MIMO PID tuning via iterated LMI restriction

We formulate multi‐input multi‐output proportional integral derivative controller design as an optimization problem that involves nonconvex quadratic matrix inequalities. We propose a simple method that replaces the nonconvex matrix inequalities with a linear matrix inequality restriction, and iterates to convergence. This method can be interpreted as a matrix extension of the convex–concave procedure, or as a particular majorization–minimization method. Convergence to a local minimum can be guaranteed. While we do not know that the resulting controller is globally optimal, the method works well in practice, and provides a simple automated method for tuning multi‐input multi‐output proportional integral derivative controllers. The method is readily extended in many ways, for example, to the design of more complex, structured controllers. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[2]  R. K. Wood,et al.  Terminal composition control of a binary distillation column , 1973 .

[3]  Heikki N. Koivo,et al.  Multivariable tuning regulators for unknown systems , 1980, Autom..

[4]  William L. Luyben,et al.  Simple method for tuning SISO controllers in multivariable systems , 1986 .

[5]  Stephen P. Boyd,et al.  Linear controller design: limits of performance , 1991 .

[6]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[7]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[8]  A. Favero,et al.  Italy , 1996, The Lancet.

[9]  Jiawen Dong,et al.  Design of robust multivariable PID controllers via IMC , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[10]  Qing-Guo Wang,et al.  Auto-tuning of multivariable PID controllers from decentralized relay feedback , 1997, Autom..

[11]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[12]  K.J. ÅSTRÖM,et al.  Design of PI Controllers based on Non-Convex Optimization , 1998, Autom..

[13]  Tore Hägglund,et al.  Design of PID controllers based on constrained optimization , 1999 .

[14]  H. Marquez,et al.  Robust Controller Design And Pid Tuning For Multivariable Processes , 2002 .

[15]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[16]  Alan L. Yuille,et al.  The Concave-Convex Procedure , 2003, Neural Computation.

[17]  Tong Heng Lee,et al.  An improvement on multivariable PID controller design via iterative LMI approach , 2004, Autom..

[18]  Tore Hägglund,et al.  Advanced PID Control , 2005 .

[19]  Masami Saeki,et al.  FIXED STRUCTURE PID CONTROLLER DESIGN FOR STANDARD H∞ CONTROL PROBLEM , 2005 .

[20]  H. Zou,et al.  Regularization and variable selection via the elastic net , 2005 .

[21]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[22]  Pierre Apkarian,et al.  Nonsmooth H∞ synthesis , 2005, IEEE Trans. Autom. Control..

[23]  Masami Saeki,et al.  Fixed structure PID controller design for standard Hinfinity control problem , 2006, Autom..

[24]  Masami Saeki,et al.  Design of multivariable H∞ PID controller using frequency response , 2007, 2007 IEEE International Conference on Control Applications.

[25]  Fernando Daniel Bianchi,et al.  Multivariable PID control with set-point weighting via BMI optimisation , 2008, Autom..

[26]  Tore Hägglund,et al.  A Software Tool for Robust PID Design , 2008 .

[27]  Stephen P. Boyd,et al.  Graph Implementations for Nonsmooth Convex Programs , 2008, Recent Advances in Learning and Control.

[28]  Masami Saeki,et al.  Low‐order H∞ controller design on the frequency domain by partial optimization , 2010 .

[29]  Fu Lin,et al.  Sparse feedback synthesis via the alternating direction method of multipliers , 2012, 2012 American Control Conference (ACC).

[30]  Ramon Vilanova,et al.  PID control in the Third Millennium : lessons learned and new approaches , 2012 .

[31]  Ramon Vilanova,et al.  PID Control in the Third Millennium , 2012 .

[32]  Tore Hägglund,et al.  Criteria and Trade-offs in PID Design , 2012 .

[33]  P. Seiler,et al.  Proceedings of the American Control Conference , 2013 .

[34]  Stephen P. Boyd,et al.  PID design by convex-concave optimization , 2013, 2013 European Control Conference (ECC).

[35]  Anton van den Hengel,et al.  Semidefinite Programming , 2014, Computer Vision, A Reference Guide.