Quadratic Performance Analysis of Switched Affine Time–Varying Systems

Abstract We analyze quadratic performance for switched systems which are composed of a finite set of affine time-varying subsystems, where both subsystem matrices and affine vectors are switched, and no single subsystem has desired quadratic performance. The quadratic performance indexes we deal with include stability, tracking and L2 gain. We show that if a linear convex combination of subsystem matrices is uniformly Hurwitz and another convex combination of affine vectors is zero, then we can design a state-dependent switching law (state feedback) and an output-dependent switching law (output feedback) such that the entire switched affine system is quadratically stable at the origin. In the case where the convex combination of affine vectors is nonzero, we show that the tracking control problem can be posed and solved using a similar switching strategy. Finally, we consider the L2gain analysis problem for the switched affine time-varying systems under state feedback.

[1]  Rafael Wisniewski,et al.  Local analysis of hybrid systems on polyhedral sets with state-dependent switching , 2014, Int. J. Appl. Math. Comput. Sci..

[2]  Arjan van der Schaft,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[3]  Guisheng Zhai,et al.  On practical asymptotic stabilizability of switched affine systems , 2006 .

[4]  S. Pettersson,et al.  Hybrid system stability and robustness verification using linear matrix inequalities , 2002 .

[5]  Hai Lin,et al.  Extended Lie Algebraic Stability Analysis for Switched Systems with Continuous-Time and Discrete-Time Subsystems , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[6]  Uri Shaked,et al.  Robust stability and stabilization of linear switched systems with dwell time , 2010, 2010 Conference on Control and Fault-Tolerant Systems (SysTol).

[7]  Guisheng Zhai,et al.  Practical stability and stabilization of hybrid and switched systems , 2004, IEEE Trans. Autom. Control..

[8]  Zhengrong Xiang,et al.  Fault tolerant control of switched nonlinear systems with time delay under asynchronous switching , 2010, Int. J. Appl. Math. Comput. Sci..

[9]  Grace S. Deaecto,et al.  Reply to: Comments on ‘State feedback H ∞ control design of continuous-time switched affine systems’ , 2017 .

[10]  G. Zhai,et al.  Quadratic stabilizability of switched linear systems with polytopic uncertainties , 2003 .

[11]  Grace S. Deaecto,et al.  Dynamic output feedback H∞ control of continuous-time switched affine systems , 2016, Autom..

[12]  Guisheng Zhai.  A generalization of the graph Laplacian with application to a distributed consensus algorithm , 2015, Int. J. Appl. Math. Comput. Sci..

[13]  A. Morse,et al.  Basic problems in stability and design of switched systems , 1999 .

[14]  Venkataramanan Balakrishnan,et al.  A Unified Algebraic Approach to Linear Control Design, R.E. Skelton, T. Iwasaki and K. Grigoriadis, Taylor & Francis, London, UK, 1998, 285 pages. ISBN 0‐7484‐0592‐5 , 2002 .

[15]  R. Decarlo,et al.  Perspectives and results on the stability and stabilizability of hybrid systems , 2000, Proceedings of the IEEE.

[16]  Emilia Fridman,et al.  Robust Sampled – Data Control of Switched Affine Systems , 2013, IEEE Transactions on Automatic Control.

[17]  Guisheng Zhai,et al.  Quadratic stabilizability and ℌ℞ disturbance attenuation of switched linear systems via state and output feedback , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[18]  Agustín Jiménez,et al.  A Novel Method for the Design of Switching Surfaces for Discretized MIMO Nonlinear Systems , 2017, Int. J. Appl. Math. Comput. Sci..

[19]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[20]  Alexandre Trofino,et al.  Switching rule design for affine switched systems using a max-type composition rule , 2014, Syst. Control. Lett..

[21]  P. Bolzern,et al.  Quadratic stabilization of a switched affine system about a nonequilibrium point , 2004, Proceedings of the 2004 American Control Conference.

[22]  Raymond A. DeCarlo,et al.  Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems , 1998, Eur. J. Control.

[23]  Guisheng Zhai,et al.  Basic consensus problems in multi-agent systems with switching interconnection graphs , 2013, 2013 CACS International Automatic Control Conference (CACS).

[24]  Weiming Xiang.  Necessary and Sufficient Condition for Stability of Switched Uncertain Linear Systems Under Dwell-Time Constraint , 2016, IEEE Transactions on Automatic Control.

[25]  Edvaldo Assunção,et al.  Observer-Based Control Design for Switched Affine Systems and Applications to DC–DC Converters , 2013 .

[26]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[27]  Hai Lin,et al.  Switched Linear Systems: Control and Design , 2006, IEEE Transactions on Automatic Control.

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

[29]  Jian Xiao,et al.  Stabilization of switched continuous-time systems with all modes unstable via dwell time switching , 2014, Autom..

[30]  Alexandre Trofino,et al.  Switching Rule Design for Switched Dynamic Systems With Affine Vector Fields , 2009, IEEE Transactions on Automatic Control.

[31]  Bo Hu,et al.  Disturbance attenuation properties of time-controlled switched systems , 2001, J. Frankl. Inst..

[32]  G. Zhai.  Quadratic stabilizability of discrete-time switched systems via state and output feedback , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).