Skip to main content
SpringerLink
Log in

(Generalized) KYP lemma and applications

  • Living reference work entry
  • First Online:
Encyclopedia of Systems and Control

  • 1080 Accesses

Abstract

Various properties of dynamical systems can be characterized in terms of inequality conditions on their frequency responses. The Kalman-Yakubovich-Popov (KYP) lemma shows equivalence of such frequency domain inequality (FDI) and a linear matrix inequality (LMI). The fundamental result has been a basis for robust and optimal control theories in the past several decades. The KYP lemma has recently been generalized to the case where an FDI on a possibly improper transfer function is required to hold in a (semi)finite frequency range. The generalized KYP lemma allows us to directly deal with practical situations where design parameters are sought to satisfy FDIs in multiple (semi)finite frequency ranges. Various design problems, including FIR filter and PID controller, reduce to LMI problems which can be solved via semidefinite programming.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Bibliography

  • Anderson B (1967) A system theory criterion for positive real matrices. SIAM J Control 5(2): 171–182

    Article  MATH  MathSciNet  Google Scholar 

  • Bachelier O, Paszke W, Mehdi D (2008) On the Kalman-Yakubovich-Popov lemma and the multidimensional models. Multidimens Syst Signal Proc 19(3–4):425–447

    Article  MATH  MathSciNet  Google Scholar 

  • Ebihara Y, Maeda K, Hagiwara T (2008) Generalized S-procedure for inequality conditions on one-vector-lossless sets and linear system analysis. SIAM J Control Optim 47(3):1547–1555

    Article  MATH  MathSciNet  Google Scholar 

  • Graham M, de Oliveira M (2010) Linear matrix inequality tests for frequency domain inequalities with affine multipliers. Automatica 46:897–901

    Article  MATH  Google Scholar 

  • Gusev S (2009) Kalman-Yakubovich-Popov lemma for matrix frequency domain inequality. Syst Control Lett 58(7):469–473

    Article  MATH  MathSciNet  Google Scholar 

  • Gusev S, Likhtarnikov A (2006) Kalman-Yakubovich-Popov lemma and the S-procedure: a historical essay. Autom Remote Control 67(11):1768–1810

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Iwasaki T, Meinsma G, Fu M (2000) Generalized S-procedure and finite frequency KYP lemma. Math Probl Eng 6:305–320

    Article  MATH  MathSciNet  Google Scholar 

  • Iwasaki T, Hara S, Yamauchi H (2003) Dynamical system design from a control perspective: finite frequency positive-realness approach. IEEE Trans Autom Control 48(8):1337–1354

    Article  MathSciNet  Google Scholar 

  • Kalman R (1963) Lyapunov functions for the problem of Lur’e in automatic control. Proc Natl Acad Sci 49(2):201–205

    Article  MATH  MathSciNet  Google Scholar 

  • Pipeleers G, Vandenberghe L (2011) Generalized KYP lemma with real data. IEEE Trans Autom Control 56(12):2940–2944

    Article  MathSciNet  Google Scholar 

  • Pipeleers G, Iwasaki T, Hara S (2013) Generalizing the KYP lemma to the union of intervals. In: Proceedings of European control conference, Zurich, pp 3913–3918

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • Scherer C (2006) LMI relaxations in robust control. Eur J Control 12(1):3–29

    Article  MATH  MathSciNet  Google Scholar 

  • Tanaka T, Langbort C (2011) The bounded real lemma for internally positive systems and H-infinity structured static state feedback. IEEE Trans Autom Control 56(9):2218–2223

    Article  MathSciNet  Google Scholar 

  • Tanaka T, Langbort C (2013) Symmetric formulation of the S-procedure, Kalman-Yakubovich-Popov lemma and their exact losslessness conditions. IEEE Trans Autom Control 58(6): 1486–1496

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Xiong J, Petersen I, Lanzon A (2012) Finite frequency negative imaginary systems. IEEE Trans Autom Control 57(11):2917–2922

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical & Aerospace Engineering, University of California, 420 Westwood Plaza, 90095, Los Angeles, CA, USA

    Ph.D Tetsuya Iwasaki

Authors
  1. Ph.D Tetsuya Iwasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tetsuya Iwasaki .

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, Minnesota, USA

    Tariq Samad

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag London

About this entry

Cite this entry

Iwasaki, T. (2013). (Generalized) KYP lemma and applications. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_160-1

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_160-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, London

  • Online ISBN: 978-1-4471-5102-9

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation