Circuits, Systems, and Signal Processing

, Volume 38, Issue 11, pp 5304–5322

# Finite-Time Interval Observers’ Design for Switched Systems

Short Paper

## Abstract

In this study, a finite-time interval observers’ design method is developed for switched systems suffering from disturbance. First, the interval observer frames for the systems are constructed. Then, sufficient conditions are derived to guarantee that the upper and lower error systems are both positive and finite-time bound. Unlike the current studies, all the conditions proposed in this paper are formulated in the form of linear programming. Finally, two numerical examples are provided to show the efficiency of designed observers.

## Keywords

Interval observers Finite-time boundedness Switched systems Linear programming

## List of Symbols

$$R^n$$

n-dimensional Euclidean space

$$R^{n\times m}$$

The set of $$n{\times }m$$ real matrices

$$x>(\ge )0$$

Its components are positive (nonnegative), i.e., $$x_{i}>(\ge )0$$

$$A>(\ge )0$$

Its components are positive (nonnegative), i.e., $$A_{ij}>(\ge )0$$

$$E^+$$

$$\max \{E,0\}$$

$$E^-$$

$$E^+-E$$

$$||x||_1$$

The 1-norm of the vector x

$$\overline{\lambda }(v)$$

The maximum value of the elements of the vector v

$$\underline{\lambda }(v)$$

The minimum value of the elements of the vector v

$$\mathbf 1 _n$$

The vector whose entries equal to 1

## References

1. 1.
A. Agresti, B. Coull, Approximate is better than exact for interval estimation of binomial proportions. Am. Stat. 52(2), 119–126 (1998)
2. 2.
F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances. Automatica 37(9), 1459–1463 (2001)
3. 3.
M. Arcak, P. Kokotovic, Nonlinear observers: a circle criterion design and robustness analysis. Automatica 37(12), 1923–1930 (2001)
4. 4.
C. Briat, M. Khammash, Simple interval observers for linear impulsive systems with applications to sampled-data and switched systems, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)
5. 5.
S. Chebotarev, D. Efimov, A. Zolghadri, Interval observers for continuous-time LPV systems with $$L_1$$/$$L_2$$ performance. Automatica 58, 82–89 (2015)
6. 6.
J. Cheng, H. Zhu, S. Zhong, F. Zheng, Y. Zeng, Finite-time filtering for switched linear systems with a mode-dependent average dwell time. Nonlinear Anal. Hybrid Syst. 15(2), 145–156 (2015)
7. 7.
K. Degue, D. Efimov, J. Ny, Interval observer approach to output stabilization of linear impulsive systems, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)Google Scholar
8. 8.
H. Du, X. Lin, S. Li, Finite-time stability and stabilization of switched linear systems, in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta (2010)Google Scholar
9. 9.
D. Efimov, T. Raissi, S. Chebotarev, A. Zolghadri, Interval state observer for nonlinear time varying systems. Automatica 49(1), 200–205 (2013)
10. 10.
H. Ethabet, T. Raissi, M. Amairi, M. Aoun, Interval observers design for continuous-time linear switched systems, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)
11. 11.
L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)
12. 12.
J. Gouze, A. Rapaport, Z. Hadj-Sadok, Interval observers for uncertain biological systems. Ecol. Model. 133(1), 45–56 (2000)
13. 13.
S. Guo, F. Zhu, Interval observer design for discrete-time switched system, in Proceedings of the 20th IFAC World Congress, Toulouse (2017)Google Scholar
14. 14.
Z. He, W. Xie, Control of non-linear switched systems with average dwell time: interval observer-based framework. IET Contr. Theory Appl. 10(1), 10–16 (2016)
15. 15.
R. Horn, C. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)
16. 16.
B. Hu, G. Zhai, A. Michel, Common quadratic Lyapunov-like functions with associated switching regions for two unstable second-order LTI systems. Int. J. Control 75(14), 1127–1135 (2002)
17. 17.
S. Ibrir, Circle-criterion approach to discrete-time nonlinear observer design. Automatica 43(8), 1432–1441 (2007)
18. 18.
D. Liberzon, Switching in Systems and Control (Springer, Berlin, 2012)
19. 19.
X. Luan, F. Liu, P. Shi, Finite-time filtering for non-linear stochastic systems with partially known transition jump rates. IET Contr. Theory Appl. 4(5), 735–745 (2010)
20. 20.
M. Moisan, O. Bernard, Robust interval observers for global Lipschitz uncertain chaotic systems. Syst. Control Lett. 59(11), 687–694 (2010)
21. 21.
T. Raissi, D. Efimov, A. Zolghadri, Interval state estimation for a class of nonlinear systems. IEEE Trans. Autom. Control 57(1), 260–265 (2011)
22. 22.
M. Rami, C. Cheng, C. Prada, Tight robust interval observers: an LP approach, in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun (2008)Google Scholar
23. 23.
Y. Su, J. Huang, Stability of a class of linear switching systems with applications to two consensus problems. IEEE Trans. Autom. Control 57(6), 1420–1430 (2012)
24. 24.
M. Wang, J. Feng, G. Dimirovski, J. Zhao, Stabilization of switched nonlinear systems using multiple Lyapunov function method, in Proceedings of the 2009 American Control Conference, St. Louis (2009)Google Scholar
25. 25.
Y. Wang, D. Bevly, R. Rajamani, Interval observer design for LPV systems with parametric uncertainty. Automatica 60(10), 79–85 (2015)
26. 26.
J. Zhang, Z. Han, F. Zhu, Finite-time control and $$L_1$$-gain analysis for positive switched systems. Optim. Control Appl. Methods 36(4), 550–565 (2015)
27. 27.
Y. Zhang, C. Liu, X. Mu, Robust finite-time stabilization of uncertain singular Markovian jump systems. Appl. Math. Model. 36(10), 5109–5121 (2012)
28. 28.
Y. Zhang, P. Shi, S. Nguang, H. Karimi, Observer-based finite-time fuzzy $$H_{\infty }$$ control for discrete-time systems with stochastic jumps and time-delays. Signal Process. 97, 252–261 (2014)
29. 29.
Y. Zhang, Y. Shi, P. Shi, Robust and non-fragile finite-time $$H_{\infty }$$ control for uncertain Markovian jump nonlinear systems. Appl. Math. Comput. 279, 125–138 (2016)
30. 30.
Y. Zhang, Y. Shi, P. Shi, Resilient and robust finite-time $$H_{\infty }$$ control for uncertain discrete-time jump nonlinear systems. Appl. Math. Model. 49, 612–629 (2017)
31. 31.
X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 57(7), 1809–1915 (2012)
32. 32.
X. Zhao, L. Zhang, P. Shi, M. Liu, Stability of a class of switched positive linear systems with average dwell time switching. Automatica 48(6), 1132–1137 (2012)
33. 33.
G. Zheng, D. Efimov, F. Bejarano, W. Perruquetti, H. Wang, Interval observer for a class of uncertain nonlinear singular systems. Automatica 71(9), 159–168 (2016)

## Authors and Affiliations

• Xiang Ma
• 1
• Jun Huang
• 1
• Liang Chen
• 1
1. 1.School of Mechanical and Electrical EngineeringSoochow UniversitySuzhouChina