Model Order Reduction by Using the Balanced Truncation and Factor Division Methods

References

• Y. Shamash , “Stable reduced-order models using Padé-type approximations,” IEEE Trans Automat. Contr. , vol. 19, pp. 615–616, Oct. 1974. doi: 10.1109/TAC.1974.1100661
   Web of Science ®Google Scholar
• N. Ashoor and V. Singh , “A note on lower order modeling,” IEEE Trans Automat. Contr. , vol. 27, no. 5, pp. 1124–1126, Oct. 1982. doi: 10.1109/TAC.1982.1103087
   Web of Science ®Google Scholar
• F. J. Alexandro Jr ., “Stable partial Padé approximations for reduced-order transfer functions,” IEEE Trans Automat. Contr. , vol. 29, no. 2, pp. 159–162, Feb. 1984. doi: 10.1109/TAC.1984.1103480
   Web of Science ®Google Scholar
• L. Fortuna , G. Nunnari , and A. Gallo , Model Order Reduction Techniques with Applications in Electrical Engineering . London : Springer, 1992.
   Google Scholar
• Y. Shamash , “Model reduction using the Routh stability criterion and the Padé approximation technique,” Int. J. Control. , vol. 21, no. 3, pp. 475–484, 1975. doi: 10.1080/00207177508922004
   Web of Science ®Google Scholar
• J. Pal , “Stable reduced-order Padé approximants using the Routh-Hurwitz array,” Electron. Lett , vol. 15, no. 8, pp. 225–226, Apr. 1979. doi: 10.1049/el:19790159
   Web of Science ®Google Scholar
• T. C. Chen , C. Y. Chang , and K. W. Han , “Stable reduced-order Padé approximants using stability-equation method,” Electron. Lett. , vol. 16 no. 9, pp. 345–346, Apr. 1980. doi: 10.1049/el:19800248
   Web of Science ®Google Scholar
• T. C. Chen , C. Y. Chang , and K. W. Han , “Model reduction using the stability-equation method and the Padé approximation method,” J. Franklin Inst. , vol. 309, no. 6, pp. 473–490, June 1980. doi: 10.1016/0016-0032(80)90096-4
   Web of Science ®Google Scholar
• B.-W. Wan , “Linear model reduction using Mihailov criterion and Padé approximation technique,” Int. J. Control. , vol. 33, no. 6, pp. 1073–1089, 1981. doi: 10.1080/00207178108922977
   Web of Science ®Google Scholar
• O. Ismail , B. Bandyopadhyay , and R. Gorez , “Discrete interval system reduction using Padé approximation to allow retention of dominant poles,” IEEE Trans Circ. Syst.—I Fund Theor Appl. , vol. 44, no. 11, pp. 1075–1078, 1997. doi: 10.1109/81.641771
   Google Scholar
• R. Prasad , “Padé type model order reduction for multivariable systems using Routh approximation,” Comput. Electr. Eng. , vol. 26, no. 6, pp. 445–459, 2000. doi: 10.1016/S0045-7906(00)00002-1
   Web of Science ®Google Scholar
• C. B. Vishwakarma and R. Prasad , “Clustering method for reducing order of linear system using Padé approximation,” IETE J. Res. , vol. 54, no. 5, pp. 326–330, 2008. doi: 10.4103/0377-2063.48531
   Web of Science ®Google Scholar
• V. Krishnamurthy and V. Seshadri , “Model reduction using the Routh stability criterion,” IEEE Trans. Automat. Contr. , vol. 23, no. 3, pp. 729–731, Aug. 1978. doi: 10.1109/TAC.1978.1101805
   Google Scholar
• A. Lepschy and U. Viaro , “An improvement in the Routh-Padé approximation techniques,” Int. J. Control. , vol. 36, no. 4, pp. 643–661, 1982. doi: 10.1080/00207178208932921
   Web of Science ®Google Scholar
• T. C. Chen , C. Y. Chang , and K. W. Han , “Reduction of transfer function by the stability-equation method,” J. Franklin Inst. , vol. 308, pp. 389–404, 1979. doi: 10.1016/0016-0032(79)90066-8
