Abstract
A new model reduction method for the simplification and design of a controller for the linear time-invariant systems is proposed. An improved generalized pole clustering algorithm is employed in the proposed technique for obtaining the denominator of the reduced model. The numerator is computed with a simple mathematical technique available in the literature. The proposed method guarantees the stability in the reduced plant given that the full-order plant is stable and also retains the fundamental characteristics of the original model in the approximated one. This reduced model has been used for the design of compensator for the large-scale original plant by using a new algorithm. The compensator obtained by using the reduced model gives the approximately same time domain specification as compensator obtained by using large-scale original system, and the design of compensator by using the reduced model is comparatively easier. The results of the proposed algorithm are compared with existing methods of reduced-order modeling which show improvement in the performance error indices, time response characteristics and time domain specifications. The validity, effectiveness and superiority of the proposed technique have been demonstrated through some standard numerical examples.
Similar content being viewed by others
References
M. Aoki, Control of large scale dynamic systems by aggregation. IEEE Trans. Autom. Control 13(3), 246–255 (1968)
Z. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43(1), 9–44 (2002)
T.C. Chen, C.Y. Chang, K.W. Han, Model reduction using the stability-equation method and the continued-fraction method. Int. J. Control 32(1), 81–94 (1980)
T.C. Chen, C.Y. Chang, K.W. Han, Model reduction using the stability-equation method and the Padé approximation method. J. Frankl. Inst. 309(6), 473–490 (1980)
T.C. Chen, C.Y. Chang, K.W. Han, Reduction of transfer functions by the stability-equation method. J. Frankl. Inst. 308(4), 389–404 (1979)
J. Cheng, C.K. Ahn, H.R. Karimi, J. Cao, W. Qi, An event-based asynchronous approach to Markov jump systems with hidden mode detections and missing measurements. IEEE Trans. Syst. Man Cybern.: Syst. 49(9), 1749–1758 (2019)
J. Cheng, J.H. Park, J. Cao, W. Qi, Hidden Markov model-based nonfragile state estimation of switched neural network with probabilistic quantized outputs. IEEE Transactions on Cybernetics (2019). https://doi.org/10.1109/TCYB.2019.2909748
A.K. Choudhary, S.K. Nagar, Order reduction in z-domain for interval system using an arithmetic operator. Circuits Syst. Process. 38(3), 1023–1038 (2019)
I. Elfadel, D.D. Ling, A block Arnoldi algorithm for multipoint passive MOR of multi-port RLC networks. IEEE Trans. Circuits Syst. 2(7), 291–299 (1997)
M. Farhood, C.L. Beck, On the balanced truncation and coprime factors reduction of Markovian jump linear systems. Syst. Control Lett. 96, 96–106 (2014)
K. Fernando, H. Nicholson, Singular perturbational model reduction of balanced systems. IEEE Trans. Autom. Control 27(2), 466–468 (1982)
R. Freund, Model reduction methods based on Krylov subspaces. Acta Numer. 12(1), 267–319 (2003)
R. Freund, Reduced-order modeling techniques based on Krylov subspaces and their use in circuit simulation. Appl. Comput. Control. Signals Circuits 1, 435–498 (1999)
R.K. Gautam, N. Singh, N.K. Choudhary, A. Narain, Model order reduction using factor division algorithm and fuzzy c-means clustering technique. Trans. Inst. Meas. Control 41(2), 468–475 (2019)
S.G. Goodhart, K.J. Burnham, D.J.G. James, A reduced order self-tuning controller. Trans. Inst. Meas. Control 13(1), 11–16 (1991)
G. Gu, All optimal Hankel-norm approximations and their error bounds in discrete-time. Int. J. Control 78(6), 408–423 (2005)
P. Gutman, C. Mannerfelt, P. Molander, Contributions to the model reduction problem. IEEE Trans. Autom. Control 27(2), 454–455 (1982)
K.S. Haider, A. Ghafoor, M. Imran, M.F. Mumtaz, Model reduction of large scale descriptor systems using time limited Gramians. Asian J. Control 19(4), 1–11 (2017)
