Abstract
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.
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References
G. Balas, J. Bokor, Z. Szabó, Invariant subspaces for LPV systems and their applications. IEEE Trans. Autom. Control 48(11), 2065–2069 (2003)
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. 235–240
J. Bokor, G. Balas, Detection filter design for LPV systems—a geometric approach. Automatica 40(3), 511–518 (2004)
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)
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. 9766–9771
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. 170–175
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. 1–8
R.E. Kalman, Mathematical description of linear dynamical systems. SIAM J. Control 1(2), 152–192 (1963)
L. Silverman, Representation and realization of time-variable linear systems. Technical Report (Department of Electrical Engineering, Columbia University, New York, 1966)
A. Isidori, A. Ruberti, State-space representation and realization of time-varying linear input–output functions. J. Franklin Inst. 301(6), 573–592 (1976)
E. Kamen, New results in realization theory for linear time-varying analytic systems. IEEE Trans. Autom. Control 24(6), 866–878 (1979)
E. Sontag, Realization theory of discrete-time nonlinear systems: part I-the bounded case. IEEE Trans. Circuits Syst. 26(5), 342–356 (1979)
P. Dewilde, A.-J. van der Veen, Time-varying state space realizations, in Time-Varying Systems and Computations (Springer, Boston, 1998), pp. 33–72
R. Tóth, Identification and Modeling of Linear Parameter-Varying Systems, in Lecture Notes in Control and Information Sciences, vol. 403 (Springer, Heidelberg, 2010)
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)
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)
L.M. Silverman, H.E. Meadows, Equivalent realizations of linear systems. SIAM J. Appl. Math. 17(2), 393–408 (1969)
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)
M. Wonham, Linear Multivariable Control: A Geometric Approach (Springer, Berlin, 1985)
G.B. Basile, G. Marro, Controlled and Conditioned Invariants in Linear System Theory (Prentice Hall, Englewood Cliffs, 1992)
A. Isidori, Nonlinear Control Systems (Springer, Berlin, 1989)
C. De Persis, A. Isidori, On the observability codistributions of a nonlinear system. Syst. Control Lett. 40(5), 297–304 (2000)
J. Bokor, Z. Szabo, Fault detection and isolation in nonlinear systems. Annu. Rev. Control 33(2), 1–11 (2009)
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. 5418–5425
A. Feintuch, Robust Control Theory in Hilbert Space, in Applied Mathematical Sciences 130 (Springer, New York, 1998)
M. Vidyasagar, Control System Synthesis: A Factorization Approach (MIT Press, Cambridge, 1985)
B.D. Anderson, A. Ilchmann, F.R. Wirth, Stabilizability of linear time-varying systems. Syst. Control Lett. 62(9), 747–755 (2013)
W. Xie, T. Eisaka, Design of LPV control systems based on Youla parameterization. IEE Proc. Control Theory Appl. 151(4), 465–472 (2004)
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)
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. 6196–6201
H. Damak, I. Ellouze, M.A. Hammami, A separation principle of time-varying nonlinear dynamical systems. Differ. Equ. Control Process, 1, 36–49 (2013)
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. 13–18
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. 4922–4927
F. Blanchini, S. Miani, Stabilization of LPV systems: state feedback, state estimation, and duality. SIAM J. Control Optim. 42(1), 76–97 (2003)
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)
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. 7547–7553
M.-C. Tsai, D. Gu, Robust and Optimal Control—A Two-Port Framework Approach (Springer, Berlin, 2014)
Acknowledgements
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.
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Bokor, J., Szabó, Z. (2020). State and Loop Equivalence for Linear Parameter Varying Systems. In: Kovács, L., Haidegger, T., Szakál, A. (eds) Recent Advances in Intelligent Engineering. Topics in Intelligent Engineering and Informatics, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-030-14350-3_1
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