Simultaneous optimization in ultra-precision motion systems

  • Jing Wang
  • Ming ZhangEmail author
  • Yu Zhu
  • Kaiming Yang
  • Xin Li
  • Leijie Wang
Research Paper


In ultra-precision motion systems, vibration has non-negligible influence on motion performance, especially when these systems are getting more lightweight and more flexible. Research on simultaneous optimization of structural and controller design has been conducted to achieve effective vibration control. However, these methods will not have an adequate performance when used in ultra-precision motion systems with a time-varying performance location. In this paper, structural sizes, actuators configuration, and controller parameters are simultaneously optimized. To realize global vibration control and facilitate simultaneous optimization, a new vibration controller with position-dependent control gains is proposed, in which the worst-case vibration magnitude across all considered performance locations is set to be the objective function. To achieve high modeling accuracy, mass and stiffness distribution of actuators is also included into structural dynamics since it plays a large role in structural dynamics. The genetic algorithm is adopted to search for a global optimum. To increase efficiency, R-functions and level-set functions are introduced to translate the complicated over-lapping constraints into a simple integral equality. Neural fitting models instead of the finite element analysis method are used to derive eigenvalues and eigenvectors of the plant. The proposed method is verified on a simplified fine stage in the wafer stage. The numerical results prove its effectiveness.


Simultaneous optimization Time-varying performance location Over-actuation Global vibration control 


Funding information

This work was supported in part by the National Natural Science Foundation of China under Grant 51677104 and 51475262.


