Skip to main content
SpringerLink
Log in

Finite Frequency Positive Real Control for Singularly Perturbed Systems

  • Chapter
  • First Online:
Finite Frequency Analysis and Synthesis for Singularly Perturbed Systems

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 78))

Abstract

With the development of high-performance mechatronics , the feedback control system is often required to exhibit some particular control system specifications such as high control bandwidth and FF disturbance suppression capability. Meanwhile, the integrated design of both mechanical plant and controller design is carried out to achieve performance specifications. In this chapter, a GKYP lemma-based algorithm is proposed for simultaneous finite frequency design of a mechanical SPS plant and controller through a convex separable parametrization. In Sect. 5.1, basic definitions and some related engineering background of passivity and positive real property are introduced. The design procedure using the classical slow-fast decomposition is represented in Sect. 5.2. To extend the results to nonstandard SPS, a descriptor-system method for SPSs are demonstrated in Sect. 5.3.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Dragan, V., Morozan, T., Shi, P.: Asymptotic properties of input-output operators norm associated with singularly perturbed systems with multiplicative white noise. SIAM J. Control Optim. 41(1), 141–163 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dragan, V., Mukaidani, H., Shi, P.: The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed ito differential equations. SIAM J. Control Optim. 50(1), 448–470 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fridman, E.: Near-optimal \(H_\infty \) control of linear singularly perturbed systems. IEEE Trans. Autom. Control 41(2), 236–240 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Haddad, W.M., Chellaboina, V.S.: Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton University Press, Princeton (2008)

    MATH  Google Scholar 

  5. Hara, S., Iwasaki, T.: Robust PID control using generalized KYP synthesis. IEEE Control Syst. Mag. 26(1), 80–91 (2006)

    Article  MathSciNet  Google Scholar 

  6. Hill, D., Moylan, P.: Stability results for nonlinear feedback systems. Automatica 13(4), 377–382 (1977)

    Article  MATH  Google Scholar 

  7. Iwasaki, T., Hara, S., Fradkov, A.L.: Time domain interpretations of frequency domain inequalities on (semi) finite ranges. Syst. Control Lett. 54(7), 681–691 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Iwasaki, T., Hara, S.: Generalized KYP lemma: unified frequency domain inequalities with design applications. IEEE Trans. Autom. Control 50(1), 41–59 (2005)

    Article  MathSciNet  Google Scholar 

  9. Khail, H.K., Chen, F.C.: \(H_\infty \) control of two-time-scale systems. Proc. Am. Control Conf. 19, 35–42 (1992)

    Google Scholar 

  10. Kokotovic, P.V., O’Malley, R.E., Sannuti, P.: Singular perturbations and order reduction in control theory-an overview. Automatica 12(2), 123–132 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kokotovic, P.V., Khalil, H.K., OReilly, J.: Singular Perturbation Methods in Control: Analysis and Design. Academic Press, New York (1986)

    Google Scholar 

  12. Luse, D.W., Ball, J.A.: Frequency-scale decomposition of \(H_\infty \) disk problems. SIAM J. Control Optim. 27(4), 814–835 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Luse, D.W., Khalil, H.K.: Frequency domain results for systems with slow and fast dynamics. IEEE Trans. Autom. Control AC–30(12), 1171–1179 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Naidu, D.S., Calise, A.J.: Singular perturbation methods and time scales in guidance and control of aerospace systems: a survey. AIAA J. Guid. Control Dyn. 24(6), 1057–1078 (2001)

    Article  Google Scholar 

  15. Oloomi, H.M., Sawan, M.E.: Suboptimal model-matching problem for two frequency scale transfer functions. In: Proceedings of the American Control Conference, pp. 2190–2191 (1989)

    Google Scholar 

  16. Pan, Z., Basar, T.: \(H_\infty \) optimal control for singularly perturbed systems Part I: perfect state measurements. In: Proceedings of American Control Conference, pp. 1850–1854 (1992)

    Google Scholar 

  17. Pan, Z., Basar, T.: \(H_\infty \) optimal control for singularly perturbed systems Part II: perfect state measurements. IEEE Trans. Autom. Control 39(2), 280–300 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rantzer, A.: On the Kalman–Yakubovich–Popov lemma. Syst. Control Lett. 28(1), 7–10 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shi, P., Shue, S., Agarwal, R.: Robust disturbance attenuation with stability for a class of uncertain singularly perturbed systems. Int. J. Control 70(6), 873–891 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shi, P., Dragan, V.: Asymptotic \(H_{\infty }\) control for singularly perturbed systems with parametric uncertainties. IEEE Trans. Autom. Control 44(9), 1738–1742 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sun, W., Khargonekar, P.P., Shim, D.: Solution to the positive real control problem for linear time-invariant systems. IEEE Trans. Autom. Control 39(10), 2034–2046 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, W., Khargonekar, P.P., Shim, D.: Robust control synthesis with general frequency domain specifications: static gain feedback case. Proc. Am. Control Conf. 5, 4613–4618 (2004)

    Google Scholar 

  23. van der Schaft, A.J.: \(L_2\)-Gain and Passivity in Nonlinear Control. Springer, New York (1999)

    Google Scholar 

  24. Verghese, G.C., Levy, B.C., Kailath, T.: A generalized state-space for singular systems: a survey. IEEE Trans. Autom. Control AC–26(4), 811–831 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Willems, J.C.: Least squares stationary optimal control and the algebraic Riccati equation. IEEE Trans. Autom. Control AC–16(6), 621–634 (1971)

    Article  MathSciNet  Google Scholar 

  26. Xu, S., Lam, J.: Robust Control and Filtering of Singular Systems. Springer, Berlin (2006)

    MATH  Google Scholar 

  27. Yakubovich, V.A.: A frequency theorem for the case in which the state and control spaces are Hilbert spaces with an application to some problems in the synthesis of optimal controls.I. Sib. Math. J. 15(3), 457–476 (1974)

    Article  MathSciNet  Google Scholar 

  28. Zames, G.: On the input-output stability of time-varying nonlinear feedback systems, part one: conditions derived using concepts of loop gain, conicity, and positivity. IEEE Trans. Autom. Control AC–11(2), 228–238 (1966)

    Article  Google Scholar 

  29. Zhong, N., Sun, M., Zou, Y.: Robust stability and stabilization for singularly perturbed control systems: method based on singular systems. Int. J. Innov. Comput. Appl. 4(7), 1761–1770 (2008)

    Google Scholar 

  30. Zhou, L., Lu, G.: Robust stability of singularly perturbed descriptor systems with nonlinear perturbation. IEEE Trans. Autom. Control 56(4), 858–863 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, Nanjing, China

    Chenxiao Cai, Jing Xu & Yun Zou

  2. Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UK

    Zidong Wang

Authors
  1. Chenxiao Cai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zidong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yun Zou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chenxiao Cai .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Cai, C., Wang, Z., Xu, J., Zou, Y. (2017). Finite Frequency Positive Real Control for Singularly Perturbed Systems. In: Finite Frequency Analysis and Synthesis for Singularly Perturbed Systems. Studies in Systems, Decision and Control, vol 78. Springer, Cham. https://doi.org/10.1007/978-3-319-45405-4_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45405-4_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45404-7

  • Online ISBN: 978-3-319-45405-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation