Kybernetika 54 no. 3, 557-575, 2018

Routh--type $L_2$ model reduction revisited

Wiesław Krajewski and Umberto ViaroDOI: 10.14736/kyb-2018-3-0557

Abstract:

A computationally simple method for generating reduced-order models that minimise the $L_2$ norm of the approximation error while preserving a number of second-order information indices as well as the steady-state value of the step response, is presented. The method exploits the energy-conservation property peculiar to the Routh reduction method and the interpolation property of the $L_2$-optimal approximation. Two examples taken from the relevant literature show that the suggested techniques may lead to approximations that are not worse than those afforded by popular more cumbersome techniques.

Keywords:

model reduction, $L_2$ norm, Routh approximation, steady-state response

Classification:

93C05, 93A15, 93B11

paper.pdf

References:

  1. O.M.K. Alsmadi and Z.S. Abo-Hammour: A robust computational technique for model order reduction of two-time-scale discrete systems via genetic algorithms. Computational Intelligence and Neuroscience 2015 (2015), 1-9.   DOI:10.1155/2015/615079
  2. A. Astolfi: Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Automat. Contr. 55 (2010), 2321-2336.   DOI:10.1109/tac.2010.2046044
  3. S. Barnett and D. D. Šiljac: Routh's algorithm: a centennial survey. SIAM Review 19 (1977), 472-489.   DOI:10.1137/1019070
  4. U. Baur, C. Beattie, P. Benner and S. Gugercin: Interpolatory projection methods for parameterized model reduction. SIAM J. Scientific Computing 33 (2011), 2489-2518.   DOI:10.1137/090776925
  5. Y. Bistritz: Optimal fraction-free Routh tests for complex and real integer polynomials. IEEE Trans. Circuits Syst. I 60 (2013), 2453-2464.   DOI:10.1109/tcsi.2013.2246232
  6. A. Beghi, A. Lepschy and U. Viaro: A property of the Routh table and its use. IEEE Trans. Automat. Control 39 (1994), 2494-2496.   DOI:10.1109/9.362839
  7. P. Benner, S. Grundel and N. Hornung: Parametric model order reduction with a small $H_2$-error using radial basis functions. Adv. Comput. Math. 41 (2015), 5, 1231-1253.   DOI:10.1007/s10444-015-9410-7
  8. A. Bultheel and B. De Moor: Rational approximation in linear systems and control. J. Comput. Appl. Math. 121 (2000), 355-378.   DOI:10.1016/s0377-0427(00)00339-3
  9. A. Bultheel and M. Van Barel: Padé techniques for model reduction in linear system theory: a survey. J. Comput. Appl. Math. 14 (1986), 401-438.   DOI:10.1016/0377-0427(86)90076-2
  10. Y. Chahlaoui and P. Van Dooren: A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working Note 2002-2: http://slicot.org/20-site/126-benchmark-examples-for-model-reduction. (2002).   CrossRef
  11. S. R. Desai and R. Prasad: A new approach to order reduction using stability equation and big bang big crunch optimization. Systems Science Control Engineering: An Open Access J. 1 (2013), 20-27.   DOI:10.1080/21642583.2013.804463
  12. S. R. Desai and R. Prasad: Generating lower order systems using modified truncation and PSO. Int. J. Computer Appl. 12 (2013), 17-21.   CrossRef
  13. P. Dorato, A. Lepschy and U. Viaro: Some comments on steady-state and asymptotic responses. IEEE Trans. Education 37 (1994), 264-268.   DOI:10.1109/13.312135
  14. A. Ferrante, W. Krajewski, A. Lepschy and U. Viaro: Convergent algorithm for $L_2$ model reduction. Automatica 35 (1999), 75-79.   DOI:10.1016/s0005-1098(98)00142-3
  15. 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
  16. P. A. Fuhrmann: A polynomial approach to Hankel norm and balanced approximations. Linear Algebra Appl. 146 (1991), 133-220.   DOI:10.1016/0024-3795(91)90025-r
  17. W. K. Gawronski: Dynamics and Control of Structures: A Modal Approach. Springer, New York, 1998.   DOI:10.1016/0024-3795(91)90025-r
  18. K. Glover: All optimal Hankel-norm approximations of linear multivariable systems and their $L_{\infty}$-error bounds. Int. J. Control 39 (1984), 1115-1193.   DOI:10.1080/00207178408933239
  19. G. Gu: All optimal Hankel-norm approximations of linear multivariable systems and their $L_{\infty}$-error bounds in discrete time. Int. J. Control 78 (2005), 408-423.   DOI:10.1080/00207170500110988
  20. S. Gugercin and A. Antoulas: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77 (2004), 748-766.   DOI:10.1080/00207170410001713448
  21. S. Gugercin, A. Antoulas and C. Beattie: A rational Krylov iteration for optimal $H_2$ model reduction. Proc. 17th Int. Symp. Mathematical Theory of Networks and Systems, Kyoto 2006, pp. 1665-1667.   CrossRef
  22. S. Gugercin, A. Antoulas and C. Beattie: $H_2$ model reduction for large-scale linear dynamical systems. SIAM J. Matrix Analysis Appl. 30 (2008), 609-638.   DOI:10.1137/060666123
  23. S. Gugercin, R. V. Polyuga, C. Beattie and A. van der Schaft: Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48 (2012), 1963-1974.   DOI:10.1016/j.automatica.2012.05.052
  24. X. X. Huang, W. Y. Yan and K. L. Teo: $H_2$ near-optimal model reduction. IEEE Trans. Automat. Control 46 (2001), 1279-1284.   DOI:10.1109/9.940934
  25. M. F. Hutton and B. Friedland: Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Automat. Control 20 (1975), 329-337.   DOI:10.1109/tac.1975.1100953
  26. R. Jeltsch, M. Mansour and eds: Stability Theory: Hurwitz Centenary Conference 1995. Birkhäuser, Basel, Switzerland, 1996.   DOI:10.1007/978-3-0348-9208-7
  27. A. M. King, U. B. Desai and R. E. Skelton: A generalized approach to q-Markov covariance equivalent realizations for discrete systems. Automatica 24 (1988), 507-515.   DOI:10.1016/0005-1098(88)90095-7
  28. W. Krajewski, A. Lepschy and U. Viaro: Compact form of the optimality conditions for multivariable $L_2$ model reduction. Atti Ist. Veneto SS.LL.AA.: Classe di Scienze Fisiche, Matematiche e Naturali Tomo CL (1991/92) (1992), 119-128.   CrossRef
  29. W. Krajewski, A. Lepschy, G. A. Mian and U. Viaro: Optimality conditions in multivariable $L_2$ model reduction. J. Franklin Inst. 330 (1993), 431-439.   DOI:10.1016/0016-0032(93)90090-h
  30. W. Krajewski, A. Lepschy and U. Viaro: Remarks on algorithms for $L_2$ model reduction. Atti Ist. Veneto SS.LL.AA.: Classe di Scienze Fisiche, Matematiche e Naturali Tomo CLII (1993/1994) (1994), 99-106.   CrossRef
  31. W. Krajewski, A. Lepschy and U. Viaro: Reduction of linear continuous-time multivariable systems by matching first- and second-order information. IEEE Trans. Automat. Control 39 (1994), 2126-2129.   DOI:10.1109/9.328814
  32. W. Krajewski, A. Lepschy and U. Viaro: Model reduction by matching Markov parameters, time moments, and impulse-response energies. IEEE Trans. Automat. Contr. 40, 949-953.   DOI:10.1109/9.384238
  33. W. Krajewski, A. Lepschy, M. Redivo-Zaglia and U. Viaro: A program for solving the L2 reduced-order problem with fixed denominator degree. Numerical Algorithms 9 (1995), 355-377.   DOI:10.1007/bf02141596
