Observer-based dynamic local piecewise control of a linear parabolic PDE using non-collocated local piecewise observation

Jun-Wei Wang; Ya-Qiang Liu; Chang-Yin Sun

Observer-based dynamic local piecewise control of a linear parabolic PDE using non-collocated local piecewise observation

View Fulltext

Author(s): Jun-Wei Wang¹ ; Ya-Qiang Liu¹ ; Chang-Yin Sun²
- Affiliations: 1: Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education School of Automation and Electrical Engineering , University of Science and Technology Beijing , Beijing 100083 , People's Republic of China ;
  2: School of Automation , Southeast University , Nanjing 210096 , People's Republic of China
Source: Volume 12, Issue 3, 13 February 2018, p. 346 – 358
DOI: 10.1049/iet-cta.2017.0797 , Print ISSN 1751-8644, Online ISSN 1751-8652

Received 02/08/2017, Accepted 14/11/2017, Revised 29/10/2017, Published 15/11/2017

This paper employs the observer-based output feedback control technique to study the problem of local piecewise control for a linear parabolic partial differential equation(PDE) using non-collocated local piecewise observation. By the non-collocated local piecewise observation, a Luenberger-type PDE observer is first constructed to exponentially track the state of the PDE in the sense of both $L 2$ norm and $H 1$ norm. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed for exponential stabilisation of the PDE. Sufficient conditions on exponential stabilisation of the PDE by the suggested feedback controller in the sense of both $L 2$ norm and $H 1$ norm are developed by utilising Lyapunov direct method, integration by parts, Wirtinger's inequality, and first mean value theorem for definite integrals, and presented in terms of standard linear matrix inequalities. The well-posedness is also analysed for both open-loop and resulting closed-loop PDE by using $C 0$ -semigroup approach. Finally, numerical simulation results are presented to show the effectiveness of the proposed design method.

