IJPAM: Volume 118, No. 2 (2018)

Title

MEASURABLE DISTURBANCE DECOUPLING FOR
HYBRID LINEAR SYSTEMS WITH PERIODIC STATE JUMPS

Authors

Elena Zattoni
Department of Electrical, Electronic, and Information Engineering ``G. Marconi''
Alma Mater Studiorum $\cdot$ University of Bologna
Viale Risorgimento 2, 40136 Bologna, ITALY

Abstract

This paper is focused on hybrid linear systems subject to periodic state jumps. The measurable disturbance decoupling problem investigated is a synthesis problem with a twofold objective: rendering the system output insensitive to a disturbance accessible for measurement and, at the same time, achieving global asymptotic stability of the compensated hybrid linear dynamics for the jump time sequence with the given period. In order to best exploit the real-time knowledge of the disturbance, a feedforward compensation scheme is adopted, under the assumption that the plant is prestabilized by a suitable feedback. Hence, a necessary and sufficient solvability condition is proven. Such condition is constructive: i.e., the proof outlines the procedure for deriving a hybrid linear feedforward compensator that satisfies the stated requirements.

History

Received: 2018-01-08
Revised: 2018-01-24
Published: March 25, 2018

AMS Classification, Key Words

AMS Subject Classification: 93C30, 93C35, 93D20
Key Words and Phrases: hybrid linear systems, periodic jumps, global asymptotic stability

Download Section

Download paper from here.
You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.

Bibliography

1
A. J. van der Schaft and H. Schumacher, An Introduction to Hybrid Dynamical Systems, ser. Lecture Notes in Control and Information Sciences, 251, Springer, Berlin Heidelberg (2000). http://www.springer.com/gp/book/9781447139164

2
Z. Li, Y. Soh, and C. Wen, Switched and Impulsive Systems: Analysis, Design and Applications, ser. Lecture Notes in Control and Information Sciences, 313, Springer-Verlag, Berlin Heidelberg (2005). http://www.springer.com/gp/book/9783540239529

3
W. M. Haddad, V. S. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, ser. Princeton Series in Applied Mathematics, 6, Princeton University Press, Princeton NJ (2006). https://press.princeton.edu/titles/8340.html

4
R. Goebel, R. G. Sanfelice, and A. R. Teel, Hybrid Dynamical Systems: Modeling, Stability, and Robustness, Princeton University Press, Princeton NJ (2012). https://press.princeton.edu/titles/9759.html

5
E. A. Medina and D. A. Lawrence, State estimation for linear impulsive systems, 2009 American Control Conference, St. Louis MO (2009), 1183-1188. DOI: 10.1109/ACC.2009.5160347

6
E. A. Medina and D. A. Lawrence, State feedback stabilization of linear impulsive systems, Automatica, 45, No. 6 (2009), 1476-1480. DOI: 10.1016/j.automatica.2009.02.003

7
E. A. Medina and D. A. Lawrence, Output feedback stabilization for linear impulsive systems, 2010 American Control Conference, Baltimore MD (2010), 4211-4216. DOI: 10.1109/ACC.2010.5530847

8
Y. Kouhi, N. Bajcinca, and R. G. Sanfelice, Suboptimality bounds for linear quadratic problems in hybrid linear systems, 2013 European Control Conference, Zürich, Switzerland (2013), 2663-2668.

9
G. Conte, A. M. Perdon, and E. Zattoni, Unknown input observers for hybrid linear systems with state jumps, 20th IFAC World Congress, ser. IFAC-PapersOnLine, 50, No. 1, Toulouse, France (2017), 6458-6464. DOI: 10.1016/j.ifacol.2017.08.1041

10
G. Conte, A. M. Perdon, and E. Zattoni, The disturbance decoupling problem for jumping hybrid systems, 54th IEEE Conference on Decision and Control, Osaka, Japan (2015), 1589-1594. DOI: 10.1109/CDC.2015.7402437

11
A. M. Perdon, E. Zattoni, and G. Conte, Disturbance decoupling with stability for linear impulsive systems, 6th IFAC Symposium on System Structure and Control, ser. IFAC-PapersOnLine, 49, No. 9, Istanbul, Turkey (2016), 1-6. DOI: 10.1016/j.ifacol.2016.07.477

12
A. M. Perdon, E. Zattoni, and G. Conte, Disturbance decoupling in hybrid linear systems with state jumps, IEEE Transactions on Automatic Control, 62, No. 12 (2017), 6552-6559. DOI: 10.1109/TAC.2017.2707536

