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Iterative learning control for linear generalized distributed parameter system

  • Emergence in Human-like Intelligence towards Cyber-Physical Systems
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Abstract

In this paper, we use the iterative learning control algorithm to deal with generalized distributed parameter system with parabolic type which described by generalized partial differential equation. Because of the particularity of the generalized system, we usually need generalized value decomposition, but the algorithm we proposed in this paper does not need to consider the impulsive solution of the generalized system, so it can simplify the calculation process. A novel generalized theoretical result is presented by using the PD-type learning law under some assumptions. The convergence conditions of algorithm are established. By the basic generalized theory, matrix theory and mapping principle, the paper gives rigorous convergence proof of the algorithm to ensure the tracking error is convergent in L2 norm. Finally, we use an example to verify the validity of the new proposed algorithm. Through this paper, we can do further study and discuss the iterative learning algorithm for generalized distributer parameter system in the future.

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References

  1. Uchiyama M (1978) Formation of high-speed motion pattern of mechanical arm by trial. Trans Soc Instrum Control Eng 19:706–712

    Article  Google Scholar 

  2. Arimoto S, Kawamura S, Miyazaki F (1984) Bettering operation of robots by learning. J Robot Syst 1(2P):123–140

    Article  Google Scholar 

  3. Zhang J, Dai X, Tian S (2015) Iterative learning control for distributed parameter switched systems. In: Control and decision conference. IEEE, pp 105–1110

  4. Lan YH, Liu X (2016) Convergence analysis of second-order P-type iterative learning control of fractional order linear systems. Control Eng China

    MathSciNet  Google Scholar 

  5. Xie S, Tian S, Fu Y (2004) A new algorithm of iterative learning control with forgetting factors. In: Control, automation, robotics and vision conference, 2004. Icarcv 2004, vol 1. IEEE, pp. 625–630

  6. Bouakrif F (2011) Iterative learning control for strictly unknown nonlinear systems subject to external disturbances. Int J Control Autom Syst 9(4):642–648

    Google Scholar 

  7. Wang H, Dong J, Wang Y (2015) High-order feedback iterative learning control algorithm with forgetting factor. Math Probl Eng 2015(3):1–7

    MathSciNet  MATH  Google Scholar 

  8. Lan T, Yan F, Lin H (2017) Iterative learning control with forgetting factor for urban road network. J Control Sci Eng 2017(4):1–7

    Article  MathSciNet  MATH  Google Scholar 

  9. Ge ZQ, Feng DX (2014) Well-posed problem of nonlinear time varying generalized distributed parameter systems. Sci China: Inf Sci 44(12):1277–1298

    Google Scholar 

  10. Ge ZQ (2013) Solvability of a time-varying generalized distributed parameter system in Banach space. Sci China: Inf Sci 43(3):386–406

    Google Scholar 

  11. Vuong J, Simeon B (2014) On finite element method–flux corrected transport stabilization for advection–diffusion problems in a partial differential–algebraic framework. J Comput Appl Math 262:115–126

    Article  MathSciNet  MATH  Google Scholar 

  12. Yang JH, Liu YQ (2000) Sliding mode control of generalized distributed parameter perturbation system (in Chinese). Control Decis 115(2):145–148

    Google Scholar 

  13. Demetriou MA (2013) Synchronization and consensus controllers for a class of parabolic distributed parameter systems. Syst Control Lett 62:70–76

    Article  MathSciNet  MATH  Google Scholar 

  14. Wu HN, Wang HD, Guo L (2016) Finite dimensional disturbance observer based control for nonlinear parabolic PDE systems via output feedback. J Process Control 48(2):25–50

    Article  Google Scholar 

  15. Huang DQ, Li XF, Xu JX, Xu C, He W (2014) Iterative learning control of inhomogeneous distributed parameter systems–frequency domain design and analysis. Syst Control Lett 72(2):22–29