References

1. 1)
  - 3. Li, H.-X., Qi, C.: ‘Modeling of distributed parameter systems for applications-A synthesized review from time-space separation’, J. Process Control, 2010, 20, (8), pp. 891–901.
2. 2)
  - 32. Wang, J.-W., Liu, Y.-Q., Hu, Y.-Y., et al: ‘A spatial domain decomposition approach to distributed H∞ observer design of a linear unstable parabolic distributed parameter system with spatially discrete sensors’, Int. J. Control, 2017, 90, (12), pp. 2772–2785.
3. 3)
  - 17. Krstic, M., Smyshlyaev, A.: ‘Boundary control of PDEs-a course on backstepping designs’ (SIAM, PA, 2008).
4. 4)
  - 8. Verica, R.-G., Karagiannis, D., Cheng, M.-B., et al: ‘Recent trends in stabilization and control of distributed parameter dynamic systems’. American Society of Mechanical Engineers (ASME) 2014 Int. Mechanical Engineering Congress and Exposition, 2014, p. V04AT04A004(8).
5. 5)
  - 18. Fridman, E., Blighovsky, A.: ‘Robust sampled-data control of a class of semilinear parabolic systems’, Automatica, 2012, 48, (5), pp. 826–836.
6. 6)
  - 14. Xu, G.-Q., Yung, S.: ‘Stabilization of Timoshenko beam by means of pointwise controls’, ESIAM, Control Optim. Calc. Var., 2003, 9, pp. 579–600.
7. 7)
  - 5. Lasiecka, I.: ‘Control of systems governed by partial differential equations: a historical perspective’. Proc. 34th IEEE Conf. Decision and Control, New Orleans, LA, USA, December 1995, pp. 2792–2796.
8. 8)
  - 28. Apostol, T.: ‘Mathematical analysis’ (Addison-Wesley Publishing Company, Massachusetts, 1974, 2nd edn.).
9. 9)
  - 9. Fridman, E., Orlov, Y.: ‘An LMI approach to H∞ boundary control of semilinear parabolic and hyperbolic systems’, Automatica, 2009, 45, (9), pp. 2060–2066.
10. 10)
  - 27. Hardy, G., Littlewood, J., Polya, G.: ‘Inequalities’ (Cambridge University Press, Cambridge, 1952, 2nd edn.).
11. 11)
  - 10. Wu, H.-N., Wang, J.-W., Li, H.-X.: ‘Fuzzy boundary control design for a class of nonlinear parabolic distributed parameter systems’, IEEE Trans. Fuzzy Syst., 2014, 22, (3), pp. 642–652.
12. 12)
  - 23. Guo, W., Guo, B.-Z.: ‘Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance’, IEEE Trans. Autom. Control, 2013, 58, (7), pp. 1631–1643.
13. 13)
  - 30. Selivanov, A., Fridman, E.: ‘Distributed event-triggered control of diffusion semilinear PDEs’, Automatica, 2016, 68, pp. 344–351.
14. 14)
  - 7. Wang, J.-W., Wu, H.-N., Sun, C.-Y.: ‘Analysis and control of distributed parameter systems for applications: research progress and prospects’, Sci. Found. China, 2013, 21, (2), pp. 49–65.
15. 15)
  - 25. Pazy, A.: ‘Semigroups of linear operators and applications to partial differential equations’ (Springer-Verlag, New York, 1983).
16. 16)
  - 16. Wang, J.-W., Wu, H.-N.: ‘Exponential pointwise stabilization of semi-linear parabolic distributed parameter systems via the Takagi-Sugeno fuzzy PDE model’, IEEE Trans. Fuzzy Syst., 2016, Available online, doi: 10.1109/TFUZZ.2016.2646745 .
17. 17)
  - 21. Guo, B.-Z., Wang, J.-M., Yang, K.-Y.: ‘Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation’, Syst. Control Lett., 2008, 57, (9), pp. 704–749.
18. 18)
  - 13. Wang, J.-W., Wu, H.-N.: ‘Lyapunov-based design of locally collocated controllers for semi-linear parabolic PDE systems’, J. Franklin Inst., 2014, 351, (1), pp. 429–441.
19. 19)
  - 31. Am, N., Fridman, E.: ‘Network-based H∞ filtering of parabolic systems’, Automatica, 2014, 50, pp. 3139–3146.
20. 20)
  - 20. Krstic, M., Guo, B.-Z., Balogh, A., et al: ‘Control of a tip-force destabilized shear beam by observer-based boundary feedback’, SIAM J. Control Optim., 2008, 47, (2), pp. 553–574.
21. 21)
  - 26. Curtain, R., Zwart, H.: ‘An introduction to infinite-dimensional linear systems theory’ (Springer-Verlag, New York, 1995).
22. 22)
  - 6. Padhi, R., Ali, S.k.: ‘An account of chronological developments in control of distributed parameter systems’, Annu. Rev. Control, 2009, 33, (1), pp. 59–68.
23. 23)
  - 11. Demetriou, M.: ‘Guidance of mobile actuator-plus-sensor networks for improved control and estimation of distributed parameter systems’, IEEE Trans. Autom. Control, 2010, 55, (7), pp. 1570–1584.
24. 24)
  - 29. Gahinet, P., Nemirovskii, A., Laub, A., et al: ‘LMI ontrol toolbox for use with matlab’ (The Math.Works Inc., Natick, MA, 1995).
25. 25)
  - 15. Demetriou, M.: ‘Adaptive control of 2-D PDEs using mobile collocated actuator/sensor pairs with augmented vehicle dynamics’, IEEE Trans. Autom. Control, 2012, 57, (12), pp. 2979–2993.
26. 26)
  - 1. Ray, W.: ‘Advanced process control’ (McGraw-Hill, New York, 1981).
27. 27)
  - 12. Fridman, E., Am, N.: ‘Sampled-data distributed H∞ control of transport reaction systems’, SIAM J. Control Optim., 2013, 51, (2), pp. 1500–1527.
28. 28)
  - 2. Christofides, P.: ‘Nonlinear and robust control of PDE systems: methods and applications to transport-reaction processes’ (Birkhäuser, Boston, MA, 2001).
29. 29)
  - 19. Wang, J.-W., Li, H.-X., Wu, H.-N.: ‘A membership-function-dependent approach to fuzzy pointwise state feedback control design for nonlinear parabolic distributed parameter systems with spatially discrete actuators’, IEEE Trans. Syst. Man Cybern. Syst., 2017, 47, (7), pp. 1486–1499.
30. 30)
  - 4. He, W., Ge, S., How, B., et al: ‘Dynamics and control of mechanical systems in offshore engineering’ (Springer-Verlag, London, 2014).
31. 31)
  - 22. Guo, B.-Z., Guo, W.: ‘The strong stabilization of a one-dimensional wave equation by non-collocated dynamic boundary feedback control’, Automatica, 2009, 45, (3), pp. 790–797.
32. 32)
  - 24. Liu, Z., Zheng, S.: ‘Semigroups associated with dissipative systems’ (Chapman and Hall/CRC, Boca Raton, 1999).

Login

Not registered yet?

Share

Tools

Login to add to favourites

Key

Observer-based dynamic local piecewise control of a linear parabolic PDE using non-collocated local piecewise observation

References

Related content