核为概周期的线性时变系统
|
摘要
概周期函数理论首先是由丹麦数学家H.Bohr在1925-1926年间发展起来的。后来,Bohr的理论有了进一步的发展,其中包括在群上的调和分析理论以及1933年由S.Bochner所建立的Banach空间的向量值概周期函数理论。紧接着H.Weyl,A.Besicovitch,J.Favard,J.Von Neumann,V.V.Stepanov,N.N.Bogolyubov等人将概周期函数理论向常微分方程、稳定性理论、动力系统等方向做了推广,使得概周期函数理论逐渐完善。
概周期函数理论在函数基本性质方面其发展过程的一个重要特点就是其函数范围不断扩大,从概周期函数、一致概周期函数、渐近概周期函数、弱概周期函数,一直到上个世纪九十年代初张传义教授提出的伪概周期函数。每一次函数的扩展大大地扩展了概周期理论的应用。同样,概周期序列的发展范围也是不断扩大,将概周期序列与系统相联系并考虑它的一些特殊的性质是本文的一个重要部分。这就引入了概周期系统这个概念,它是一类特殊的时变系统。我们知道时变系统是比时不变系统更为复杂的一种系统,往往在讨论时变系统的某一性质时具有很大困难,而研究概周期系统则比较容易,概周期系统在时变系统中起重要作用。
本文针对上述情况,对概周期系统及伪概周期函数作了两方面的具体工作:
1对时变系统与概周期系统的性质作了讨论
本文主要是给出了概周期系统新的性质,它是一般的时变系统所不具有的,以及考虑了一些前人对概周期系统研究的性质,并将某些性质推广到了概周期类系统,这就促进了概周期系统性质的完善。
2对伪概周期函数的窗口富里埃变换的性质作了研究
本文主要研究了伪概周期函数的窗口富里埃变换,并给出了它的一些好的性质。
The theory of almost periodic functions was first developed by the Dan-ish mathematician H.Bohr during 1925-1926.Soon after,there are further researcheson Bohr’s theory,these include theory of harmonic analysis on Banach spaces in-troduced by S.Bochner in 1933.Then H.Weyl,A.Besicovitch,J.Favard,J.Von Neu-mann,V.V.Stepanov,N.N.Bogolyubo did further researches of the theory of stabil-ity,dynamics systems,which consummated the almost periodic function’s theory.
In developing the theory of the almost periodic type functions,a main feature isthat the scope is expanding:From Almost periodic function,Uniform almost periodicfunction,Asymptotically almost periodic function,Weakly almost periodic function toPseudo almost periodic function which was proposed by Professor Zhang in 1992.Ev-ery extension consumedly enlarges the application of the almost periodic functionaltheory.In the same,in the development of the theory of the almost periodic sequence,the scope is also expanding .To connect the almost periodic sequence with the systemand investigate the properties is the most important part in this paper.Then it is nessaryto introduce the concept of almost periodic system,which is a special time-varyingsystem.We know that time-varying system is a system which is more complicatedthan time-invariant system,so it is difficulty to talk over.But it is easier to investigatethe almost periodic system,so it is very important.In this paper,in allusion to above-motioned situations,two aspects of concrete work are done to almost periodic systemand the Pseudo almost periodic function.
1 The properties of time-varying system and almost periodic system are dis-cussed
This paper discussed the new properties of almost periodic system,which is notpossessed in normal time-varying system.This paper also talk over some properties ofthe almost periodic functions which have been studied by predecessors.Here,we ex-tend them to the type of almost periodic functions,which will consummate the almostperiodic functions.
2 And the properties of the Windowed fourier transform of Pseudo almost peri-odic function is investigated
概周期函数理论在函数基本性质方面其发展过程的一个重要特点就是其函数范围不断扩大,从概周期函数、一致概周期函数、渐近概周期函数、弱概周期函数,一直到上个世纪九十年代初张传义教授提出的伪概周期函数。每一次函数的扩展大大地扩展了概周期理论的应用。同样,概周期序列的发展范围也是不断扩大,将概周期序列与系统相联系并考虑它的一些特殊的性质是本文的一个重要部分。这就引入了概周期系统这个概念,它是一类特殊的时变系统。我们知道时变系统是比时不变系统更为复杂的一种系统,往往在讨论时变系统的某一性质时具有很大困难,而研究概周期系统则比较容易,概周期系统在时变系统中起重要作用。
本文针对上述情况,对概周期系统及伪概周期函数作了两方面的具体工作:
1对时变系统与概周期系统的性质作了讨论
本文主要是给出了概周期系统新的性质,它是一般的时变系统所不具有的,以及考虑了一些前人对概周期系统研究的性质,并将某些性质推广到了概周期类系统,这就促进了概周期系统性质的完善。
2对伪概周期函数的窗口富里埃变换的性质作了研究
本文主要研究了伪概周期函数的窗口富里埃变换,并给出了它的一些好的性质。
The theory of almost periodic functions was first developed by the Dan-ish mathematician H.Bohr during 1925-1926.Soon after,there are further researcheson Bohr’s theory,these include theory of harmonic analysis on Banach spaces in-troduced by S.Bochner in 1933.Then H.Weyl,A.Besicovitch,J.Favard,J.Von Neu-mann,V.V.Stepanov,N.N.Bogolyubo did further researches of the theory of stabil-ity,dynamics systems,which consummated the almost periodic function’s theory.
