Skip to main content
SpringerLink
Log in

Newton iterative algorithm based modeling and proportional derivative controller design for second-order systems

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper proposes an identification method for estimating the parameters of a stable second-order system based on the impulse response experiment. From the impulse response experiment, the measured data are collected for implementing parameter estimation. By defining and minimizing a cost function, a Newton iterative algorithm is derived for estimating the parameters of the system. The multi-point identification method is used to show the effectiveness of the proposed Newton iterative algorithm. The results show that the estimated model by the proposed Newton iterative estimation method has higher accuracy. Based on the estimated model, a design method of the proportional derivative controller is presented according to the system dynamical performance. The simulation test shows that the proposed controller design method can meet the desired dynamic specifications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Rosa, P.V., Josep, M.F.L., Urbano, L., Javier, C.: Parameter identification method for a three-dimensional footCground contact model. Mech. Mach. Theory 75(5), 107–116 (2014)

    Google Scholar 

  2. Liu, C.H.: Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation. Appl. Math. Model. 37(10–11), 6899–6908 (2013)

    Article  MathSciNet  Google Scholar 

  3. Upadhyay, R.K., Agrawal, R.: Modeling the effect of mutual interference in a delay-induced predator-prey system. J. Appl. Math. Comput. 1–27 (2014). doi:10.1007/s12190-014-0822-1

  4. Li, C.H., Feng, E.M.: Existence of optimal solution and optimality condition for parameter identification of an ecological species system. J. Appl. Math. Comput. 18(1–2), 273–286 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ding, F.: System Identification-Performances Analysis for Identification Methods. Science Press, Beijing (2014)

    Google Scholar 

  6. Shi, Y., Fang, H.: Kalman filter based identification for systems with randomly missing measurements in a network environment. Int. J. Control 83(3), 538–551 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Zhang, Y.: Unbiased identification of a class of multi-input single-optput systems with correlated disturbances using bias compensation methods. Math. Comput. Model. 53(9–10), 1810–1819 (2011)

    Article  MATH  Google Scholar 

  8. Liu, Y.J., Ding, F., Shi, Y.: An efficient hierarchical identification method for general dual-rate sampled-data systems. Automatica 50(3), 962–970 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  9. Samanta, G.P., Manna, D., Maiti, A.: Bioeconomic modelling of a three-species fishery with switching effect. J. Appl. Math. Comput. 12(1–2), 219–231 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Boutayeb, A., Chetouani, A., Achouyab, A., Twizell, E.H.: A non-linear population model of diabetes mellitus. J. Appl. Math. Comput. 21(1–2), 127–139 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, T., Wang, Q.G., Huang, H.P.: A tutorial review on process identification from step or relay feedback test. J. Proc. Control 23(10), 1597–1623 (2013)

    Article  MathSciNet  Google Scholar 

  12. Ahmed, S., Huang, B., Shah, S.L.: Novel identification method from step response. Control Eng. Pract. 15(5), 545–556 (2007)

    Article  Google Scholar 

  13. Mei, H., Li, S.Y., Cai, W.J., Xiong, Q.: Decentralized closed-loop parameter identification for multivariable processes from step responses. Math. Comput. Simul. 68(2), 171–192 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Huang, H.P., Lee, M.W., Chen, C.L.: A system of procedures for identification of simple models using transient step response. Ind. Eng. Chem. Res. 40(8), 1903–1915 (2001)

    Article  Google Scholar 

  15. Ahemd, S., Huang, B., Shah, S.L.: Identification from step responses with transient initial conditions. J. Process Control 18(2), 121–130 (2008)

    Article  Google Scholar 

  16. Chen, L., Li, J.H., Ding, R.F.: Identification for the second-order systems based on the step response. Math. Comput. Model. 53(5–6), 1074–1083 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shi, Y., Yu, B.: Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Autom. Control 54(7), 1668–1674 (2009)

    Article  MathSciNet  Google Scholar 

  18. Hidayat, E., Medvedev, A.: Laguerre domain identification of continuous linear time-delay systems from impulse response data. Automatica 48(11), 2902–2907 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Deb, A., Sarkar, G., Mandal, P., Biswas, A., Ganguly, A., Biswas, D.: Transfer function identification from impulse response via a new set of orthogonal hybrid functions (HF). Appl. Math. Comput. 218(9), 4760–4787 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu, T., Gao, F.R.: A frequency domain step response identification method for continuous-time processes with time delay. J. Process Control 20(7), 800–809 (2010)

