Bounded real and positive real balanced truncation for infinite-dimensional systems

Chris Guiver; Mark R. Opmeer

doi:10.3934/mcrf.2013.3.83

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March 2013, 3(1): 83-119. doi: 10.3934/mcrf.2013.3.83

Bounded real and positive real balanced truncation for infinite-dimensional systems

Chris Guiver ^1, and Mark R. Opmeer ^2,

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

Received February 2012 Revised January 2013 Published February 2013

Abstract
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Bounded real balanced truncation for infinite-dimensional systems is considered. This provides reduced order finite-dimensional systems that retain bounded realness. We obtain an error bound analogous to the finite-dimensional case in terms of the bounded real singular values. By using the Cayley transform a gap metric error bound for positive real balanced truncation is subsequently obtained. For a class of systems with an analytic semigroup, we show rapid decay of the bounded real and positive real singular values. Together with the established error bounds, this proves rapid convergence of the bounded real and positive real balanced truncations.

Keywords: positive real., well-posed linear system, Model reduction, infinite-dimensional system, balanced truncation, bounded real.

Mathematics Subject Classification: Primary: 93B11, 93C05, 47N70, 49N10, 93B28, 93C2.

Citation: Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control & Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83

References:

[1]	Brian D. O. Anderson and Sumeth Vongpanitlerd, "Network Analysis and Synthesis: A Modern Systems Theory Approach," Prentice Hall, 1973. Google Scholar
[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	Chris Guiver, "Model Reduction by Balanced Truncation," Ph.D thesis, University of Bath, 2012. Available from: http://opus.bath.ac.uk/32863/. Google Scholar
[12]	Chris Guiver and Mark R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, submitted, 2011. Google Scholar
[13]	Chris Guiver and Mark R. Opmeer, An error bound in the gap metric for dissipative balanced approximations, submitted, 2012. Google Scholar
[14]	P. Harshavardhana, "Model Reduction Methods in Control and Signal Processing," Ph.D thesis, Univ. South. Calif. EE Dept., Los Angeles, CA, 1984. Google Scholar
[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. Google Scholar
[16]	Tosio Kato, "Perturbation Theory for Linear Operators," Classics in Mathematics, Springer-Verlag, Berlin, 1995. Google Scholar
[17]	Kalle Mikkola, "Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati Equations," Thesis (D.Sc.(Tech.))–Teknillinen Korkeakoulu, Helsinki, Finland, 2002. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	Mark R. Opmeer, Model reduction for distributed parameter systems: A functional analytic view, American Control Conference, (2012), 1418-1423. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	Marvin Rosenblum and James Rovnyak, "Hardy Classes and Operator Theory," Dover Publications, Inc., Mineola, NY, 1997. Google Scholar
[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. Google Scholar
[26]	Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), 147-164. doi: 10.1007/BF02088011. Google Scholar
[27]	Olof J. Staffans, Quadratic optimal control of regular well-posed linear systems, with applications to parabolic equations,, 1997., (). Google Scholar
[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. Google Scholar
[29]	Olof J. Staffans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim., 37 (1999), 131-164. doi: 10.1137/S0363012996314257. Google Scholar
[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. Google Scholar
[31]	Olof J. Staffans, "Well-Posed Linear Systems," Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. Google Scholar
[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. Google Scholar
[33]	George Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545. doi: 10.1137/0327028. Google Scholar
[34]	George Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43. doi: 10.1007/BF02788172. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

show all references

References:

[1]	Brian D. O. Anderson and Sumeth Vongpanitlerd, "Network Analysis and Synthesis: A Modern Systems Theory Approach," Prentice Hall, 1973. Google Scholar
[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. Google Scholar
[3]	Vitold Belevitch, "Classical Network Theory," Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1968. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	Chris Guiver, "Model Reduction by Balanced Truncation," Ph.D thesis, University of Bath, 2012. Available from: http://opus.bath.ac.uk/32863/. Google Scholar
[12]	Chris Guiver and Mark R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, submitted, 2011. Google Scholar
[13]	Chris Guiver and Mark R. Opmeer, An error bound in the gap metric for dissipative balanced approximations, submitted, 2012. Google Scholar
[14]	P. Harshavardhana, "Model Reduction Methods in Control and Signal Processing," Ph.D thesis, Univ. South. Calif. EE Dept., Los Angeles, CA, 1984. Google Scholar
[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. Google Scholar
[16]	Tosio Kato, "Perturbation Theory for Linear Operators," Classics in Mathematics, Springer-Verlag, Berlin, 1995. Google Scholar
[17]	Kalle Mikkola, "Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati Equations," Thesis (D.Sc.(Tech.))–Teknillinen Korkeakoulu, Helsinki, Finland, 2002. Google Scholar
[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. Google Scholar
[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. Google Scholar
[20]	Mark R. Opmeer, Model reduction for distributed parameter systems: A functional analytic view, American Control Conference, (2012), 1418-1423. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[24]	Marvin Rosenblum and James Rovnyak, "Hardy Classes and Operator Theory," Dover Publications, Inc., Mineola, NY, 1997. Google Scholar
[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. Google Scholar
[26]	Dietmar Salamon, Realization theory in Hilbert space, Math. Systems Theory, 21 (1989), 147-164. doi: 10.1007/BF02088011. Google Scholar
[27]	Olof J. Staffans, Quadratic optimal control of regular well-posed linear systems, with applications to parabolic equations,, 1997., (). Google Scholar
[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. Google Scholar
[29]	Olof J. Staffans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim., 37 (1999), 131-164. doi: 10.1137/S0363012996314257. Google Scholar
[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. Google Scholar
[31]	Olof J. Staffans, "Well-Posed Linear Systems," Encyclopedia of Mathematics and its Applications, 103, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. Google Scholar
[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. Google Scholar
[33]	George Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545. doi: 10.1137/0327028. Google Scholar
[34]	George Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43. doi: 10.1007/BF02788172. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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