Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks

被引：0
作者：
Dong-Xia Zhao
Jun-Min Wang
机构：
[1] Beijing Institute of Technology,Department of Mathematics
[2] North University of China,Department of Mathematics
来源：
Journal of Dynamical and Control Systems | 2012年 / 18卷
关键词：
Pendulum system; delay feedbacks; spectrum; asymptotic analysis; stability; Riesz spectrum projection; 93D20; 47E05;
D O I：
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中图分类号：
学科分类号：
摘要：
In this paper, we examine the stability of a linearized inverted pendulum system with two delayed position feedbacks. The semigroup approach is adopted in investigation for the well-posedness of the closed loop system. We prove that the spectrum of the system is located in the left complex half-plane and its real part tends to − ∞ when the feedback gains satisfy some additional conditions. The asymptotic eigenvalues of the system is presented. By estimating the norm of the Riesz spectrum projection of the system operator that does not have the uniformly upper bound, we show that the eigenfunctions of the system do not form a basis in the state Hilbert space. Furthermore, the spectrum determined growth condition of the system is concluded and the exponential stability of the system is then established. Finally, numerical simulation is presented by applying the MATLAB software.
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页码：269 / 295
页数：26
相关论文
共 28 条
  • [1] Atay FM(1999)Balancing the inverted pendulum using position feedback Appl. Math. Lett. 12 51-56
  • [2] Datko R(1986)An example on the effect of time delays in boundary feedback stabilization of wave equations SIAM J. Control Optim. 24 152-156
  • [3] Lagnese J(2008)Dynamic stabilization of an Euler–Bernoulli beam under boundary control and non-collocated observation Systems Control Lett. 57 740-749
  • [4] Polis MP(2005)Asymptotic solutions of characteristic equations Nonlinear Anal. Real World Appl. 6 429-446
  • [5] Guo BZ(2011)Dynamical behavior of a hybrid system of nonhomogeneous Timoshenko beam with partial non-collocated inputs J. Dynam. Control Systems 17 77-121
  • [6] Wang JM(2004)Using delayed state feedback to stabilize periodic motions of an oscillator J. Sound Vibration 275 1009-1025
  • [7] Yang KY(2009)Control of an unstable reaction-diffusion PDE with long input delay Systems Control Lett. 58 773-782
  • [8] Hagen T(1931)On the zeros of exponential sum and integrals Bull. Am. Math. Soc. 7 213-239
  • [9] Han ZJ(2008)Stabilization of linear undamped systems via position and delayed position feedbacks J. Sound Vibration 312 509-528
  • [10] Xu GQ(2003)Time-delay systems: an overview of some recent advances and open problems Automat. J. IFAC 39 1667-1694
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