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Iterative LMI Approach to Robust State-feedback Control of Polynomial Systems with Bounded Actuators

  • Tanagorn Jennawasin
  • Michihiro Kawanishi
  • Tatsuo Narikiyo
  • David BanjerdpongchaiEmail author
Regular Papers Control Theory and Applications
  • 28 Downloads

Abstract

This paper presents a novel approach to state-feedback stabilization of polynomial systems with bounded actuators. To overcome limitation of the existing approaches, we introduce additional variables that separate the system matrices and the Lyapunov matrices. Therefore, parameterization of the state-feedback controllers is independent of the Lyapunov matrices. The proposed design condition is bilinear in the decision variables, and hence we provide an iterative algorithm to solve the design problem. At each iteration, the design condition is cast as convex optimization using the sum-of-squares technique and can be efficiently solved. In addition, the novel parameter-dependent Lyapunov functions are readily applied to robust state-feedback stabilization of polynomial systems subject to parametric uncertainty. Effectiveness of the proposed approach is demonstrated by numerical examples.

Keywords

Bounded actuators convex optimization parameter-dependent LMI polynomial systems robust state-feedback stabilization sum-of-squares technique 

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References

  1. [1]
    Y. Wu and R. Lu, “Event-based control for network systems via integral quadratic constraints,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 65, no. 4, pp. 1386–1394, 2018.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Y. Wu, R. Lu, P. Shi, H. Su, and Z. G. Wu, “Sampled data synchronization of complex networks with partial couplings and T-S fuzzy nodes,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 2, pp. 782–793, 2018.CrossRefGoogle Scholar
  3. [3]
    I. Ghous and Z. Xiang, “H¥ stabilization of 2-D discrete switched delayed systems represented by the Roesser model subject to actuator saturation,” Transactions of the Institute of Measurement and Control, vol. 37, no. 10, pp. 1242–1253, 2015.CrossRefGoogle Scholar
  4. [4]
    I. Ghous, Z. Xiang, and H. R. Karimi, “State feedback H¥ control for 2-D switched delay systems with actuator saturation in the second FM model,” Circuits, Systems and Signal Processing, vol. 34, no. 7, pp. 2167–2192, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Y. Qian, Z. Xiang, and H. R. Karimi, “Disturbance tolerance and rejection of discrete switched systems with timevarying delay and saturating actuator,” Nonlinear Analysis: Hybrid Systems, vol. 16, pp. 81–92, 2015.MathSciNetzbMATHGoogle Scholar
  6. [6]
    C. W. Scherer and C. W. J. Hol, “Matrix sum-of-squares relaxations for robust semi-definite programs, Mathematical Programming Series B, vol. 107, Nos. 1–2, pp. 189–211, 2006.Google Scholar
  7. [7]
    S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, 1994.CrossRefzbMATHGoogle Scholar
  8. [8]
    C. Albea and F. Gordillo, “Estimating the attraction domain for the boost inverter,” Asian Journal of Control, vol. 15, no. 1, pp. 169–176, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    G. Chesi, “LMI techniques for optimization over polynomials in control: a survey,” IEEE Transactions on Automatic Control, vol. 55, no. 11, pp. 2500–2510, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    P. A. Parrilo, “Semidefinite programming relaxations for semialgebraic problems,” Mathematical Programming Series B, vol. 96, no. 2, pp. 293–320, 2003.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. M. Belhaouane and N. B. Braiek, “Design of stabilizing control for synchronous machines via polynomial modelling and linear matrix inequalities approach,” International Journal of Control, Automation, and Systems, vol. 9, no. 3, pp. 425–436, 2011.CrossRefGoogle Scholar
  12. [12]
    C. Ebenbauer and F. Allgower, “Analysis and design of polynomial control systems using dissipation inequalities and sum of squares,” Journal of Computers and Chemical Engineering, vol. 30, no. 11, pp. 1601–1614, 2006.Google Scholar
  13. [13]
    H. Ichihara, “Optimal control for polynomial systems using matrix sum of squares relaxation,” IEEE Transactions on Automatic Control, vol. 54, no. 5, pp. 1048–1053, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    H. Ichihara, “A Convex approach to state feedback synthesis for polynomial nonlinear systems with input saturation,” SICE Journal of Control, Measurement, and System Integration, vol. 6, no.3, pp. 186–193, 2013.Google Scholar
