Abstract
In this contribution, R-method based approximation of continuous systems to reduced order models (ROMs) is presented by utilizing the grey wolf optimization (GWO) algorithm. The approximation is done by minimizing the errors between time moments (TMs) and Markov parameters (MPs) of the higher-order system (HOS) and desired ROM. So, the TMs and MPs of HOS and ROM are utilized to frame the objective function. The weights associated with objective function are determined using R-method. These weights are further utilized to convert the multi-objective problem into single-objective problem. In objective function, the normalized errors are minimized using GWO algorithm to obtain desired ROM. To ensure the steady-state matching between HOS and its ROM, first TMs of HOS and desired ROM are matched. The stability of obtained ROM is ensured by Hurwitz stability criterion. The superiority of proposed method is determined with the help of two test systems. The results of proposed technique are compared with results of other already obtained ROM available in the literature. For comparative analysis, tabulated values are given for both test cases by considering time domain specifications. These time domain specifications are rise time, settling time, overshoot, undershoot, peak and peak time. The error indices are also provided to validate the proposed method. The step responses, impulse responses, Bode plots and Nyquist plots of the system and ROMs are also presented. The provided results prove the efficacy and effectiveness of the proposed method.
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A.L. Alberti, T.S. Palmer, Reduced-order modeling of nuclear reactor kinetics using proper generalized decomposition. Nucl. Sci. Eng. 194(10), 837–858 (2020)
H.R. Ali, B.C. Pal, Model order reduction of multi-terminal direct-current grid systems. IEEE Trans. Power Syst. 36(1), 699–711 (2020)
S. Arun, T. Manigandan, P. Mariaraja, Pole clustering-based modified reduced-order model for boiler system. IETE J. Res. pp. 1–7 (2020)
J.K. Bokam, V. Singh, Improved Routh-Padé approximants based on matching of Markov parameters and time moments for continuous interval systems. Int. J. Pure Appl. Math. 119(12), 12755–12766 (2018)
D. Botto, S. Zucca, M. Gola, Reduced-order models for the calculation of thermal transients of heat conduction/convection FE models. J. Therm. Stresses 30(8), 819–839 (2007)
A. Davoudi, J. Jatskevich, P.L. Chapman, A. Bidram, Multi-resolution modeling of power electronics circuits using model-order reduction techniques. IEEE Trans. Circuits Syst. I Regul. Pap. 60(3), 810–823 (2012)
X. Dong, A. Griffo, J. Wang, Multiparameter model order reduction for thermal modeling of power electronics. IEEE Trans. Power Electron. 35(8), 8550–8558 (2020)
M.F. Far, F. Martin, A. Belahcen, P. Rasilo, H.A.A. Awan, Real-time control of an IPMSM using model order reduction. IEEE Trans. Ind. Electron. 68(3), 2005–2014 (2020)
L. Fortuna, G. Nunnari, A. Gallo, Model Order Reduction Techniques with Applications in Electrical Engineering (Springer Science & Business Media, Cham, 2012)
M. Garg. Model order reduction and approximation analysis for control system design. In 2017 4th International Conference on Signal Processing, Computing and Control (ISPCC), pp. 473–476. IEEE (2017)
Y. Gu, N. Bottrell, T.C. Green, Reduced-order models for representing converters in power system studies. IEEE Trans. Power Electron. 33(4), 3644–3654 (2017)
W. Hu, Z. Wu, V. Dinavahi, Dynamic analysis and model order reduction of virtual synchronous machine based microgrid. IEEE Access 8, 106585–106600 (2020)
M. Hutton, B. Friedland, Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Autom. Control 20(3), 329–337 (1975)
M. Juneja, S. Nagar, Comparative study of model order reduction using combination of PSO with conventional reduction techniques. In 2015 International Conference on Industrial Instrumentation and Control (ICIC), pp. 406–411. IEEE (2015)
S. Kalra, M. Nabi, Model order reduction of finite element model of pacemaker electrode. In TENCON 2019-2019 IEEE Region 10 Conference (TENCON), pp. 1076–1081. IEEE (2019)
H.J. Kelley, Aircraft maneuver optimization by reduced-order approximation, in Control and Dynamic Systems. volume 10. (Elsevier, New York, 1973), pp.131–178