J. Hickin, N.K. Sinha, Aggregation matrices for a class of low-order models for large-scale systems. Electron. Lett. 11(9), 186 (1975)
C. Huang, K. Zhang, X. Dai, W. Tan, A modified balanced truncation method and its application to model reduction of power system, in IEEE Power and Energy Society General Meeting, Vancouver, BC, Canada, Jul. 2013
M.F. Hutton, B. Friedland, Routh approximations for reducing order of linear, time-invariant systems. IEEE Trans. Autom. Control 20(3), 329–337 (1975)
M. Imran, A. Ghafoor, Model reduction of descriptor systems using frequency limited Gramians. J. Frankl. Inst. 352(1), 33–51 (2015)
O. Ismail, B. Bandyopadhyay, R. Gorez, Discrete interval system reduction using Padé approximation to allow retention of dominant poles. IEEE Trans Circuits Syst Fundam Theory Appl 44(11), 1075–1078 (1997)
A. Jazlan, P. Houlis, V. Sreeram, R. Togneri, An improved parameterized controller reduction technique via new frequency weighted model reduction formulation. Asian J. Control 19(6), 1920–1930 (2017)
P.V. Kokotovik, R.E.O. Malley, P. Sannuti, Singular perturbation and order reduction in control theory-an overview. Automatica 12, 123–132 (1976)
R. Komarasamy, N. Albhonso, G. Gurusamy, Order reduction of linear systems with an improved pole clustering. J. Vib. Control 18(12), 1876–1885 (2011)
E.D. Koronaki, P.A. Gkinis, L. Beex, S.P.A. Bordas, C. Theodoropoulos, A.G. Boudouvis, Classification of states and model order reduction of large scale chemical vapor deposition processes with solution multiplicity. Comput. Chem. Eng. 121, 148–157 (2019)
V. Krishnamurthy, V. Seshadri, Model reduction using the Routh stability criterion. IEEE Trans. Autom. Control 23(3), 729–731 (1978)
D. Kumar, S.K. Nagar, Model reduction by extended minimal degree optimal Hankel norm approximation. Appl. Math. Model. 38, 2922–2933 (2014)
D.K. Kumar, S.K. Nagar, J.P. Tiwari, A new algorithm for model order reduction of interval systems. Bonfring Int. J. Data Min. 3(1), 6–11 (2013)
B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–32 (1981)
A. Narwal, R. Prasad, A novel order reduction approach for LTI systems using cuckoo search optimization and stability equation. IETE J. Res. 62(2), 154–163 (2015)
A. Narwal, R. Prasad, Optimization of LTI systems using modified clustering algorithm. IETE Tech. Rev. 34(2), 201–213 (2016)
Y. Ni, C. Li, Z. Du, G. Zhang, Model order reduction based dynamic equivalence of a wind farm. Electr. Power Energy Syst. 83, 96–103 (2016)
J. Pal, Stable reduced-order Padé approximants using the Routh-Hurwitz array. Electron. Lett. 15(8), 225–226 (1979)
G. Parmar, R. Prasad, S. Mukherjee, Order reduction of linear dynamic systems using stability equation method and GA. Int. J. Electr. Comput. Eng. 1(2), 236–242 (2007)
G. Parmar, S. Mukherjee, R. Prasad, System reduction using factor division algorithm and eigen spectrum analysis. Appl. Math. Model. 31, 2542–2552 (2007)
S. Paul, J. Chang, Fast numerical analysis of electric motor using nonlinear model order reduction. IEEE Trans. Magnet. 54(3), 1–4 (2018)
L. Pernebo, L.M. Silverman, Model reduction via balanced state space representations. IEEE Trans. Autom. Control 27(2), 382–387 (1982)
W.C. Peterson, A.H. Nassar, On the synthesis of optimum linear feedback control systems. J. Frankl. Inst. 306(3), 237–256 (1978)
A. Pierquin, T. Henneron, S. Clénet, Data-driven model-order reduction for magnetostatic problem coupled with circuit equations. IEEE Trans. Magnet. 54(3), 1–4 (2018)
A.K. Prajapati, R. Prasad, Model order reduction by using the balanced truncation method and the factor division algorithm. IETE J. Res. (2018). https://doi.org/10.1080/03772063.2018.1464971
A.K. Prajapati, R. Prasad, Order reduction of linear dynamic systems by improved Routh approximation method. IETE J. Res (2018). https://doi.org/10.1080/03772063.2018.1452645
A.K. Prajapati, R. Prasad, Reduced order modelling of LTI systems by using Routh approximation and factor division methods. Circuits Syst. Signal Process. 38(7), 3340–3355 (2019)