In developing the theory of the almost periodic type functions,a main feature isthat the scope is expanding:From Almost periodic function,Uniform almost periodicfunction,Asymptotically almost periodic function,Weakly almost periodic function toPseudo almost periodic function which was proposed by Professor Zhang in 1992.Ev-ery extension consumedly enlarges the application of the almost periodic functionaltheory.In the same,in the development of the theory of the almost periodic sequence,the scope is also expanding .To connect the almost periodic sequence with the systemand investigate the properties is the most important part in this paper.Then it is nessaryto introduce the concept of almost periodic system,which is a special time-varyingsystem.We know that time-varying system is a system which is more complicatedthan time-invariant system,so it is difficulty to talk over.But it is easier to investigatethe almost periodic system,so it is very important.In this paper,in allusion to above-motioned situations,two aspects of concrete work are done to almost periodic systemand the Pseudo almost periodic function.
1 The properties of time-varying system and almost periodic system are dis-cussed
This paper discussed the new properties of almost periodic system,which is notpossessed in normal time-varying system.This paper also talk over some properties ofthe almost periodic functions which have been studied by predecessors.Here,we ex-tend them to the type of almost periodic functions,which will consummate the almostperiodic functions.
2 And the properties of the Windowed fourier transform of Pseudo almost peri-odic function is investigated
引文
1. H. Bohr. Almost Periodic Functions. Chelsea. New York, 1951:20～28
2. Zhang Chuanyi. Almost Periodic Type Functions and Ergodicity. Science Press,2003:10～59
3. C. Zhang. Ergodicity and Its Applications-Part one: Basic Properties. ActaAnalysis Functionalis Applicata, 1999,(1):28～39
4. C. Zhang. Ergodicity and Its Applications-Part two: Averging Method of SomeDynamical Systems. Acta Analysis Functionalis Applicata, 1999,(1):146～159
5. C. Zhang. Erodicity and Its Applications in Regularity and Solutions of Pseudoalmost Periodic Equations. Nonlinear Analysis, TMA, 2001,(46):511～523
6. C. Zhang. Erodicity and Asymptotically Almost Periodic Solutions of SomeDifferential Equations. International J. Math. Math. Sci., 2001, (15):787～801
7. M. Frechet. Les Function Sasumptotiquement. C. R. Acad. Sci. Paris. 1941,(213):520～522
8. M. Frcchet. Les Functions Asumptotiquement Presque Periociques. rev. sci.1941, (79):341～354
9. W. F. Eberlein. Abstract Ergodic Theorems and Weakly Almost Periodic Func-tions. Trans. Amer. Math. Soc. 1949, (67):217～240
10. W. F. Eberlein. The Point Spectrum of Weakly Almost Periodic Functions.Michigan Math. J. 1955, (3):137～139
11. C. Zhang. Pseudo Almost Periodic Functions and Their Applications. Ph. D.Thesis. University of Western Ontario. 1992:52～58
12. C. Zhang. Pseudo Almost Solutions of Some Differential Equations. J. Math.Anal . Appl. 1994, (181):62～76
13. C. Zhang. Pseudo Almost Periodic Solutions of Some Differential EquationsII.J. Math. Anal. Appl. 1995,(192):543～561
14. C. Zhang. Vector Valued Almost Periodic Functions. Czech. Math. J. 1994,47(12): 385～394
15. P. M. Makila, J. R. Partington, T. Norlander. Bounded Power Signal Space forRobust Control and Modeling. SLAM J. Contr. 1998, 37(1): 92～117
