State and Loop Equivalence for Linear Parameter Varying Systems

  • József Bokor
  • Zoltán SzabóEmail author
Part of the Topics in Intelligent Engineering and Informatics book series (TIEI, volume 14)


In the last decades the LPV modelling paradigm grew up from the desire of having a gain scheduling method with guaranteed stability and performance bound by using as much as possible from the classical design techniques. LPV design becomes a proven method of the field of robust control through a series of applications. While system equivalence, state transformation and loop transformation are fundamental concepts and efficient tools of the linear time invariant (LTI) theory, in the context of the LPV framework some basic modelling issues still evades the attention of the researchers. The main goal of the paper is to provide an initialization in LPV modelling and to review the fundamental concepts in order to eliminate the possible pitfalls that often occur in the related literature. The work provides an opportunity for pointing out some research topics that might be interesting for a much larger audience, too.



This work has been supported by the GINOP-2.3.2-15-2016-00002 grant of the Ministry of National Economy of Hungary and by the European Commission through the H2020 project EPIC under grant No. 739592.


  1. 1.
    G. Balas, J. Bokor, Z. Szabó, Invariant subspaces for LPV systems and their applications. IEEE Trans. Autom. Control 48(11), 2065–2069 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Z. Szabó, J. Bokor, G. Balas, Inversion of LPV systems and its application to fault detection, in Proceedings of the 5th IFAC Symposium on fault detection supervision and safety for technical processes (SAFEPROCESS’03), Washington, USA (2003), pp. 235–240Google Scholar
  3. 3.
    J. Bokor, G. Balas, Detection filter design for LPV systems—a geometric approach. Automatica 40(3), 511–518 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Luspay, T. Péni, I. Gőzse, Z. Szabó, B. Vanek, Model reduction for LPV systems based on approximate modal decomposition. Int. J. Numer. Meth. Eng. 113(6), 891–909 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Z. Alkhoury, M. Petreczky, G. Mercere, comparing global input-output behavior of frozen-equivalent LPV state-space models, in IFAC-PapersOnLine, vol. 50, no. 1, 20th IFAC World Congress (2017), pp. 9766–9771Google Scholar
  6. 6.
    Z. Szabó, T. Péni, J. Bokor, Null-space computation for qLPV systems, in IFAC-PapersOnLine, 1st IFAC Workshop on Linear Parameter Varying Systems, Grenoble, France, vol. 48, no. 26 (2015), pp. 170–175Google Scholar
  7. 7.
    T. Péni, B. Vanek, G. Lipták, Z. Szabó, J. Bokor, Nullspace-based input reconfiguration architecture for over actuated aerial vehicles. IEEE Trans. Control Syst. Technol. no. 99 (2017), pp. 1–8Google Scholar
  8. 8.
    R.E. Kalman, Mathematical description of linear dynamical systems. SIAM J. Control 1(2), 152–192 (1963)MathSciNetzbMATHGoogle Scholar
  9. 9.
    L. Silverman, Representation and realization of time-variable linear systems. Technical Report (Department of Electrical Engineering, Columbia University, New York, 1966)Google Scholar
  10. 10.
    A. Isidori, A. Ruberti, State-space representation and realization of time-varying linear input–output functions. J. Franklin Inst. 301(6), 573–592 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    E. Kamen, New results in realization theory for linear time-varying analytic systems. IEEE Trans. Autom. Control 24(6), 866–878 (1979)MathSciNetCrossRefGoogle Scholar
  12. 12.
    E. Sontag, Realization theory of discrete-time nonlinear systems: part I-the bounded case. IEEE Trans. Circuits Syst. 26(5), 342–356 (1979)CrossRefGoogle Scholar
  13. 13.
    P. Dewilde, A.-J. van der Veen, Time-varying state space realizations, in Time-Varying Systems and Computations (Springer, Boston, 1998), pp. 33–72Google Scholar
  14. 14.
    R. Tóth, Identification and Modeling of Linear Parameter-Varying Systems, in Lecture Notes in Control and Information Sciences, vol. 403 (Springer, Heidelberg, 2010)Google Scholar
  15. 15.
    R. Toth, H.S. Abbas, H. Werner, On the state-space realization of LPV input-output models: practical approaches. IEEE Trans. Control Syst. Technol. 20(1), 139–153 (2012)Google Scholar
