Subspace identification of Bilinear and LPV systems for open- and closed-loop data

In this paper we present a novel algorithm to identify LPV systems with affine parameter dependence operating under open- and closed-loop conditions. A factorization is introduced which makes it possible to form a predictor that predicts the output, which is based on past inputs, outputs, and scheduling data. The predictor contains the LPV equivalent of the Markov parameters. Using this predictor, ideas from closed-loop LTI identification are developed to estimate the state sequence from which the LPV system matrices can be constructed. A numerically efficient implementation is presented using the kernel method. It turns out that if structure is present in the scheduling sequence the computational complexity reduces even more.

[1]  P. Heuberger,et al.  Discrete time LPV I/O and state space representations, differences of behavior and pitfalls of interpolation , 2007, 2007 European Control Conference (ECC).

[2]  Michel Verhaegen,et al.  Subspace identification of multivariable linear parameter-varying systems , 2002, Autom..

[3]  Lawton H. Lee,et al.  Identification of Linear Parameter-Varying Systems Using Nonlinear Programming , 1999 .

[4]  Michael Athans,et al.  Guaranteed properties of gain scheduled control for linear parameter-varying plants , 1991, Autom..

[5]  R. Shah,et al.  Least Squares Support Vector Machines , 2022 .

[6]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[7]  Michel Verhaegen,et al.  Subspace identification of multivariable LPV systems: a novel approach , 2008, 2008 IEEE International Conference on Computer-Aided Control Systems.

[8]  Torben Knudsen.  Consistency analysis of subspace identification methods based on a linear regression approach , 2001, Autom..

[9]  Fernando D. Bianchi,et al.  Gain scheduling control of variable-speed wind energy conversion systems using quasi-LPV models , 2005 .

[10]  Bart De Moor,et al.  Subspace identification of bilinear systems subject to white inputs , 1999, IEEE Trans. Autom. Control..

[11]  Bassam Bamieh,et al.  LPV model identification for gain scheduling control: An application to rotating stall and surge control problem , 2006 .

[12]  Michel Verhaegen,et al.  Subspace IDentification of MIMO LPV systems: The PBSID approach , 2008, 2008 47th IEEE Conference on Decision and Control.

[13]  Bassam Bamieh,et al.  Identification of linear parameter varying models , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[14]  Johan A. K. Suykens,et al.  Fixed-size Least Squares Support Vector Machines: A Large Scale Application in Electrical Load Forecasting , 2006, Comput. Manag. Sci..

[15]  M. Lovera,et al.  Identification of non-linear parametrically varying models using separable least squares , 2004 .

[16]  Michel Verhaegen,et al.  Subspace identification of bilinear systems using a dedicated input sequence , 2007, 2007 46th IEEE Conference on Decision and Control.

[17]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[18]  Kazutaka Takahashi,et al.  Neuroengineering model of human limb control - Gainscheduled feedback control approach , 2007, 2007 46th IEEE Conference on Decision and Control.

[19]  Marco Lovera,et al.  Identification of Nonlinear Parametrically Varying Models Using Separable Least Squares , 2003 .

[20]  Pierre Apkarian,et al.  Advanced gain-scheduling techniques for uncertain systems , 1998, IEEE Trans. Control. Syst. Technol..

[21]  Joe Brewer,et al.  Kronecker products and matrix calculus in system theory , 1978 .

[22]  Alessandro Chiuso,et al.  Consistency analysis of some closed-loop subspace identification methods , 2005, Autom..

[23]  M. Lovera,et al.  Identification for gain-scheduling: a balanced subspace approach , 2007, 2007 American Control Conference.

[24]  L. Giarréa,et al.  LPV model identification for gain scheduling control : An application to rotating stall and surge control problem , 2005 .

[25]  David M. Eggleston,et al.  Wind Turbine Engineering Design , 1987 .

[26]  Alessandro Chiuso,et al.  The role of vector autoregressive modeling in predictor-based subspace identification , 2007, Autom..

[27]  Magnus Jansson.  A NEW SUBSPACE IDENTIFICATION METHOD FOR OPEN AND CLOSED LOOP DATA , 2005 .

[28]  Gary J. Balas,et al.  Comparing Linear Parameter-Varying Gain-Scheduled Control Techniques for Active Flutter Suppression , 2000 .

[29]  J.L. Martins de Carvalho,et al.  Identification of Bilinear Systems Using an Iterative Deterministic-Stochastic Subspace Approach , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[30]  Bernhard Schölkopf,et al.  Sparse Greedy Matrix Approximation for Machine Learning , 2000, International Conference on Machine Learning.

[31]  Roy M. Howard,et al.  Linear System Theory , 1992 .

[32]  Okko H. Bosgra,et al.  LPV control for a wafer stage: beyond the theoretical solution , 2005 .

[33]  Diana Maria Sima,et al.  Regularization Techniques in Model Fitting and Parameter Estimation (Regularisatietechnieken in modellering en parameterschatting) , 2006 .

[34]  Michel Verhaegen,et al.  Closed‐loop identification of the time‐varying dynamics of variable‐speed wind turbines , 2009 .

[35]  Carsten W. Scherer,et al.  LPV control and full block multipliers , 2001, Autom..

[36]  B. Moor,et al.  Subspace identification for linear systems , 1996 .

[37]  Vincent Verdult,et al.  Kernel methods for subspace identification of multivariable LPV and bilinear systems , 2005, Autom..

[38]  Roland Tóth,et al.  LPV system identification with globally fixed orthonormal basis functions , 2007, 2007 46th IEEE Conference on Decision and Control.

[39]  Michel Verhaegen,et al.  Subspace identification of MIMO LPV systems using a periodic scheduling sequence , 2007, Autom..

[40]  Kefu Liu,et al.  IDENTIFICATION OF LINEAR TIME-VARYING SYSTEMS , 1997 .

[41]  R. Ravikanth,et al.  Identification of linear parametrically varying systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[42]  J. W. Van Wingerden,et al.  Control of wind turbines with 'Smart' rotors : Proof of concept & LPV subspace identification , 2008 .

[43]  Mario Garcia-Sanz,et al.  Wind turbines: New challenges and advanced control solutions , 2009 .

[44]  A. Laub,et al.  Numerical solution of the discrete-time periodic Riccati equation , 1994, IEEE Trans. Autom. Control..

[45]  A. Varga.  ON SOLVING DISCRETE-TIME PERIODIC RICCATI EQUATIONS , 2005 .

[46]  Michel Verhaegen,et al.  Subspace identification of MIMO LPV systems using a piecewise constant scheduling sequence with hard/soft switching , 2007, 2007 European Control Conference (ECC).