16. P. M. Makila. On Three Puzzles in Robust Control. IEEETransactions on Au-tomatic Control. 2000,45(3):552～556
17. Birgit Jacob, Jonathan R. Partington. On Discrete-Time Linear Systems withAlmost-Periodic Kernels. Journal of Mathematical Analysis and Application2000,25(2):549～570
18. Jonathan R. Partington, Baru Unalmns. On the Windowed Fourier Transformand Wavelet Transform of Almost Periodic Functions. Applied and Computa-tional Harmonic Analysis. 2001,(10):45～60
19. A. S. Besicovitch. Almost Periodic Functions. Cambridge Univ. Press Cam-bridge UK, 1932:120～128
20. N. Wiener. Nonlinear Problems in Random Theory. MIT Press Cambrige. MA,1958:90～96
21. S. Boyd, L. O. Chua. Fading Memory and the Problem of Approximating Non-linear Operators with Volterra Series. IEEE Trans. Circ. Syst., 1985, (32):1150～1161
22. Birgit Jacob, Mikael Larser, Hans Zwarz. Correcttions and Extensions of ”Opti-mal Control of Linear Systems with Almost Periodic Input”by G. Da Prato andA. Ichikawa. SLAM J. CONTROL OPTIM. 1998,36(4):1473～1480
23. W. F. Eberlein. Abstract Ergodic Theorems and Weakly Almost Periodic Func-tions. Trans. Amer. Math. Soc. 1949,(67):217～240
24. K. Poola, P. Khargonekar, A. Tikku, J. Krause and K. Nagpal. A Time Do-main Approach to Model Validation.IEEE Trans. Automat. Contr., 1994,(39):951～959
25. R. S. Smith and J. C. Dole. Model Validation Connection Between Robust-Control and Indentification. IEEE Trans, Automat. Contr. Conf. 1992,(37):942～952
26. R. S. Smith and H.Kimura. The Modeling of Uncertainty in Control Systems.in Proc, 1992:60～64
27. P. M. Makila. Approximation and Identification of Continuous Time System.Int. J. Contr., 1990,(52):669～687
28. 何崇佑.概周期微分方程.北京:高等教育出版社, 1992:32～40
29. 韩 京 清.控 制 理 论 ― 模 型 论 还 是 控 制 论.系 统 科 学 与 数学,1989,(4):328～335
30. 冯旭,孙忧贤.鲁棒辩识问题评述.控制理论与应用.1993,(6):609～616
31. P.艾克霍夫.系统辩识-参数和状态估计.北京:科学出版社, 1980:50～54
32. 方崇智,萧德云著.过程辩识.北京:清华大学出版社,1989:72～78
33. 蔡金狮.动力学系统辩识与建模.北京:国防工业出版社, 1984:36～42
34. 韩光文.辩识与参数估计.北京:国防工业出版社,1980:67～86
35. 相良节夫,秋月影雄等.系统辩识.北京:化工工业出版社, 1988:91～95
36. J.Mari. A Counterexample in Power Signals Space. IEEE Trans. Automat.Contr. 1996, 41(1):115～116
37. G. E. Dullerud and S. Lall. A New Approach for Analysis and Synthesis ofTime-varying Systems. IEEE Trans.Automat. Control. 1999,(44):1486～1497
38. J. S. Shamma and R. Zhao. Fading Memory Feed Back Systems and RobustStability. Automatica. 1993,29(1):191～200
39. J. C. Willems. Stability,Instability,Invertibility and Causality. SLAM J. Contr.1969, 7(4):645～671
40. I. Daubechies. The Wavelet Transform, Time-frequency Localisation and Sig-nal Analysis. IEEE Trans. Inform. Theory. 1990, 36(5):961～1005
2. Zhang Chuanyi. Almost Periodic Type Functions and Ergodicity. Science Press,2003:10～59
3. C. Zhang. Ergodicity and Its Applications-Part one: Basic Properties. ActaAnalysis Functionalis Applicata, 1999,(1):28～39
4. C. Zhang. Ergodicity and Its Applications-Part two: Averging Method of SomeDynamical Systems. Acta Analysis Functionalis Applicata, 1999,(1):146～159
5. C. Zhang. Erodicity and Its Applications in Regularity and Solutions of Pseudoalmost Periodic Equations. Nonlinear Analysis, TMA, 2001,(46):511～523
6. C. Zhang. Erodicity and Asymptotically Almost Periodic Solutions of SomeDifferential Equations. International J. Math. Math. Sci., 2001, (15):787～801
7. M. Frechet. Les Function Sasumptotiquement. C. R. Acad. Sci. Paris. 1941,(213):520～522
8. M. Frcchet. Les Functions Asumptotiquement Presque Periociques. rev. sci.1941, (79):341～354