13
A. Kaldmäe, Ü. Kotta, A. Shumsky, and A. Zhirabok, Disturbance decoupling in nonlinear hybrid systems, Nonlinear Analysis: Hybrid Systems, 28, Supplement C (2018), 42-53. DOI: 10.1016/j.nahs.2017.11.001

14
E. A. Medina and D. A. Lawrence, Controlled and conditioned invariants for linear impulsive systems, 45th IEEE Conference on Decision and Control, San Diego CA (2006), 2753-2758. DOI: 10.1109/CDC.2006.377071

15
E. Zattoni, A. M. Perdon, and G. Conte, Output regulation by error dynamic feedback in linear time-invariant hybrid dynamical systems, 14th European Control Conference, Linz, Austria (2015), 1438-1443. DOI: 10.1109/ECC.2015.7330741

16
D. Carnevale, S. Galeani, L. Menini, and M. Sassano, Hybrid output regulation for linear systems with periodic jumps: Solvability conditions, structural implications and semi-classical solutions, IEEE Transactions on Automatic Control, 61, No. 9 (2016), 2416-2431. DOI: 10.1109/TAC.2015.2496258

17
E. Zattoni, A. M. Perdon, and G. Conte, Output regulation by error dynamic feedback in hybrid systems with periodic state jumps, Automatica, 81, No. 7 (2017), 322-334. DOI: 10.1016/j.automatica.2017.03.037

18
D. Carnevale, S. Galeani, L. Menini, and M. Sassano, Robust hybrid output regulation for linear systems with periodic jumps: Semi-classical internal model design, IEEE Transactions on Automatic Control, 62, No. 12 (2017), 6649-6656. DOI: 10.1109/TAC.2017.2716107

19
E. Zattoni, A. M. Perdon, and G. Conte, Output regulation by error dynamic feedback in hybrid linear systems with state jumps, 20th IFAC World Congress, ser. IFAC-PapersOnLine, 50, No. 1, Toulouse, France (2017), 10808-10815. DOI: 10.1016/j.ifacol.2017.08.2352

20
E. Zattoni, A. M. Perdon, and G. Conte, Measurement dynamic feedback output regulation in hybrid linear systems with state jumps, International Journal of Robust and Nonlinear Control, 28, No. 2 (2018), 416-436. DOI: 10.1002/rnc.3875

21
C. Yuan and F. Wu, Hybrid almost output regulation of linear impulsive systems with average dwell time, Nonlinear Analysis: Hybrid Systems, 20, 82-94, 2016. DOI: 10.1016/j.nahs.2015.11.001

22
E. Zattoni, Structural model matching in hybrid linear systems with state jumps, 2017 American Control Conference, Seattle WA (2017), 511-516. DOI: 10.23919/ACC.2017.7963004

23
W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd ed. Springer-Verlag, New York (1985).

24
G. Basile and G. Marro, Controlled and Conditioned Invariants in Linear System Theory. Prentice Hall, Englewood Cliffs NJ (1992).

25
H. Nijmeijer and A. van der Schaft, The disturbance decoupling problem for nonlinear control systems, IEEE Transactions on Automatic Control, 28, No. 5 (1983), 621-623. DOI: 10.1109/TAC.1983.1103268

26
L. van der Wegen and H. Nijmeijer, The local disturbance decoupling problem with stability for nonlinear systems, Systems & Control Letters, 12, No. 2 (1989), 139-149. DOI: 10.1016/0167-6911(89)90007-8

27
M. Velasco, J. Alvarez, and R. Castro, Disturbance decoupling for time delay systems, International Journal of Robust and Nonlinear Control, 7, No. 9 (1997), 847-864. DOI: 10.1002/(SICI)1099-1239(199709)7:9$<$847::AID-RNC228$>$3.0.CO;2-B

28
A. Kaldmäe and C. H. Moog, Disturbance decoupling of time delay systems, Asian Journal of Control, 18, No. 3 (2016), 1130-1134. DOI: 10.1002/asjc.1169

29
G. Conte and A. M. Perdon, Disturbance decoupling problem for systems over a ring, SIAM Journal on Control and Optimization, 33, No. 3 (1995), 750-764. DOI: 10.1137/S0363012992235638

30
Q. Zou and S. Devasia, Preview-based stable-inversion for output tracking of linear systems, Journal of Dynamic Systems, Measurement and Control, 121, No. 4 (1999), 625-630. DOI: 10.1115/1.2802526

