Your browser does not support JavaScript!
http://iet.metastore.ingenta.com
1887

access icon free Robust pole placement for second-order linear systems using velocity-plus-acceleration feedback

  • Author(s): Taha Helmy S. Abdelaziz 1, 2
    • View affiliations
    • Affiliations: 1: Department of Mechanical Engineering, Faculty of Engineering, Helwan University, 1 Sherif Street, 11792 Helwan, Cairo, Egypt;
      2: Department of Industrial Engineering, Faculty of Engineering, Northern Border University, Arar, KSA
  • Source: Volume 7, Issue 14, 19 September 2013, p. 1843 – 1856
    DOI:  10.1049/iet-cta.2013.0039 , Print ISSN 1751-8644, Online ISSN 1751-8652
© The Institution of Engineering and Technology
Received 19/05/2012, Accepted 10/06/2013, Revised 05/06/2013, Published

This paper considers the robust pole assignment problem using combined velocity and acceleration feedbacks for matrix second-order linear systems. The necessary and sufficient conditions which ensure solvability are derived. Based on recently developed parametric solution to the eigenstructure assignment problem by velocity-plus-acceleration feedback, a new technique is described to perform robust pole placement for second-order systems. The available degrees-of-freedom offered by the velocity-plus-acceleration feedback in selecting the associated eigenvectors are utilised to improve robustness of the closed-loop system. The main advantage of the described approach is that the problem is tackled in the second-order form without transformation into the first-order form. Finally, several examples are introduced to demonstrate the effectiveness of the proposed approach.

Inspec keywords: feedback; pole assignment; robust control; closed loop systems; eigenstructure assignment; linear systems; matrix algebra

Other keywords: necessary and sufficient condition; closed loop system; velocity feedback; robust pole placement problem; acceleration feedback; degree of freedom; eigenvectors; matrix second-order linear system; eigenstructure assignment problem

Subjects: Linear algebra (numerical analysis); Specific control systems; Control system analysis and synthesis methods