9. W. F. Eberlein. Abstract Ergodic Theorems and Weakly Almost Periodic Func-tions. Trans. Amer. Math. Soc. 1949, (67):217～240
10. W. F. Eberlein. The Point Spectrum of Weakly Almost Periodic Functions.Michigan Math. J. 1955, (3):137～139
11. C. Zhang. Pseudo Almost Periodic Functions and Their Applications. Ph. D.Thesis. University of Western Ontario. 1992:52～58
12. C. Zhang. Pseudo Almost Solutions of Some Differential Equations. J. Math.Anal . Appl. 1994, (181):62～76
13. C. Zhang. Pseudo Almost Periodic Solutions of Some Differential EquationsII.J. Math. Anal. Appl. 1995,(192):543～561
14. C. Zhang. Vector Valued Almost Periodic Functions. Czech. Math. J. 1994,47(12): 385～394
15. P. M. Makila, J. R. Partington, T. Norlander. Bounded Power Signal Space forRobust Control and Modeling. SLAM J. Contr. 1998, 37(1): 92～117
16. P. M. Makila. On Three Puzzles in Robust Control. IEEETransactions on Au-tomatic Control. 2000,45(3):552～556
17. Birgit Jacob, Jonathan R. Partington. On Discrete-Time Linear Systems withAlmost-Periodic Kernels. Journal of Mathematical Analysis and Application2000,25(2):549～570
18. Jonathan R. Partington, Baru Unalmns. On the Windowed Fourier Transformand Wavelet Transform of Almost Periodic Functions. Applied and Computa-tional Harmonic Analysis. 2001,(10):45～60
19. A. S. Besicovitch. Almost Periodic Functions. Cambridge Univ. Press Cam-bridge UK, 1932:120～128
20. N. Wiener. Nonlinear Problems in Random Theory. MIT Press Cambrige. MA,1958:90～96
21. S. Boyd, L. O. Chua. Fading Memory and the Problem of Approximating Non-linear Operators with Volterra Series. IEEE Trans. Circ. Syst., 1985, (32):1150～1161
22. Birgit Jacob, Mikael Larser, Hans Zwarz. Correcttions and Extensions of ”Opti-mal Control of Linear Systems with Almost Periodic Input”by G. Da Prato andA. Ichikawa. SLAM J. CONTROL OPTIM. 1998,36(4):1473～1480
23. W. F. Eberlein. Abstract Ergodic Theorems and Weakly Almost Periodic Func-tions. Trans. Amer. Math. Soc. 1949,(67):217～240
24. K. Poola, P. Khargonekar, A. Tikku, J. Krause and K. Nagpal. A Time Do-main Approach to Model Validation.IEEE Trans. Automat. Contr., 1994,(39):951～959
25. R. S. Smith and J. C. Dole. Model Validation Connection Between Robust-Control and Indentification. IEEE Trans, Automat. Contr. Conf. 1992,(37):942～952
26. R. S. Smith and H.Kimura. The Modeling of Uncertainty in Control Systems.in Proc, 1992:60～64
27. P. M. Makila. Approximation and Identification of Continuous Time System.Int. J. Contr., 1990,(52):669～687
28. 何崇佑.概周期微分方程.北京:高等教育出版社, 1992:32～40
29. 韩 京 清.控 制 理 论 ― 模 型 论 还 是 控 制 论.系 统 科 学 与 数学,1989,(4):328～335
30. 冯旭,孙忧贤.鲁棒辩识问题评述.控制理论与应用.1993,(6):609～616
31. P.艾克霍夫.系统辩识-参数和状态估计.北京:科学出版社, 1980:50～54
32. 方崇智,萧德云著.过程辩识.北京:清华大学出版社,1989:72～78
33. 蔡金狮.动力学系统辩识与建模.北京:国防工业出版社, 1984:36～42
34. 韩光文.辩识与参数估计.北京:国防工业出版社,1980:67～86
35. 相良节夫,秋月影雄等.系统辩识.北京:化工工业出版社, 1988:91～95
36. J.Mari. A Counterexample in Power Signals Space. IEEE Trans. Automat.Contr. 1996, 41(1):115～116
37. G. E. Dullerud and S. Lall. A New Approach for Analysis and Synthesis ofTime-varying Systems. IEEE Trans.Automat. Control. 1999,(44):1486～1497
38. J. S. Shamma and R. Zhao. Fading Memory Feed Back Systems and RobustStability. Automatica. 1993,29(1):191～200
39. J. C. Willems. Stability,Instability,Invertibility and Causality. SLAM J. Contr.1969, 7(4):645～671
40. I. Daubechies. The Wavelet Transform, Time-frequency Localisation and Sig-nal Analysis. IEEE Trans. Inform. Theory. 1990, 36(5):961～1005