Performance analysis of switched linear systems under arbitrary switching via generalized coordinate transformations
|
摘要
For a continuous-time switched linear system, the spectral abscissa is defined as the worst-case divergence rate under arbitrary switching, which is critical for characterizing the asymptotic performance of the switched system. In this study, based on the generalized coordinate transformations approach, we develop a computational scheme that iteratively produces sequences of minimums of matrix set μ_1 measures,where the limits of the sequences are upper bound estimates of the spectral abscissa. A simulation example is presented to illustrate the effectiveness of the proposed scheme.
For a continuous-time switched linear system, the spectral abscissa is defined as the worst-case divergence rate under arbitrary switching, which is critical for characterizing the asymptotic performance of the switched system. In this study, based on the generalized coordinate transformations approach, we develop a computational scheme that iteratively produces sequences of minimums of matrix set μ_1 measures,where the limits of the sequences are upper bound estimates of the spectral abscissa. A simulation example is presented to illustrate the effectiveness of the proposed scheme.
For a continuous-time switched linear system, the spectral abscissa is defined as the worst-case divergence rate under arbitrary switching, which is critical for characterizing the asymptotic performance of the switched system. In this study, based on the generalized coordinate transformations approach, we develop a computational scheme that iteratively produces sequences of minimums of matrix set μ_1 measures,where the limits of the sequences are upper bound estimates of the spectral abscissa. A simulation example is presented to illustrate the effectiveness of the proposed scheme.
引文
1 Williams S M,Hoft R G.Adaptive frequency domain control of PWM switched power line conditioner.IEEE Trans Power Electron,1991,6:665-670
2 Zhao J,Spong M W.Hybrid control for global stabilization of the cart-pendulum system.Automatica,2001,37:1941-1951
3 Zhang W,Hu J H.Dynamic buffer management using optimal control of hybrid systems.Automatica,2008,44:1831-1840
4 Branicky M S.Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.IEEE Trans Autom Control,1998,43:475-482
5 Hespanha J P,Morse A S.Stability of switched systems with average dwell-time.In:Proceedings of the 38th IEEEConference on Decision and Control,Phoenix,2002.2655-2660
6 Liberzon D,Morse A S.Basic problems in stability and design of switched systems.IEEE Control Syst Mag,1999,19:59-70
7 Liberzon D,Hespanha J P,Morse A S.Stability of switched systems:a Lie-algebraic condition.Syst Control Lett,1999,37:117-122
8 Wang Y,Wang W,Liu G P.Stability of linear discrete switched systems with delays based on average dwell time method.Sci China Inf Sci,2010,53:1216-1223
9 Shorten R,Wirth F,Mason O,et al.Stability criteria for switched and hybrid systems.SIAM Rev,2007,49:545-592
10 Lin H,Antsaklis P J.Stability and stabilizability of switched linear systems:a survey of recent results.IEEE Trans Autom Control,2009,54:308-322
11 Sun Z D,Ge S S.Stability Theory of Switched Dynamical Systems.Berlin:Springer,2011
12 Zhu Y Z,Zhang L X,Lin W Y,et al.Benefits of redundant channels in observer-based H∞control for discrete-time switched linear systems.Sci China Tech Sci,2016,59:55-62
13 Blanchini F.Nonquadratic Lyapunov functions for robust control.Automatica,1995,31:451-461
14 Blanchini F,Miani S.A new class of universal Lyapunov functions for the control of uncertain linear systems.IEEETrans Autom Control,1996,44:641-647
15 Sun Z D.Recent advances on analysis and design of switched linear systems.Control Theory Technol,2017,15:242-244
16 Boscain U.Stability of planar switched systems:the linear single input case.SIAM J Control Optim,2002,41:89-112
17 Mason P,Boscain U,Chitour Y.Common polynomial Lyapunov functions for linear switched systems.SIAM J Control Optim,2006,45:226-245
18 Boscain U,Balde M.Stability of planar switched systems:the nondiagonalizable case.Commun Pure Appl Anal,2008,7:1-21
19 Molchanov A P,Pyatnitskiy Y S.Criteria of asymptotic stability of differential and difference inclusions encountered in control theory.Syst Control Lett,1989,13:59-64
20 Mancilla-Aguilar J L,Garc′?a R A.A converse Lyapunov theorem for nonlinear switched systems.Syst Control Lett,2000,41:67-71
21 Peleties P,Decarlo R.Asymptotic stability of m-switched systems using Lyapunov-like functions.In:Proceedings of the American Control Conference,Boston,1991.1679-1684
22 Pettersson S,Lennartson B.Stability and robustness for hybrid systems.In:Proceedings of the 35th IEEE Conference on Decision and Control,Kobe,1996.1202-1207
23 Elsner L.The generalized spectral-radius theorem:an analytic-geometric proof.Linear Algebra Appl,1995,220:151-159
24 Blondel V D,Nesterov Y.Computationally efficient approximations of the joint spectral radius.SIAM J Matrix Anal Appl,2005,27:256-272
25 Sun Z D.A note on marginal stability of switched systems.IEEE Trans Autom Control,2008,53:625-631
26 Chitour Y,Mason P,Sigalotti M.On the marginal instability of linear switched systems.Syst Control Lett,2011,61:7322-7327
27 Barabanov N E.Ways to compute the Lyapunov index for differential nesting.Avtomatika I Telemekhanika,1989,50:475-479
28 Sun Z D.Matrix measure approach for stability of switched linear systems.In:Proceedings of the 7th IFAC Symposium Nonlinear Control System,Pretoria,2007.557-560
29 Xiong J D,Sun Z D.Approximation of extreme measure for switched linear systems.In:Proceedings of the 9th IEEEInternational Conference on Control and Automation,Santiago,2011.722-725
30 Lin M L,Sun Z D.Approximating the spectral abscissa for switched linear systems via coordinate transformations.JSyst Sci Complex,2016,29:350-366
31 Lin M L,Sun Z D.Approximation of the spectral abscissa for switched linear systems via generalized coordinate transformations.In:Proceedings of the 11th World Congress on Intelligent Control and Automation,Shenyang,2014.2208-2212
32 Protasov V Y,Jungers R M.Analysing the stability of linear systems via exponential Chebyshev polynomials.IEEETrans Autom Control,2016,61:795-798
33 Gurvits L.Stability of discrete linear inclusion.Linear Algebra Appl,1995,231:47-85
34 Shih M H,Wu J W,Pang C T.Asymptotic stability and generalized Gelfand spectral radius formula.Linear Algebra Appl,1997,252:61-70
35 Parrilo P A,Jadbabaie A.Approximation of the joint spectral radius using sum of squares.Linear Algebra Appl,2008,428:2385-2402
36 Sun Z D,Shorten R N.On convergence rates of simultaneously triangularizable switched linear systems.IEEE Trans Autom Control,2005,50:1224-1228
37 Vidyasagar M.Nonlinear Systems Analysis.Englewood Cliffs:Prentice-Hall,1993
38 Blanchini F.The gain scheduling and the robust state feedback stabilization problems.IEEE Trans Autom Control,2000,45:2061-2070
39 Macduffee C.The Theory of Matrices.New York:Chelsea,1946
40 Hartwig R E.The resultant and the matrix equation AX=XB.SIAM J Appl Math,1972,22:538-544
41 Zahreddine Z.Matrix measure and application to stability of matrices and interval dynamical systems.Int J Math Math Sci,2003,2003:75-85
2 Zhao J,Spong M W.Hybrid control for global stabilization of the cart-pendulum system.Automatica,2001,37:1941-1951
3 Zhang W,Hu J H.Dynamic buffer management using optimal control of hybrid systems.Automatica,2008,44:1831-1840
4 Branicky M S.Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.IEEE Trans Autom Control,1998,43:475-482
5 Hespanha J P,Morse A S.Stability of switched systems with average dwell-time.In:Proceedings of the 38th IEEEConference on Decision and Control,Phoenix,2002.2655-2660
6 Liberzon D,Morse A S.Basic problems in stability and design of switched systems.IEEE Control Syst Mag,1999,19:59-70
7 Liberzon D,Hespanha J P,Morse A S.Stability of switched systems:a Lie-algebraic condition.Syst Control Lett,1999,37:117-122
8 Wang Y,Wang W,Liu G P.Stability of linear discrete switched systems with delays based on average dwell time method.Sci China Inf Sci,2010,53:1216-1223
9 Shorten R,Wirth F,Mason O,et al.Stability criteria for switched and hybrid systems.SIAM Rev,2007,49:545-592
