Abstract
A difficulty arises in the implementation of a predictor feedback law with the distributed delay term in it. To overcome this implementation difficulty, Lin and Fang (IEEE Trans. Autom. Control 52(6), 998–1013 (2007)) proposed to truncate the distributed delay term from the predictor feedback law, implementing only the remaining static state feedback term . This leads to a truncated predictor feedback (TPF) law, which can be readily implemented. In this chapter, we introduce the intuition, expression, and construction of the TPF law and its output feedback counterpart. In particular, the construction of the feedback gain matrix of the truncated predictor feedback law is first carried out by using an eigenstructure assignment based low gain feedback design technique. An alternative design is then presented by using the Lyapunov equation based low gain design technique. Stabilization of linear systems with input delay by the TPF is established under both the eigenstructure assignment based TPF and the Lyapunov equation based TPF.
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References
H. I. Ansoff, “Stability for linear oscillating systems with constant time lag,” Journal of Applied Mechanics, Vol. 16, pp. 158–164, 1949.
H. I. Ansoff and J. A. Krumhansl, “A general stability criterion for linear oscillating systems with constant time lag,” Quarterly of Applied Mathematics, Vol. 6, pp. 337–341, 1948.
Y. Y. Cao, Z. Lin and T. Hu, “Stability analysis of linear time-delay systems subject to input saturation,” IEEE Transactions on Circuits and Systems, Vol. 49, pp. 233–240, 2002.
B. S. Chen, S. S. Wang and H. C. Lu, “Stabilization of time-delay system containing saturating actuators,” International Journal of Control, Vol. 47, pp. 867–881, 1988.
J. Chen and A. Latchman, “Frequency sweeping tests for stability independent of delay,” IEEE Transactions on Automatic Control, Vol. 40, No. 9, pp. 1640–1645, 1995.
J. Chen, G. Gu and C. N. Nett, “A new method for computing delay margins for stability of linear delay systems,” Systems & Control Letters, Vol. 26, No. 2, pp. 107–117, 1995.
J. H. Chou, I. R. Horng and B. S. Chen, “Dynamical feedback compensator for uncertain time-delay systems containing saturating actuators,” International Journal of Control, Vol. 49, pp. 961–968, 1989.
K. Engelborghs, T. Luzyanina and G. Samaey, “DDE-BIFTOOL v.2.00: A Matlab package for bifurcation analysis of delay differential equation,” Dept. Comput. Sci., K.U. Leuven, T.W. Rep. 330, 2001.
H. Fang and Z. Lin, “A further result on global stabilization of oscillators with bounded delayed input,” IEEE Transactions on Automatic Control, Vol. 51, No. 1, pp. 121–128, 2006.
K. Gu, “An integral inequality in the stability problem of time-delay systems,” Proc. 39th IEEE Conference on Decision and Control, Sydney, Australia, pp. 2805–2810, 2000.
K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Boston, MA: Birkhäuser, 2003.
J. K. Hale, Theory of Functional Differential Equations, New York: Springer, 1977.
Q. L. Han and B. Ni, “Delay-dependent robust stabilization for uncertain constrained systems with pointwise and distributed time-varying delays,” Proc. 38th IEEE Conference on Decision and Control, pp. 215–220, 1999.
M. Krstic, “Lyapunov stability of linear predictor feedback for time-varying input delay,” IEEE Transactions on Automatic Control, Vol. 55, No. 2, pp. 554–559, 2010.
Z. Lin, Low Gain Feedback, London. U.K.: Springer-Verlag, 1998.
Z. Lin, B. M. Chen and X. Liu, Linear Systems Toolkit, Technical Report, Department of Electrical and Computer Engineering, University of Virginia, 2004.
Z. Lin and H. Fang, “On asymptotic stabilizability of linear systems with delayed input,” IEEE Transactions on Automatic Control, Vol. 52, No. 6, pp. 998–1013, 2007.