   Web of Science ®Google Scholar
• M. F. Hutton and B. Friedland , “Routh approximations for reducing order of linear, time-invariant systems,” IEEE Trans. Automat. Contr. , vol. 20, no. 3, pp. 329–337, June 1975. doi: 10.1109/TAC.1975.1100953
   Web of Science ®Google Scholar
• A. K. Sinha and J. Pal , “Simulation based reduced order modelling using a clustering technique,” Comput. Electr. Eng. , vol. 16, no. 3, pp. 159–169, May 1990. doi: 10.1016/0045-7906(90)90020-G
   Web of Science ®Google Scholar
• J. Singh , C. B. Vishwakarma , and K. Chatterjee , “Biased reduction method by combining improved modified pole clustering and improved Padé approximations,” Appl. Math. Model. , vol. 40, pp. 1418–1426, 2016. doi: 10.1016/j.apm.2015.07.014
   Web of Science ®Google Scholar
• T. N. Lucas , “Factor division: a useful algorithm in model reduction,” IEE Proc. D – Control Theory Appl. A. , vol. 130, no. 6 , pp. 362–364, 1983. doi: 10.1049/ip-d.1983.0060
   Google Scholar
• T. N. Lucas , “Biased model reduction by factor division,” Electron. Lett , vol. 20, no. 14, pp. 582–583, 1984. doi: 10.1049/el:19840402
   Web of Science ®Google Scholar
• T. N. Lucas , “Linear system reduction by the modified factor division method,” IEE Proc. D – Control Theory A. , vol. 133, no. 6, pp. 293–296, 1986. doi: 10.1049/ip-d.1986.0047
   Google Scholar
• N. Singh , R. Prasad , and H. O. Gupta , “Reduction of linear dynamic systems using Routh Hurwitz array and factor division method,” IETE J. Educ. , vol. 47, no 1, pp. 25–29, 2006. doi: 10.1080/09747338.2006.11415859
   Google Scholar
• G. Parmar , S. Mukherjee , and R. Prasad , “System reduction using factor division algorithm and Eigen spectrum analysis,” Appl. Math. Model. , vol. 31, pp. 2542–2552, 2007. doi: 10.1016/j.apm.2006.10.004
   Web of Science ®Google Scholar
• D. Kranthi Kumar , S. K. Nagar , and J.P. Tiwari , “A new algorithm for model order reduction of interval systems,” Bonfring Int. J. Data Min. , vol. 3, no. 1, pp. 6–11, March 2013. doi: 10.9756/BIJDM.10131
   Google Scholar
• A. Sikander and R. Prasad , “Linear time-invariant system reduction using a mixed methods approach,” Appl. Math. Model. , vol. 39, pp. 4848–4858, 2015. doi: 10.1016/j.apm.2015.04.014
   Web of Science ®Google Scholar
• B. C. Moore , “Principal component analysis in control system: controllability, observability, and model reduction,” IEEE Trans. Automat. Contr. , vol. 26, no. 1, pp. 17–36, 1981. doi: 10.1109/TAC.1981.1102568
   Web of Science ®Google Scholar
• L. Pernebo and L. M. Silverman , “Model reduction via balanced state space representations,” IEEE Trans. Automat. Contr. , vol. 27, no. 2, pp. 382–387, Apr. 1982. doi: 10.1109/TAC.1982.1102945
   Web of Science ®Google Scholar
• U. M. Al. Saggaf , “Approximate balanced-truncation model reduction for large scale systems,” Int. J. Control , vol. 56, no. 6, pp. 1263–1273, 1992. doi: 10.1080/00207179208934366
   Web of Science ®Google Scholar
• V. Sreeram , S. Sahlan , W. M. Wan Muda , T. Fernando , and H. H. C. Iu , “A generalized partial-fraction-expansion based frequency weighted balanced truncation technique,” Int. J. Control , vol. 86, no. 5, pp. 833–843, 2013. doi: 10.1080/00207179.2013.764017
