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Model Reduction of Nonlinear Systems by Trajectory Piecewise Linear Based on Output-Weighting Models: A Balanced-Truncation Methodology

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Abstract

This paper extends some methods in model order reduction (MOR) of nonlinear models by trajectory piecewise-linear models based on output weighting (TPWLOW). The devised methods generalize the balanced-truncation method as the core for the proposed algorithms. These algorithms preserve the stability in some mentioned conditions. An error bound of reduction is calculated for these algorithms. In addition, a Krylov–TBR combinational MOR is also casted which benefits from the advantages of both balanced-truncation and Krylov subspace methods. The results show that in balanced-truncation-based algorithms, the efficiency of the first proposed algorithm fairly exceeds the second one. In addition, it is shown that the Krylov–TBR algorithm can reach the same efficiency despite remarkable lower order, provided that the designer can find a stable subspace. The robustness of TPWLOW-reduced models to linearization point movement is also evaluated, which shows the less sensitivity of the proposed methods in comparison with formerly introduced trajectory piecewise-linear models.

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References

  • Antoulas AC (2005) Approximation of large-scale dynamical systems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Bond BN (2010) Stability-preserving model reduction for linear and nonlinear systems arising in analog circuit applications. PhD thesis, Mass. Inst. Technol

  • Bond BN, Daniel L (2009) Stable reduced models for nonlinear descriptor systems through piecewise-linear approximation and projection. IEEE Trans Comput Aided Des Int Circ Syst 28:1467–1480

    Article  Google Scholar 

  • Casciati S, Faravelli L (2014) Quantity vs quality in the model order reduction (MOR) of a linear system. Smart Struct Syst 13(1):99–109

    Article  Google Scholar 

  • Chen Y (1999) Model order reduction for nonlinear systems. M.S. Thesis, Mass Inst Technol

  • Faravelli L, Casciati F (2017) Model order reduction in nonlinear systems. Adv Sci Technol 101:71–78

    Article  Google Scholar 

  • Ferranti F, Antonini G, Dhaene T, Knockaert L (2011) Passivity—preserving interpolation-based parameterized model order reduction of PEEC models based on scattered grids. Int J Numer Model Electron 24:478–495

    Article  MATH  Google Scholar 

  • Filev D (1991) Fuzzy modeling of complex systems. Int J Approx Reason 5:281–290

    Article  MATH  Google Scholar 

  • Gugercin S, Antoulas AC (2004) A survey of model reduction by balanced truncation and some new results. Int J Control 77:748–766

    Article  MathSciNet  MATH  Google Scholar 

  • Khademi G, Mohammadi H, Dehghani M (2015) Order reduction of linear systems with keeping the minimum phase characteristic of the system: LMI based approach, Iranin. J Sci Technol Electron Eng 39(E2):217–227

    Google Scholar 

  • Kuure-Kinsey M, Bequette BW (2009) Nonlinear model predictive control towards new challenging applications. In: Magni L, Raimondo DM, Allgower F (eds) Lecture Notes in Control and Information Sciences, vol 384. Springer, Berlin, pp 153–165

    Google Scholar 

  • Lohmann B, Eid R (2009) Efficient order reduction of parametric and nonlinear models by superposition of locally reduced models. in Methoden und Anwendungen der Regelungstechnik. Erlangen Munchener Workshops, 2007 und 2008, Shaker 1–20

  • Mohseni SS, Yazdanpanah MJ, Ranjbar Noei A (2016) New strategies in model order reduction of trajectory piecewise-linear models. Int J Numer Model Electron Wily 29(4):707–725

    Article  Google Scholar 

  • Mohseni SS, Yazdanpanah MJ, Ranjbar Noei A (2017) Model order reduction of nonlinear models based on decoupled multi-model via trajectory piecewise linearization. Int J Control Autom Syst 15(5):2088–2098

    Article  Google Scholar 

  • Nahvi SA, Nabi M, Janardhanan S (2012) Adaptive sampling of nonlinear system trajectory for model order reduction. In: Proceedings of IEEE International Conference Model, Identification Control (ICMIC), pp 1314–1320

  • Nahvi SA, Nabi M, Janardhanan S (2013) Nonlinearity-aware sub-model combination in trajectory based methods for nonlinear Mor. Math Comput Simul 94:127–144

    Article  MathSciNet  Google Scholar 

  • Nasiri SH, Hajmohammadi R, Farsangi MM (2015) Model order reduction based on moment matching using Legendre wavelet and harmony search algorithm, Iranin. J Sci Technol Electron Eng 39(E1):39–54

    Google Scholar 

  • Nelles O (1999) Nonlinear system identification with local linear neuro-fuzzy models. Shaker, Herzogenrath

    Google Scholar 

  • Orjuela R, Marx B, Ragot J, Maquin D (2008) State estimation for nonlinear systems using a decoupled multiple mode. Int J Model Identif Control 4(1):59–67

    Article  Google Scholar 

  • Rewienski M. (2003) A trajectory piecewise-linear approach to model order reduction of nonlinear dynamical systems. PhD thesis, Mass Inst Technol

  • Rewienski M, White J (2001) Improving trajectory piecewise-linear approach to nonlinear model order reduction for micromachined devices using an aggregated projection basis. In proceedings of 5th international conference on modelling simulation microsystems, pp 128–131

  • Salimbahrami B (2010) A sequential reduction scheme in reduced order modelling of large scale systems. Tech. Rep. Autom. Control., TRAC-5, pp 1–16

  • Tiwary S, Rutenbar RA (2005) Scalable trajectory methods for on-demand analog macromodel extraction. ACM/IEEE Design Automation Conference, pp 403–408

  • Vasilyev D (2008) Theoretical and practical aspects of linear and nonlinear model order reduction techniques. PhD thesis, Massachusetts Institute of Technology

  • Vasilyev D, Rewienski M, White J (2003) A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and MEMS. Proceedings of ACM/IEEE design automation, conference, pp 490–495

  • Voβ T, Pulch R, ter Maten EJW, El Guennouni A (2007) Trajectory piecewise linear approach for nonlinear differential-algebraic equations in circuit simulation. Sci Comput Electr Eng Math Ind 11:167–173

    MATH  Google Scholar 

  • Yu Z, Yu J, Karimi HR (2014) Model reduction of fuzzy logic systems. Math Probl Eng 2014:874530

    MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Electrical Engineering Department, Islamic Azad University, Nour Branch, Mazandaran, Iran

    Seyed Saleh Mohseni

  2. Control and Intelligent Processing Center of Excellence (CIPCE), School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran

    Mohamad Javad Yazdanpanah

  3. Control Engineering Department, Babol University of Technology, Babol, Iran

    Abolfazl Ranjbar Noei

Authors
  1. Seyed Saleh Mohseni
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  2. Mohamad Javad Yazdanpanah
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  3. Abolfazl Ranjbar Noei
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Correspondence to Seyed Saleh Mohseni.

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Mohseni, S.S., Yazdanpanah, M.J. & Ranjbar Noei, A. Model Reduction of Nonlinear Systems by Trajectory Piecewise Linear Based on Output-Weighting Models: A Balanced-Truncation Methodology. Iran J Sci Technol Trans Electr Eng 42, 195–206 (2018). https://doi.org/10.1007/s40998-018-0058-4

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  • DOI: https://doi.org/10.1007/s40998-018-0058-4

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