# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2019016

## The generalised singular perturbation approximation for bounded real and positive real control systems

 Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

Received  June 2017 Published  December 2018

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

Citation: Chris Guiver. The generalised singular perturbation approximation for bounded real and positive real control systems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019016
##### References:
 [1] U. M. Al-Saggaf and G. F. Franklin, Model reduction via balanced realizations: An extension and frequency weighting techniques, IEEE Trans. Automat. Control, 33 (1988), 687-692. doi: 10.1109/9.1280. [2] B. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach, Prentice Hall, 1973. [3] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898718713. [4] O. Brune, Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency, Stud. Appl. Math., 10 (1931), 191-236. doi: 10.1002/sapm1931101191. [5] X. Chen and J. T. Wen, Positive realness preserving model reduction with $\mathcal{H}^ \infty$ norm error bounds, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 23-29. doi: 10.1109/81.350793. [6] R. F. Curtain, Reciprocals of regular linear systems: A survey, in Electronic Proceedings of the 15th International Symposium on the Mathematical Theory of Networks and Systems. (University of Notre Dame, South Bend, Indiana), 2002. [7] R. F. Curtain, Regular linear systems and their reciprocals: Applications to Riccati equations, Systems Control Lett., 49 (2003), 81-89. doi: 10.1016/S0167-6911(02)00302-X. [8] R. F. Curtain and K. Glover, Balanced realisations for infinite-dimensional systems, in Operator Theory and Systems, Birkhäuser, Basel, 19 (1986), 87-104. [9] U. B. Desai and D. Pal, A transformation approach to stochastic model reduction, IEEE Trans. Automat. Control, 29 (1984), 1097-1100. doi: 10.1109/TAC.1984.1103438. [10] D. Enns, Model reduction with balanced realizations: An error bound and a frequency weighted generalization, in Proc. CDC, 1984,127-132 doi: 10.1109/CDC.1984.272286. [11] K. V. Fernando and H. Nicholson, Singular perturbational model reduction of balanced systems, IEEE Trans. Automat. Control, 27 (1982), 466-468. [12] K. V. Fernando and H. Nicholson, Singular perturbational model reduction in the frequency domain, IEEE Trans. Automat. Control, 27 (1982), 969-970. [13] L. Fortuna, G. Nunnari and A. Gallo, Model Order Reduction Techniques with Applications in Electrical Engineering, Springer-Verlag, London, 1992. doi: 10.1007/978-1-4471-3198-4. [14] K. Glover, J. Lam and J. R. Partington, Rational approximation of a class of infinite-dimensional systems. I. Singular values of Hankel operators, Math. Control Signals Systems, 3 (1990), 325-344. doi: 10.1007/BF02551374. [15] K. 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. [16] K. Glover, R. F. Curtain and J. 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. [17] M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Inc., Upper Saddle River, 1995. [18] S. Gugercin and A. 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. [19] C. Guiver, Model Reduction by Balanced Truncation, PhD thesis, University of Bath, UK, 2012. [20] C. Guiver, H. Logemann and M. R. Opmeer, Transfer functions of infinite-dimensional systems: Positive realness and stabilization, Math. Control Signals Systems, 29 (2017), Art. 2, 61 pp, available from www.maths.bath.ac.uk/~mashl/research.html doi: 10.1007/s00498-017-0203-z. [21] C. Guiver and M. R. Opmeer, A counter-example to "positive realness preserving model reduction with $\mathcal{H}_\infty$ norm error bounds", IEEE Trans. Circuits Syst. I. Regul. Pap. I, 58 (2011), 1410-1411. doi: 10.1109/TCSI.2010.2097750. [22] C. Guiver and M. R. Opmeer, Bounded real and positive real balanced truncation for infinite-dimensional systems, Math. Control Relat. Fields, 3 (2013), 83-119. doi: 