    Article  Google Scholar 

  21. Mei, H., Li, S.Y.: Decentralized identification for multivariable integrating processes with time delays from closed-loop step tests. ISA Trans. 46(2), 189–198 (2007)

    Article  Google Scholar 

  22. Panda, R.C., Vijayan, V., Sujatha, V., Deepa, P., Manamali, D., Mandal, A.B.: Parameter estimation of integrating and time delay processes using single relay feedback test. ISA Trans. 50(4), 529–537 (2011)

    Article  Google Scholar 

  23. Xu, L.: A proportional differential control nethod for a time delay system using the Taylor approximation. Appl. Math. Comput. 236(6), 391–399 (2014)

    Article  MathSciNet  Google Scholar 

  24. Malek, H., Luo, Y., Chen, Y.Q.: Identification and tuning fractional order proportional integral controllers for time delayed systems with a fractional pole. Mechatronics 23(7), 746–754 (2013)

    Article  Google Scholar 

  25. Baran, N., Wozny, G., Arellano-Garcia, H.: Model-based system identification and PI controller tuning using closed-loop set-point response. Comput. Aided Chem. Eng. 31, 755–759 (2012)

    Article  Google Scholar 

  26. Mohideen, K.A., Saravanakumar, G., Valarmathi, K., Devaraj, D., Radhakrishnan, T.K.: Real-coded genetic algorithm for system identification and tuning of a modified model reference adaptive controller for a hybrid tank system. Appl. Math. Model. 37(6), 3829–3847 (2013)

    Article  MathSciNet  Google Scholar 

  27. Vörös, J.: Iterative algorithm for parameter identification of Hammerstein systems with two-segment nonlinearities. IEEE Trans. Autom. Control 44(11), 2145–2149 (1999)

    Article  MATH  Google Scholar 

  28. Liu, M.M., Xiao, Y.S., Ding, R.F.: Iterative identification algorithm for Wiener nonlinear systems using the Newton method Appl. Math. Model. 37(9), 6584–6591 (2013)

    Article  MathSciNet  Google Scholar 

  29. Ramadan, M.A., Naby, M.A.A., Bayoumi, A.M.E.: Iterative algorithm for solving a class of general sylvester-conjugate matrix equation \(\sum _{i = 1}^{s} A_{i}V + \sum _{j = 1}^{t} B_{j}W = \sum _{l = 1}^{m} E_{l}\overline{V}F_{l} + C\). J. Appl. Math. Comput. 44, 99–118 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ding, F., Liu, X.G., Chu, J.: Gradient-based and least-squares-based iterative algorithms for Hammerstein systems using the hierarchical identification principle. IET Control Theory Appl. 7(2), 176–184 (2013)

    Article  MathSciNet  Google Scholar 

  31. Ding, F.: Two-stage least squares based iterative estimation algorithm for CARARMA system modeling. Appl. Math. Model. 37(7), 4798–4808 (2013)

    Article  MathSciNet  Google Scholar 

  32. Yun, B.I.: Iterative methods for solving nonlinear equations with finitely many roots in an interval. J. Comput. Appl. Math. 236(13), 3308–3318 (2013)

    Article  Google Scholar 

  33. Chidume, C.E., Djitté, N.: An iterative method for solving nonlinear integral equations of Hammerstein type. Appl. Math. Comput. 219(10), 5613–5621 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  34. Noor, M.A., Khan, W.A.: New iterative methods for solving nonlinear equation by using homotopy perturbation method. Appl. Math. Comput. 219(8), 3565–3574 (2012)

    Article  MathSciNet  Google Scholar 

  35. Wu, A.G., Lv, L.L., Duan, G.R.: Iterative algorithms for solving a class of complex conjugate and transpose matrix equations. Appl. Math. Comput. 217(21), 8343–8353 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sharma, J.R., Arora, H.: On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222(1), 497–506 (2013)

    Article  MathSciNet  Google Scholar 

  37. Li, J.H.: Parameter estimation for Hammerstein CARARMA systems based on the Newton iteration. Appl. Math. Lett. 26(1), 91–96 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  38. Bruschetta, M., Picci, G., Saccon, A.: A variational integrators approach to second order modeling and identification of linear mechanical systems. Automatica 50(3), 727–736 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  39. Delis, A.I., Nikolos, I.K., Papageorgiou, M.: High-resolution numerical relaxation approximations to second-order macroscopic traffic flow models. Transp. Res. Part C Emerg. Technol. 44(7), 318–349 (2014)

