Design of parameter-scheduled state-feedback controllers using shifting specifications

Abstract In this paper, the problem of designing a parameter-scheduled state-feedback controller is investigated. The paper presents an extension of the classical regional pole placement, H 2 control and H ∞ control problems, so as to satisfy new specifications, that will be referred to as shifting pole placement control, shifting H 2 control and shifting H ∞ control, respectively. By introducing some parameters, or using the existing ones, the controller can be designed in such a way that different values of these parameters imply different regions where the closed-loop poles are situated, or different performances in the H 2 or H ∞ sense. The proposed approach is derived within the so-called Lyapunov Shaping Paradigm, where a single quadratic Lyapunov function is used for ensuring stability and desired performances in spite of arbitrary parameter time variation. The problem is analyzed in the continuous-time LPV case, even though the developed theory could be applied to LTI systems in cases when it is desired to vary the control system performances online. Results obtained in simulation demonstrate the effectiveness and the relevant features of the proposed approach.

[1]  Philippe Weber,et al.  Fault-tolerant control design with respect to actuator health degradation: An LMI approach , 2011, 2011 IEEE International Conference on Control Applications (CCA).

[2]  Zhenyu Yang.  Incorporating Performance Degradation in Fault Tolerant Control System Design with Multiple Actuator Failures , 2004 .

[3]  Pedro Luis Dias Peres,et al.  Robust H∞ filter design with pole constraints for discrete-time systems , 2000, J. Frankl. Inst..

[4]  M. Fu,et al.  Piecewise Lyapunov functions for robust stability of linear time-varying systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

[6]  I.I. Kaminer,et al.  Linear parametrically varying systems with brief instabilities: an application to vision/inertial navigation , 2004, IEEE Transactions on Aerospace and Electronic Systems.

[7]  ShiPeng,et al.  Robust H∞ output feedback control design for fuzzy dynamic systems with quadratic D stability constraints , 2006 .

[8]  Young Hoon Joo,et al.  Lifted versions of robust D-stability and D-stabilisation conditions for uncertain polytopic linear systems , 2012 .

[9]  M. J. Khosrowjerdi,et al.  Multiobjective H2/H∞ control design for a VSTOL flight model , 2010, 2010 18th Iranian Conference on Electrical Engineering.

[10]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[11]  Valter J. S. Leite,et al.  An improved LMI condition for robust D-stability of uncertain polytopic systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[12]  Jürgen Ackermann,et al.  Sampled-Data Control Systems , 1985 .

[13]  Pascal Gahinet,et al.  H/sub /spl infin// design with pole placement constraints: an LMI approach , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[14]  Fuwen Yang,et al.  Fixed-Order Robust $H_{\infty}$ Controller Design With Regional Pole Assignment , 2007, IEEE Transactions on Automatic Control.

[15]  Rainer Palm,et al.  Observers for Takagi-Sugeno fuzzy systems , 2002, IEEE Trans. Syst. Man Cybern. Part B.

[16]  Michio Sugeno,et al.  Fuzzy identification of systems and its applications to modeling and control , 1985, IEEE Transactions on Systems, Man, and Cybernetics.

[17]  R. Takahashi,et al.  Improved optimisation approach to the robust H2/H∞ control problem for linear systems , 2005 .

[18]  Robert Babuska,et al.  Fuzzy Modeling for Control , 1998 .

[19]  Carsten W. Scherer,et al.  Multi-Objective Output-Feedback Control via LMI Optimization , 1996 .

[20]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[21]  Xiang Chen,et al.  Multiobjective \boldmathHt/Hf Control Design , 2001, SIAM J. Control. Optim..

[22]  J. Schur,et al.  Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. , 1917 .

[23]  C. Scherer,et al.  Multiobjective output-feedback control via LMI optimization , 1997, IEEE Trans. Autom. Control..

[24]  Damiano Rotondo,et al.  A shifting pole placement approach for the design of parameter-scheduled state-feedback controllers , 2013, 2013 European Control Conference (ECC).

[25]  Pierre Apkarian,et al.  Self-scheduled H∞ control of linear parameter-varying systems: a design example , 1995, Autom..