10.3934/mcrf.2013.3.83. [23] C. Guiver and M. R. Opmeer, Error bounds in the gap metric for dissipative balanced approximations, Linear Algebra Appl., 439 (2013), 3659-3698. doi: 10.1016/j.laa.2013.09.032. [24] C. Guiver and M. R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, SIAM J. Control Optim., 52 (2014), 1366-1401. doi: 10.1137/110846981. [25] S. V. Gusev and A. Likhtarnikov, Kalman-Popov-Yakubovich Lemma and the S-procedure: A historical essay, Automat. Rem. Contr., 67 (2006), 1768-1810. doi: 10.1134/S000511790611004X. [26] P. Harshavardhana, E. A. Jonckheere and L. M. Silverman, Stochastic balancing and approximation-stability and minimality, IEEE Trans. Automat. Control, 29 (1984), 744-746. doi: 10.1109/TAC.1984.1103631. [27] P. Heuberger, A family of reduced order models based on open-loop balancing, in Selected Topics in Identification, Modelling and Control, Delft University Press, 1990, 1–10. [28] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [29] W. Liu, V. Sreeram and K. L. Teo., Model reduction for state-space symmetric systems, System. Control Lett., 34 (1998), 209-215. doi: 10.1016/S0167-6911(98)00024-3. [30] Y. Liu and B. D. O. Anderson, Singular perturbation approximation of balanced systems, Internat. J. Control, 50 (1989), 1379-1405. doi: 10.1080/00207178908953437. [31] B. 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. [32] G. Muscato and G. Nunnari, On the $\sigma$-reciprocal system for model order reduction, Math. Model. Systems, 1 (1995), 261-271. doi: 10.1080/13873959508837022. [33] G. Muscato, G. Nunnari and L. Fortuna, Singular perturbation approximation of bounded real balanced and stochastically balanced transfer matrices, Internat. J. Control, 66 (1997), 253-269. doi: 10.1080/002071797224739. [34] R. W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966. [35] R. Ober and S. Montgomery-Smith, Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions, SIAM J. Control Optim., 28 (1990), 438-465. doi: 10.1137/0328024. [36] G. Obinata and B. D. Anderson, Model Reduction for Control System Design, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-0283-0. [37] P. C. Opdenacker and E. 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. [38] M.R. Opmeer and T. Reis, A lower bound for the balanced truncation error for MIMO systems, IEEE Trans. Automat. Control, 60 (2015), 2207-2212. doi: 10.1109/TAC.2014.2368232. [39] J. R. Partington, An Introduction to Hankel Operators, Cambridge University Press, Cambridge, 1988. [40] L. Pernebo and L. M. Silverman, Model reduction via balanced state space representations, IEEE Trans. Automat. Control, 27 (1982), 382-387. doi: 10.1109/TAC.1982.1102945. [41] A. Rantzer, On the Kalman-Yakubovich-Popov Lemma, System. Control Lett., 28 (1996), 7-10. doi: 10.1016/0167-6911(95)00063-1. [42] T. Reis and T. Stykel, Positive real and bounded real balancing for model reduction of descriptor systems, Int. J. Control, 83 (2010), 74-88. doi: 10.1080/00207170903100214. [43] E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer-Verlag, New York, 1998 doi: 10.1007/978-1-4612-0577-7. [44] O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. [45] O. 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. [46] O. J. Staffans, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), in Mathematical Systems Theory in Biology, Communications, Computation, and Finance (Notre Dame, IN, 2002), vol. 134 of IMA Vol. Math. Appl., Springer, New York, 2003,375–413. doi: 10.1007/978-0-387-21696-6_14. [47] J. C. Willems, Dissipative dynamical systems part Ⅰ: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493. [48] J. C. Willems, Dissipative dynamical systems part Ⅱ: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 352-393. doi: 10.1007/BF00276494. [49] N. Young, Balanced realizations in infinite dimensions, Operator Theory: Advances and Applications, 19 (1986), 449-471. [50] K. Zhou, J. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall Englewood Cliffs, 1996.

