The output regulation problem with stability for linear switching systems: A geometric approach

Abstract This work presents a solution for the output regulation problem with quadratic stability under arbitrary switching in linear switching systems. The extension to other stability requirements, like asymptotic stability in particular, is also considered and it is shown to affect only few specific features of the proposed solution. The main reason is that the geometric approach, which is at the basis of the developed methodology, establishes a neat separation between the structural aspects and the stability aspects of the problem. For the same reason, continuous-time systems and discrete-time systems are given a unified treatment as far as the structural issues are concerned, while different technicalities characterize the discussion of the stability issues.

[1]  Michael Margaliot,et al.  Analysis of Discrete-Time Linear Switched Systems: A Variational Approach , 2011, SIAM J. Control. Optim..

[2]  Daniel Liberzon,et al.  Common Lyapunov functions for families of commuting nonlinear systems , 2005, Syst. Control. Lett..

[3]  Zhendong Sun.  Robust switching of discrete-time switched linear systems , 2012, Autom..

[4]  Hai Lin,et al.  Switching Stabilizability for Continuous-Time Uncertain Switched Linear Systems , 2007, IEEE Transactions on Automatic Control.

[5]  G. Basile,et al.  Controlled and conditioned invariants in linear system theory , 1992 .

[6]  Pramod P. Khargonekar,et al.  Optimal Output Regulation for Discrete-Time Switched and Markovian Jump Linear Systems , 2008, SIAM J. Control. Optim..

[7]  Giovanni Marro,et al.  Multivariable regulation in geometric terms: Old and new results , 1996 .

[8]  Franco Blanchini,et al.  A separation principle for linear switching systems and parametrization of all stabilizing controllers , 2008, 2008 47th IEEE Conference on Decision and Control.

[9]  Michael Margaliot,et al.  Stability analysis of switched systems using variational principles: An introduction , 2006, Autom..

[10]  Michael Margaliot,et al.  Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems , 2012, Autom..

[11]  D. Xie,et al.  Stabilisability and observer-based switched control design for switched linear systems , 2008 .

[12]  Masood Dehghan,et al.  Discrete-time switching linear system with constraints: Characterization and computation of invariant sets under dwell-time consideration , 2012, Autom..

[13]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Short Survey of Recent Results , 2005, Proceedings of the 2005 IEEE International Symposium on, Mediterrean Conference on Control and Automation Intelligent Control, 2005..

[14]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in System and Control Theory , 1994, Studies in Applied Mathematics.

[15]  Hai Lin,et al.  Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results , 2009, IEEE Transactions on Automatic Control.

[16]  J. M. Schumacher.  Regulator synthesis using (C,A,B)-pairs , 1982 .

[17]  S. Dasgupta,et al.  Parametric Lyapunov functions for uncertain systems: the multiplier approach , 1999 .

[18]  Yu Huang,et al.  Criteria of Stability for Continuous-Time Switched Systems by Using Liao-Type Exponents , 2009, SIAM J. Control. Optim..

[19]  Liang Lu.  Output regulation of a class of switched linear systems with saturated continuous feedback , 2011, Proceedings of the 30th Chinese Control Conference.

[20]  Daizhan Cheng,et al.  Stabilizer design of planar switched linear systems , 2008, Syst. Control. Lett..

[21]  J. Bernussou,et al.  Stabilization of an uncertain linear dynamic system by state and output feedback : a quadratic stabilizability approach , 1996 .

[22]  Pramod P. Khargonekar,et al.  Detectability and Stabilizability of Discrete-Time Switched Linear Systems , 2009, IEEE Transactions on Automatic Control.

[23]  A. Morse,et al.  Stability of switched systems: a Lie-algebraic condition ( , 1999 .

[24]  G. Leitmann.  Stability without Eigenspaces in the Geometric Approach : The Regulator Problem ~ , .

[25]  Zhendong Sun,et al.  Stabilizing switching design for switched linear systems: A state-feedback path-wise switching approach , 2009, Autom..

[26]  Giovanni Marro,et al.  On the robust controlled invariant , 1987 .

[27]  Jun Zhao,et al.  Output regulation of a class of switched linear systems with disturbances , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[28]  Elena Zattoni,et al.  A Numerical Example about the Geometric Approach to the Output Regulation Problem with Stability for Linear Switching Systems , 2013, ArXiv.

[29]  Zheng Guo. Li.  Stability of switched systems. , 2001 .

[30]  Patrizio Colaneri,et al.  A Nonconservative LMI Condition for Stability of Switched Systems With Guaranteed Dwell Time , 2012, IEEE Transactions on Automatic Control.

[31]  K. Narendra,et al.  A result on common quadratic Lyapunov functions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..