F. Mazenc, S. Mondie and S. I. Niculescu, “Global asymptotic stabilization for chain of integrators with a delay in the input,” IEEE Transactions on Automatic Control, Vol. 48, No. 1, pp. 57–63, 2003.
F. Mazenc, S. Mondie and S. I. Niculescu, “Global stabilization of oscillators with bounded delayed input,” Systems and Control Letters, Vol. 53, pp. 415–422, 2004.
N. Minorsky, “Self-excited oscillations in dynamical systems possessing retarded actions,” Journal of Applied Mechanics, Vol. 9, pp. A65-A71, 1942.
S. Mondié, R. Lozano and F. Mazenc, “Semiglobal stabilization of continuous systems with bounded delayed input,” Proc. the 15th IFAC World Congress, Barcelona, Spain, 2002.
S. I. Niculescu, Delay Effects on Stability: a Robust Control Approach, Vol. 269. Springer Science & Business Media, 2001.
S. Oucheriah, “Global stabilization of a class of linear continuous time-delay systems containing saturating controls,” IEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 43, pp. 1012–1015, 1996.
S. Ruan and J. Wei, “On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,” Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, Vol. 10, pp. 863–874, 2003.
S. Tarbouriech and J. M. Gomes da Silva Jr., “Synthesis of controllers for continuous-time delay systems with saturating controls via LMI’s,” IEEE Transactions on Automatic Control, Vol. 45, pp. 105–111, 2000.
Y. Wei and Z. Lin, “Stability criteria of linear systems with multiple input delays under truncated predictor feedback,” Systems & Control Letters, Vol. 111, pp. 9–17, 2018.
W. M. Wonham, Linear Multivariable Control: A Geometric Approach, Springer-Verlag, New York, 1979.
M. Wu, Y. He and J. H. She, “New delay-dependent stability criteria and stabilizing method for neutral systems,” IEEE Transactions on Automatic Control, Vol. 49, pp. 2266–2271, 2004.
L. Xie, E. Fridman and U. Shaked, “Robust control of distributed delay systems with application to combustion control,” IEEE Transactions on Automatic Control, Vol. 46, No. 12, pp. 1930–1935, 2001.
S. Xie and L. Xie, “Stabilization of a class of uncertain large-scale stochastic systems with time delays,” Automatica, Vol. 36, pp. 161–167, 2000.
K. Yakoubi and Y. Chitour, “Linear systems subject to input saturation and time delay: stabilization and L p-stability,” Proceedings of the 2nd Symposium on Structure, System and Control, Oaxaca, Mexico, 2004.
S. Y. Yoon and Z. Lin, “Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay,” Systems & Control Letters, Vol. 62, No. 10, pp. 312–317, 2013.
H. Zhang, D. Zhang and L. Xie, “An innovation approach to H ∞ prediction with application to systems with delayed measurements,” Automatica, Vol. 40, No. 7, pp. 1253–1261, 2004.
B. Zhou, G. Duan and Z. Lin, “A parametric Lyapunov equation approach to the design of low gain feedback,” IEEE Transactions on Automatic Control, Vol. 53, No. 6, pp. 1548–1554, 2008.
B. Zhou, Z. Lin and G. Duan, “Properties of the parametric Lyapunov equation based low gain design with application in stabilizing of time-delay systems,” IEEE Transactions on Automatic Control, Vol. 54, No. 7, pp. 1698–1704, 2009.
B. Zhou, Z. Lin and G. Duan, “Truncated predictor feedback for linear systems with long time-varying input delays,” Automatica, Vol. 48, No. 10, pp. 2387–2399, 2012.
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Wei, Y., Lin, Z. (2021). Truncated Predictor Feedback for Continuous-Time Linear Systems. In: Truncated Predictor Based Feedback Designs for Linear Systems with Input Delay. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-53429-5_2
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DOI: https://doi.org/10.1007/978-3-030-53429-5_2
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