References

    1. 1)
      • 37. Abdelaziz, T.H.S.: ‘Robust pole assignment for linear time-invariant systems using state-derivative feedback’, Proc. IMechE, I, J. Syst. Control Eng., 2009, 223, (3), pp. 187–199, DOI: 10.1243/09596518JSCE632.
    2. 2)
      • 2. Laub, A.J., Arnold, W.F.: ‘Controllability and observability criteria for multivariable linear second-order models’, IEEE Trans. Autom. Control, 1984, 29, (2), pp. 163–165 (doi: 10.1109/TAC.1984.1103470).
    3. 3)
      • 9. Chan, H.C., Lam, J., Ho, D.W.C.: ‘Robust eigenvalue assignment in second-order systems: a gradient flow approach’, Optim. Control Appl. Methods, 1997, 18, pp. 283–296 (doi: 10.1002/(SICI)1099-1514(199707/08)18:4<283::AID-OCA603>3.0.CO;2-Q).
    4. 4)
      • 26. Necsulescu, D.S., De Carufel, J., De Wit, C.C.: ‘Investigation on the efficiency of acceleration feedback in servomechanism with friction’, Dyn. Control, 1997, 7, pp. 377–397 (doi: 10.1023/A:1008224527981).
    5. 5)
      • 28. Han, J.D., Wang, Y.C., Tan, D.L., Xu, W.L.: ‘Acceleration feedback control for direct-drive motor system’. Proc. 2000 IEEE/RSJ Int. Conf. Intelligent Robots and Systems (IROS 2000), October 2000, pp. 1068–1074.
    6. 6)
      • 33. Reithmeier, E., Leitmann, G.: ‘Robust vibration control of dynamical systems based on the derivative of the state’, Arch. Appl. Mech., 2003, 72, pp. 856–864.
    7. 7)
      • 24. Schmidt, P.B., Lorenz, R.D.: ‘Design principles and implementation of acceleration feedback to improve performance of dc drives’, IEEE Trans. Ind. Appl., 1992, 28, (3), pp. 594–599, (doi: 10.1109/28.137444).
    8. 8)
      • 43. Lam, J., Yan, W.Y.: ‘Pole assignment with optimal spectral conditioning’, Syst. Control Lett., 1997, 29, (5), pp. 241–253 (doi: 10.1016/S0167-6911(97)90007-4).
    9. 9)
      • 41. Lam, J., Yan, W.Y.: ‘A gradient flow approach to the robust pole placement problem’, Int. J. Nonlinear Robust Control, 1995, 5, (3), pp. 175–185 (doi: 10.1002/rnc.4590050303).
    10. 10)
      • 38. Abdelaziz, T.H.S.: ‘Optimal control using derivative feedback for linear systems’, Proc. IMechE, I, J. Syst. Control Eng., 2010, 224, (2), pp. 185–202, DOI: 10.1243/09596518JSCE886.
    11. 11)
      • 6. Chu, E.K.: ‘Pole assignment for second-order systems’, Mech. Syst. Signal Process., 2002, 16, (1), pp. 39–59, (doi: 10.1006/mssp.2001.1439).
    12. 12)
      • 23. Kwak, S.-K., Washington, G., Yedavalli, R.K.: ‘Acceleration-based vibration control of distributed parameter systems using the ‘reciprocal state-space framework’’, J. Sound Vib., 2002, 251, (3), pp. 543–557 (doi: 10.1006/jsvi.2001.3842).
    13. 13)
      • 15. Bai, Z.J., Datta, B.N., Wang, J.: ‘Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems: a new optimization approach’, Mech. Syst. Signal Process., 2010, 24, pp. 766–783 (doi: 10.1016/j.ymssp.2009.09.014).
    14. 14)
      • 8. Chu, E.K., Datta, B.N.: ‘Numerically robust pole assignment for second-order systems’, Int. J. Control, 1996, 64, (4), pp. 1113–1127 (doi: 10.1080/00207179608921677).
    15. 15)
      • 11. Tehrani, M.G., Mottershead, J.E., Shenton, A.T., Ram, Y.M.: ‘Robust pole placement in structures by the method of receptances’, Mech. Syst. Signal Process., 2011, 25, (1), pp. 112–122, (doi: 10.1016/j.ymssp.2010.04.005).
    16. 16)
      • 44. Zhou, B., Duan, G.-R., Li, Z.-Y.: ‘A Stein matrix equation approach for computing coprime matrix fraction description’, IET Control Theory Appl., 2009, 3, (6), pp. 691–700, (doi: 10.1049/iet-cta.2008.0128).
    17. 17)
      • 40. Kautsky, J., Nichols, N.K., Van Dooren, P.: ‘Robust pole assignment in linear state feedback’, Int. J. Control, 1985, 41, (5), pp. 1129–1155 (doi: 10.1080/0020718508961188).
    18. 18)
      • 17. Xu, S., Qian, J.: ‘Orthogonal basis selection method for robust partial eigenvale assignment problem in second-order control systems’, J. Sound Vib., 2008, 317, (1), pp. 1–19 (doi: 10.1016/j.jsv.2008.03.002).
    19. 19)
      • 12. Inman, D.J., Kress, A.: ‘Eigenstructure assignment using inverse eigenvalue methods’, J. Guid. Control Dyn., 1995, 18, pp. 625–627 (doi: 10.2514/3.21433).
    20. 20)
      • 35. Abdelaziz, T.H.S., Valášek, M.: ‘Direct algorithm for pole placement by state-derivative feedback for multi-input linear systems – nonsingular case’, Kybernetika, 2005, 41, (5), pp. 637–660.
    21. 21)
      • 25. De Jager, B.: ‘Acceleration assisted tracking control’, IEEE Control Syst., 1994, 14, (5), pp. 20–27, (doi: 10.1109/37.320883).
    22. 22)
      • 46. Zhang, B.: ‘Parametric eigenstructure assignment by state feedback in descriptor systems’, IET Control Theory Appl., 2008, 2, (4), pp. 303–309, (doi: 10.1049/iet-cta:20070337).
    23. 23)
      • 19. Zhang, J.F.: ‘Optimal control for mechanical vibration systems based on second-order matrix equations’, Mech. Syst. Signal Process., 2002, 16, (1), pp. 61–67, (doi: 10.1006/mssp.2001.1441).
    24. 24)