  16. 16.
    M. Petreczky, R. Tóth, G. Mercere, Realization theory for LPV state-space representations with affine dependence. IEEE Trans. Autom. Control 62(9), 4667–4674 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    L.M. Silverman, H.E. Meadows, Equivalent realizations of linear systems. SIAM J. Appl. Math. 17(2), 393–408 (1969)MathSciNetCrossRefGoogle Scholar
  18. 18.
    F. Blanchini, D. Casagrande, S. Miani, Stable LPV realization of parametric transfer functions and its application to gain-scheduling control design. IEEE Trans. Autom. Control 55(10), 2271–2281 (2010)MathSciNetCrossRefGoogle Scholar
  19. 19.
    M. Wonham, Linear Multivariable Control: A Geometric Approach (Springer, Berlin, 1985)Google Scholar
  20. 20.
    G.B. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory (Prentice Hall, Englewood Cliffs, 1992)Google Scholar
  21. 21.
    A. Isidori, Nonlinear Control Systems (Springer, Berlin, 1989)Google Scholar
  22. 22.
    C. De Persis, A. Isidori, On the observability codistributions of a nonlinear system. Syst. Control Lett. 40(5), 297–304 (2000)MathSciNetCrossRefGoogle Scholar
  23. 23.
    J. Bokor, Z. Szabo, Fault detection and isolation in nonlinear systems. Annu. Rev. Control 33(2), 1–11 (2009)CrossRefGoogle Scholar
  24. 24.
    R. Tóth, F. Felici, P.S.C. Heuberger, P.M.J.V. den Hof, Discrete time LPV I/O and state space representations, differences of behavior and pitfalls of interpolation, in 2007 European Control Conference (ECC) (2007), pp. 5418–5425Google Scholar
  25. 25.
    A. Feintuch, Robust Control Theory in Hilbert Space, in Applied Mathematical Sciences 130 (Springer, New York, 1998)Google Scholar
  26. 26.
    M. Vidyasagar, Control System Synthesis: A Factorization Approach (MIT Press, Cambridge, 1985)zbMATHGoogle Scholar
  27. 27.
    B.D. Anderson, A. Ilchmann, F.R. Wirth, Stabilizability of linear time-varying systems. Syst. Control Lett. 62(9), 747–755 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    W. Xie, T. Eisaka, Design of LPV control systems based on Youla parameterization. IEE Proc. Control Theory Appl. 151(4), 465–472 (2004)CrossRefGoogle Scholar
  29. 29.
    I. Ellouze, M.A. Hammami, A separation principle of time-varying dynamical systems: a practical stability approach. Math. Modelling Anal. 12(3), 297–308 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Shiriaev, R. Johansson, A. Robertsson, L. Freidovich, Separation principle for a class of nonlinear feedback systems augmented with observers, in IFAC Proceedings Volumes, 17th IFAC World Congress Seoul, Korea, vol. 41, no. 2 (2008), pp. 6196–6201Google Scholar
  31. 31.
    H. Damak, I. Ellouze, M.A. Hammami, A separation principle of time-varying nonlinear dynamical systems. Differ. Equ. Control Process, 1, 36–49 (2013)Google Scholar
  32. 32.
    G.I. Bara, J. Daafouz, J. Ragot, Gain scheduling techniques for the design of observer state feedback controllers, in IFAC Proceedings Volumes, 15th IFAC World Congress Barcelona, Spain, vol. 35, no. 1 (2002), pp. 13–18Google Scholar
  33. 33.
    A. Bouali, M. Yagoubi, P. Chevrel, Gain scheduled observer state feedback controller for rational LPV systems, in IFAC Proceedings Volumes, 17th IFAC World Congress Seoul, Korea, vol. 41, no. 2 (2008), pp. 4922–4927Google Scholar
  34. 34.
    F. Blanchini, S. Miani, Stabilization of LPV systems: state feedback, state estimation, and duality. SIAM J. Control Optim. 42(1), 76–97 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    J.A. Ball, J.W. Helton, M. Verma, A factorization principle for stabilization of linear control systems. J. Nonlinear Robust Control 1, 229–294 (1991)CrossRefGoogle Scholar
  36. 36.
    Z. Szabó, P. Seiler, J. Bokor, Internal stability and loop-transformations: an overview on LFTs, Möbius transforms and chain scattering, in 20th IFAC World Congress Toulouse, France, IFAC-PapersOnLine, vol. 50, no. 1 (2017), pp. 7547–7553Google Scholar
  37. 37.
    M.-C. Tsai, D. Gu, Robust and Optimal Control—A Two-Port Framework Approach (Springer, Berlin, 2014)Google Scholar

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Authors and Affiliations

  1. 1.Institute for Computer Science and ControlHungarian Academy of SciencesBudapestHungary

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