  34. W. Krajewski and U. Viaro: On MIMO model reduction by the weighted equation-error approach. Numerical Algorithms 44 (2007), 83-98.   DOI:10.1007/s11075-007-9086-2
  35. W. Krajewski and U. Viaro: Iterative-interpolation algorithms for $L_2$ model reduction. Control and Cybernetics 38 (2009), 543-554.   CrossRef
  36. V. Krishnamurthy and V. Seshadri: Model reduction using the Routh stability criterion. IEEE Trans. Automat. Contr. 23 (1978), 729-731.   DOI:10.1109/tac.1978.1101805
  37. A. Lepschy, G. A. Mian and U. Viaro: A geometrical interpretation of the Routh test. J. Franklin Inst. 325 (1988, 695-703.   DOI:10.1016/0016-0032(88)90003-8
  38. A. Lepschy, G. A. Mian and U. Viaro: Splitting of some $s$-domain stability test algorithms. Int. J. Control 50 (1989), 2237-2247.   DOI:10.1080/00207178908953495
  39. D. G. Luenberger: Optimization by Vector Space Methods. Wiley, New York 1969.   DOI:10.1137/1012072
  40. L. Meier and D. G. Luenberger: Approximation of linear constant systems. IEEE Trans. Automat. Contr. 12 (1967), 585-588.   DOI:10.1109/tac.1967.1098680
  41. W. Mi, T. Qian and F. Wan: A fast adaptive model reduction method based on Takenaka-Malmquist systems. Systems Control Lett. 61 (2012), 223-230.   DOI:10.1016/j.sysconle.2011.10.016
  42. S. K. Mittal, D. Chandra and B. Dwivedi: A computer-aided approach for Routh-Padé approximation of SISO systems based on multi-objective optimization. Int. J. Eng. Technol. 2 (2010), 204-210.   CrossRef
  43. 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
  44. S. Mukherjee, Satakshi and R .C. Mittal: Model order reduction using response-matching technique. J. Franklin Inst. 342 (2005), 503-519.   DOI:10.1016/j.jfranklin.2005.01.008
  45. C. T. Mullis and R. A. Roberts: The use of second-order data in the approximation of discrete-time linear systems. IEEE Trans. Acoust. Speech Signal Process. 24 (1976), 226-238.   DOI:10.1109/tassp.1976.1162795
  46. V.Y. Pan: Solving a polynomial equation: some history and recent progress. SIAM Rev. 39 (1997), 187-220.   DOI:10.1137/s0036144595288554
  47. S. Panda, S. K. Tomar, R. Prasad and C. Ardil: Reduction of linear time-invariant systems using Routh-approximation and PSO. Int. J. Electrical Computer Electronics and Communication Eng. 3 (2009), 20-27.   CrossRef
  48. G. Parmar, S. Mukherjee and R. Prasad: System reduction using factor division algorithm and eigen spectrum analysis. Appl. Math. Modelling 31 (2007), 2542-2552.   DOI:10.1016/j.apm.2006.10.004
  49. D. Petersson and J. Löfberg: Model reduction using a frequency-limited $H_2$ cost. Systems Control Lett. 67 (2012), 32-39.   DOI:10.1016/j.sysconle.2014.02.004
  50. L. Qiu: What can Routh table offer in addition to stability?. J. Contr. Theory Appl. 1 (2003), 9-16.   DOI:10.1007/s11768-003-0003-5
  51. K. Ramawat and A. Kumar: Improved Padé-pole clustering approach using genetic algorithm for model order reduction. Int. J. Computer Appl. 114 (2015), 24-285.   DOI:10.5120/19943-1737
  52. J. S. Rana, R. Prasad and R. Singh: Order reduction using modified pole clustering and factor division method. Int. J. Innovative Tech. Exploring Eng. 3 (2014), 134-136.   CrossRef
  53. V. S. Ryaben'kii and S. V. Tsynkov: A Theoretical Introduction to Numerical Analysis. Chapman and Hall/CRC (Taylor and Francis Group), Boca Raton 2006.   CrossRef
  54. M. Rydel and W. Stanisławski: Selection of fitness functions for evolutionary algorithms and their influence on the properties of a reduced mathematical MIMO model. In: Proc. IEEE Int. Conf. Methods and Models in Automation and Robotics, Miedzyzdroje 2015.   DOI:10.1109/mmar.2015.7283941
