Interpolation-Based Modeling of MIMO LPV Systems

This paper presents State-space Model Interpolation of Local Estimates (SMILE), a technique to estimate linear parameter-varying (LPV) state-space models for multiple-input multiple-output (MIMO) systems whose dynamics depends on multiple time-varying parameters, called scheduling parameters. The SMILE technique is based on the interpolation of linear time-invariant models estimated for constant values of the scheduling parameters. As the linear time-invariant models can be either continuous- or discrete-time, both continuous- and discrete-time LPV models can be obtained. The underlying interpolation technique is formulated as a linear least-squares problem that can be efficiently solved. The proposed technique yields homogeneous polynomial LPV models in the multi-simplex that are numerically well-conditioned and therefore suitable for LPV control synthesis. The potential of the SMILE technique is demonstrated by computing a continuous-time interpolating LPV model for an analytic mass-spring-damper system and a discrete-time interpolating LPV model for a mechatronic -motion system based on experimental data.

[1]  Paul Van Dooren,et al.  Computation of zeros of linear multivariable systems , 1980, Autom..

[2]  Wilson J. Rugh,et al.  Research on gain scheduling , 2000, Autom..

[3]  Jan Swevers,et al.  Interpolating model identification for SISO linear parameter-varying systems , 2009 .

[4]  William Leithead,et al.  Survey of gain-scheduling analysis and design , 2000 .

[5]  A. Laub,et al.  Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms , 1987 .

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

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

[8]  R. van de Molengraft,et al.  Experimental modelling and LPV control of a motion system , 2003, Proceedings of the 2003 American Control Conference, 2003..

[9]  Rik Pintelon,et al.  System Identification: A Frequency Domain Approach , 2012 .

[10]  Jamal Daafouz,et al.  Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties , 2001, Syst. Control. Lett..

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

[12]  Fen Wu,et al.  Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions , 2005, Proceedings of the 2005, American Control Conference, 2005..

[13]  W. Xie.  H2 gain scheduled state feedback for LPV system with new LMI formulation , 2005 .

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

[15]  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).

[16]  Bassam Bamieh,et al.  Identification of linear parameter varying models , 2002 .

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

[18]  Jan Swevers,et al.  Identification of Interpolating Affine LPV Models for Mechatronic Systems with one Varying Parameter , 2008, Eur. J. Control.

[19]  Valter J. S. Leite,et al.  Robust control through piecewise Lyapunov functions for discrete time-varying uncertain systems , 2004 .

[20]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

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

[22]  Gary J. Balas,et al.  Linear, parameter‐varying control and its application to a turbofan engine , 2002 .

[23]  Jan Swevers,et al.  Gain-Scheduled H∞-Control for Discrete-Time Polytopic LPV Systems Using Homogeneous Polynomially Parameter-Dependent Lyapunov Functions , 2009 .

[24]  Ricardo C. L. F. Oliveira,et al.  Linear matrix inequality characterisation for H ∞ and H 2 guaranteed cost gain-scheduling quadratic stabilisation of linear time-varying polytopic systems , 2007 .

[25]  Jan Swevers,et al.  Gain-scheduled H∞-control of discrete-time polytopic time-varying systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[26]  Jan Swevers,et al.  AN APPLICATION OF INTERPOLATING GAIN-SCHEDULING CONTROL , 2007 .

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

[28]  Jan Swevers,et al.  Identification of MIMO LPV models based on interpolation , 2008 .

[29]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

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

[31]  Jan Swevers,et al.  Gain-scheduled H infinity -control of discrete-time polytopic time-varying systems. , 2008, CDC 2008.

[32]  Massimiliano Mattei,et al.  Gain scheduled control for discrete‐time systems depending on bounded rate parameters , 2005 .

[33]  C. de Souza,et al.  Gain‐scheduled ℋ︁2 controller synthesis for linear parameter varying systems via parameter‐dependent Lyapunov functions , 2006 .

[34]  Jan Swevers,et al.  Gain-scheduled H 2 and H ∞ control of discrete-time polytopic time-varying systems , 2010 .

[35]  Ricardo C. L. F. Oliveira,et al.  Robust LMIs with parameters in multi-simplex: Existence of solutions and applications , 2008, 2008 47th IEEE Conference on Decision and Control.