This article was downloaded by: [National Kaohsiung University of Applied Sciences]
On: 09 September 2014, At: 21:37
Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,
37-41 Mortimer Street, London W1T 3JH, UK
Journal of the Chinese Institute of Engineers
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/tcie20
Robust controllability of linear time-invariant interval
systems
Shinn-Horng Chen
a& Jyh-Horng Chou
aba
Department of Mechanical Engineering , National Kaohsiung University of Applied
Sciences , 415 Chien-Kung Road, Kaohsiung 807 , Taiwan , Republic of China
b
Institute of System Information and Control, National Kaohsiung First University of Science
and Technology , 1 University Road, Yenchao, Kaohsiung 824 , Taiwan , Republic of China
Published online: 19 Oct 2012.
To cite this article: Shinn-Horng Chen & Jyh-Horng Chou (2013) Robust controllability of linear time-invariant interval
systems, Journal of the Chinese Institute of Engineers, 36:5, 672-676, DOI:
10.1080/02533839.2012.734624
To link to this article: http://dx.doi.org/10.1080/02533839.2012.734624
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained
in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no
representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the
Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and
are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and
should be independently verified with primary sources of information. Taylor and Francis shall not be liable for
any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever
or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of
the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic
reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any
form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at
http://
www.tandfonline.com/page/terms-and-conditions
and Zhang (2004). Hence, the condition of Cheng and Zhang (2004) is not satisfied. Then, no conclusion can be made. That is, the sufficient condition of Cheng and Zhang (2004) cannot be applied in this example.
Next, we use the method proposed by Ahn et al. (2005) and Chen et al. (2006) to test the controllability. The controllability matrix D is calculated as
D ¼ 1þ0:9 0:5 0 1þ2:610:75 0 0 0 0 1þ4:70:75 0 1þ0:9 0:5 0 4þ12:722:95 2 6 4 3 7 5: So, we have four sub-square matrices:
S1¼ 1þ0:90:5 0 1þ2:610:75 0 0 0 0 1þ0:90:5 0 2 6 4 3 7 5, S2¼ 1þ0:9 0:5 0 0 0 0 1þ4:70:75 0 1þ0:90:5 4þ12:722:95 2 6 4 3 7 5, S3¼ 1þ0:90:5 1þ2:610:75 0 0 0 1þ4:70:75 0 0 4þ12:722:95 2 6 4 3 7 5 and S4 ¼ 0 1þ2:610:75 0 0 0 1þ2:61 0:75 1þ0:90:5 0 4þ12:722:95 2 6 4 3 7 5: Since, from S2 and S4, S2
0 and S40 are non-singular, ðkðS20Þ1kDS2Þ ¼55:256 65 1 and ðkðS4
0Þ
1kDS4Þ ¼ 54:169 65 1, the condition of Ahn et al. (2005) and Chen et al. (2006) is not satisfied either. Then, no conclusion can be made. That is, the sufficient condi-tion of Ahn et al. (2005) and Chen et al. (2006) can also not be applied in this example.
Now, applying the sufficient criterion in the pro-posed theorem for the robust controllability, we have rankðE0Þ ¼9 and we get
S1UHD1V In2, 0n2nðm1Þ T ¼0:92892 5 1 ð17aÞ and S1UHD2V In2, 0n2nðm1Þ T ¼0:66438 5 1: ð17bÞ Then, from the results obtained above, we can conclude that the linear time-invariant interval system is robustly controllable. From this example, it is clear that our proposed approach is much simpler and gives less conservative results than the existing methods of
Cheng and Zhang (2004), Ahn et al. (2005) and Chen et al. (2006).
4. Conclusions
The robust controllability problem for the linear time-invariant interval system has been investigated. The rank preservation problem for robust controllability of the linear time-invariant interval system is converted into the non-singularity analysis problem. Based on some essential properties of matrix measures, a new, sufficient, algebraically elegant criterion for the robust controllability of linear time-invariant interval systems is established. A numerical example has been given to illustrate the application of the proposed sufficient algebraic criterion, and it has also been shown that the proposed sufficient criterion can obtain less conserva-tive results than the existing sufficient criteria given by Cheng and Zhang (2004), Ahn et al. (2005), and Chen et al. (2006). The approach proposed in this paper can be extended to deal with the robust controllability of uncertain linear time-delay systems (Chen et al. 2012a,b).