   Web of Science ®Google Scholar
• K. Perev and B. Shafai , “Balanced realization and model reduction of singular systems,” Int. J. Sys. Sci. , vol. 25, no. 6, pp. 1039–1052, 1994. doi: 10.1080/00207729408929014
   Web of Science ®Google Scholar
• M. Tahavori and H. R. Shaker , “Model reduction via time-interval balanced stochastic truncation for linear time invariant systems,” Int. J. Sys. Sci. , vol. 44, no. 3, pp. 493–501, March 2013. doi: 10.1080/00207721.2011.604741
   Web of Science ®Google Scholar
• H. Sandberg and A. Rantzer , “Balanced truncation of linear time-varying systems,” IEEE Trans. Automat. Contr. , vol. 49, no. 2, pp. 217–229, Feb. 2004. doi: 10.1109/TAC.2003.822862
   Web of Science ®Google Scholar
• H. S. Abbas and H. Werner , “Frequency-weighted discrete-time LPV model reduction using structurally balanced truncation,” IEEE Trans. Contr. Syst. T , vol. 19, no. 1, pp. 140–147, Jan. 2011. doi: 10.1109/TCST.2010.2080274
   Web of Science ®Google Scholar
• P. Wittmuess , C. Tarin , A. Keck , E. Arnold , and O. Sawodny , “Parametric model order reduction via balanced truncation with Taylor series representation,” IEEE Trans. Automat. Contr. , vol. 61, no. 11, pp. 3438–3451, Nov. 2016. doi: 10.1109/TAC.2016.2521361
   Web of Science ®Google Scholar
• B. Besselink , N. Van De Wouw , J. M. A. Scherpen , and H. Nijmeijer , “Model reduction for nonlinear systems by incremental balanced truncation,” IEEE Trans. Automat. Contr. , vol. 59, no. 10, pp. 2739–2753, 2014. doi: 10.1109/TAC.2014.2326548
   Web of Science ®Google Scholar
• J. Qi , J. Wang , H. Liu , and A. D. Dimitrovski , “Nonlinear model reduction in power systems by balancing of empirical controllability and observability covariances,” IEEE Trans. Power Syst. , vol. 32, no. 1, pp. 114–126, Jan. 2017.
   Google Scholar
• G. Kotsalis and A. Rantzer , “Balanced truncation for discrete time Markov jump linear systems,” IEEE Trans. Automat. Control. , vol. 55, no. 11, pp. 2606–2611, 2010. doi: 10.1109/TAC.2010.2060241
   Web of Science ®Google Scholar
• M. Farhood and C. L. Beck , “On the balanced truncation and coprime factors reduction of Markovian jump linear systems,” Syst. Control Lett. , vol. 96, pp. 96–106, 2014. doi: 10.1016/j.sysconle.2013.12.004
   Google Scholar
• Y. Zhu , L. Zhang , V. Sreeram , W. Shammakh , and B. Ahmad , “Resilient model approximation for Markov jump time-delay systems via reduced model with hierarchical Markov chains,” Int. J. Syst. Sci. , vol. 47, no. 14, pp. 3496–3507, 2016. doi: 10.1080/00207721.2015.1089450
   Web of Science ®Google Scholar
• S. Gugercin and A. C. Antoulas , “A survey of model reduction by balanced truncation and some new results,” Int. J. Control. , vol. 77, no. 8, pp. 748–766, May 2004. doi: 10.1080/00207170410001713448
   Web of Science ®Google Scholar
• B. N. Datta , Numerical Methods for Linear Control Systems . San Diego, CA: Elsevier, 2005.
   Google Scholar
• K. Glover , “All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds,” Int. J. Control. , vol. 39, pp. 1115–1193, 1984. doi: 10.1080/00207178408933239
   Web of Science ®Google Scholar
• Y. Saad , Numerical Methods for Large Eigenvalue Problems , 2nd ed. Manchester: Society for Industrial and Applied Mathematics, 2011.