A.K. Prajapati, R. Prasad, Reduced order modelling of linear time invariant systems by using improved modal method. Int. J. Pure Appl. Math. 119(12), 13011–13023 (2018)
A.K. Prajapati, R. Prasad, Reduced order modelling of linear time invariant systems using the factor division method to allow retention of dominant modes. IETE Tech. Rev. (2018). https://doi.org/10.1080/02564602.2018.1503567
R. Prasad, Analysis and design of control systems using reduced order models. Ph.D. Thesis, University of Roorkee, Roorkee, India, 1989
R. Prasad, Padé type model order reduction for multivariable systems using Routh approximation. Comput. Electr. Eng. 26(6), 445–459 (2000)
M.G. Safonov, R.Y. Chiang, A Schur method for balanced-truncation model reduction. IEEE Trans. Autom. Control 34(7), 729–733 (1989)
V.R. Saksena, J.O. Reillly, P.V. Kokotovik, Singular perturbations and time scale methods in control theory: survey. Automatica 20, 273–293 (1988)
Y. Sato, T. Shimotani, H. Igarashi, Synthesis of Cauer-equivalent circuit based on model order reduction considering nonlinear magnetic property. IEEE Trans. Magnet. 53(6), 1–4 (2017)
Y. Shamash, Model reduction using the Routh stability criterion and the Padé approximation technique. Int. J. Control 21(3), 475–484 (1975)
Y. Shamash, Stable reduced-order models using Padé-type approximations. IEEE Trans. Autom. Control 19, 615–616 (1974)
Y. Shamash, Truncation method of reduction: a viable alternative. Electron. Lett. 17(2), 97–98 (1981)
A. Sikander, R. Prasad, A new technique for reduced-order modelling of linear time-invariant system. IETE J. Res. 63(3), 316–324 (2017)
A. Sikander, R. Prasad, Linear time-invariant system reduction using a mixed methods approach. Appl. Math. Model. 39, 4848–4858 (2015)
N. Singh, R. Prasad, H.O. Gupta, Reduction of linear dynamic systems using Routh Hurwitz array and factor division method. IETE J. Educ. 47(1), 25–29 (2006)
J. Singh, C.B. Vishwakarma, K. Chattterjee, Biased reduction method by combining improved modified pole clustering and improved Padé approximations. Appl. Math. Model. 40, 1418–1426 (2016)
A.K. Sinha, J. Pal, Simulation based reduced order modelling using a clustering technique. Comput. Electr. Eng. 16(3), 159–169 (1990)
D.R. Towill, Transfer function techniques for control engineers (Illiffebooks ltd., London, 1970)
C.B. Vishwakarma, Order reduction using modified pole clustering and Padé approximations. Int. J. Electr. Comput. Energ. Electron. Commun. Eng. 5(8), 998–1002 (2011)
C.B. Vishwakarma, R. Prasad, Clustering method for reducing order of linear system using Padé approximation. IETE J. Res. 54(5), 326–330 (2008)
C.B. Vishwakarma, R. Prasad, MIMO system reduction using modified pole clustering and genetic algorithm. Model. Simul. Eng. 2009(1), 1–5 (2009)
C.B. Vishwakarma, R. Prasad, Time domain model order reduction using Hankel matrix approach. J. Frankl. Inst. 351(6), 3445–3456 (2014)
B. Wang, D. Zhang, J. Cheng, J.H. Park, Fuzzy model-based nonfragile control of switched discrete-time systems. Nonlinear Dyn. 93(4), 2461–2471 (2018)
V. Zakian, Simplification of linear time invariant systems by moment approximations. Int. J. Control 18, 455–460 (1973)
Z. Zhang, Y. Zheng, X. Xiao, W. Yan, Improved order-reduction method for cooperative tracking control of time-delayed multi-spacecraft network. J. Frankl. Inst. 355, 2849–2873 (2018)
Y. Zhu, L. Zhang, V. Sreeram, W. Shammakh, B. Ahmad, Resilient model approximation for Markov jump time-delay systems via reduced model with hierarchical Markov chains. Int. J. Syst. Sci. 47(14), 3496–3507 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Prajapati, A.K., Prasad, R. A New Model Reduction Method for the Linear Dynamic Systems and Its Application for the Design of Compensator. Circuits Syst Signal Process 39, 2328–2348 (2020). https://doi.org/10.1007/s00034-019-01264-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-019-01264-1