Nonstationary LPV control for trajectory tracking: a double pendulum example

This article focusses on the implementation of recently developed nonstationary linear-parameter varying (NSLPV) control algorithms for the regulation of nonlinear systems about pre-specified trajectories. The trajectories of interest eventually settle into periodic orbits, and hence are duly called eventually periodic trajectories. Parameterising the nonlinear system equations about such trajectories results in eventually periodic NSLPV models, and then NSLPV controllers are designed for these models to ensure accurate trajectory tracking despite various disturbances and uncertainties. These control algorithms will be applied to control a double pendulum where a vessel containing fluid is rigidly attached to the end of the second link of the pendulum. The mass of the fluid varies in time and, together with its rate of variation, is available for measurement during plant operation.

[1]  Geir E. Dullerud,et al.  Duality and eventually periodic systems , 2005 .

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

[3]  Mazen Farhood,et al.  LPV Control of Nonstationary Systems: A Parameter-Dependent Lyapunov Approach , 2012, IEEE Transactions on Automatic Control.

[4]  A. Packard.  Gain scheduling via linear fractional transformations , 1994 .

[5]  M. Laurent.  Sums of Squares, Moment Matrices and Optimization Over Polynomials , 2009 .

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

[7]  Carsten W. Scherer,et al.  Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs , 2006, Math. Program..

[8]  Didier Henrion,et al.  GloptiPoly 3: moments, optimization and semidefinite programming , 2007, Optim. Methods Softw..

[9]  Geir E. Dullerud,et al.  Control of nonstationary LPV systems , 2008, Autom..

[10]  P. Apkarian,et al.  Relaxations of parameterized LMIs with control applications , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[11]  Mi-Ching Tsai,et al.  Robust and Optimal Control , 2014 .

[12]  P. Parrilo,et al.  From coefficients to samples: a new approach to SOS optimization , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[13]  P. Parrilo.  Exploiting Algebraic Structure in Sum of Squares Programs , 2005 .

[14]  Geir E. Dullerud,et al.  Control of Systems With Uncertain Initial Conditions , 2008, IEEE Transactions on Automatic Control.

[15]  E. Feron,et al.  Obstacle-Sensitive Trajectory Regulation via Gain Scheduling and Semidefinite Programming , 2012, IEEE Transactions on Control Systems Technology.

[16]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[17]  P. Apkarian,et al.  Relaxations of parameterized LMIs with control applications , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[18]  Carsten W. Scherer,et al.  LMI Relaxations in Robust Control , 2006, Eur. J. Control.

[19]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[20]  A. Plastino,et al.  On the use and abuse of Newton's second law for variable mass problems , 1992 .

[21]  V. Ionescu,et al.  Time-Varying Discrete Linear Systems , 1994 .

[22]  Pascal Gahinet,et al.  Explicit controller formulas for LMI-based H∞ synthesis , 1996, Autom..

[23]  Mark W. Spong,et al.  Robot dynamics and control , 1989 .

[24]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[25]  Pierre Apkarian,et al.  Parameterized LMIs in Control Theory , 2000, SIAM J. Control. Optim..

[26]  Karolos M. Grigoriadis,et al.  Anti-windup controller design using linear parameter-varying control methods , 2000 .

[27]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[28]  P. Gahinet,et al.  A linear matrix inequality approach to H∞ control , 1994 .

[29]  Pierre Apkarian,et al.  Monotonic relaxations for robust control: new characterizations , 2002, IEEE Trans. Autom. Control..

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

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

[32]  S. Prajna,et al.  SOS-based solution approach to polynomial LPV system analysis and synthesis problems , 2005 .

[33]  Geir E. Dullerud,et al.  A new approach for analysis and synthesis of time-varying systems , 1999, IEEE Trans. Autom. Control..

[34]  Andres Marcos,et al.  Linear Parameter Varying Modeling of the Boeing 747-100/200 Longitudinal Motion , 2001 .

[35]  Jean B. Lasserre,et al.  A semidefinite programming approach to the generalized problem of moments , 2007, Math. Program..

[36]  A. Sommerfeld,et al.  Mechanics: Lectures on Theoretical Physics , 1964 .

[37]  P. Gahinet,et al.  A convex characterization of gain-scheduled H∞ controllers , 1995, IEEE Trans. Autom. Control..

[38]  P. Gahinet,et al.  A convex characterization of gain-scheduled H∞ controllers , 1995, IEEE Trans. Autom. Control..

[39]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[40]  Geir E. Dullerud,et al.  LMI tools for eventually periodic systems , 2002, Syst. Control. Lett..

[41]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

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

[43]  Michael Athans,et al.  Analysis of gain scheduled control for nonlinear plants , 1990 .

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