Linear Fractional Transformations

  • Mi-Ching Tsai
  • Da-Wei Gu
Part of the Advances in Industrial Control book series (AIC)


This chapter introduces the linear fractional transformation (LFT), which is a convenient and powerful formulation in control system analysis and controller synthesis. The LFT formulation employs a two-port matrix description linked by a terminator to represent a closed-loop feedback system with two individual open-loop systems. This representation is inherently suitable for MIMO systems. Several examples are given to show how to locate the interconnected transfer function for a given system by using LFT and also how to formulate a control design problem into LFT. Additionally, in order to understand the benefit of utilizing LFT, the relationship between Mason’s gain formulae and LFT will be discussed in this chapter. Inner and co-inner systems are relevant to various aspects of control theory, especially H control. Definitions of inner and co-inner functions are thus introduced in the last section of this chapter.


Transfer Function MIMO System Star Product Linear Fractional Transformation Controller Synthesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Dorf RC (1995) Modern control systems. Addison-Wesley, New YorkGoogle Scholar
  2. 2.
    Doyle JC, Francis BA, Tannenbaum AR (1992) Feedback control theory. Macmillan, New YorkGoogle Scholar
  3. 3.
    Franco S (1995) Electric circuits fundamentals. Saunders College Publishing, OrlandozbMATHGoogle Scholar
  4. 4.
    Franklin GF, Powell JD, Emami-Naeini A (2009) Feedback control of dynamic systems, 6th edn. Addison Wesley, New YorkGoogle Scholar
  5. 5.
    Kimura H (1997) Chain-scattering approach to H control. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  6. 6.
    Kuo BC (1976) Automatic control systems, 6th edn. Prentice Hall, Englewood CliffsGoogle Scholar
  7. 7.
    Mason SJ (1953) Feedback theory-some properties of signal flow graphs. Proc IRE 41:1144–1156CrossRefGoogle Scholar
  8. 8.
    Mason SJ (1956) Feedback theory-further properties of signal flow graphs. Proc IRE 44:920–926CrossRefGoogle Scholar
  9. 9.
    Redheffer RM (1960) On a certain linear fractional transformation. J Math Phys 39:269–286MathSciNetGoogle Scholar
  10. 10.
    Youla DC, Jabr HA, Bongiorno JJ (1976) Modern Wiener-Hopf design of optimal controllers part II: the multivariable case. IEEE Trans Autom Control 21:319–338CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Mi-Ching Tsai
    • 1
  • Da-Wei Gu
    • 2
  1. 1.Department of Mechanical EngineeringNational Cheng Kung UniversityTainanTaiwan
  2. 2.Department of EngineeringUniversity of LeicesterLeicesterUK

Personalised recommendations