June  2019, 9(2): 313-350. 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, 2019, 9 (2) : 313-350. 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. Google Scholar

[2]

B. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach, Prentice Hall, 1973.Google Scholar

[3]

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898718713. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[8]

R. F. Curtain and K. Glover, Balanced realisations for infinite-dimensional systems, in Operator Theory and Systems, Birkhäuser, Basel, 19 (1986), 87-104. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

K. V. Fernando and H. Nicholson, Singular perturbational model reduction of balanced systems, IEEE Trans. Automat. Control, 27 (1982), 466-468. Google Scholar

[12]

K. V. Fernando and H. Nicholson, Singular perturbational model reduction in the frequency domain, IEEE Trans. Automat. Control, 27 (1982), 969-970. Google Scholar

[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. Google Scholar

[14]

K. GloverJ. 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. Google Scholar

[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. Google Scholar

[16]

K. GloverR. 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. Google Scholar

[17]

M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Inc., Upper Saddle River, 1995.Google Scholar

[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. Google Scholar

[19]

C. Guiver, Model Reduction by Balanced Truncation, PhD thesis, University of Bath, UK, 2012.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

P. HarshavardhanaE. 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. Google Scholar

[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.Google Scholar

[28]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. Google Scholar

[29]

W. LiuV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

G. MuscatoG. 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. Google Scholar

[34]

R. W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39] J. R. Partington, An Introduction to Hankel Operators, Cambridge University Press, Cambridge, 1988. Google Scholar
[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. Google Scholar

[41]

A. Rantzer, On the Kalman-Yakubovich-Popov Lemma, System. Control Lett., 28 (1996), 7-10. doi: 10.1016/0167-6911(95)00063-1. Google Scholar

[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. Google Scholar

[43]

E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer-Verlag, New York, 1998 doi: 10.1007/978-1-4612-0577-7. Google Scholar

[44] O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. Google Scholar
[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. Google Scholar

[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. Google Scholar

[47]

J. C. Willems, Dissipative dynamical systems part Ⅰ: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493. Google Scholar

[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. Google Scholar

[49]

N. Young, Balanced realizations in infinite dimensions, Operator Theory: Advances and Applications, 19 (1986), 449-471. Google Scholar

[50]

K. Zhou, J. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall Englewood Cliffs, 1996.Google Scholar

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. Google Scholar

[2]

B. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern Systems Theory Approach, Prentice Hall, 1973.Google Scholar

[3]

A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898718713. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[8]

R. F. Curtain and K. Glover, Balanced realisations for infinite-dimensional systems, in Operator Theory and Systems, Birkhäuser, Basel, 19 (1986), 87-104. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

K. V. Fernando and H. Nicholson, Singular perturbational model reduction of balanced systems, IEEE Trans. Automat. Control, 27 (1982), 466-468. Google Scholar

[12]

K. V. Fernando and H. Nicholson, Singular perturbational model reduction in the frequency domain, IEEE Trans. Automat. Control, 27 (1982), 969-970. Google Scholar

[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. Google Scholar

[14]

K. GloverJ. 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. Google Scholar

[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. Google Scholar

[16]

K. GloverR. 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. Google Scholar

[17]

M. Green and D. J. N. Limebeer, Linear Robust Control, Prentice-Hall, Inc., Upper Saddle River, 1995.Google Scholar

[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. Google Scholar

[19]

C. Guiver, Model Reduction by Balanced Truncation, PhD thesis, University of Bath, UK, 2012.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

P. HarshavardhanaE. 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. Google Scholar

[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.Google Scholar

[28]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. Google Scholar

[29]

W. LiuV. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

G. MuscatoG. 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. Google Scholar

[34]

R. W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39] J. R. Partington, An Introduction to Hankel Operators, Cambridge University Press, Cambridge, 1988. Google Scholar
[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. Google Scholar

[41]

A. Rantzer, On the Kalman-Yakubovich-Popov Lemma, System. Control Lett., 28 (1996), 7-10. doi: 10.1016/0167-6911(95)00063-1. Google Scholar

[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. Google Scholar

[43]

E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer-Verlag, New York, 1998 doi: 10.1007/978-1-4612-0577-7. Google Scholar

[44] O. Staffans, Well-posed Linear Systems, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511543197. Google Scholar
[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. Google Scholar

[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. Google Scholar

[47]

J. C. Willems, Dissipative dynamical systems part Ⅰ: General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351. doi: 10.1007/BF00276493. Google Scholar

[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. Google Scholar

[49]

N. Young, Balanced realizations in infinite dimensions, Operator Theory: Advances and Applications, 19 (1986), 449-471. Google Scholar

[50]

K. Zhou, J. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall Englewood Cliffs, 1996.Google Scholar

Figure 5.1.  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)
Figure 5.2.  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)
Figure 5.3.  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)
Figure 5.4.  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
Figure 5.5.  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 $
[1]

Belinda A. Batten, Hesam Shoori, John R. Singler, Madhuka H. Weerasinghe. Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 83-107. doi: 10.3934/dcdsb.2018162

[2]

Martin Redmann, Melina A. Freitag. Balanced model order reduction for linear random dynamical systems driven by Lévy noise. Journal of Computational Dynamics, 2018, 5 (1&2) : 33-59. doi: 10.3934/jcd.2018002

[3]

Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621

[4]

Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259

[5]

Jann-Long Chern, Zhi-You Chen, Yong-Li Tang. Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2299-2318. doi: 10.3934/dcds.2013.33.2299

[6]

Charles Fefferman. Interpolation by linear programming I. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477

[7]

Ilona Gucwa, Peter Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 783-806. doi: 10.3934/dcdss.2009.2.783

[8]

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

[9]

Marc Massot. Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 433-456. doi: 10.3934/dcdsb.2002.2.433

[10]

Martin Redmann, Peter Benner. Approximation and model order reduction for second order systems with Levy-noise. Conference Publications, 2015, 2015 (special) : 945-953. doi: 10.3934/proc.2015.0945

[11]

Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619

[12]

A. Lehikoinen, S. Finsterle, A Voutilainen, L. M. Heikkinen, M. Vauhkonen, J. P. Kaipio. Approximation errors and truncation of computational domains with application to geophysical tomography. Inverse Problems & Imaging, 2007, 1 (2) : 371-389. doi: 10.3934/ipi.2007.1.371

[13]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457

[14]

Chunyan Ji, Daqing Jiang. Persistence and non-persistence of a mutualism system with stochastic perturbation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 867-889. doi: 10.3934/dcds.2012.32.867

[15]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883

[16]

Rogério Martins. One-dimensional attractor for a dissipative system with a cylindrical phase space. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 533-547. doi: 10.3934/dcds.2006.14.533

[17]

Eduard Feireisl, Antonin Novotny, Yongzhong Sun. Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 121-143. doi: 10.3934/dcds.2014.34.121

[18]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[19]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[20]

Weihua Liu, Andrew Klapper. AFSRs synthesis with the extended Euclidean rational approximation algorithm. Advances in Mathematics of Communications, 2017, 11 (1) : 139-150. doi: 10.3934/amc.2017008

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (42)
  • HTML views (434)
  • Cited by (0)

Other articles
by authors

[Back to Top]