[26]  S. Gutman,et al.  A general theory for matrix root-clustering in subregions of the complex plane , 1981 .

[27]  Tetsuya Iwasaki,et al.  All controllers for the general H∞ control problem: LMI existence conditions and state space formulas , 1994, Autom..

[28]  Ricardo Salvador Sánchez Peña,et al.  LPV control of a 6-DOF vehicle , 2002, IEEE Trans. Control. Syst. Technol..

[29]  IwasakiT.,et al.  All controllers for the general H control problem , 1994 .

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

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

[32]  B. Francis,et al.  A Course in H Control Theory , 1987 .

[33]  Damiano Rotondo,et al.  Robust state-feedback control of uncertain LPV systems: An LMI-based approach , 2014, J. Frankl. Inst..

[34]  Juan Wang,et al.  Adaptive Clustering with Dynamic Cluster Radius for Wireless Sensor Network , 2013 .

[35]  P. Khargonekar,et al.  State-space solutions to standard H/sub 2/ and H/sub infinity / control problems , 1989 .

[36]  Kazuo Tanaka,et al.  Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach , 2008 .

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

[38]  Peng Shi,et al.  Hinfinity fuzzy state-feedback control design for nonlinear systems with I-stability constraints: An LMI approach , 2008, Math. Comput. Simul..

[39]  Olivier Bachelier,et al.  Pole placement in a union of regions with prespecified subregion allocation , 2006, Math. Comput. Simul..

[40]  P. Gahinet,et al.  H∞ design with pole placement constraints: an LMI approach , 1996, IEEE Trans. Autom. Control..

[41]  Roderick Murray-Smith,et al.  Multiple Model Approaches to Modelling and Control , 1997 .

[42]  Vincent Pilaud,et al.  Polytopality and Cartesian products of graphs , 2010, 1009.1499.

[43]  Pedro Luis Dias Peres,et al.  An improved LMI condition for robust D-stability of uncertain polytopic systems , 2003, IEEE Trans. Autom. Control..

[44]  S. Bittanti,et al.  Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty , 1996 .

[45]  P. Khargonekar,et al.  State-space solutions to standard H2 and H∞ control problems , 1988, 1988 American Control Conference.

[46]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[47]  Vladimir A. Yakubovich,et al.  Linear Matrix Inequalities in System and Control Theory (S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan) , 1995, SIAM Rev..

[48]  Sing Kiong Nguang,et al.  H∞ filtering for fuzzy singularly perturbed systems with pole placement constraints: an LMI approach , 2004, IEEE Trans. Signal Process..

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

[50]  K. Glover,et al.  State-space formulae for all stabilizing controllers that satisfy an H(infinity)-norm bound and relations to risk sensitivity , 1988 .

[51]  T. Başar.  Feedback and Optimal Sensitivity: Model Reference Transformations, Multiplicative Seminorms, and Approximate Inverses , 2001 .

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

[53]  J. Geromel,et al.  Extended H 2 and H norm characterizations and controller parametrizations for discrete-time systems , 2002 .

[54]  Pierre Apkarian,et al.  Robust pole placement in LMI regions , 1999, IEEE Trans. Autom. Control..

[55]  J. Bernussou,et al.  A new robust D-stability condition for real convex polytopic uncertainty , 2000 .

[56]  G. W. Irwin,et al.  LMI-based control design for discrete polytopic LPV systems , 2003, 2003 European Control Conference (ECC).

[57]  Carsten W. Scherer,et al.  The Riccati inequality and state-space H∞-optimal control , 1990 .

[58]  Jeff S. Shamma,et al.  A Linear Parameter Varying Approach to Gain Scheduled Missile Autopilot Design , 1992, 1992 American Control Conference.

[59]  G. Zames.  Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses , 1981 .

[60]  Peng Shi,et al.  Robust Hinfinity output feedback control design for fuzzy dynamic systems with quadratic D stability constraints: An LMI approach , 2006, Inf. Sci..

[61]  ApkarianPierre,et al.  Self-scheduled H control of linear parameter-varying systems , 1995 .

[62]  Athanasios Sideris,et al.  H ∞ control with parametric Lyapunov functions , 1997 .