Skip to main content Accessibility help

Output Feedback Admissible Control for Singular Systems: Delta Operator (Discretised) Approach

  • Xin-zhuang Dong (a1) and Mingqing Xiao (a2)


Singular systems simultaneously capture the dynamics and algebraic constraints in many practical applications. Output feedback admissible control for singular systems through a delta operator method is considered in this article. Two novel admissibility conditions, derived for the singular delta operator system (SDOS) from a singular continuous system through sampling, can not only produce unified admissibility for both continuous and discrete singular systems but also practical procedures. To solve the problem of output feedback admissible control for the SDOS, an existence condition and design procedure is given for the determination of a physically realisable observer for the state estimation, and then a suitable state-feedback-like admissible controller design based on the observer is developed. All of the conditions presented are necessary and sufficient, involving strict linear matrix inequalities (LMI) with feasible solutions obtained with low computational costs. Numerical examples illustrate our approach.


Corresponding author

*Corresponding author. Email address: (M. Xiao)


Hide All
[1] Luenberger, D.G. and Arbel, A., Singular dynamical Leotief systems, Econometrica 5, 991995 (1997).
[2] Newcomb, R.W., The semistate description of nonlinear time variable circuits, IEEE Trans. Circuits Syst. 28, 6267 (1981).
[3] Kumar, A. and Daoutidis, P., Feedback control of nonlinear differential-algebraic equation systems, AIChE J. 41, 619636 (1995).
[4] Dai, L., Singular Control Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin (1989).
[5] Xu, S. and Lam, J., Robust Control and Filtering of Singular Systems, Springer, Berlin (2006).
[6] Lu, R., Su, H., Xue, A. and Chu, J., Robust Control Theory of Singular Systems, Science Press, Beijing (2008).
[7] Duan, G., Analysis and Design of Descriptor Linear Systems, Science Press, Beijing (2012).
[8] Yang, C., Zhang, Q. and Zhou, L., Stability Analysis and Design for Nonlinear Singular Systems, Springer, Berlin (2013).
[9] Wu, Z., Su, H., Shi, P and Chu, J., Analysis and Synthesis of Singular Systems with Time-Delays, Springer, Berlin (2013).
[10] Fridman, E. and Shaked, U., A descriptor system approach to H control of linear time- delay systems, IEEE Trans. Automat. Contr. 47, 253270 (2002).
[11] Cao, Y. and Lin, Z., A descriptor system approach to robust stability analysis and controller synthesis, IEEE Trans. Automat. Contr. 49, 20812084 (2004).
[12] Middleton, R. and Goodwin, G.C., Improved finite word length characteristics in digital control using delta operators, IEEE Trans. Automat. Contr. 31, 10151021 (1986).
[13] Goodwin, G.C., Lozano Leal, R., Mayne, D.Q. and Middleton, R.H., Rapproachement between continuous and discrete model reference adaptive control, Automatica 22, 199207 (1986).
[14] Li, G. and Gevers, M., Comparative study of finite wordlength effects in shift and delta operator parameterisations, IEEE Trans. Automat. Contr. 38, 803807 (1993).
[15] Wu, J., Li, G., Istepanian, R.H. and Chu, J., Shift and delta operator realisation for digital controllers with finite word length consideration, IEE Proc. - Control Theory and Applications 147, 664672 (2000).
[16] Li, H., Wu, B., Li, G. and Yang, C., Basic Theory of Delta Operator Control and Robust Control, National Defence Industry Press, Beijing (2005).
[17] Yang, H., Xia, Y. and Shi, P., Observer-based sliding mode control for a class of discrete systems via delta operator approach, J. Franklin Institute 347, 11991213 (2010).
[18] Yang, H. and Xia, Y., Low frequency positive real control for delta operator systems, Automatica 48, 17911795 (2012).
[19] Yang, H., Xia, Y., Shi, P. and Zhao, L., Analysis and Synthesis of Delta Operator Systems, Springer, Berlin (2012).
[20] Dong, X., Controllability analysis of linear singular delta operator systems, Proc. 12th Int. Conf. Control, Automation, Robotics and Vision, Guangzhou, China, pp. 1199-1204 (2012).
[21] Dong, X., Mao, Q., Tian, W. and Wang, D., Observability analysis of linear singular delta operator systems, Proc. 10th IEEE Int. Conf. Control and Automation, Hangzhou, China, pp. 10-15 (2013).
[22] Dong, X., Admissibility analysis of linear singular systems via a delta operator method, Int. J. Systems Science 45, 23662375 (2014).
[23] Dong, X. and Xiao, M., Admissible control of linear singular delta operator systems, Circuits, Systems, Signal Processing 33, 20432064 (2014).
[24] Dong, X., Xiao, M., Wang, Y. and He, W., Observer-based admissible control for singular delta operator systems, Proc. 14th Int. Conf. Control, Automation and Systems, Korea, pp. 1117-1122 (2014).
[25] Dong, X., LMI-based robust admissible control for uncertain singular delta operator systems, Int. J. Systems Sc 46, 21012110 (2015).
[26] Dong, X. and Xiao, M., D-admissible hybrid control of a class of singular systems, Nonlinear Analysis: Hybrid Systems 17, 94110 (2015).
[27] Koenig, D. and Mammar, S., Design of proportional-integral observer for unknown input descriptor systems, IEEE Trans. Automat. Contr. 47, 20572062 (2002).
[28] Koenig, D., Observer design for unknown input nonlinear descriptor systems via convex optimisation, IEEE Trans. Automat. Contr. 51, 10471052 (2006).
[29] Lu, G. and Ho, D.W.C., Full-order and reduced-order observers for Lipschitz desriptor systems: the unified LMI approach, IEEE Trans. Circuits and Systems 53, 563567 (2006).
[30] Wang, Z., Shen, Y., Zhang, X. and Wang, Q., Observer design for discrete-time descriptor systems: an LMI approach, Systems Control Letts. 61, 683687 (2012).


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed