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On the minimum time function around the origin
Bounded real and positive real balanced truncation for infinite-dimensional systems
1. | Environment & Sustainability Institute, College of Engineering, Mathematics and Physical Sciences, University of Exeter Cornwall Campus, Cornwall, TR10 9EZ, United Kingdom |
2. | Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom |
References:
[1] |
Brian D. O. Anderson and Sumeth Vongpanitlerd, "Network Analysis and Synthesis: A Modern Systems Theory Approach," Prentice Hall, 1973. |
[2] |
Athanasios C. Antoulas, "Approximation of Large-Scale Dynamical Systems," Advances in Design and Control, 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
doi: 10.1137/1.9780898718713. |
[3] |
Vitold Belevitch, "Classical Network Theory," Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1968. |
[4] |
Ronald R. Coifman and Richard Rochberg, Representation theorems for holomorphic and harmonic functions in $L^p$, in "Representation Theorems for Hardy Spaces," Astérisque, 77, Soc. Math. France, Paris, (1980), 11-66. |
[5] |
Ruth Curtain, Kalle Mikkola and Amol Sasane, The Hilbert-Schmidt property of feedback operators, J. Math. Anal. Appl., 329 (2007), 1145-1160.
doi: 10.1016/j.jmaa.2006.07.037. |
[6] |
Uday B. Desai and Debajyoti Pal, A transformation approach to stochastic model reduction, IEEE Trans. Automat. Control, 29 (1984), 1097-1100.
doi: 10.1109/TAC.1984.1103438. |
[7] |
Dale F. Enns, Model reduction with balanced realizations: An error bound and a frequency weighted generalization, Proc. CDC, (1984), 127-132.
doi: 10.1109/CDC.1984.272286. |
[8] |
Keith Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^{\infty} $-error bounds, Internat. J. Control, 39 (1984), 1115-1193.
doi: 10.1080/00207178408933239. |
[9] |
Keith Glover, Ruth F. Curtain and Jonathan R. Partington, Realisation and approximation of linear infinite-dimensional systems with error bounds, SIAM J. Control Optim., 26 (1988), 863-898.
doi: 10.1137/0326049. |
[10] |
Serkan Gugercin and Athanasios C. Antoulas, A survey of model reduction by balanced truncation and some new results, Internat. J. Control, 77 (2004), 748-766.
doi: 10.1080/00207170410001713448. |
[11] |
Chris Guiver, "Model Reduction by Balanced Truncation," Ph.D thesis, University of Bath, 2012. Available from: http://opus.bath.ac.uk/32863/. |
[12] |
Chris Guiver and Mark R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, submitted, 2011. |
[13] |
Chris Guiver and Mark R. Opmeer, An error bound in the gap metric for dissipative balanced approximations, submitted, 2012. |
[14] |
P. Harshavardhana, "Model Reduction Methods in Control and Signal Processing," Ph.D thesis, Univ. South. Calif. EE Dept., Los Angeles, CA, 1984. |
[15] |
P. Harshavardhana, Edmond A. Jonckheere and Leonard M. Silverman, Stochastic balancing and approximation-stability and minimality, IEEE Trans. Automat. Control, 29 (1984), 744-746.
doi: 10.1109/TAC.1984.1103631. |
[16] |
Tosio Kato, "Perturbation Theory for Linear Operators," Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
[17] |
Kalle Mikkola, "Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati Equations," Thesis (D.Sc.(Tech.))–Teknillinen Korkeakoulu, Helsinki, Finland, 2002. |
[18] |
Bruce C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32.
doi: 10.1109/TAC.1981.1102568. |
[19] |
Philippe C. Opdenacker and Edmond A. Jonckheere, A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds, IEEE Trans. Circuits and Systems, 35 (1988), 184-189.
doi: 10.1109/31.1720. |
[20] |
Mark R. Opmeer, Model reduction for distributed parameter systems: A functional analytic view, American Control Conference, (2012), 1418-1423. |
[21] |
Mark R. Opmeer, Decay of Hankel singular values of analytic control systems, Systems Control Lett., 59 (2010), 635-638.
doi: 10.1016/j.sysconle.2010.07.009. |
[22] |
Lars Pernebo and Leonard M. Silverman, Model reduction via balanced state space representations, IEEE Trans. Automat. Control, 27 (1982), 382-387.
doi: 10.1109/TAC.1982.1102945. |
[23] |
Richard Rebarber, Conditions for the equivalence of internal and external stability for distributed parameter systems, IEEE Trans. Automat. Control, 38 (1993), 994-998.
doi: 10.1109/9.222318. |
[24] |
Marvin Rosenblum and James Rovnyak, "Hardy Classes and Operator Theory," Dover Publications, Inc., Mineola, NY, 1997. |
[25] |
Dietmar Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.
doi: 10.2307/2000351. |
[26] |
Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), 147-164.
doi: 10.1007/BF02088011. |
[27] |
Olof J. Staffans, Quadratic optimal control of regular well-posed linear systems, with applications to parabolic equations, 1997. |
[28] |
Olof J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc., 349 (1997), 3679-3715.
doi: 10.1090/S0002-9947-97-01863-1. |
[29] |
Olof J. Staffans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim., 37 (1999), 131-164.
doi: 10.1137/S0363012996314257. |
[30] |
Olof J. Staffans, Passive and conservative continuous-time impedance and scattering systems. I. Well-posed systems, Math. Control Signals Systems, 15 (2002), 291-315.
doi: 10.1007/s004980200012. |
[31] |
Olof J. Staffans, "Well-Posed Linear Systems," Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197. |
[32] |
Weiqian Sun, Pramod P. Khargonekar and Duksun Shim, Solution to the positive real control problem for linear time-invariant systems, IEEE Trans. Automat. Control, 39 (1994), 2034-2046.
doi: 10.1109/9.328822. |
[33] |
George Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.
doi: 10.1137/0327028. |
[34] |
George Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.
doi: 10.1007/BF02788172. |
[35] |
George Weiss, Representation of shift-invariant operators on $L^2$ by $H^\infty$ transfer functions: An elementary proof, a generalization to $L^p$, and a counterexample for $L^\infty$, Math. Control Signals Systems, 4 (1991), 193-203.
doi: 10.1007/BF02551266. |
[36] |
George Weiss, Transfer functions of regular linear systems. I. Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.
doi: 10.2307/2154655. |
[37] |
Martin Weiss and George Weiss, Optimal control of stable weakly regular linear systems, Math. Control Signals Systems, 10 (1997), 287-330.
doi: 10.1007/BF01211550. |
[38] |
John T. Wen, Time domain and frequency domain conditions for strict positive realness, IEEE Trans. Automat. Control, 33 (1988), 988-992.
doi: 10.1109/9.7263. |
show all references
References:
[1] |
Brian D. O. Anderson and Sumeth Vongpanitlerd, "Network Analysis and Synthesis: A Modern Systems Theory Approach," Prentice Hall, 1973. |
[2] |
Athanasios C. Antoulas, "Approximation of Large-Scale Dynamical Systems," Advances in Design and Control, 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
doi: 10.1137/1.9780898718713. |
[3] |
Vitold Belevitch, "Classical Network Theory," Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1968. |
[4] |
Ronald R. Coifman and Richard Rochberg, Representation theorems for holomorphic and harmonic functions in $L^p$, in "Representation Theorems for Hardy Spaces," Astérisque, 77, Soc. Math. France, Paris, (1980), 11-66. |
[5] |
Ruth Curtain, Kalle Mikkola and Amol Sasane, The Hilbert-Schmidt property of feedback operators, J. Math. Anal. Appl., 329 (2007), 1145-1160.
doi: 10.1016/j.jmaa.2006.07.037. |
[6] |
Uday B. Desai and Debajyoti Pal, A transformation approach to stochastic model reduction, IEEE Trans. Automat. Control, 29 (1984), 1097-1100.
doi: 10.1109/TAC.1984.1103438. |
[7] |
Dale F. Enns, Model reduction with balanced realizations: An error bound and a frequency weighted generalization, Proc. CDC, (1984), 127-132.
doi: 10.1109/CDC.1984.272286. |
[8] |
Keith Glover, All optimal Hankel-norm approximations of linear multivariable systems and their $L^{\infty} $-error bounds, Internat. J. Control, 39 (1984), 1115-1193.
doi: 10.1080/00207178408933239. |
[9] |
Keith Glover, Ruth F. Curtain and Jonathan R. Partington, Realisation and approximation of linear infinite-dimensional systems with error bounds, SIAM J. Control Optim., 26 (1988), 863-898.
doi: 10.1137/0326049. |
[10] |
Serkan Gugercin and Athanasios C. Antoulas, A survey of model reduction by balanced truncation and some new results, Internat. J. Control, 77 (2004), 748-766.
doi: 10.1080/00207170410001713448. |
[11] |
Chris Guiver, "Model Reduction by Balanced Truncation," Ph.D thesis, University of Bath, 2012. Available from: http://opus.bath.ac.uk/32863/. |
[12] |
Chris Guiver and Mark R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, submitted, 2011. |
[13] |
Chris Guiver and Mark R. Opmeer, An error bound in the gap metric for dissipative balanced approximations, submitted, 2012. |
[14] |
P. Harshavardhana, "Model Reduction Methods in Control and Signal Processing," Ph.D thesis, Univ. South. Calif. EE Dept., Los Angeles, CA, 1984. |
[15] |
P. Harshavardhana, Edmond A. Jonckheere and Leonard M. Silverman, Stochastic balancing and approximation-stability and minimality, IEEE Trans. Automat. Control, 29 (1984), 744-746.
doi: 10.1109/TAC.1984.1103631. |
[16] |
Tosio Kato, "Perturbation Theory for Linear Operators," Classics in Mathematics, Springer-Verlag, Berlin, 1995. |
[17] |
Kalle Mikkola, "Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati Equations," Thesis (D.Sc.(Tech.))–Teknillinen Korkeakoulu, Helsinki, Finland, 2002. |
[18] |
Bruce C. Moore, Principal component analysis in linear systems: Controllability, observability, and model reduction, IEEE Trans. Automat. Control, 26 (1981), 17-32.
doi: 10.1109/TAC.1981.1102568. |
[19] |
Philippe C. Opdenacker and Edmond A. Jonckheere, A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds, IEEE Trans. Circuits and Systems, 35 (1988), 184-189.
doi: 10.1109/31.1720. |
[20] |
Mark R. Opmeer, Model reduction for distributed parameter systems: A functional analytic view, American Control Conference, (2012), 1418-1423. |
[21] |
Mark R. Opmeer, Decay of Hankel singular values of analytic control systems, Systems Control Lett., 59 (2010), 635-638.
doi: 10.1016/j.sysconle.2010.07.009. |
[22] |
Lars Pernebo and Leonard M. Silverman, Model reduction via balanced state space representations, IEEE Trans. Automat. Control, 27 (1982), 382-387.
doi: 10.1109/TAC.1982.1102945. |
[23] |
Richard Rebarber, Conditions for the equivalence of internal and external stability for distributed parameter systems, IEEE Trans. Automat. Control, 38 (1993), 994-998.
doi: 10.1109/9.222318. |
[24] |
Marvin Rosenblum and James Rovnyak, "Hardy Classes and Operator Theory," Dover Publications, Inc., Mineola, NY, 1997. |
[25] |
Dietmar Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.
doi: 10.2307/2000351. |
[26] |
Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), 147-164.
doi: 10.1007/BF02088011. |
[27] |
Olof J. Staffans, Quadratic optimal control of regular well-posed linear systems, with applications to parabolic equations, 1997. |
[28] |
Olof J. Staffans, Quadratic optimal control of stable well-posed linear systems, Trans. Amer. Math. Soc., 349 (1997), 3679-3715.
doi: 10.1090/S0002-9947-97-01863-1. |
[29] |
Olof J. Staffans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim., 37 (1999), 131-164.
doi: 10.1137/S0363012996314257. |
[30] |
Olof J. Staffans, Passive and conservative continuous-time impedance and scattering systems. I. Well-posed systems, Math. Control Signals Systems, 15 (2002), 291-315.
doi: 10.1007/s004980200012. |
[31] |
Olof J. Staffans, "Well-Posed Linear Systems," Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9780511543197. |
[32] |
Weiqian Sun, Pramod P. Khargonekar and Duksun Shim, Solution to the positive real control problem for linear time-invariant systems, IEEE Trans. Automat. Control, 39 (1994), 2034-2046.
doi: 10.1109/9.328822. |
[33] |
George Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.
doi: 10.1137/0327028. |
[34] |
George Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.
doi: 10.1007/BF02788172. |
[35] |
George Weiss, Representation of shift-invariant operators on $L^2$ by $H^\infty$ transfer functions: An elementary proof, a generalization to $L^p$, and a counterexample for $L^\infty$, Math. Control Signals Systems, 4 (1991), 193-203.
doi: 10.1007/BF02551266. |
[36] |
George Weiss, Transfer functions of regular linear systems. I. Characterizations of regularity, Trans. Amer. Math. Soc., 342 (1994), 827-854.
doi: 10.2307/2154655. |
[37] |
Martin Weiss and George Weiss, Optimal control of stable weakly regular linear systems, Math. Control Signals Systems, 10 (1997), 287-330.
doi: 10.1007/BF01211550. |
[38] |
John T. Wen, Time domain and frequency domain conditions for strict positive realness, IEEE Trans. Automat. Control, 33 (1988), 988-992.
doi: 10.1109/9.7263. |
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