices Q 0, R 0, S 0 in (18) such that the constrained algebraic matrix equation0A0 P + P A Ei (P eAh C + L0 )R1 (C0 eA h P + L)EieA0 h QeAh S0 eAh
26、 R1 eA0 h S 0,+ eA0 h QeAh = 0,(29)0ieA0 h S = LE C0 eA h P Ekhas an unique solution P 0, then the system (17) is stabilizable.Proof Comparing (22), (25) with (26), (27) and by lemma 1, we have that the system (22) is output feedback stabilizable. From theorem 1, we get that the theorem is true. Thi
27、s completes the proof.参考文献（References）1 Jiang Wei, Song WenZhong and Fei Shumin. The function-controllability of the nonlinear control systems with state and control delayJ. London, World Scientific, 2000, 143148.2 Jiang Wei. The controllability of fractional control systems with control delayJ. Com
28、put- ers and Mathematics with Applications, 2012, 64(10):3153-3159.3 Jiang Wei. The constant variation formulae for singular fractional differential systems with delayJ. Computers and Mathematics with Applications, 2010, 59(3): 1184-1190.4 J. Hale. Introduction to Functional Differential EquationsM.
29、 Springer-Verlay, Berlin, NewYork, 1992.5 Zheng Zuxiu. Theory of Functional Differential EquationsM. AnHui Education Press, Hefei, 1994.6 L.Dugard and E.I.Verriest(Eds). Stability and Control of Time-delay SystemsM. Springer- Verlag, Berlin, Heidelberg New York, 1998.7 Jie Chen. On computing the max
30、imal delay intervals for stability of linear delay systemsJ.IEEE Transactions on Automatic Control, 40(1995):1087-1093.8 Jiang Wei and WenZhong Song. Controllability of sinular systems with control delayJ.Automatica, 2001, 37(11):1873-1877.9 Jiang Wei. Eigenvalue and stability of singular differenti
31、al delay systemsJ. Journal ofMathematical Analysis and Applications, 2004, 297:305- 316.10 B. Ahmad, S. Sivasundaram. Instability of nonautonomous state-dependent delay integro- differential systemsJ. Nonlinear Analysis: RealWorld Applications, 2006, 7:662 673.11 W.H. Kwon and A.W. Pearson. Feedback
32、 stabilization of linear systems with delayed controlJ. IEEE Transactions on Automatic Control, 1980, 25(2):266-269.12 A.Trofino-Neto and V.Kucera. Stabilization via static output feedbackJ. IEEE Transac- tions on Automatic Control, 1993, 38(5):764-765.13 Yong-Yan Cao, You-Xian, and Wei-Jie Mao. A new necessary and sufficient condition for static output feedback stabilizability and comments on ”stabilization via static output feedback”J. IEEE Transactions on Automatic Control, 1998, 43(8):1110-1111.
