Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Linear Matrix Inequality Techniques in Optimal Control

  • Robert E. SkeltonEmail author
Living reference work entry

Latest version View entry history

DOI: https://doi.org/10.1007/978-1-4471-5102-9_207-2


LMI (linear matrix inequality) techniques offer more flexibility in the design of dynamic linear systems than techniques that minimize a scalar functional for optimization. For linear state space models, multiple goals (performance bounds) can be characterized in terms of LMIs, and these can serve as the basis for controller optimization via finite-dimensional convex feasibility problems. LMI formulations of various standard control problems are described in this entry, including dynamic feedback stabilization, covariance control, LQR, H control, and L control. The integration of control and information architecture design (i.e., adding sensor/actuator precision selection to the control problem) is also shown to be a convex problem by formulating the feasibility constraints as LMIs.


Control system design Covariance control H control L control LQR/LQG Matrix inequalities Sensor/actuator design Integrated plant and control design 
This is a preview of subscription content, log in to check access.


  1. Balakrishnan V, Huang Y, Packard A, Doyle JC (1994) Linear matrix inequalities in analysis with multipliers. In: Proceedings of the 1994 American control conference, Baltimore, vol 2, pp 1228–1232Google Scholar
  2. Boyd SP, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  3. Boyd SP, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaCrossRefGoogle Scholar
  4. Camino JF, Helton JW, Skelton RE, Ye J (2001) Analysing matrix inequalities systematically: how to get Schur complements out of your life. In: Proceedings of the 5th SIAM conference on control & its applications, San DiegoGoogle Scholar
  5. Camino JF, Helton JW, Skelton RE, Ye J (2003a) Matrix inequalities: a symbolic procedure to determine convexity automatically. Integral Equ Oper Theory 46(4):399–454MathSciNetCrossRefGoogle Scholar
  6. Camino JF, de Oliveira MC, Skelton RE (2003b) Convexifying linear matrix inequality methods for integrating structure and control design. J Struct Eng 129(7): 978–988CrossRefGoogle Scholar
  7. Goyal R, Skelton RE (2019) Joint Optimization of plant, controller, and sensor/actuator design. In: Proceedings of the 2019 American control conference(ACC), Philadelphia. 1507–1512Google Scholar
  8. de Oliveira MC, Skelton RE (2001) Stability tests for constrained linear systems. In: Reza Moheimani SO (ed) Perspectives in robust control. Lecture notes in control and information sciences. Springer, New York, pp 241–257. ISBN:1852334525Google Scholar
  9. de Oliveira MC, Geromel JC, Bernussou J (2002) Extended H2 and H norm characterizations and controller parametrizations for discrete-time systems. Int J Control 75(9):666–679CrossRefGoogle Scholar
  10. Dullerud G, Paganini F (2000) A course in robust control theory: a convex approach. Texts in applied mathematics. Springer, New YorkCrossRefGoogle Scholar
  11. Gahinet P, Apkarian P (1994) A linear matrix inequality approach to H control. Int J Robust Nonlinear Control 4(4):421–448MathSciNetCrossRefGoogle Scholar
  12. Gahinet P, Nemirovskii A, Laub AJ, Chilali M (1995) LMI control toolbox user’s guide. The Mathworks Inc., NatickGoogle Scholar
  13. Hamilton WR (1834) On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function. Philos Trans R Soc (Part II): 247–308Google Scholar
  14. Hamilton WR (1835) Second essay on a general method in dynamics. Philos Trans R Soc (Part I):95–144Google Scholar
  15. Iwasaki T, Skelton RE (1994) All controllers for the general H control problem – LMI existence conditions and state-space formulas. Automatica 30(8):1307–1317MathSciNetCrossRefGoogle Scholar
  16. Iwasaki T, Meinsma G, Fu M (2000) Generalized S-procedure and finite frequency KYP lemma. Math Probl Eng 6:305–320MathSciNetCrossRefGoogle Scholar
  17. Khargonekar PP, Rotea MA (1991) Mixed H2H control: a convex optimization approach. IEEE Trans Autom Control 39:824–837CrossRefGoogle Scholar
  18. Li F, de Oliveira MC, Skelton RE (2008) Integrating information architecture and control or estimation design. SICE J Control Meas Syst Integr 1(2):120–128CrossRefGoogle Scholar
  19. Scherer CW (1995) Mixed H2H control. In: Isidori A (ed) Trends in control: a European perspective, pp 173–216. Springer, BerlinCrossRefGoogle Scholar
  20. Scherer CW, Gahinet P, Chilali M (1997) Multiobjective output-feedback control via LMI optimization. IEEE Trans Autom Control 42(7):896–911MathSciNetCrossRefGoogle Scholar
  21. Skelton RE (1988) Dynamics Systems Control: linear systems analysis and synthesis. Wiley, New YorkGoogle Scholar
  22. Skelton RE, Iwasaki T, Grigoriadis K (1998) A unified algebraic approach to control design. Taylor & Francis, LondonGoogle Scholar
  23. Vandenberghe L, Boyd SP (1996) Semidefinite programming. SIAM Rev 38:49–95MathSciNetCrossRefGoogle Scholar
  24. Youla DC, Bongiorno JJ, Jabr HA (1976) Modern Wiener-Hopf design of optimal controllers, part II: the multivariable case. IEEE Trans Autom Control 21: 319–338CrossRefGoogle Scholar
  25. Zhu G, Skelton R (1992) A two-Riccati, feasible algorithm for guaranteeing output l constraints. J Dyn Syst Meas Control 114(3):329–338CrossRefGoogle Scholar
  26. Zhu G, Rotea M, Skelton R (1997) A convergent algorithm for the output covariance constraint control problem. SIAM J Control Optim 35(1):341–361MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  1. 1.Texas A&M UniversityCollege StationUSA

Section editors and affiliations

  • Michael Cantoni
    • 1
  1. 1.Department of Electrical & Electronic EngineeringThe University of MelbourneParkvilleAustralia