S. Keyumarsi, A. Nobakhti, M.S. Tavazoei, Non-fragile h order reduction of LTI controllers. IEEE Control Systems Letters 5(1), 163–168 (2020)
V. Krishnamurthy, V. Seshadri, Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(4), 729–731 (1978)
B. K. Kushwaha, A. Narain, Controller design for CUK converter using model order reduction. In 2012 2nd International Conference on Power, Control and Embedded Systems, pp. 1–5. IEEE (2012)
S. Lavania, D. Nagaria, Pade approximation based moment matching technique for model order reduction. In 2015 International Conference on Computer, Communication and Control (IC4), pp. 1–4. IEEE (2015)
M.S. Mahmoud, S. MG, Large scale systems modelling (1981)
E. Malekshahi, S.-M.-A. Mohammadi, The model order reduction using LS, RLS and MV estimation methods. Int. J. Control Autom. Syst. 12(3), 572–581 (2014)
D. Mandelli, A. Alfonsi, P. Talbot, C. Wang, D. Maljovec, C. Smith, C. Rabiti, J. Cogliati, An overview of reduced order modeling techniques for safety applications (2016)
H. Manohar, D. Sambariya, Model order reduction of MIMO system using differentiation method. In 2016 10th International Conference on Intelligent Systems and Control (ISCO), pp. 1–5. IEEE (2016)
G. Mendonça, F. Afonso, F. Lau, Model order reduction in aerodynamics: review and applications. Proc. Inst. Mech. Eng. Part G: J. Aerospace Eng. 233(15), 5816–5836 (2019)
S. Mirjalili, S.M. Mirjalili, A. Lewis, Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)
P. Munaswamy, R. Satyanarayana, Pupillary light reflex system order reduction by RMS model order reduction matlab tool (2011)
S.A. Nahvi, M.A. Bazaz, H. Khan, Model order reduction in power electronics: issues and perspectives. In 2017 International Conference on Computing, Communication and Automation (ICCCA), pp. 1417–1421. IEEE (2017)
D.S. Naidu, A.J. Calise, Singular perturbations and time scales in guidance and control of aerospace systems: a survey. J. Guid. Control. Dyn. 24(6), 1057–1078 (2001)
S. Naqvi, S. A. Naqvi, et al. A reduced order model for AQC system using Routh approximation technique. In 2014 Innovative Applications of Computational Intelligence on Power, Energy and Controls with their impact on Humanity (CIPECH), pp. 248–252. IEEE (2014)
I. Necoara, T.C. Ionescu, \( h_2 \) model reduction of linear network systems by moment matching and optimization. IEEE Trans. Autom. Control 65(12), 5328–5335 (2020)
Y. Nie, Z. Wang, L. Gao, J. Li, Y. Zhao, Frequency-weighted residual truncation and Padé approximant for model reduction of power system with constant time delay. IEEE Access 8, 63018–63026 (2020)
G. Parmar, S. Mukherjee, R. Rasad, Reduced order modelling of linear mimo systems using genetic algorithm. Int. J. Simul. Model. (IJSIMM), 6(3) (2007)
J.R. Phillips, Model reduction of time-varying linear systems using approximate multipoint Krylov-subspace projectors. In 1998 IEEE/ACM International Conference on Computer-Aided Design. Digest of Technical Papers (IEEE Cat. No. 98CB36287) pp. 96–102. IEEE (1998)
A.K. Prajapati, R. Prasad, A new model order reduction method for the design of compensator by using moment matching algorithm. Trans. Inst. Meas. Control. 42(3), 472–484 (2020)
R. Rao, J. Lakshmi, R-method: a simple ranking method for multi-attribute decision-making in the industrial environment. J. Project Manag. 6(4), 223–230 (2021)
R. Salim, M. Bettayeb, H2 and h optimal model reduction using genetic algorithms. J. Franklin Inst. 348(7), 1177–1191 (2011)
D. Sambariya, G. Arvind, High order diminution of LTI system using stability equation method. J. Adv. Math. Comput. Sci. pp. 1–15 (2016)
D.K. Sambariya, A.K. Sharma, An efficient approach for stability preservation in model order reduction using moment matching technique. In 2017 International Conference on Computer, Communications and Electronics (Comptelix), pp. 478–483. IEEE (2017)
P. Saraswat, G. Parmar, Model order reduction of transformer linear section model using simulated annealing. In 2015 Communication, Control and Intelligent Systems (CCIS), pp. 272–276. IEEE (2015)
S. Saxena, Y.V. Hote, Load frequency control in power systems via internal model control scheme and model-order reduction. IEEE Trans. Power Syst. 28(3), 2749–2757 (2013)
W.H. Schilders, H.A. Van der Vorst, J. Rommes, Model Order Reduction: Theory, Research Aspects and Applications, vol. 13 (Springer, Cham, 2008)