    Article  Google Scholar 

  40. Yuan, K., Li, H.X., Cao, J.D.: Robust stabilization of the distributed parameter system with time delay via fuzzy control. IEEE Trans. Fuzzy Syst. 16(3), 567–584 (2008)

    Article  Google Scholar 

  41. Yuan, K., Cao, J.D., Deng, M.: Exponential stability and periodic solutions of fuzzy cellular neural networks with time-varying delays. Nerocomputing 69(13–15), 1619–1627 (2006)

    Article  Google Scholar 

  42. Sondhi, S., Hote, Y.V.: Fractional order PID controller for load frequency control. Energy Conserv. Manage. 85(9), 343–353 (2014)

    Article  Google Scholar 

  43. Sommer, S., Nguyen, H.N., Kienle, A.: Auto-tuning of multivariable PI/PID controllers using iterative feedback tuning: design examples. Comput. Aided Chem. Eng. 33, 721–726 (2014)

    Article  Google Scholar 

  44. Zhang, R., Wu, S., lu, R., Gao, F.R.: Predictive control optimization based PID control for temperature in an industrial surfactant reactor. Chemometr. Intell. Lab. Syst. 135(15), 48–62 (2014)

    Article  Google Scholar 

  45. Wu, S., Zhang, R., Lu, R.Q., Gao, F.R.: Design of dynamic matrix control based PID for residual oil outlet temperature in a coke furnace. Chemometr. Intell. Lab. Syst. 134(15), 110–117 (2014)

    Article  Google Scholar 

  46. Jeng, J.C., Tseng, W.L., Chiu, M.S.: A one-step tuning method for PID controllers with robustness specification using plant step-response data. Chem. Eng. Res. Des. 92(3), 545–558 (2014)

    Article  Google Scholar 

  47. Tran, H.D., Guan, Z.H., Dang, X.K., Cheng, X.M., Yuan, F.S.: A normalized PID controller in networked control systems with varying time delays. ISA Trans. 52(5), 592–599 (2013)

    Article  Google Scholar 

  48. Vijayan, V., Panda, R.C.: Design of PID controllers in double feedback loops for SISO systems with set-point filters. ISA Trans. 51(4), 514–521 (2012)

    Article  Google Scholar 

  49. Ramasamy, M., Sundaramoorthy, S.: PID controller tuning for desired closed-loop responses for SISO systems using impulse response. Comput. Chem. Eng. 32(8), 1773–1788 (2008)

    Article  Google Scholar 

  50. Chen, Z.H., Yuan, X.H., Ji, B., Wang, P.T., Tian, H.: Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Conserv. Manage. 84(8), 390–404 (2014)

    Article  Google Scholar 

  51. Yuan, K., Alofi, A., Cao J.D., Al-Mazrooei A., Elaiw a.: Synchronization of the coupled distributed parameter system with time delay via proportional spatial derivative control. Discret. Dyn. Nat. Soc. Article ID 418258 2014, DOI:10.1155/2014/418258

  52. Chen, L.: Identification for transfer function models of continuous-time systems, Jiangnan University Master Degree Thesis (2011)

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 21276111), and the Ph.D. Candidate Scientific Research Foundation of Jiangnan University. The authors are grateful to their supervisor Professor Feng Ding at the Jiangnan University (Wuxi, China) for his helpful suggestions and instructions.

Author information

Authors and Affiliations

  1. School of Internet of Things Engineering, Jiangnan University, Wuxi, 214122, People’s Republic of China

    Kai Ji, Ling Xu, Weili Xiong & Lei Chen

  2. School of Internet of Things Technology, Wuxi Institute of Commerce, Wuxi, 214153, People’s Republic of China

    Ling Xu

Authors
  1. Kai Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ling Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Weili Xiong
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ling Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ji, K., Xu, L., Xiong, W. et al. Newton iterative algorithm based modeling and proportional derivative controller design for second-order systems. J. Appl. Math. Comput. 49, 557–572 (2015). https://doi.org/10.1007/s12190-014-0853-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-014-0853-7

Keywords

Mathematical Subject Classification

Search

Navigation