  15. [15]
    S. Prajna, A. Papachristodoulou, and F. Wu, “Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach,” Proc. of the Asian Control Conference, Melbourne, Australia, 2004. pp. 157–165.Google Scholar
  16. [16]
    T. Jennawasin, M. Kawanishi, and T. Narikiyo, “Stabilization of nonlinear systems with bounded actuators using convex optimization,” Proc. of the 18th IFAC World Congress, Milano, Italy, pp. 6745–6750, August 2011.Google Scholar
  17. [17]
    H. J. Ma and G. H. Yang, “Fault-tolerant control synthesis for a class of nonlinear systems: sum of squares optimization approach,” International Journal of Robust and Nonlinear Control, vol. 19, no. 5, pp. 591–610, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Q. Zheng and F. Wu, “Regional stabilisation of polynomial nonlinear systems using rational Lyapunov functions,” International Journal of Control, vol. 82, no. 9, pp. 1605–1615, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Jennawasin, M. Kawanishi, T. Narikiyo, and C. L. Lin, “An improved stabilizing condition for polynomial systems with bounded actuators: an SOS-based approach,” Proc. of the IEEE Multi-Conference on Systems and Control, Dubrovnik, Croatia, pp. 258–263, October 2012.Google Scholar
  20. [20]
    J. B. Lasserre, “Global optimization with polynomials and the problems of moments,” SIAM Journal on Optimization, vol. 11, no. 3, pp. 796–817, 2001.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Y. Ebihara and T. Hagiwara, “New dilated LMI characterizations for continuous-time multi-objective controller synthesis,” Automatica, vol. 40, no. 11, pp. 2003–2009, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    M. C. de Oliveira, J. C. Geromel, and J. Bernoussou, “Extended H2 and H¥ norm characterizations and controller parameterizations for discrete-time systems,” International Journal of Control, vol. 75, no. 9, pp. 666–679, 2002.MathSciNetCrossRefGoogle Scholar
  23. [23]
    D. Peaucelle, D. Arzelier, O. Bachelier, and J. Bernussou, “A new robusr D-stability condition for real convex polytopic uncertainty,” Systems & Control Letters, vol. 40, no. 1, pp. 21–30, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Sajjadi-Kia and F. Jabbari, “Dilated-matrix inequalities for control design in systems with actuator constraint,” Proc. of 46th IEEE Conference on Decision and Control, New Orleans, USA, pp. 5678–5683, December 2007.Google Scholar
  25. [25]
    T. Jennawasin and D. Banjerdpongchai, “Design of statefeedback control for polynomial systems with quadratic performance criterion and control input constraints,” Systems & Control Letters, vol. 117, pp. 53–59, 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Y. Wei, J. Qiu, H. R. Karimi, and M. Wang, “Model approximation for two-dimensional Markovian jump systems with state-delays and imperfect mode information,” Multidimensional Systems and Signal Processing, vol. 26, no. 3, pp. 575–597, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Y. Wei, J. Qiu, P. Shi, and M. Chadli, “Fixed-order piecewise-affine output feedback controller for fuzzyaffine-model-based nonlinear systems with time-varying delay,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 64, no. 4, pp. 945–958, 2017.CrossRefGoogle Scholar
  28. [28]
    Y. Wei, J. H. Park, J. Qiu, L. Wu, and H. Y. Jung, “Sliding mode control for semi-Markovian jump systems via output feedback,” Automatica, vol. 81, pp. 133–141, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    T. Hu and Z. Lin, Control Systems with Actuator Saturation: Analysis and Design, Birkhauser, 2001.CrossRefzbMATHGoogle Scholar
  30. [30]
    K. Sawada, T. Kiyama, and T. Iwasaki, “Generalized sector synthesis of output feedback control with anti-windup structure,” Systems & Control Letters, vol. 58, no. 6, pp. 421–428, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Tarbouriech, G. Garcia, J. M. Gomes da Silva Jr., and I. Queinnec, Stability and Stabilization of Linear Systems with Saturating Actuators, Springer, 2011.CrossRefzbMATHGoogle Scholar

Copyright information

© ICROS, KIEE and Springer 2019

Authors and Affiliations

  1. 1.Department of Control System and Instrumentation Engineering, Faculty of EngineeringKing Mongkut’s University of Technology ThonburiBangkokThailand
  2. 2.Control System LaboratoryToyota Technological InstituteNagoya 468Japan
  3. 3.Department of Electrical Engineering, Faculty of EngineeringChulalongkorn UniversityBangkokThailand

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