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Fixed-structure LPV-IO controllers: an implicit representation based approach. (English) Zbl 1373.93124
Summary: In this note, novel Linear Matrix Inequality (LMI) analysis conditions for the stability of Linear Parameter-Varying (LPV) systems in Input-Output (IO) representation form are proposed together with Bilinear Matrix Inequality (BMI) conditions for fixed-structure LPV-IO controller synthesis. Both the LPV-IO plant model and the controller are assumed to depend affinely and statically on the scheduling variables. By using an implicit representation of the plant and the controller interaction, an exact representation of the closed-loop behavior with affine dependence on the scheduling variables is achieved. This representation allows to apply Finsler’s lemma for deriving exact stability as well as exact quadratic performance conditions. A DK-iteration based solution is carried out to synthesize the controller. The main results are illustrated by a numerical example.

93B50 Synthesis problems
93C05 Linear systems in control theory
93D99 Stability of control systems
Full Text: DOI
[1] Adegas, F. D.; Stoustrup, J., Structured control of affine linear parameter varying systems, (Proc. of the American control conference, ACC, 2011, San Francisco, California, USA, (2011)), 739-744
[2] Ali, M.; Abbas, H.; Werner, H., Controller synthesis for input-output LPV models, (Proc. of the 49th IEEE conference on decision and control, Atlanta, Gorgia, USA, (2010)), 7694-7699
[3] Apkarian, P.; Gahinet, P.; Becker, G., Self-scheduled \(\mathcal{H}_\infty\) control of linear parameter-varying systems: a design example, Automatica, 31, 9, 1251-1261, (1995) · Zbl 0825.93169
[4] Burke, J.; Henrion, D.; Lewis, A.; Overton, M., HIFOO — A Matlab package for fixed-order controller design and \(H_\infty\) optimization, (Proc. of the IFAC symposium on robust control design, Toulouse, France, (2006))
[5] Cerone, V.; Piga, D.; Regruto, D.; Tóth, R., Fixed order LPV controller design for LPV models in input-output form, (Proc. of the 51st IEEE conference on decision and control, Maui, Hawaii, (2012)), 6297-6302
[6] Coutinho, D. F.; de Souza, C. E.; Trofino, Alexandre, Stability analysis of implicit polynomial systems, IEEE Transactions on Automatic Control, 54, 5, 1012-1018, (2009) · Zbl 1367.93429
[7] de Oliveira, M. C.; Skelton, R., Stability tests for constrained linear systems, (Moheimani, R. O., Perspectives in robust control, Vol. 268, (2001), Springer Berlin), 241-257 · Zbl 0997.93086
[8] Gilbert, W.; Henrion, D.; Bernussou, J.; Boyer, D., Polynomial LPV synthesis applied to turbofan engine, Control Engineering Practice, 18, 9, 1077-1083, (2010)
[9] Henrion, D.; Arzelier, D.; Peaucelle, D., Positive polynomial matrices and improved LMI robustness condtions, Automatica, 39, 8, 1479-1485, (2003) · Zbl 1037.93027
[10] Hoffmann, C.; Werner, H., A survey of linear parameter-varying control applications validated by experiments or high-fidelity simulations, IEEE Transactions on Control Systems Technology, 23, 2, 416-433, (2015)
[11] Horn, R.; Johnson, C., Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001
[12] Packard, A., Gain scheduling via linear fractional transformations, Systems & Control Letters, 22, 2, 79-92, (1994) · Zbl 0792.93043
[13] Polderman, J. W.; Willems, J. C., Introduction to mathematical systems theory, a behavioral approach, (1991), Springer
[14] Popov, A.; Abbas, H., On eigenvalues and their derivatives, (2009), Internal report, Insitute of Control Systems, Hamburg University of Technology
[15] Popov, A.; Werner, H.; Millstone, M., Fixed-structure discrete-time \(H_\infty\) controller synthesis with HIFOO, (Proc. the 49th IEEE conference on decision and control, Atlanta, Gorgia, USA, (2010)), 3152-3155
[16] Roffel, B.; Betlem, B., Process dynamics and control: modeling for control and prediction, (2007), Wiley
[17] Rugh, W. J.; Shamma, J. S., A survey of research on gain-scheduling, Automatica, 36, 1401-1425, (2000) · Zbl 0976.93002
[18] Scherer, C., LPV control and full block multipliers, Automatica, 27, 3, 325-485, (2001) · Zbl 0982.93060
[19] Tóth, R., (Modeling and identification of linear parameter-varying systems, Lecture notes in control and information sciences, Vol. 403, (2010), Springer Heidelberg)
[20] Toth, R.; Abbas, H.; Werner, H., On the state-space realization of LPV input-output models: practical approaches, IEEE Trans. on Control System Technology, 20, 1, 139-153, (2012)
[21] Tóth, R.; Van den Hof, P.; Ludlage, J.; Heuberger, P., Identitifcation of nonlinear process models in an LPV framework, (Proc. of the 9th international symposium on dynmaics and control of process systems, Leuven, Belgium, (2010)), 869-874
[22] Tóth, R.; Willems, J. C.; Heuberger, P. S.C.; Van den Hof, P. M.J., The behavioral approach to linear parameter-varying systems, IEEE Transactions on Automatic Control, 56, 2499-2514, (2011) · Zbl 1368.93103
[23] van Wingerden, J. W.; Verhaegen, M., Subspace identification of bilinear and LPV systems for open- and closed-loop data, Automatica, 45, 2, 372-381, (2009) · Zbl 1158.93324
[24] Willems, J. C., Dissipative dynamical systems, part ii: linear systems with quadratic supply rates, Archive for Rational Mechanics and Analysis, 45, 352-393, (1972) · Zbl 0252.93003
[25] Willems, J. C., Models for dynamics, Dynamics Reported, 2, 171-269, (1989)
[26] Wollnack, S.; Abbas, H. S.; Werner, H.; Tóth, R., Fixed-structure LPV controller synthesis based on implicit input output representations, (Proc. of the 52nd IEEE conference on decision and control, Florence, Italy, (2013)), 2103-2108
[27] Wollnack, S.; Werner, H., LPV-IO controller design: an LMI approach, (2016 American control conference, ACC, (2016)), 4617-4622
[28] Zhou, K.; Doyle, J. C.; Glover, K., Robust and optimal control, (1996), Prentice-Hall · Zbl 0999.49500
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