    Article  MathSciNet  MATH  Google Scholar 

  16. Xiao TF, Li HX (2017) Eigenspectrum-based iterative learning control for a class of distributed parameter systems. IEEE Trans Autom Control 62(2):834–836

    Article  MathSciNet  Google Scholar 

  17. Dai XS, Tian SP, Peng YJ, Luo WG (2014) Closed-loop P-type iterative learning control of uncertain linear distributed parameter systems. IEEE/CAA J Sin Autom 1(3):267–273

    Article  Google Scholar 

  18. Dai XS, Xu C, Tian SP, Li ZL (2016) Iterative learning control for MIMO second-order hyperbolic distributed parameter systems with uncertainties. Adv Differ Equ 1:94–106

    Article  MathSciNet  MATH  Google Scholar 

  19. Piao FX, Zhang QL, Wang ZF (2007) Iterative learning control for a class of generalized systems. Acta Autom Sin 33(6):658–659

    Google Scholar 

  20. Tian SP, Liu Q, Dai XS, Zhang JX (2016) A PD-type iterative learning control algorithm for generalized discrete systems. Adv Differ Equ 1:321–329

    Article  MATH  Google Scholar 

  21. Kaczorek T (1987) Generalized Rosser model and reduction to its canonical form. Bull Pol Acad Sci Tech Sci 35:645–652

    MathSciNet  MATH  Google Scholar 

  22. Kaczorek T (1988) The generalized general model of 2-D systems and its solution. IEEE Trans Autom Control 33:1060–1061

    Article  MATH  Google Scholar 

  23. Lewis FL (1987) Recent work in generalized systems. In: Proceedings of the International Symposium on generalized systems. Atlants, GA, pp 20–24

  24. Lewis FL, Mertzios BG (1992) On the analysis of two-dimensional discrete singular systems. Birkhauser Boston Inc

  25. Kaczorek T (1989) Decomposition of 2-d linear systems into 1-d systems in the frequency domain. IEEE Trans Autom Control 34(6):639–641

    Article  MathSciNet  MATH  Google Scholar 

  26. Joder L (1990) Computing accurate solutions for coupled systems of second order partial differential equations. Int J Comput Math 37:201–212

    Article  Google Scholar 

  27. Yuan Y, Ge Z (2010) Exponential stability of the first order generalized distributed parameter systems. Syst Sci Math 30(3):315–322

    MATH  Google Scholar 

  28. Zhang Y, Li Y, Su J (2016) Singular distributed parameter system iterative learning control with forgetting factor with time-delay. Int J u- e- Serv Sci Technol 9(7):1–8

    Article  Google Scholar 

  29. Liu Q, Yang H-J (1999) Variable structure control method of structural stability for generalized distributed parameter system. J South China Univ Technol (Nat Sci Edn) 4:9–12

    Google Scholar 

  30. Yang J, Wang HO (2004) Dynamic output variable structure control of generalized distributed parameter system. In: Control, automation, robotics and vision conference, 2004. ICARCV 2004, vol 2. IEEE, pp 1217–1221

  31. Liu F, Shi G, Luo Y (2010) An equivalent proposition for stability by feedback of the second order generalized distributed parameter system. In: International conference on advanced computer theory and engineering. IEEE, V3-71–V3-74

  32. Jiang YS, Zhang QL (2016) State feedback control on generalized distributed parameter system with parabolic-elliptic type. In: Proceedings of the 35th Chinese control and decision conference. pp 261–265

  33. Luo B, Wu HN, Li HX (2015) Adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro-dynamic programming. IEEE Trans. Neural Netw Learn Syst 26(4):684–696

    Article  MathSciNet  Google Scholar 

  34. Ge ZQ, Feng DX (2013) Well-posed problem of nonlinear generalized distributed parameter systems and nonlinear GE-semigroup. Sci China Inf Sci 56(12):1–14