  1. Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alujević N, Zhao G, Depraetere B, Sas P, Pluymers B, Desmet W (2014) H2 optimal vibration control using inertial actuators and a comparison with tuned mass dampers. J Sound Vib 333(18):4073–4083CrossRefGoogle Scholar
  3. Balas MJ (1979) Direct velocity feedback control of large space structures. J Guid Control Dyn 2(3):252–253CrossRefGoogle Scholar
  4. Begg DW, Liu X (2000) On simultaneous optimization of smart structures – part ii: algorithms and examples. Comput Methods Appl Mech Eng 184(1):25–37CrossRefGoogle Scholar
  5. Bendsøe MP, Sigmund O (2004) Topology optimization by distribution of isotropic material. In: Topology optimization. Springer, HeidelbergCrossRefGoogle Scholar
  6. Biglar M, Gromada M, Stachowicz F, Trzepieciński T (2015) Optimal configuration of piezoelectric sensors and actuators for active vibration control of a plate using a genetic algorithm. Acta Mech 226(10):3451–3462MathSciNetCrossRefzbMATHGoogle Scholar
  7. Borrvall T, Petersson J (2001) Large-scale topology optimization in 3D using parallel computing. Comput Methods Appl Mech Eng 190(46):6201–6229CrossRefzbMATHGoogle Scholar
  8. Butler H (2011) Position control in lithographic equipment [applications of control]. IEEE Control Syst 31(5):28–47MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cazzulani G, Resta F, Ripamonti F, Zanzi R (2012) Negative derivative feedback for vibration control of flexible structures. Smart Mater Struct 21(21):75024–75033(10)CrossRefGoogle Scholar
  10. Chattopadhyay A, Seeley CE, Jha R (2015) Aeroelastic tailoring using piezoelectric actuation and hybrid optimization. Smart Structures and Materials 1998. Smart Struct Integr Syst 3329(9):83–91Google Scholar
  11. Chhabra D, Bhushan G, Chandna P (2016) Optimal placement of piezoelectric actuators on plate structures for active vibration control via modified control matrix and singular value decomposition approach using modified heuristic genetic algorithm. Mech Adv Mater Struct 23(3):272–280CrossRefGoogle Scholar
  12. Darivandi N, Morris K, Khajepour A (2013) An algorithm for LQ optimal actuator location. Smart Mater Struct 22(3):035001CrossRefGoogle Scholar
  13. Dhingra AK, Lee BH (2010) Multiobjective design of actively controlled structures using a hybrid optimization method. Int J Numer Methods Eng 38(20):3383–3401CrossRefzbMATHGoogle Scholar
  14. Dong HK, Kim D, Chang S, Jung HY (2008) Active control strategy of structures based on lattice type probabilistic neural network. Probab Eng Mech 23(1):45–50CrossRefGoogle Scholar
  15. Dong HK (2009) Neuro-control of fixed offshore structures under earthquake. Eng Struct 31(2):517–522CrossRefGoogle Scholar
  16. Duc ND, Vu NL, Tran DT, Bui HL (2012) A study on the application of hedge algebras to active fuzzy control of a seism-excited structure. J Vib Control 18(14):2186–2200CrossRefGoogle Scholar
  17. Fang JQ, Li QS, Jeary AP (2003) Modified independent modal space control of m.d.o.f. systems. J Sound Vib 261(3):421–441MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hać A, Liu L (1993) Sensor and actuator location in motion control of flexible structures. J Sound Vib 167(2):239–261CrossRefzbMATHGoogle Scholar
  19. Herpen RV, Oomen T, Kikken E, Wal MVD, Aangenent W, Steinbuch M (2014) Exploiting additional actuators and sensors for nano-positioning robust motion control. Am Control Confer 24:619–631Google Scholar
  20. Laro D, Boshuizen R, Van Eijk J (2010) Design and control of over-actuated lightweight 450 mm fine stage. Proceedings of the ASPE control topical meeting. Cambridge, MA, USA, 48:141–144Google Scholar
  21. Lindberg RE, Longman RW (2012) On the number and placement of actuators for independent modal space control. J Guid Control Dyn 7(7):215–221zbMATHGoogle Scholar
  22. Liu P, Rao VS (2002) Active control of smart structures with optimal actuator and sensor locations. Spie’s, international symposium on smart. Struct Mater 4693:1–12Google Scholar
  23. Liu X, Begg DW, Matravers DR (1997) Optimal topology/actuator placement design of structures using sa. J Aerosp Eng 10(10):119–125CrossRefGoogle Scholar
  24. Oomen T, Herpen RV, Quist S, Wal MVD, Bosgra O, Steinbuch M (2012) Next-generation wafer stage motion control: connecting system identification and robust control. Proceedings of the American Control Conference, Montreal, QC, Canada, 2012:2455–2460Google Scholar
  25. Oomen T, Herpen RV, Quist S, Wal MVD, Bosgra O, Steinbuch M (2013) Connecting system identification and robust control for next-generation motion control of a wafer stage. IEEE Trans Control Syst Technol 22(1):102–118CrossRefGoogle Scholar
  26. Ou JS, Kikuchi N (1996) Integrated optimal structural and vibration control design. Structural Optimization 12(4):209–216CrossRefGoogle Scholar
  27. Ronde MJC, Molengraft RVD, Steinbuch M (2012b) Model-based feedforward for inferential motion systems, with application to a prototype lightweight motion system. Am Control Confer 2012:5324–5329Google Scholar
  28. Ronde MJC, Bulk JVD, Molengraft RVD, Steinbuch M (2012a) Feedforward for flexible systems with time-varying performance locations. Am Control Confer 2012:6033–6038Google Scholar
  29. Ronde MJC, Leenknegt GAL, Molengraft RVD, Steinbuch M (2014a) Data-based spatial feedforward for over-actuated motion systems. Mechatronics 24(4):307–317CrossRefGoogle Scholar
  30. Ronde MJC, Schneiders MGE, Kikken E, Molengraft RVD, Steinbuch M (2014b) Model-based spatial feedforward for over-actuated motion systems. Mechatronics 24(4):307–317CrossRefGoogle Scholar
  31. Roover DD (1997) Motion control of a wafer stage: a design approach for speeding up ic production. Mechanical Maritime & Materials Engineering. Ph.D. dissertation, Dept Mech Eng. TU Delft, NL, Delft, The NetherlandsGoogle Scholar
  32. Schmidhuber J (2015) Deep learning in neural networks: an overview. Neural Netw 61:85–117CrossRefGoogle Scholar
  33. Schneiders MGE, Molengraft RVD, Steinbuch M (2004) Modal framework for closed-loop analysis of over-actuated motion systems. Proceedings of 2004 ASME International Mechanical Engineering Congress, Anaheim, California, USA. Hal Arch 53(7)Google Scholar
  34. Shapiro V (1991) Theory of r-functions and applications: a primer. Technical report Cornell UniversityGoogle Scholar
  35. Sluijk B, Castenmiller T, de Jongh RDC, Jasper H, Modderman T (2001) Performance results of a new generation of 300-mm lithography systems. Proceedings of SPIE - The International Society for Optical Engineering, Santa Clara, CA, USA 4346(1):544–557Google Scholar
  36. Sontag ED (1988) Mathematical control theory: deterministic finite dimensional systems. Of Texts Appl Math 40(92):28–30Google Scholar
  37. Veen GVD, Langelaar M, Meulen SVD, Laro D, Aangenent W, Keulen FV (2017) Integrating topology optimization in precision motion system design for optimal closed-loop control performance. Mechatronics 47:1–13CrossRefGoogle Scholar
  38. Wang MY, Wang X, Guo D (2003) Level set method for structural topology optimization. Comput Methods Appl Mech Eng 192:227–246MathSciNetCrossRefzbMATHGoogle Scholar
  39. Xia L, Zhu J, Zhang W (2012) A superelement formulation for the efficient layout design of complex multi-component system. Struct Multidiscip Optim 45(5):643–655MathSciNetCrossRefzbMATHGoogle Scholar
  40. Xia L, Zhu J, Zhang W, Breitkopf P (2013) An implicit model for the integrated optimization of component layout and structure topology. Comput Methods Appl Mech Eng 257(257):87–102MathSciNetCrossRefzbMATHGoogle Scholar
  41. Zhai J, Zhao G, Shang L (2016) Integrated design optimization of structure and vibration control with piezoelectric curved shell actuators. J Intell Mater Syst Struct 27(19):2672–2691Google Scholar
  42. Zhao Y, Zheng S, Wang H, Yang L (2015) Simultaneous optimization of photostrictive actuator locations, numbers and light intensities for structural shape control using hierarchical genetic algorithm. Adv Eng Softw 88(C):21–29CrossRefGoogle Scholar
  43. Zhu J, Zhang W, Beckers P, Chen Y, Guo Z (2008) Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique. Struct Multidiscip Optim 36(1):29–41MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Jing Wang
    • 1
  • Ming Zhang
    • 1
    Email author
  • Yu Zhu
    • 1
  • Kaiming Yang
    • 1
  • Xin Li
    • 1
  • Leijie Wang
    • 1
  1. 1.State Key Laboratory of Tribology and the Beijing Laboratory of Precision/Ultra-Precision Manufacture Equipment and Control, Department of Mechanical EngineeringTsinghua UniversityBeijingChina

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