Acknowledgments
This work was in part supported by the National Science Council, Taiwan, Republic of China, under grant nos NSC 99-2221-E151-009 and NSC 99-2221-E151-072-MY2.
Nomenclature
xðtÞ state vector uðtÞ input vector A 2 ½ A, A, B 2 ½ B, B interval matrices
ð ~WÞ eigenvalue of ~W ð ~WÞ spectral radius of ~W ð ~WÞ matrix measure of ~W
References
Ahn, H.S., Moore, K.I., and Chen, Y.Q., 2005. Linear independency of interval vectors and its applications to robust controllability tests. In: Proceedings of the 44th IEEE conference on decision and control, and European control conference, 12–15 December 2005, Seville, Spain: IEEE Control Society Press, 8070–8075.
Ashchepkov, L.T., 2007. The controllability of an interval linear discrete system. Journal of computer and systems sciences international, 46 (3), 399–406.
Ashchepkov, L.T., 2008. External bounds and step control-lability of the linear interval system. Automation and remote control, 69 (4), 590–596.
675
Journal of the Chinese Institute of Engineers
Chen, Y.Q., Ahn, H.S., and Xue, D., 2006. Robust control-lability of interval fractional order linear time invariant systems. Signal processing, 86 (10), 2794–2802.
Chen, S.H., Chou, F.I., and Chou, J.H., 2012a. On robust controllability of linear systems with multiple delays in control. IET control theory and applications, 6 (10), 1552–1556.
Chen, S.H., Chou, F.I., and Chou, J.H., 2012b. Robust controllability of linear interval systems with multiple control delays. International journal of systems science, (in press).
Cheng, B. and Zhang, J., 2004. Robust controllability for a class of uncertain linear time-invariant MIMO systems. IEEE transactions on automatic control, 49 (11), 2022–2027.
Desoer, C.A. and Vidyasagar, M., 1975. Feedback systems: input–output properties. New York: Academic Press. Li, X.H., Yu, H.B., Yuan, M.Z., and Wang, J., 2010. Robust
stability of interval polynomials and matrices for linear systems. In: Proceedings of the IASTED international conference on modelling, identification and control, 15–17 February 2010, Innsbruck, Austria: ACTA Press, 138–147. Liang, Y.X. and Wang, R, 2010. Method of the stability control for a class of the generalized interval systems. In: Proceedings of the 5th international
conference on computer science and education, 24–27 August 2010, Anhui, China: IEEE Computer Society Press, 1305–1308.
Ling, Q., 2010. Robust stability analysis of discrete interval systems. In: Proceedings of the international conference on electrical and control engineering, 25–27 June 2010, Wuhan, China: IEEE Computer Society Press, 1848–1850. Pastravanu, O. and Matcovschi, M.H., 2010. Diagonal stability of interval matrices and applications. Linear algebra and its applications, 433 (8–10), 1646–1658. Rosenbrock, H.H., 1970. State-space and multivariable
theory. New York: John Wiley and Sons.
Shao, J. and Hou, X., 2010. Stability analysis of uncertain systems based on interval analysis. In: Proceedings of the international conference on intelligent computing and integrated systems, 22–24 October 2010, Guilin, China: IEEE Computer Society Press, 383–386.
Zhang, D.Q., Zhang, Q.L., and Chen, Y.P., 2006. Controllability and quadratic stability quadratic stabiliza-tion of discrete-time interval systems – an LMI approach. IMA Journal of mathematical control and information, 23 (4), 413–431.
Zhirabok, A.N., 2007. Analysis of controllability degree of discrete dynamic system. Journal of computer and systems sciences international, 46 (2), 169–176.
