IJPAM: Volume 109, No. 4 (2016)

$\gH_2$-OPTIMAL DISTURBANCE REJECTION
BY MEASUREMENT FEEDBACK: THE SINGULAR CASE

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 work concerns a new methodology to solve the $\gH_2$-optimal disturbance rejection problem by measurement feedback in the singular case: namely, when the plant has no feedthrough terms from the control input and the disturbance input to the controlled output and the measured output, respectively. A necessary and sufficient condition for problem solvability is expressed as the inclusion of two subspaces -- a controlled-invariant subspace and a conditioned-invariant subspace. Such subspaces are directly derived from the Hamiltonian systems associated to the $\gH_2$-optimal control problem and, respectively, to the $\gH_2$-optimal filtering problem. The proof of sufficiency, which is constructive, provides the computational tools for the synthesis of the feedback regulator. A numerical example is worked out in order to illustrate how to implement the devised procedure.

Received: August 12, 2016

Revised: September 20, 2016

Published: October 9, 2016

AMS Subject Classification: 93C05, 93C35, 49N05

Key Words and Phrases: $\gH_2$-optimal control, disturbance rejection, linear systems
Download paper from here.

Bibliography

1
A. Saberi, P. Sannuti, and B. M. Chen, $H_2$ Optimal Control, ser. Systems and Control Engineering. Prentice Hall International, Englewood Cliffs, New Jersey (1995).

2
V. Ionescu, C. Oară, and M. Weiss, Generalized Riccati Theory and Robust Control: A Popov Function Approach. John Wiley & Sons, Chichester, England (1999).

3
H. L. Trentelman, A. A. Stoorvogel, and M. Hautus, Control Theory for Linear Systems, ser. Communications and Control Engineering. Springer, Great Britain (2001), https://www.springer.com/us/book/9781852333164

4
R. Fraanje, M. Verhaegen, N. Doelman, and A. Berkhoff, Optimal and robust feedback controller estimation for a vibrating plate, Control Engineering Practice, 12, No. 8 (2004), 1017-1027, DOI:10.1016/j.conengprac.2003.09.007.

5
E. Zattoni, Detection of incipient failures using an $H_2$-norm criterion: Application to electric point machines, in 16th IFAC World Congress, Prague, Czech Republic, ser. IFAC Proceedings Volumes (IFAC-PapersOnline), 16 (2005), 250-255, DOI:10.3182/20050703-6-CZ-1902.01770.

6
E. Zattoni, Detection of incipient failures by using an $H_2$-norm criterion: Application to railway switching points, Control Engineering Practice, 14, No. 8 (2006), 885-895, DOI:10.1016/j.conengprac.2005.05.004.

7
F. Claveau, P. Chevrel, and K. Knittel, A 2DOF gain-scheduled controller design methodology for a multi-motor web transport system, Control Engineering Practice, 16, No. 5 (2008), 609-622, DOI:10.1016/j.conengprac.2007.07.002.

8
A. Zakharov, E. Zattoni, L. Xie, O. Garcia, and S.-L. Jämsä-Jounela, An autonomous valve stiction detection system based on data characterization, Control Engineering Practice, 21, No. 11 (2013), 1507-1518, DOI:10.1016/j.conengprac.2013.07.004.

9
A. Zakharov, E. Zattoni, L. Xie, O. Pozo, and S.-L. Jämsä-Jounela, Data characterization for automatic selection of valve stiction detection algorithms, in 52nd IEEE Conference on Decision and Control, Florence, Italy (2013), 4355-4360, DOI:10.1109/CDC.2013.6760559.

10
B.-S. Chen, H.-C. Lee, and C.-F. Wu, Pareto optimal filter design for nonlinear stochastic fuzzy systems via multiobjective $H_2/H_\infty$ optimization, IEEE Transactions on Fuzzy Systems, 23, No. 2 (2015), 387-399, DOI:10.1109/TFUZZ.2014.2312985.

11
K. Morris, M. Demetriou, and S. Yang, Using $H_2$-control performance metrics for the optimal actuator location of distributed parameter systems, IEEE Transactions on Automatic Control, 60, No. 2 (2015), 450-462, DOI:10.1109/TAC.2014.2346676.

12
A. Zakharov, E. Zattoni, M. Yu, and S.-L. Jämsä-Jounela, A performance optimization algorithm for controller reconfiguration in fault tolerant distributed model predictive control, in IEEE International Conference on Automation Science and Engineering, Gothenburg, Sweden (2015), 886-891, DOI:10.1109/CoASE.2015.7294193.