  55. D. K. Saini and R. Prasad: Order reduction of linear interval systems using genetic algorithm. Int. J. Eng. Technol. 2 (2010), 316-319.   CrossRef
  56. A. Sikander and R. Prasad: A novel order reduction method using cuckoo search algorithm. IETE J. Research 61 (2015), 83-90.   DOI:10.1080/03772063.2015.1009396
  57. A. Sikander and R. Prasad: Linear time invariant system reduction using mixed method approach. Appl. Math. Modelling 39 (2015), 4848-4858.   DOI:10.1016/j.apm.2015.04.014
  58. C. B. Soh: Generalization of the Hermite-Biehler theorem and applications. IEEE Trans. Automat. Control 35 (1990), 222-225.   DOI:10.1109/9.45186
  59. H. N. Soloklo and M. M. Farsangi: Order reduction by minimizing integral square error and $H_{\infty}$ norm of error. J. Adv. Computer Res. 5 (2014), 29-42.   CrossRef
  60. V. Sreeram and P. Agathoklis: Model reduction of linear discrete systems via weighted impulse response Gramians. Int. J. Control 53 (1991), 129-144.   DOI:10.1080/00207179108953613
  61. N. Tanguy, N. Iassamen, M. Telescu and P. Cloastre: Parameter optimization of orthonormal basis functions for efficient rational approximations. Appl. Math. Modelling 39 (2015), 4963-4970.   DOI:10.1016/j.apm.2015.04.017
  62. P. Van Dooren, K. A. Gallivan and P. A. Absil: $H_2$-optimal model reduction of MIMO systems. Appl. Math. Lett. 21 (2008), 1267-1273.   DOI:10.1016/j.aml.2007.09.015
  63. A. Varga and B. D. O. Anderson: Accuracy-enhancing methods for balancing-related frequency-weighted model and controller reduction. Automatica 39 (2003), 919-927.   DOI:10.1016/s0005-1098(03)00030-x
  64. U. Viaro: Stability tests revisited. In: A Tribute to Antonio Lepschy (G. Picci and M. E. Valcher, eds.), Edizioni Libreria Progetto, Padova 2007, pp. 189-199.   CrossRef
  65. C.B. Vishwakarma and R. Prasad: Order reduction using the advantages of differentiation method and factor division algorithm. Indian J. Engineering \& Materials Sciences 15 (2008), 447-451.   CrossRef
  66. Y. Xu and T. Zeng: Optimal $H_2$ model reduction for large scale MIMO systems via tangential interpolation. Int. J. Numerical Analysis and Modeling 8 (2011), 174-188.   CrossRef
  67. Y. Wang, D. S. Bernstein and L. T. Watson: Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order $H _2/H_{\infty}$ modeling, estimation, and control. Appl. Math Comput. 123 (2001), 155-185.   DOI:10.1016/s0096-3003(00)00059-x
  68. W. Y. Yan and J. Lam: An approximate approach to $H^2$ optimal model reduction. IEEE Trans. Automat. Control 44 (1999), 1341-1358.   DOI:10.1109/9.774107
  69. C. Zeng and Y. Q. Chen: Global Padé approximations of the generalized Mittag-Leffler function and its inverse. Fractional Calculus and Applied Analysis (arXiv:1310.5592) 18 (2015), 1-15.   DOI:10.1515/fca-2015-0086
  70. T. Zeng and C. Lu: Two-sided Grassmann manifold algorithm for optimal $H_2$ model reduction. Int. J. Numerical Methods Engrg. 104 (2015), 10, 928-943.   DOI:10.1515/fca-2015-0086
  71. J. G. Ziegler and N. B. Nichols: Optimum settings for automatic controllers. Trans. ASME 64 (1942), 759-765.   DOI:10.1109/tit.1972.1054906
  72. D. Žigić, L. T. Watson, E. G. Collins, Jr. and D. S. Bernstein: Homotopy methods for solving the optimal projection equations for the $H_2$ reduced order model problem. Int. J. Control 56 (1992), 173-191.   DOI:10.1080/00207179208934308