research-article

Design of functional interval observers for non‐linear fractional‐order systems

Author:

Dinh Cong HuongAuthors Info & Claims

Asian Journal of Control, Volume 22, Issue 3

Pages 1127 - 1137

https://doi.org/10.1002/asjc.1984

Published: 28 May 2020 Publication History

Abstract

This paper considers the problem of designing functional interval observers for a class of non‐linear fractional‐order systems with bounded uncertainties. First, interval observers for linear functions of the state vector of the considered system are designed. Then, conditions for the existence of such interval observers are established and an effective algorithm for computing unknown observer matrices is provided in this paper. Finally, numerical examples and simulation results are given to illustrate the effectiveness of the proposed design method.

References

[1]

C. P. Li and F. R. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J. Spec. Top. 193 (2011), 27–47.

[2]

R. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), 299–307.

[3]

J. A. Gallegos and M. A. Duarte, On the Lyapunov theory for fractional order systems, Appl. Math. Comput. 287‐288 (2016), 161–170.

[4]

R.‐J. Liu et al., Robust disturbance rejection for uncertain fractional order systems, Appl. Math. Comput. 322 (2018), 79–88.

[5]

A. L. Wu and Z. G. Zeng, Boundedness, Mittag‐Leffler stability and asymptotical ω‐periodicity of fractional‐order fuzzy neural networks, Neural Netw. 74 (2016), 73–84.

Digital Library

[6]

A. L. Wu, Z. G. Zeng, and X. G. Song, Global Mittag‐Leffler stabilization of fractional‐order bidirectional associative memory neural networks, Neurocomputing 177 (2016), 489–496.

Digital Library

[7]

X. J. Yang, Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat‐transfer problems, Therm. Sci. 21 (2017), 161–171.

[8]

X. J. Yang, H. M. Srivastava, and J. A. Tenreiro Machado, A new fractional derivative without singular kernel: Application to the modelling of the steady heat flow, Therm. Sci. 20 (2016), 753–756.

[9]

X. J. Yang and J. A. Tenreiro Machado, A new fractional operator of variable order: application in the description of anomalous diffusion, Phys. A, Stat. Mech. Appl. 481 (2017), 276–283.

[10]

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations North‐Holland Mathematics Studies, Vol. 204, Elsevier Science BV, Amsterdam, 2006.

Digital Library

[11]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[12]

V. Lakshmikantham, Fractional Differential Equations, Academic Press, San Diego, 1999.

[13]

M. V. Thuan and D. C. Huong, New results on stabilization of fractional‐order nonlinear systems via an LMI approachs, Asian J. Control 20 (2018), 1–10.

[14]

D. C. Huong and M. V. Thuan, Design of unknown input reduced‐order observers for a class of nonlinear fractional‐order time‐delay systems, Int. J. Adapt. Control Signal Process. 32 (2018), 412–423.

[15]

E. A. Boroujeni and H. R. Momeni, Non‐fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems, Signal Process. 92 (2012), 2365–2370.

Digital Library

[16]

T. Kaczorek, Reduced‐order fractional descriptor observers for fractional descriptor continuous‐time linear system, Bull. Polish Acad. Sci.: Tech. Sci. 62 (2014), 889–895.

[17]

T. Kaczorek, Reduced‐order perfect nonlinear observers of fractional descriptor discrete‐time nonlinear systems, Int. J. Appl. Math. Comput. Sci. 27 (2017), 245–251.

[18]

H. Trinh, D. C. Huong, and S. Nahavandi, Observers design for positive fractional‐order interconnected time‐delay, Systems. Trans. Inst. Meas. Control. (2018). https://doi.org/10.1177/0142331218757864

[19]

Y. H. Lan, H. X. Huang, and Y. Zhou, Observer‐based robust control of α(1 ≤ α < 2) fractional‐order uncertain systems: a linear matrix inequality approach, IET Contr. Theory Appl. 6 (2012), 229–234.

[20]

Y. Li, Y. Q. Chen, and S. H. Ahn, Fractional‐order iterative learning control for fractional‐order linear systems, Asian J. Control. 13 (2011), 54–63.

[21]

I. N'doye, M. Darouach, and M. Zasadzinski, Design of unkown input fractional‐order observers for fractional‐order systems, Int. J. Appl. Math. Comput. Sci. 23 (2013), 491–500.

Digital Library

[22]

I. N'doye et al., H_∞ adaptive observer for nonlinear fractional‐order systems, Int. J. Adapt. Control Signal Process. 31 (2017), 314–331.

Digital Library

[23]

T. Kaczorek, Perfect nonlinear observers of fractional descriptor continuous‐time nonlinear systems, Fract. Calc. Appl. Anal. 19 (2016), 775–784.

[24]

T. Kaczorek, Reduced‐order fractional descriptor observers for a class of fractional descriptor continuous‐time nonlinear systems, Int. J. Appl. Math. Comput. Sci. 26 (2016), 277–283.

Digital Library

[25]

D. Efimov et al., Interval observers for time‐varying discrete‐time systems, IEEE Trans. Autom. Control 58 (2013), 3218–3224.

[26]

D. Efimov, W. Perruquetti, and A. Zolghadri, Control of nonlinear and LPV systems: Interval observer‐based framework, IEEE Trans. Autom. Control 58 (2013), 773–782.