13
W.-D. Sun, P. Li, and Z. Fang, D-stable $H_2/H_\infty$ mixed cruise control of unmanned helicopter based on LPV dynamic inversion, Shanghai Jiaotong Daxue Xuebao/Journal of Shanghai Jiaotong University, 49, No. 11 (2015), 1647-1654, DOI:10.16183/j.cnki.jsjtu.2015.11.010.

14
A. Zakharov, E. Zattoni, M. Yu, and S.-L. Jämsä-Jounela, A performance optimization algorithm for controller reconfiguration in fault tolerant distributed model predictive control, Journal of Process Control, 34 (2015), 56-69, DOI:10.1016/j.jprocont.2015.07.006.

15
W. Jiang, H. Wang, J. Lu, and Z. Xie, HOSVD-based LPV modeling and mixed robust $H_2/H_\infty$ control design for air-breathing hypersonic vehicle, Journal of Systems Engineering and Electronics, 27, No. 1 (2016), 183-191, DOI:10.1109/JSEE.2016.00018.

16
C.-C. Ku and M.-D. Li, A mixed $H_2/H_\infty$ passivity performance controller design for a drum-boiler system, Journal of Marine Engineering and Technology, 14, No. 3 (2016), 137-145, DOI:10.1080/20464177.2015.1115296.

17
G. Marro, D. Prattichizzo, and E. Zattoni, $H_2$ optimal decoupling of previewed signals in the discrete-time case, Kybernetika, 38, No. 4 (2002), 479-492, https://www.kybernetika.cz/content/2002/4/479

18
G. Marro, D. Prattichizzo, and E. Zattoni, $H_2$ optimal decoupling FIRS, in 15th IFAC World Congress, Barcelona, Spain, ser. IFAC Proceedings Volumes (IFAC-PapersOnline), 15, No. 1 (2002), 395-400, DOI:10.3182/20020721-6-ES-1901.00315.

19
G. Marro and E. Zattoni, Regulation transients in discrete-time linear parameter varying systems: $l_2$ optimization with preview, in 2007 American Control Conference, New York, NY (2007), 6097-6102, DOI:10.1109/ACC.2007.4282856.

20
G. Marro and E. Zattoni, A nested computational approach for $l_2$ optimization of regulation transients in discrete-time linear parameter varying systems, in 2007 European Control Conference, Kos, Greece (2007), 4889-4895.

21
E. Zattoni, Regulation transients in DT-LPV systems: $\ell_2$-optimal approach via Hamiltonian system structural invariant subspaces, in 46th IEEE Conference on Decision and Control, New Orleans, LA (2007), 2761-2766, DOI:10.1109/CDC.2007.4434498.

22
G. Marro and E. Zattoni, A nested computational approach to $l_2$-optimization of regulation transients in discrete-time LPV systems, European Journal of Control, 14, No. 1 (2008), 30-46, DOI:10.3166/ejc.14.30-46.

23
E. Zattoni, $H_2$-optimal rejection with preview: Geometric constraints and dynamic feedforward solutions via spectral factorization, Kybernetika, 44, No. 1 (2008), 3-16, https://www.kybernetika.cz/content/2008/1/3

24
A. A. Stoorvogel, The singular ${H}_2$ control problem, Automatica, 28, No. 3 (1992), 627-631, DOI:10.1016/0005-1098(92)90189-M.

25
A. A. Stoorvogel, A. Saberi, and B. M. Chen, Full and reduced-order observer-based controller design for ${H}_2$-optimization, International Journal of Control, 58, No. 4 (1993), 803-834, DOI:10.1080/00207179308923030.

26
A. Saberi, P. Sannuti, and A. A. Stoorvogel, ${H}_2$ optimal controllers with measurement feedback for continuous-time systems--Flexibility in closed-loop pole placement, Automatica, 32, No. 8 (1996), 1201-1209, DOI:10.1016/0005-1098(96)00052-0.

27
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA (1994) https://web.stanford.edu/~boyd/lmibook/lmibook.pdf.

28
B. M. Chen, Z. Lin, and Y. Shamash, Linear Systems Theory: A Structural Decomposition Approach, ser. Control Engineering. Birkhäuser, Boston (2004) https://www.springer.com/us/book/9780817637798.

29
A. Saberi, A. A. Stoorvogel, and P. Sannuti, Filtering Theory with Applications to Fault Detection, Isolation, and Estimation, ser. Systems & Control: Foundations & Applications. Birkhäuser, Boston (2007) https://www.springer.com/us/book/9780817643010.

30
G. Marro and E. Zattoni, A geometric perspective on $H_2$-optimal rejection by measurement feedback in strictly-proper systems: The continuous-time case, in 50th IEEE Conference on Decision and Control, Orlando, FL (2011), 1195-1200, DOI:10.1109/CDC.2011.6160729.

