Theoretical Foundation of Finite Frequency Control

  • Chenxiao CaiEmail author
  • Zidong Wang
  • Jing Xu
  • Yun Zou
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 78)


Finite frequency control strategy has been proven to be an important method for modern control system. Combined with the particular frequency characteristics of the plant, many control specifications in the full frequency domain can be simplified into finite frequency ones. Commonly used tools in the frequency division are the weighting function and general Kalman-Yakubovich-Popov (GKYP) Lemma. In this chapter, some background information and useful lemmas in the field of finite frequency control have been investigated in detail.


Weighting Function Frequency Division Finite Frequency Modern Control System Integral Quadratic Constraint 
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  1. 1.
    Davidson, T., Luo, Z., Sturm, J.: Linear matrix inequality formulation of spectral mask constraints. In: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 6, pp. 3813–3816 (2001)Google Scholar
  2. 2.
    Fradkov, A.L.: Duality theorems for certain nonconvex extremal problems. Siberian Math. J. 14(2), 357–383 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Genin, Y., Hachez, Y., Nesterov, Y., Van Dooren, P.: Convex optimization over positive polynomials and filter design. In: Proceedings of UKACC International Conference on Control (2000)Google Scholar
  4. 4.
    Hara, S., Iwasaki, T.: Robust PID control using generalized KYP synthesis. IEEE Control Syst. Mag. 26(1), 80–91 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Iwasaki, T., Hara, S., Yamauchi, H.: Dynamical system design from a control perspective: finite frequency positive realness approach. IEEE Trans. Autom. Control 48(8), 1337–1354 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Iwasaki, T., Hara, S., Fradkov, A.L.: Time domain interpretations of frequency domain inequalities on finite ranges. Syst. Control Lett. 54(7), 681–691 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Iwasaki, T., Meinsma, G., Fu, M.: Generalized S-procedure and finite frequency KYP lemma. Math. Prob. Eng. 6(2–3), 305–320 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Iwasaki, T., Hara, S.: Robust control synthesis with general frequency domain specifications: static gain feedback case. Proc. Am. Control Conf. 5, 4613–4618 (2004)Google Scholar
  9. 9.
    Iwasaki, T., Hara, S.: Generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Trans. Autom. Control 50(1), 41–59 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jonsson, U.: Robustness analysis of uncertain and nonlinear systems. Ph.D. Dissertation, Department of Automatic Control, Lund Institute of Technology (1996)Google Scholar
  11. 11.
    Luse, D.W., Ball, J.A.: Frequency-scale decoposition of \(H_\infty \) disk problems. SIAM J. Control Optimization 27, 814–835 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Megretski, A., Rantzer, A.: System analysis via integral quadratic constraints. IEEE Trans. Autom. Control 42(6), 819–830 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Megretski, A., Treil, S.: Power distribution inequalities in optimization and robustness of uncertain systems. J. Math. Syst. Estim. Control 3(3), 301–319 (1993)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk, H., et al. (eds.) High Performance Optimization, pp. 405–440. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  15. 15.
    Oloomi, H., Shafai, B.: A system theory criterion for positive real matrices. SIAM J. Control 5(171–182), 2008 (1967)MathSciNetGoogle Scholar
  16. 16.
    Pipeleers, G., Vandenberghe, L.: Generalized KYP lemma with real data. IEEE Trans. Autom. Control 56(12), 2942–2946 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rantzer, A.: On the Kalman–Yakubovich–Popov lemma. Systems and Control Letters 28(1), 7–10 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Scherer, C.W.: LMI relaxations in robust control. Eur. J Control 12(1), 3–29 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Willems, J.C.: Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Autom. Control 16(6), 621–634 (1971)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Yakubovich, V.A.: The S-procedure in nonlinear control theory. Vestn. Leningrad Univ. 1, 62–77 (1971)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Yakubovich, V.A.: Nonconvex optimization problem: the infinite-horizon linear-quadratic control problem with quadratic constraints. Syst. Control Lett. 19(1), 13–22 (1992). (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.School of AutomationNanjing University of Science and TechnologyNanjingChina
  2. 2.Department of Computer ScienceBrunel University LondonUxbridgeUK

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