Bayes Control of Hammerstein Systems

Abstract In this paper, we consider data driven control of Hammerstein systems. For such systems a common control structure is a transfer function followed by a static output nonlinearity that tries to cancel the input nonlinearity of the system, which is modeled as a polynomial or piece-wise linear function. The linear part of the controller is used to achieve desired disturbance rejection and tracking properties. To design a linear part of the controller, we propose a weighted average risk criterion with the risk being the average of the squared L2 tracking error. Here the average is with respect to the observations used in the controller and the weighting is with respect to how important it is to have good control for different impulse responses. This criterion corresponds to the average risk criterion leading to the Bayes estimator and we therefore call this approach Bayes control. By parametrizing the weighting function and estimating the corresponding hyperparameters we tune the weighting function to the information regarding the true impulse response contained in the data set available to the user for the control design. The numerical results show that the proposed methods result in stable controllers with performance comparable to the optimal controller, designed using the true input nonlinearity and true plant.

[1]  T. Söderström,et al.  Instrumental-variable methods for identification of Hammerstein systems , 1982 .

[2]  Giulio Bottegal,et al.  A nonparametric kernel-based approach to Hammerstein system identification , 2017, Autom..

[3]  Håkan Hjalmarsson,et al.  Iterative feedback tuning—an overview , 2002 .

[4]  Matthew A. Franchek,et al.  Robust SISO H 8 controller design for nonlinear systems , 2005 .

[5]  Mohamed Darouach,et al.  A robust and recursive identification method for MISO Hammerstein model , 1996 .

[6]  Simone Formentin,et al.  Bayesian Kernel-Based Linear Control Design , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[7]  A. Karimi,et al.  Non-iterative data-driven controller tuning using the correlation approach , 2007, 2007 European Control Conference (ECC).

[8]  Johan A. K. Suykens,et al.  Subspace identification of Hammerstein systems using least squares support vector machines , 2005, IEEE Transactions on Automatic Control.

[9]  Giulio Bottegal,et al.  A kernel-based approach to Hammerstein system identification , 2014, ArXiv.

[10]  H. Bloemen,et al.  Model-based predictive control for Hammerstein systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[11]  Sergio M. Savaresi,et al.  Virtual reference feedback tuning: a direct method for the design of feedback controllers , 2002, Autom..

[12]  Brian D. O. Anderson,et al.  I design to generalize internal model control , 2006, Autom..

[13]  Svante Gunnarsson,et al.  Iterative feedback tuning: theory and applications , 1998 .

[14]  B. Anderson,et al.  Controller Reduction: Concepts and Approaches , 1987, 1987 American Control Conference.

[15]  Ioan Doré Landau,et al.  A Flexible Transmission System as a Benchmark for Robust Digital Control , 1995, Eur. J. Control.

[16]  Sergio M. Savaresi,et al.  Virtual reference direct design method: an off-line approach to data-based control system design , 2000, IEEE Trans. Autom. Control..

[17]  Henrik Ohlsson,et al.  On the estimation of transfer functions, regularizations and Gaussian processes - Revisited , 2012, Autom..

[18]  Giuseppe De Nicolao,et al.  A new kernel-based approach for linear system identification , 2010, Autom..

[19]  Er-Wei Bai,et al.  Decoupling the linear and nonlinear parts in Hammerstein model identification , 2004, Autom..

[20]  A. Karimi,et al.  Iterative correlation‐based controller tuning , 2004 .