show all references

##### References:
 [1] U. M. Al-Saggaf and G. F. Franklin, Model reduction via balanced realizations: An extension and frequency weighting techniques, IEEE Trans. Automat. Control, 33 (1988), 687-692. doi: 10.1109/9.1280. [2] B. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach, Prentice Hall, 1973. [3] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898718713. [4] O. Brune, Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency, Stud. Appl. Math., 10 (1931), 191-236. doi: 10.1002/sapm1931101191. [5] X. Chen and J. T. Wen, Positive realness preserving model reduction with $\mathcal{H}^ \infty$ norm error bounds, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 23-29. doi: 10.1109/81.350793. [6] R. F. Curtain, Reciprocals of regular linear systems: A survey, in Electronic Proceedings of the 15th International Symposium on the Mathematical Theory of Networks and Systems. (University of Notre Dame, South Bend, Indiana), 2002. [7] R. F. Curtain, Regular linear systems and their reciprocals: Applications to Riccati equations, Systems Control Lett., 49 (2003), 81-89. doi: 10.1016/S0167-6911(02)00302-X. [8] R. F. Curtain and K. Glover, Balanced realisations for infinite-dimensional systems, in Operator Theory and Systems, Birkhäuser, Basel, 19 (1986), 87-104. [9] U. B. Desai and D. Pal, A transformation approach to stochastic model reduction, IEEE Trans. Automat. Control, 29 (1984), 1097-1100. doi: 10.1109/TAC.1984.1103438. [10] D. Enns, Model reduction with balanced realizations: An error bound and a frequency weighted generalization, in Proc. CDC, 1984,127-132 doi: 10.1109/CDC.1984.272286. [11] K. V. Fernando and H. Nicholson, Singular perturbational model reduction of balanced systems, IEEE Trans. Automat. Control, 27 (1982), 466-468. [12] K. V. Fernando and H. Nicholson, Singular perturbational model reduction in the frequency domain, IEEE Trans. Automat. Control, 27 (1982), 969-970. [13] L. Fortuna, G. Nunnari and A. Gallo, Model Order Reduction Techniques with Applications in Electrical Engineering, Springer-Verlag, London, 1992. doi: 10.1007/978-1-4471-3198-4. [14] K. Glover, J. Lam and J. R. Partington, Rational approximation of a class of infinite-dimensional systems. I. Singular values of Hankel operators, Math. Control Signals Systems, 3 (1990), 325-344. doi: 10.1007/BF02551374. [15] K. 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. [16] K. Glover, R. F. Curtain and J. 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. [17] M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Inc., Upper Saddle River, 1995. [18] S. Gugercin and A. 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. [19] C. Guiver, Model Reduction by Balanced Truncation, PhD thesis, University of Bath, UK, 2012. [20] C. Guiver, H. Logemann and M. R. Opmeer, Transfer functions of infinite-dimensional systems: Positive realness and stabilization, Math. Control Signals Systems, 29 (2017), Art. 2, 61 pp, available from www.maths.bath.ac.uk/~mashl/research.html doi: 10.1007/s00498-017-0203-z. [21] C. Guiver and M. R. Opmeer, A counter-example to "positive realness preserving model reduction with $\mathcal{H}_\infty$ norm error bounds", IEEE Trans. Circuits Syst. I. Regul. Pap. I, 58 (2011), 1410-1411. doi: 10.1109/TCSI.2010.2097750. [22] C. Guiver and M. R. Opmeer, Bounded real and positive real balanced truncation for infinite-dimensional systems, Math. Control Relat. Fields, 3 (2013), 83-119. doi: 10.3934/mcrf.2013.3.83. [23] C. Guiver and M. R. Opmeer, Error bounds in the gap metric for dissipative balanced approximations, Linear Algebra Appl., 439 (2013), 3659-3698. doi: 10.1016/j.laa.2013.09.032. [24] C. Guiver and M. R. Opmeer, Model reduction by balanced truncation for systems with nuclear Hankel operators, SIAM J. Control Optim., 52 (2014), 1366-1401. doi: 10.1137/110846981. [25] S. V. Gusev and A. Likhtarnikov, Kalman-Popov-Yakubovich Lemma and the S-procedure: A historical essay, Automat. Rem. Contr., 67 (2006), 1768-1810. doi: 10.1134/S000511790611004X. [26] P. Harshavardhana, E. A. Jonckheere and L. M. Silverman, Stochastic balancing and approximation-stability and minimality, IEEE Trans. Automat. Control, 29 (1984), 744-746. doi: 10.1109/TAC.1984.1103631. [27] P. Heuberger, A family of reduced order models based on open-loop balancing, in Selected Topics in Identification, Modelling and Control, Delft University Press, 1990, 1–10. [28] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [29] W. Liu, V. Sreeram and K. L. Teo., Model reduction for state-space symmetric systems, System. Control Lett., 34 (1998), 209-215. doi: 10.1016/S0167-6911(98)00024-3. [30] Y. Liu and B. D. O. Anderson, Singular perturbation approximation of balanced systems, Internat. J. Control, 50 (1989), 1379-1405. doi: 10.1080/00207178908953437. [31] B. 