31
W. M. Wonham, Linear Multivariable Control: A Geometric Approach, 3rd ed., Springer-Verlag, USA (1985), https://www.springer.com/us/book/9780387960715

32
G. Basile and G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice Hall, USA (1992), https://www3.deis.unibo.it/Staff/FullProf/GiovanniMarro/gm_books.htm#book5

33
A. M. Perdon, E. Zattoni, and G. Conte, Model matching with strong stability in switched linear systems, Systems & Control Letters (2016), DOI:10.1016/j.sysconle.2016.09.009 (in press).

34
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), DOI:10.1109/TAC.2015.2498123 (in press).

35
A. M. Perdon, G. Conte, and E. Zattoni, Necessary and sufficient conditions for asymptotic model matching of switching linear systems, Automatica, 64, 294-304 (2016), DOI:10.1016/j.automatica.2015.11.017.

36
E. Zattoni, A. M. Perdon, and G. Conte, Disturbance decoupling with stability in continuous-time switched linear systems under dwell-time switching, in 19th IFAC World Congress, Cape Town, South Africa, ser. IFAC Proceedings Volumes (IFAC-PapersOnline), 19 (2014), 164-169, DOI:10.3182/20140824-6-ZA-1003.00908.

37
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.

38
E. Zattoni, A. M. Perdon, and G. Conte, A geometric approach to output regulation for linear switching systems, in 5th IFAC Symposium on System Structure and Control, Grenoble, France (2013), 172-177, DOI:10.3182/20130204-3-FR-2033.00007.

39
G. Marro and E. Zattoni, Unknown-state, unknown-input reconstruction in discrete-time nonminimum-phase systems: geometric methods, Automatica, 46, No. 5 (2010), 815-822, DOI:10.1016/j.automatica.2010.02.012.

40
S. Kirtikar, H. Palanthandalam-Madapusi, E. Zattoni, and D. S. Bernstein, $l$-delay input reconstruction for discrete-time linear systems, in 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, Shanghai, People's Republic of China (2009), 1848-1853, DOI:10.1109/CDC.2009.5400498.

41
G. Marro, D. Prattichizzo, and E. Zattoni, Geometric insight into discrete-time cheap and singular linear quadratic Riccati (LQR) problems, IEEE Transactions on Automatic Control, 47, No. 1 (2002), 102-107, DOI:10.1109/9.981727.

42
G. Marro, D. Prattichizzo, and E. Zattoni, A nested computational approach to the discrete-time finite-horizon LQ control problem, SIAM Journal on Control and Optimization, 42, No. 3 (2003), 1002-1012, DOI:10.1137/S0363012901384429.

43
E. Zattoni, A multi-level algorithm for the finite horizon LQ optimal control problem with assigned final state: Additive and multiplicative procedures, in 14th Mediterranean Conference on Control and Automation, Ancona, Italy (2006), DOI:10.1109/MED.2006.328747.

44
E. Zattoni, An improved algorithm for the non-iterative solution of the discrete-time finite-horizon LQ control problem with fixed final state, in 45th IEEE Conference on Decision and Control, San Diego, CA (2006), 1364-1368, DOI:10.1109/CDC.2006.377213.

45
G. Marro and E. Zattoni, Structural invariants of the singular Hamiltonian system and non-iterative solution of finite-horizon optimal control problems, in 2007 American Control Conference, New York, NY (2007), 5153-5157, DOI:10.1109/ACC.2007.4282196.

46
E. Zattoni, Structural invariant subspaces of singular Hamiltonian systems and nonrecursive solutions of finite-horizon optimal control problems, IEEE Transactions on Automatic Control, 53, No. 5 (2008), 1279-1284, DOI:10.1109/TAC.2008.921040.

47
G. Marro, F. Morbidi, and D. Prattichizzo, A geometric solution to the cheap spectral factorization problem, in European Control Conference 2009, Budapest, Hungary (2009), 814-819.

.



DOI: 10.12732/ijpam.v109i4.18 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: 2016
Volume: 109
Issue: 4
Pages: 975 - 992


$\gH_2$-OPTIMAL DISTURBANCE REJECTION BY MEASUREMENT FEEDBACK: THE SINGULAR CASE%22&as_occt=any&as_epq=&as_oq=&as_eq=&as_publication=&as_ylo=&as_yhi=&as_sdtAAP=1&as_sdtp=1" title="Click to search Google Scholar for this entry" rel="nofollow">Google Scholar; DOI (International DOI Foundation); WorldCAT.

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