Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Optimal Control via Factorization and Model Matching

  • Michael CantoniEmail author
Living reference work entry

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DOI: https://doi.org/10.1007/978-1-4471-5102-9_206-2


One approach to linear control system design involves the matching of input-output models with respect to a quantification of performance. The approach is based on a parametrization of all stabilizing feedback controllers for the given plant model. This parametrization, constructed from coprime factorizations of the plant, and spectral factorization methods for solving model-matching problems, are described in this article. Both impulse-response energy and worst-case energy-gain measures of controller performance are considered.


Coprime factorization \(\mathscr {H}_2\) control \(\mathscr {H}_\infty \) control Spectral factorization Youla-Kučera controller parametrization 
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  1. Ball JA, Ran ACM (1987) Optimal Hankel norm model reductions and Wiener-Hopf factorization I: the canonical case. SIAM J Control Optim 25(2):362–382MathSciNetCrossRefGoogle Scholar
  2. Ball JA, Helton JW, Verma M (1991) A factorization principle for stabilization of linear control systems. Int J Robust Nonlinear Control 1(4):229–294CrossRefGoogle Scholar
  3. Boyd SP, Barratt CH (1991) Linear controller design: limits of performance. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  4. Curtain RF, Zwart HJ (1995) An introduction to infinite-dimensional linear systems theory. Volume 21 of texts in applied mathematics. Springer, New YorkGoogle Scholar
  5. Dahleh MA, Diaz-Bobillo IJ (1995) Control of uncertain systems: a linear programming approach. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  6. DeSantis RM, Saeks R, Tung LJ (1978) Basic optimal estimation and control problems in Hilbert space. Math Syst Theory 12(1):175–203MathSciNetCrossRefGoogle Scholar
  7. Desoer C, Liu R-W, Murray J, Saeks R (1980) Feedback system design: the fractional representation approach to analysis and synthesis. IEEE Trans Autom Control 25(3):399–412MathSciNetCrossRefGoogle Scholar
  8. Feintuch A (1998) Robust control theory in Hilbert space. Applied mathematical sciences. Springer, New YorkCrossRefGoogle Scholar
  9. Francis BA (1987) A course in H control theory. Lecture notes in control and information sciences. Springer, Berlin/New YorkGoogle Scholar
  10. Francis BA, Doyle JC (1987) Linear control theory with an H optimality criterion. SIAM J Control Optim 25(4):815–844MathSciNetCrossRefGoogle Scholar
  11. Glover K, Limebeer DJN, Doyle JC, Kasenally EM, Safonov MG (1991) A characterization of all solutions to the four block general distance problem. SIAM J Control Optim 29(2):283–324MathSciNetCrossRefGoogle Scholar
  12. Green M, Glover K, Limebeer DJN, Doyle JC (1990) A J-spectral factorization approach to \(\mathscr {H}_\infty \) control. SIAM J Control Optim 28(6):1350–1371MathSciNetCrossRefGoogle Scholar
  13. Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  14. Kimura H (1989) Conjugation, interpolation and model-matching in H . Int J Control 49(1):269–307MathSciNetzbMATHGoogle Scholar
  15. Kimura H (1997) Chain-scattering approach to H control. Systems and control. Birkhäuser, BostonCrossRefGoogle Scholar
  16. Kucera V (1975) Stability of discrete linear control systems. In: 6th IFAC world congress, Boston. Paper 44.1Google Scholar
  17. Qi X, Salapaka MV, Voulgaris PG, Khammash M (2004) Structured optimal and robust control with multiple criteria: a convex solution. IEEE Trans Autom Control 49(10):1623–1640MathSciNetCrossRefGoogle Scholar
  18. Quadrat A (2006) On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems. Math Control Signals Syst 18(3):199–235MathSciNetCrossRefGoogle Scholar
  19. Vidyasagar M (1985) Control system synthesis: a factorization approach. Signal processing, optimization and control. MIT, CambridgeGoogle Scholar
  20. Youla D, Jabr H, Bongiorno J (1976) Modern Wiener-Hopf design of optimal controllers – Part II: the multivariable case. IEEE Trans Autom Control 21(3):319–338CrossRefGoogle Scholar
  21. Zhou K, Doyle JC, Glover K (1996) Robust and optimal control. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar

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

Authors and Affiliations

  1. 1.Department of Electrical and Electronic EngineeringThe University of MelbourneParkvilleAustralia

Section editors and affiliations

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