    Article  MATH  Google Scholar 

  35. Qin FU, Panpan GU, Xiangdong LI et al (2017) Iterative learning control approach for consensus of multi-agent systems with regular linear dynamics. Sci China 60(7):079202

    Google Scholar 

  36. Qin FU, Panpan GU, Jianrong WU (2017) Iterative learning control for one-dimensional fourth order distributed parameter systems. Sci China 60(1):012204

    Google Scholar 

  37. Qin FU, Weiguo GU, Panpan GU et al (2016) Feedback control for a class of second order hyperbolic distributed parameter systems. Sci China 59(9):092206

    Google Scholar 

  38. Gu W, Fu Q, Wu J (2016) Iterative learning control for quasi-one-sided Lipschitz non-linear systems. IMA J Math Control Inf 34(3):821–830

    MathSciNet  MATH  Google Scholar 

  39. Gu P, Fu Q, Li X (2016) Iterative learning control for switched linear parabolic systems in space W1, 2. In: Control and decision conference. IEEE

  40. Wu J, Fu Q, Li Z (2015) H∞ control via state observer feedback for the T–S fuzzy generalized system. Int J Mach Learn Cybernet 8(2):1–8

    Google Scholar 

  41. Chen C, Liu Z, Zhang Y et al (2017) Coordinated motion/force control of multiarm robot with unknown sensor nonlinearity and manipulated object’s uncertainty. IEEE Trans Syst Man Cybernet Syst 47(7):1123–1134

    Article  Google Scholar 

  42. Chen C, Liu Z, Zhang Y et al (2016) Adaptive control of MIMO mechanical systems with unknown actuator nonlinearities based on the Nussbaum gain approach. Automation 3(1):26–34

    Article  MathSciNet  Google Scholar 

  43. Lai G, Liu Z, Zhang Y et al (2017) Adaptive inversion-based fuzzy compensation control of uncertain pure-feedback systems with asymmetric actuator backlash. IEEE Trans Fuzzy Syst 25(1):141–155

    Article  Google Scholar 

  44. Trzaska Z, Marszalek W (1993) Singular distributed parameter systems. IEE Proc D Control Theory Appl 140(5):305–308

    Article  MATH  Google Scholar 

  45. Sauer N, Singleton JE (1989) Evolution operators in empathy with a semigroup. Semigroup Forum 39:85–94

    Article  MathSciNet  MATH  Google Scholar 

  46. Macionis J (1985) Solvability of a degenerate differential equation with spectral parameter. Lith Math J 25(2):138–142

    MathSciNet  Google Scholar 

  47. Ge ZQ, Zhu GT, Feng DX (2008) Degenerate semi-group methods for the exponential stability of the first order generalized distributed parameter systems. J Syst Sci Complex 21(2):260–266

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The work was supported by the Hechi University Foundation (XJ2016ZD004), Hechi University Youth teacher Foundation (XJ2017QN08), the Projection of Environment Master Foundation (2017HJA001, 2017HJB001), the important Project of the New Century Teaching Reform Project in Guangxi (2010JGZ033) and Guangxi Youth teacher Foundation (2018KY0495).

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Authors and Affiliations

  1. Aeronautics and Astronautics Engineering Institute, Air Force Engineering University, Xi’an, 710038, Shanxi, China

    Yingjun Zhang & Yinghui Li

  2. School of Physics and Electrical Engineering, Hechi University, Yizhou, 546300, Guangxi, China

    Yingjun Zhang & Mengji Chen

Authors
  1. Yingjun Zhang
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  2. Yinghui Li
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  3. Mengji Chen
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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Yinghui Li is the corresponding author.

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Correspondence to Yinghui Li.

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Zhang, Y., Li, Y. & Chen, M. Iterative learning control for linear generalized distributed parameter system. Neural Comput & Applic 31, 4503–4512 (2019). https://doi.org/10.1007/s00521-018-3835-0

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