Y. Shamash, Linear system reduction using Pade approximation to allow retention of dominant modes. Int. J. Control 21(2), 257–272 (1975)
N. Shrivastava, P. Varshney, Comparative analysis of order reduction techniques. In 2016 Second International Innovative Applications of Computational Intelligence on Power, Energy and Controls with their Impact on Humanity (CIPECH), pp. 46–50. IEEE, (2016)
A. Singh, S. Yadav, N. Singh, K. K. Deveerasetty, Model order reduction of power plant system by balanced realization method. In 2018 International Conference on Computing, Power and Communication Technologies (GUCON), pp. 1014–1018. IEEE (2018)
N. Singh. Reduced order modeling and controller design. Unpublished Ph. D. Thesis, IIT Roorkee (2007)
S. Singh, V. Singh, V. Singh, Symbiotic organisms search algorithm based model reduction of higher order continuous systems. Int. J. Math. Oper. Res. 18(1), 115–125 (2021)
V. Singh, Obtaining Routh-Pade approximants using the Luus-jaakola algorithm. IEE Proc. Control Theory Appl. 152(2), 129–132 (2005)
V. Singh, Sine cosine algorithm based reduction of higher order continuous systems. In 2017 International Conference on Intelligent Sustainable Systems (ICISS), pp. 649–653. IEEE (2017)
V. Singh, D. Chandra, H. Kar, Improved Routh-Pade/spl acute/approximants: a computer-aided approach. IEEE Trans. Autom. Control 49(2), 292–296 (2004)
V. Singh, D.P.S. Chauhan, S.P. Singh, T. Prakash, On time moments and Markov parameters of continuous interval systems. J. Circuits Syst. Comput. 26(03), 1750038 (2017)
V.P. Singh, Sine cosine algorithm based reduction of higher order continuous systems. In 2017 International Conference on Intelligent Sustainable Systems (ICISS), pp. 649–653 (2017)
V.P. Singh, P. Chaubey, D. Chandra, Model order reduction of continuous time systems using pole clustering and chebyshev polynomials. In 2012 Students Conference on Engineering and Systems, pp. 1–4. IEEE (2012)
N.K. Sinha, Reduced-order models for linear systems. In Proceedings 1992 IEEE International Conference on Systems, Man, and Cybernetics, pp. 537–542. IEEE (1992)
H.N. Soloklo, M.M. Farsangi, Multi-objective weighted sum approach model reduction by Routh-Pade approximation using harmony search. Turk. J. Electr. Eng. Comput. Sci. 21(2), 2283–2293 (2013)
D. Srivastava and D. Chandra, Model order reduction of finite element model. In 2016 2nd International Conference on Advances in Electrical, Electronics, Information, Communication and Bio-Informatics (AEEICB), pp. 27–31. IEEE (2016)
S.K. Sunori, P.K. Juneja, M. Chaturvedi, P. Saini. Model order reduction of a higher order model of pH neutralizer of sugar mill. In 2016 8th International Conference on Computational Intelligence and Communication Networks (CICN), pp. 653–656. IEEE (2016)
D. Trudnowski, Order reduction of large-scale linear oscillatory system models. IEEE Trans. Power Syst. 9(1), 451–458 (1994)
P. Verma, P.K. Juneja, M. Chaturvedi. Reduction of SISO system using different mixed reduction methods. In 2016 8th International Conference on Computational Intelligence and Communication Networks (CICN), pp. 657–661. IEEE (2016)
C. Vishwakarma, Model order reduction of linear dynamic systems for control system design. Indian Institute of Technology Roorkee (2009)
C. Yuan, S. Kreß, G. Sadashivaiah, E.B. Rudnyi, D. Hohlfeld, T. Bechtold, Towards efficient design optimization of a miniaturized thermoelectric generator for electrically active implants via model order reduction and submodeling technique. Int. J. Numer. Methods Biomed. Eng. 36(4), e3311 (2020)
M. Zarei, On a reduced order modeling of the nuclear reactor dynamics. Appl. Math. Comput. 393, 125819 (2021)
A. Zhang, J. Liu, Economic MPC of wastewater treatment plants based on model reduction. Processes 7(10), 682 (2019)
U. Zulfiqar, V. Sreeram, X. Du, Time-and frequency-limited h2-model order reduction of bilinear control systems. arXiv e-prints, pp. arXiv–2001 (2020)
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Yadav, U.K., Singh, V.P. R-method-Based Reduction of Continuous Systems Using Grey Wolf Optimization Algorithm. Circuits Syst Signal Process 42, 1389–1418 (2023). https://doi.org/10.1007/s00034-022-02144-x
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DOI: https://doi.org/10.1007/s00034-022-02144-x