   Google Scholar
• Y. Liu and B. D. O. Anderson , “Singular perturbation approximation of balanced systems,” Int. J. Control. , vol. 50, no. 4, pp. 1379–1405, 1989. doi: 10.1080/00207178908953437
   Web of Science ®Google Scholar
• A. Buscarino , L. Fortuna , M. Frasca , and M. G. Xibilia , “Continuous time LTI systems under lossless positive real transformations: open-loop balanced representation and truncated reduced-order models,” Int. J. Control , 2016. http://www.tandfonline.com/loi/tcon20
   Web of Science ®Google Scholar
• M. Lal and R. Mitra , “Simplification of large system dynamics using a moment evaluation algorithm,” IEEE Trans. Automat. Contr. , vol. 19, pp. 602–603, 1974. doi: 10.1109/TAC.1974.1100671
   Web of Science ®Google Scholar
• B. K. Kushwaha and A. Narain , “Controller design for Cuk converter using model order reduction,” 2nd International Conference on Power, Control and Embedded Systems, 2012.
   Google Scholar
• E. Vuthchhay , P. Unnat , and C. Bunlaksananusorn , “Modeling of a sepic converter operating in continuous conduction mode,” 6th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information, pp. 136–139, 2009.
   Google Scholar
• R. D. Middlebook and S. Cuk , “A general unified approach to modeling switching-converter power stages,” Int. J. Electr. , vol. 42, pp. 521–550, 1977. doi: 10.1080/00207217708900678
   Web of Science ®Google Scholar
• B. C. Kuo , Automatic Control Systems , 7th ed., Upper Saddle River, NJ: Prentice Hall Inc, 1995.
   Google Scholar
• A. Narwal and R. Prasad , “A novel order reduction approach for LTI systems using cuckoo search optimization and stability equation,” IETE J. Res. , vol. 62, no. 2, pp. 154–163, Aug. 2016. doi: 10.1080/03772063.2015.1075915
   Web of Science ®Google Scholar
• A. Sikander and R. Prasad , “A new technique for reduced-order modelling of linear time-invariant system,” IETE J. Res. , vol. 63, no. 3, pp. 316–324, Jan. 2017. doi: 10.1080/03772063.2016.1272436
   Web of Science ®Google Scholar
• Y. Shamash , “Linear system reduction using Padé approximation to allow retention of dominant modes,” Int. J. Control , vol. 21, no. 2, pp. 257–272, 1975. doi: 10.1080/00207177508921985
   Web of Science ®Google Scholar
• Y. Shamash , “Truncation method of reduction: a viable alternative,” Electron. Lett. , vol. 17, pp. 79–98, 1981. doi: 10.1049/el:19810057
   Web of Science ®Google Scholar
• A. Antoulas , Approximation of Large Scale Dynamical Systems , Philadelphia : SIAM Publ., 2005.
   Google Scholar
• K. Fernando and H. Nicholson , “Singular perturbational model reduction of balanced systems,” IEEE Trans. Automat. Contr. , vol. 27, no. 2, pp. 466–468, 1982. doi: 10.1109/TAC.1982.1102932
   Web of Science ®Google Scholar
• P. Gutman , C. Mannerfelt , and P. Molander , “Contributions to the model reduction problem,” IEEE Trans. Automat. Contr. , vol. 27, no. 2, pp. 454–455, 1982. doi: 10.1109/TAC.1982.1102930
   Web of Science ®Google Scholar
• W. Tan , “Unified tuning of PID load frequency controller for power systems via IMC,” IEEE Trans. Power Syst. , vol. 25, no. 1, pp. 341–350, Feb. 2010. doi: 10.1109/TPWRS.2009.2036463
   Web of Science ®Google Scholar
• P. V. Kokotovic , R. E. O'Malley , and P. Sannuti , “Singular perturbations and order reduction in control theory an overview,” Automatica , vol. 12, pp. 123–132, 1976. doi: 10.1016/0005-1098(76)90076-5
   Web of Science ®Google Scholar
• K. V. Fernando and H. Nicholson , “Singular perturbation model reduction in frequency domain,” IEEE Trans. Automat. Contr. , vol. 27, no 4, pp. 969–970, Aug. 1984. doi: 10.1109/TAC.1982.1103037
   Web of Science ®Google Scholar
• V. Zakian , “Simplification of linear time invariant systems by moment approximations,” Int. J. Control , vol. 18, pp. 455–460, 1973. doi: 10.1080/00207177308932525
   Web of Science ®Google Scholar