31
L. Mirkin, On the ${H}^\infty$ fixed-lag smoothing: How to exploit the information preview, Automatica, 39, No. 8 (2003), 1495-1504, DOI: 10.1016/S0005-1098(03)00141-9

32
G. Marro, D. Prattichizzo, and E. Zattoni, A unified setting for decoupling with preview and fixed-lag smoothing in the geometric context, IEEE Transactions on Automatic Control, 51, No. 5 (2006), 809-813. DOI: 10.1109/TAC.2006.875020

33
E. Zattoni, Decoupling of measurable signals via self-bounded controlled invariant subspaces: Minimal unassignable dynamics of feedforward units for prestabilized systems, IEEE Transactions on Automatic Control, 52, No. 1 (2007), 140-143. DOI: 10.1109/TAC.2006.886499

34
N. Otsuka, Disturbance decoupling with quadratic stability for switched linear systems, Systems & Control Letters, 59, No. 6 (2010), 349-352. DOI: 10.1016/j.sysconle.2010.03.010

35
E. Zattoni and G. Marro, A constructive condition for inaccessible signal rejection with quadratic stability in discrete-time linear switching systems, 52nd IEEE Conference on Decision and Control, Florence, Italy (2013), 4650-4655. DOI: 10.1109/CDC.2013.6760617

36
E. Zattoni, A. M. Perdon, and G. Conte, Disturbance decoupling with stability in continuous-time switched linear systems under dwell-time switching, 19th IFAC World Congress, ser. IFAC Proceedings Volumes (IFAC-PapersOnLine), 47, No. 3, Cape Town, South Africa (2014), 164-169. DOI: 10.3182/20140824-6-ZA-1003.00908

37
E. Zattoni, A. M. Perdon, and G. Conte, Disturbance decoupling with closed-loop modes stability in switched linear systems, IEEE Transactions on Automatic Control, 61, No. 10 (2016), 3115-3121. DOI: 10.1109/TAC.2015.2498123

38
A. R. F. Everts and M. K. Camlibel, When is a linear multi-modal system disturbance decoupled? Systems & Control Letters, 101, No. 3 (2017), 50-57. DOI: 10.1016/j.sysconle.2016.05.005

39
E. Zattoni, A. M. Perdon, and G. Conte, A geometric approach to output regulation for linear switching systems, 5th IFAC Symposium on System Structure and Control, ser. IFAC Proceedings Volumes (IFAC-PapersOnLine), 46, No. 2, Grenoble, France (2013), 172-177. DOI: 10.3182/20130204-3-FR-2033.00007

40
E. Zattoni, A. M. Perdon, and G. Conte, The output regulation problem with stability for linear switching systems: a geometric approach, Automatica, 49, No. 10 (2013), 2953-2962. DOI: 10.1016/j.automatica.2013.07.005

41
G. Conte, A. M. Perdon, and E. Zattoni, Model matching problems for switching linear systems, 19th IFAC World Congress, ser. IFAC Proceedings Volumes (IFAC-PapersOnLine), 47, No. 3, Cape Town, South Africa (2014), 1501-1506. DOI: 10.3182/20140824-6-ZA-1003.00162

42
A. M. Perdon, G. Conte, and E. Zattoni, Necessary and sufficient conditions for asymptotic model matching of switching linear systems, Automatica, 64, No. 2 (2016), 294-304. DOI: 10.1016/j.automatica.2015.11.017

43
A. M. Perdon, E. Zattoni, and G. Conte, Model matching with strong stability in switched linear systems, Systems & Control Letters, 97, No. 11 (2016), 98-107. DOI: 10.1016/j.sysconle.2016.09.009

44
G. Conte, A. M. Perdon, and E. Zattoni, The unknown input observation problem for switching systems with dwell-time, 56th IEEE Conference on Decision and Control, Melbourne, Australia (2017), 3626-3633. DOI: 10.1109/CDC.2017.8264192

45
G. Conte, A. M. Perdon, and E. Zattoni, Geometric tools and methods in the study of switching systems, Hybrid Systems. Performance, Applications and Technology, ser. Physics Research and Technology, J. Li, Ed., Nova Science Publishers, New York (2017), 83-110. https://www.novapublishers.com/catalog/product_info.php?cPath=23_29&products_id=62988

How to Cite?

DOI: 10.12732/ijpam.v118i2.27 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 2
Pages: 451 - 464


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).