Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

LMI Approach to Robust Control

  • Kang-Zhi LiuEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_161-1


In the analysis and design of robust control systems, LMI method plays a fundamental role. This article gives a brief introduction to this topic. After the introduction of LMI, it is illustrated how a control design problem is related with matrix inequality. Then, two methods are explained on how to transform a control problem characterized by matrix inequalities to LMIs, which is the core of the LMI approach. Based on these knowledge, the LMI solutions to various kinds of robust control problems are illustrated. Included are \(\mathcal{H}_{\infty }\) and \(\mathcal{H}_{2}\) control, regional pole placement, and gain-scheduled control.


LMI Robust control \(\mathcal{H}_{\infty }\) and \(\mathcal{H}_{2}\) control Regional pole placement Gain-scheduled control Multi-objective control 
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  1. Apkarian P, Gahinet P (1995) A convex characterization of gain-scheduled \(\mathcal{H}_{\infty }\) controllers. IEEE Trans Autom Control 40(5):853–864CrossRefzbMATHMathSciNetGoogle Scholar
  2. Boyd SP et al (1994) Linear matrix inequalities in system and control. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  3. Chilali M, Gahinet P (1996) H design with pole placement constraints: an LMI approach. IEEE Trans Autom Control 41(3):358–367CrossRefzbMATHMathSciNetGoogle Scholar
  4. Chilali M, Gahinet P, Apkarian P (1999) Robust pole placement in LMI regions. IEEE Trans Autom Control 44(12):2257–2270CrossRefzbMATHMathSciNetGoogle Scholar
  5. Gahinet P (1996) Explicit controller formulas for LMI-based \(\mathcal{H}_{\infty }\) synthesis. Automatica 32(7):1007–1014CrossRefzbMATHMathSciNetGoogle Scholar
  6. Gahinet P, Apkarian P (1994) A linear matrix inequality approach to \(\mathcal{H}_{\infty }\) control. Int J Robust Nonlinear Control 4:421–448CrossRefzbMATHMathSciNetGoogle Scholar
  7. Gahinet P, Nemirovski A, Laub AJ, Chilali M (1995) LMI control toolbox. The MathWorks, Inc., NatickGoogle Scholar
  8. Liu KZ, Yao Y (2014, to appear) Robust control: theory and applications. Wiley, New YorkGoogle Scholar
  9. Nesterov Y, Nemirovskii A (1994) Interior-point polynomial methods in convex programming. SIAM, PhiladelphiaCrossRefGoogle Scholar
  10. Packard A (1994) Gain scheduling via linear fractional transformations. Syst Control Lett 22: 79–92CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  1. 1.Department of Electrical and Electronic EngineeringChiba UniversityChibaJapan