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. [32] G. Muscato and G. Nunnari, On the $\sigma$-reciprocal system for model order reduction, Math. Model. Systems, 1 (1995), 261-271. doi: 10.1080/13873959508837022. [33] G. Muscato, G. Nunnari and L. Fortuna, Singular perturbation approximation of bounded real balanced and stochastically balanced transfer matrices, Internat. J. Control, 66 (1997), 253-269. doi: 10.1080/002071797224739. [34] R. W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966. [35] R. Ober and S. Montgomery-Smith, Bilinear transformation of infinite-dimensional state-space systems and balanced realizations of nonrational transfer functions, SIAM J. Control Optim., 28 (1990), 438-465. doi: 10.1137/0328024. [36] G. Obinata and B. D. Anderson, Model Reduction for Control System Design, Springer Science & Business Media, 2012. doi: 10.1007/978-1-4471-0283-0. [37] P. C. Opdenacker and E. 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. [38] M.R. Opmeer and T. Reis, A lower bound for the balanced truncation error for MIMO systems, IEEE Trans. Automat. Control, 60 (2015), 2207-2212. doi: 10.1109/TAC.2014.2368232. [39] J. R. Partington, An Introduction to Hankel Operators, Cambridge University Press, Cambridge, 1988. [40] L. Pernebo and L. M. Silverman, Model reduction via balanced state space representations, IEEE Trans. Automat. Control, 27 (1982), 382-387. doi: 10.1109/TAC.1982.1102945. [41] A. Rantzer, On the Kalman-Yakubovich-Popov Lemma, System. Control Lett., 28 (1996), 7-10. doi: 10.1016/0167-6911(95)00063-1. [42] T. Reis and T. Stykel, Positive real and bounded real balancing for model reduction of descriptor systems, Int. J. Control, 83 (2010), 74-88. doi: 10.1080/00207170903100214. [43] E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer-Verlag, New York, 1998 doi: 10.1007/978-1-4612-0577-7. [44] O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. [45] O. 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. [46] O. J. Staffans, Passive and conservative infinite-dimensional impedance and scattering systems (from a personal point of view), in Mathematical Systems Theory in Biology, Communications, Computation, and Finance (Notre Dame, IN, 2002), vol. 134 of IMA Vol. Math. Appl., Springer, New York, 2003,375–413. doi: 10.1007/978-0-387-21696-6_14. [47] J. C. Willems, Dissipative dynamical systems part Ⅰ: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493. [48] J. C. Willems, Dissipative dynamical systems part Ⅱ: Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 352-393. doi: 10.1007/BF00276494. [49] N. Young, Balanced realizations in infinite dimensions, Operator Theory: Advances and Applications, 19 (1986), 449-471. [50] K. Zhou, J. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall Englewood Cliffs, 1996.
Semi-log plot of combined errors on the real axis for the bounded real GSPA from Example 5.1, with $r = 2$. The lines numbered 1-4 correspond to $\xi_1 = 0.1$, $\xi_2 = 1$, $\xi_3 = 10$ and $\xi_4 = 100$, respectively. Note the interpolation properties (2.7) and (3.7) hold and are highlighted with vertical dotted lines. The dashed dotted line is the bound (3.3)
Semi-log plot of combined errors on the real axis for the bounded real GSPA from Example 5.1, with $r = 1$. The lines numbered 1-4 correspond to $\xi_1 = 0.1$, $\xi_2 = 1$, $\xi_3 = 10$ and $\xi_4 = 100$, respectively. Note the interpolation properties (2.7) and (3.7) hold and are highlighted with vertical dotted lines. The dashed dotted line is the error bound (3.3)
Plots of errors on the imaginary axis for the bounded real GSPA from Example 5.1, with $r = 1$ and $r = 2$ in panels (a) and (b), respectively. The lines numbered 1-4 correspond to $\xi_1 = 0.1$, $\xi_2 = 1$, $\xi_3 = 10$ and $\xi_4 = 100$, respectively, and are symmetric around $\omega = 0$. The dashed dotted lines are the bounds (3.3)
Semi-log plot of combined errors on the real axis for the positive real GSPA from Example 5.2, with $\xi = 10$. The lines numbered 1-3 correspond to $r \in \{1, 2, 3\}$ respectively. Note the interpolation property (2.7) holds
Semi-log plot of gap metric error $\hat \delta( \mathbf G , \mathbf G _r^\xi)$ (crosses) and error bounds (4.4) (circles) for extended circuit model from Example 5.2. Here $\xi = 10$
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