10 Lin H,Antsaklis P J.Stability and stabilizability of switched linear systems:a survey of recent results.IEEE Trans Autom Control,2009,54:308-322
11 Sun Z D,Ge S S.Stability Theory of Switched Dynamical Systems.Berlin:Springer,2011
12 Zhu Y Z,Zhang L X,Lin W Y,et al.Benefits of redundant channels in observer-based H∞control for discrete-time switched linear systems.Sci China Tech Sci,2016,59:55-62
13 Blanchini F.Nonquadratic Lyapunov functions for robust control.Automatica,1995,31:451-461
14 Blanchini F,Miani S.A new class of universal Lyapunov functions for the control of uncertain linear systems.IEEETrans Autom Control,1996,44:641-647
15 Sun Z D.Recent advances on analysis and design of switched linear systems.Control Theory Technol,2017,15:242-244
16 Boscain U.Stability of planar switched systems:the linear single input case.SIAM J Control Optim,2002,41:89-112
17 Mason P,Boscain U,Chitour Y.Common polynomial Lyapunov functions for linear switched systems.SIAM J Control Optim,2006,45:226-245
18 Boscain U,Balde M.Stability of planar switched systems:the nondiagonalizable case.Commun Pure Appl Anal,2008,7:1-21
19 Molchanov A P,Pyatnitskiy Y S.Criteria of asymptotic stability of differential and difference inclusions encountered in control theory.Syst Control Lett,1989,13:59-64
20 Mancilla-Aguilar J L,Garc′?a R A.A converse Lyapunov theorem for nonlinear switched systems.Syst Control Lett,2000,41:67-71
21 Peleties P,Decarlo R.Asymptotic stability of m-switched systems using Lyapunov-like functions.In:Proceedings of the American Control Conference,Boston,1991.1679-1684
22 Pettersson S,Lennartson B.Stability and robustness for hybrid systems.In:Proceedings of the 35th IEEE Conference on Decision and Control,Kobe,1996.1202-1207
23 Elsner L.The generalized spectral-radius theorem:an analytic-geometric proof.Linear Algebra Appl,1995,220:151-159
24 Blondel V D,Nesterov Y.Computationally efficient approximations of the joint spectral radius.SIAM J Matrix Anal Appl,2005,27:256-272
25 Sun Z D.A note on marginal stability of switched systems.IEEE Trans Autom Control,2008,53:625-631
26 Chitour Y,Mason P,Sigalotti M.On the marginal instability of linear switched systems.Syst Control Lett,2011,61:7322-7327
27 Barabanov N E.Ways to compute the Lyapunov index for differential nesting.Avtomatika I Telemekhanika,1989,50:475-479
28 Sun Z D.Matrix measure approach for stability of switched linear systems.In:Proceedings of the 7th IFAC Symposium Nonlinear Control System,Pretoria,2007.557-560
29 Xiong J D,Sun Z D.Approximation of extreme measure for switched linear systems.In:Proceedings of the 9th IEEEInternational Conference on Control and Automation,Santiago,2011.722-725
30 Lin M L,Sun Z D.Approximating the spectral abscissa for switched linear systems via coordinate transformations.JSyst Sci Complex,2016,29:350-366
31 Lin M L,Sun Z D.Approximation of the spectral abscissa for switched linear systems via generalized coordinate transformations.In:Proceedings of the 11th World Congress on Intelligent Control and Automation,Shenyang,2014.2208-2212
32 Protasov V Y,Jungers R M.Analysing the stability of linear systems via exponential Chebyshev polynomials.IEEETrans Autom Control,2016,61:795-798
33 Gurvits L.Stability of discrete linear inclusion.Linear Algebra Appl,1995,231:47-85
34 Shih M H,Wu J W,Pang C T.Asymptotic stability and generalized Gelfand spectral radius formula.Linear Algebra Appl,1997,252:61-70
35 Parrilo P A,Jadbabaie A.Approximation of the joint spectral radius using sum of squares.Linear Algebra Appl,2008,428:2385-2402
36 Sun Z D,Shorten R N.On convergence rates of simultaneously triangularizable switched linear systems.IEEE Trans Autom Control,2005,50:1224-1228
37 Vidyasagar M.Nonlinear Systems Analysis.Englewood Cliffs:Prentice-Hall,1993
38 Blanchini F.The gain scheduling and the robust state feedback stabilization problems.IEEE Trans Autom Control,2000,45:2061-2070
39 Macduffee C.The Theory of Matrices.New York:Chelsea,1946
40 Hartwig R E.The resultant and the matrix equation AX=XB.SIAM J Appl Math,1972,22:538-544
41 Zahreddine Z.Matrix measure and application to stability of matrices and interval dynamical systems.Int J Math Math Sci,2003,2003:75-85