      • 3. Diwekar, A.M., Yedavalli, R.K.: ‘Stability of matrix second-order systems: new conditions and perspectives’, IEEE Trans. Autom. Control, 1999, 44, (9), pp. 1773–1777 (doi: 10.1109/9.788551).
    25. 25)
      • 36. Abdelaziz, T.H.S.: ‘Pole assignment by state-derivative feedback for single-input linear systems’, Proc. IMechE, I, J. Syst. Control Eng., 2007, 221, (7), pp. 991–1000, DOI: 10.1243/09596518JSCE409.
    26. 26)
      • 42. Tits, L., Yang, Y.: ‘Globally convergent algorithms for robust pole assignment by state feedback’, IEEE Trans. Autom. Control, 1996, 41, (10), pp. 1432–1452 (doi: 10.1109/9.539425).
    27. 27)
      • 5. Ram, Y.M., Elhay, S.: ‘Pole assignment in vibratory systems by multi-input control’, J. Sound Vib., 2000, 230, (2), pp. 309–321 (doi: 10.1006/jsvi.1999.2622).
    28. 28)
      • 18. Ram, Y.M., Inman, D.J.: ‘Optimal control for vibrating systems’, Mech. Syst. Signal Process., 1999, 13, (6), pp. 879–892 (doi: 10.1006/mssp.1999.1239).
    29. 29)
      • 45. Zhou, B., Li, Z.-Y., Duan, G.-R., Wang, Y.: ‘Optimal pole assignment for discrete-time systems via Stein equations’, IET Control Theory Appl., 2009, 3, (8), pp. 983–994, (doi: 10.1049/iet-cta.2007.0371).
    30. 30)
      • 27. Deur, J., Peric, N.: ‘A comparative study of servosystems with acceleration feedback’. Proc. Industry Applications Conf., October 2000, vol. 3, pp. 1533–1540.
    31. 31)
      • 16. Qian, J., Xu, S.: ‘Robust partial eigenvalue assignment problem for the second-order system’, J. Sound Vib., 2005, 282, pp. 937–948 (doi: 10.1016/j.jsv.2004.03.046).
    32. 32)
      • 31. Zhu, W.-H., Tryggvason, B., Piedboeuf, J.-C.: ‘On active acceleration control of vibration isolation systems’, Control Eng. Pract., 2006, 14, pp. 863–873 (doi: 10.1016/j.conengprac.2005.04.016).
    33. 33)
      • 34. Abdelaziz, T.H.S., Valášek, M.: ‘Pole-placement for SISO linear systems by state-derivative feedback’, IEE Proc. D, Control Theory Appl., 2004, 151, (4), pp. 377–385, (doi: 10.1049/ip-cta:20040660).
    34. 34)
      • 20. Yang, J.N., Li, Z., Liu, S.C.: ‘Instantaneous optimal control with acceleration and velocity feedback’, Probabilistic Eng. Mech., 1991, 6, pp. 204–211 (doi: 10.1016/0266-8920(91)90011-R).
    35. 35)
      • 4. Nichols, N.K., Kautsky, J.: ‘Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular case’, SIAM J. Matrix Anal. Appl., 2001, 23, (1), pp. 77–102 (doi: 10.1137/S0895479899362867).
    36. 36)
      • 7. Ouyang, H.: ‘A hybrid control approach for pole assignment to second-order asymmetric systems’, Mech. Syst. Signal Process., 2011, 25, (1), pp. 123–132, (doi: 10.1016/j.ymssp.2010.07.020).
    37. 37)
      • 14. Abdelaziz, T.H.S., Valášek, M.: ‘Eigenstructure assignment by proportional-plus-derivative feedback for second-order linear control systems’, Kybernetika, 2005, 41, (5), pp. 661–676.
    38. 38)
      • 32. Dyke, S.J., Spencer, B.F., Quast, P., Sain, M.K., Kaspari, D.C., Soong, T.T.: ‘Acceleration feedback control of MDOF structures’, J. Eng. Mech., 1996, 122, pp. 907–917 (doi: 10.1061/(ASCE)0733-9399(1996)122:9(907)).
    39. 39)
      • 21. Rofooei, F.R., Tadjbakhsh, I.G.: ‘Optimal control of structures with acceleration, velocity, and displacement feedback’, J. Eng. Mech., 1993, 119, (10), pp. 1993–2010 (doi: 10.1061/(ASCE)0733-9399(1993)119:10(1993)).
    40. 40)
      • 13. Kim, Y., Kim, H.S., Junkins, J.L.: ‘Eigenstructure assignment algorithm for mechanical second-order systems’, J. Guid. Control Dyn., 1999, 22, (5), pp. 729–731 (doi: 10.2514/2.4444).
    41. 41)
      • 29. Liu, K., Ma, J.: ‘Improving the servo performance of mechatronic systems by acceleration feedback’. Proc. Seventh Int. Conf. Machine Learning and Cybernetics, Kunming, July 2008, vol. 4, pp. 2004–2009.
    42. 42)
      • 1. Abdelaziz, T.H.S.: ‘Eigenstructure assignment for second-order systems using velocity-plus-acceleration feedback’, Struct. Control Health Monit., 2013, 20, (4), pp. 465–482, (doi: 10.1002/stc.503).
    43. 43)
      • 10. Henrion, D., Sebek, M., Kučera, V.: ‘Robust pole placement for second-order systems an LMI approach’, Kybernetika, 2005, 41, (1), pp. 1–14.
    44. 44)
      • 39. Abdelaziz, T.H.S.: ‘Parametric eigenstructure assignment using state-derivative feedback for linear systems’, J. Vib. Control, 2012, 18, (12), pp. 1809–1827, (doi: 10.1177/1077546311423549).
    45. 45)
      • 22. Preumont, A.: ‘Vibration control of active structures: an introduction’, (Kluwer Academic Publishers, The Netherlands, 2002, 2nd edn.).
    46. 46)
      • 30. Shin, C., Hong, C., Jeong, W.B.: ‘Active vibration control of beam structures using acceleration feedback control with piezoceramic actuators’, J. Sound Vib., 2012, 331, pp. 1257–1269 (doi: 10.1016/j.jsv.2011.11.004).
http://iet.metastore.ingenta.com/content/journals/10.1049/iet-cta.2013.0039
Loading

Related content

content/journals/10.1049/iet-cta.2013.0039
pub_keyword,iet_inspecKeyword,pub_concept
6
6
Loading
Print this page
This is a required field
Please enter a valid email address