Transition and control of nonlinear systems by combining the loop shaping design procedure and the gap metric theory

In this paper, in order to synthesize a control law we propose a new approach that enables identification of the intermediate equilibrium points of a nonlinear system, knowing the first and the last ones. These points are those around which the nonlinear system is linearized and therefore yields local models (sub-models) that contribute to forming the multimodel describing the nonlinear system. This approach is based on the transition from a given point (source) to the next by varying a scheduling parameter (SP) defining the source point sub-model. The variation of this parameter is limited by the maximum value of the stability margin determined by the loop shaping design procedure approach (LSDP) applied to such a sub-model. Hence, the new equilibrium point is defined by the new obtained value of the SP for which the gap metric between this sub-model and the one corresponding to the new value of SP is larger than the given stability margin. The different robust controllers synthesized for the different equilibrium points will be used to synthesize the robust control of the nonlinear system, by applying the gain-scheduling technique. The proposed transition approach as well as the robust control algorithm were validated on the continuous stirred tank reactor (CSTR) system.

[1]  Petko H. Petkov,et al.  Robust control design with MATLAB , 2005 .

[2]  Zhong-Ping Jiang,et al.  Global Stabilization of Nonlinear Systems Based on Vector Control Lyapunov Functions , 2012, IEEE Transactions on Automatic Control.

[3]  Thomas F. Edgar,et al.  Robust feedback linearization and fuzzy control , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

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

[5]  Keith Glover,et al.  A loop-shaping design procedure using H/sub infinity / synthesis , 1992 .

[6]  Diyi Chen,et al.  Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization , 2013 .

[7]  Wilson J. Rugh,et al.  Stability preserving interpolation methods for the synthesis of gain scheduled controllers , 2000, Autom..

[8]  Jose A. Romagnoli,et al.  Gap Metric Concept and Implications for Multilinear Model-Based Controller Design , 2003 .

[9]  Xinzhi Liu,et al.  Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks , 2014, Commun. Nonlinear Sci. Numer. Simul..

[10]  T. Georgiou,et al.  Optimal robustness in the gap metric , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[11]  G. Cohen,et al.  A J-spectral factorization approach to control , 1990 .

[12]  William Leithead,et al.  Gain-scheduled controller design: An analytic framework directly incorporating non-equilibrium plant dynamics , 1998 .

[13]  Clement Festila,et al.  Feedback linearization control design for the 13C cryogenic separation column , 2010, 2010 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR).

[14]  Thomas F. Edgar,et al.  Process Dynamics and Control , 1989 .

[15]  E. Armstrong.  Robust controller design for flexible structures using normalized coprime factor plant descriptions , 1993 .

[16]  Henri Bourlès,et al.  Robust nonlinear control associating robust feedback linearization and H∞ control , 2006, IEEE Trans. Autom. Control..

[17]  Sourav Patra,et al.  Design of static Hinfinity loop shaping controller in four-block framework using LMI approach , 2008, Autom..

[18]  Miroslav Halás,et al.  An algebraic framework generalizing the concept of transfer functions to nonlinear systems , 2008, Autom..

[19]  S. Mammar,et al.  Loop shaping H∞ design: Application to the robust stabilization of a helicopter , 1993 .

[20]  Spilios Theodoulis,et al.  Missile Autopilot Design: Gain-Scheduling and the Gap Metric , 2007 .

[21]  Mark Beale,et al.  Neural Network Toolbox™ User's Guide , 2015 .

[22]  Seddik M. Djouadi,et al.  On the computation of the gap metric for LTV systems , 2007, Syst. Control. Lett..

[23]  Michael Green,et al.  A J-Spectral Factorization Approach to H∞ Control , 1990 .

[24]  Keith Glover,et al.  A loop shaping design procedure , 1990 .

[25]  Ian Postlethwaite,et al.  Multivariable Feedback Control: Analysis and Design , 1996 .

[26]  Marcus Reble,et al.  Model predictive control for nonlinear continuous-time systems with and without time-delays , 2013 .

[27]  W. Leithead,et al.  Gain-scheduled and nonlinear systems : dynamic analysis by velocity-based linearization families , 1998 .