[27]

Y. Wang, M. B. David, and R. Rajesh, Interval observer design for LPV systems with parametric uncertainty, Automatica 60 (2015), 79–85.

Digital Library

[28]

Z. He and W. Xie, Control of nonlinear switched systems with average dwell time: Interval observer‐based framework, IET Contr. Theory Appl. 10 (2016), 10–16.

[29]

J. L. Gouzé, A. Rapaport, and M. Z. Hadj‐Sadok, Interval observers for uncertain biological systems, Ecol. Model. 133 (2000), 45–56.

[30]

L. Farina and S. Rinaldi, Positive Linear Systems Theory and Applications, John Wiley, New York, 2000.

[31]

F. Mazenc and O. Bernard, Interval observers for linear time‐invariant systems with disturbances systems, Automatica 47 (2011), 140–147.

Digital Library

[32]

Z. Shu et al., Positive observers and dynamic outputfeedback controllers for interval positive linear systems, IEEE Trans. Circuits Syst. I 55 (2008), 3209–3222.

[33]

M. Moisan and O. Bernard, Robust interval observers for global Lipschitz uncertain chaotic systems, Syst. Control Lett. 59 (2010), 687–694.

[34]

D. Efimov et al., Interval estimation for LPV systems applying high order sliding mode techniques, Automatica 48 (2012), 2365–2371.

Digital Library

[35]

Z. He and W. Xie, Control of non‐linear systems based on interval observer design, IET Contr. Theory Appl. 4 (2018), 543–548.

[36]

D.K. Gu, L.W. Liu and G.R. Duan, Functional interval observer for the linear systems with disturbances, IET Contr. Theory Appl. (2018). https://doi.org/10.1049/iet-cta.2018.5113

[37]

S. E. Hamdi et al., Interval state observer design for fractional systems, 10th Int. Multi‐conf. Syst. Signals Devices (SSD), Hammamet, Tunisia, 2013, pp. 1–6.

[38]

T. Raïssi and M. Aoun, On robust pseudo state estimation of fractional order systems, 5th Int. Symp. Positive Systems, Rome, Italy, 2016, pp. 97–111.

[39]

F. Cacace et al., Positive System, Lecture Notes in Control and Information Sciences. Rome: Springer, 2017.

[40]

I. Petras, Fractional‐Order Nonlinear Systems, Springer, Berlin, 2011.

[41]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

[42]

T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer‐Verlag, Berlin Heidelberg, 2011.

[43]

D. G. Luenberger, Introduction to Dynamic Systems: Theory, Models and Applications, John Wiley, New York, 1979.

[44]

F. Cacace, A. Germani, and C. Manes, A new approach to design interval observers for linear systems, IEEE Trans. Autom. Control 60 (2015), 1665–1670.

[45]

G. R. Duan, Generalized Sylvester equations: Unified parametric solutions. Boca Raton:CRC Press, Taylor & Francis Group, (2014).

[46]

B. Zhou and G. R. Duan, A new solution to the generalized Sylvester matrix equation AV − EVF = BW, Syst. Control Lett. 55 (2006), 193–198.

[47]

J. D. Murray, Mathematical Biology. New York:Springer‐Verlag, 1990.

Cited By

Mondal RDey J(2023)Fractional order internal model control of non‐minimum phase and time‐delayed plantsAsian Journal of Control10.1002/asjc.301525:4(3208-3229)Online publication date: 2-Jul-2023
https://dl.acm.org/doi/10.1002/asjc.3015
Gu DLiu QLiu Y(2022)Parametric Design of Functional Interval Observer for Time-Delay Systems with Additive DisturbancesCircuits, Systems, and Signal Processing10.1007/s00034-021-01906-341:5(2614-2635)Online publication date: 1-May-2022
https://dl.acm.org/doi/10.1007/s00034-021-01906-3
Messaoud R(2020)Reduced nonlinear unknown inputs observer using mean value theorem and patternsearch algorithmAutomatica (Journal of IFAC)10.1016/j.automatica.2019.108708112:COnline publication date: 1-Feb-2020
https://dl.acm.org/doi/10.1016/j.automatica.2019.108708
Show More Cited By

Index Terms

Design of functional interval observers for non‐linear fractional‐order systems

Index terms have been assigned to the content through auto-classification.

Recommendations

On Reduced-Order Linear Functional Interval Observers for Nonlinear Uncertain Time-Delay Systems with External Unknown Disturbances

In this paper, we consider the problem of designing reduced-order linear functional interval observers for nonlinear uncertain time-delay systems with external unknown disturbances. Given bounds on the uncertainties, we design two reduced-order linear ...
Robust Tracking and Model Following Control with Zero Tracking Error for Uncertain Dynamical Systems

The problem of robust tracking and model following for a class of linear dynamical systems with time-varying uncertain parameters and disturbances is considered. A class of continuous (nonlinear) state feedback controllers is proposed for robust ...
Brief Adaptive output feedback tracking with almost disturbance decoupling for a class of nonlinear systems

For a class of single-input, single-output uncertain nonlinear systems affected both by uncertain constant parameters (with known bounds) and unknown time-varying bounded disturbances, a new robust adaptive output-feedback tracking control is presented. ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Asian Journal of Control

Asian Journal of Control Volume 22, Issue 3

May 2020

364 pages

ISSN:1561-8625

EISSN:1934-6093

DOI:10.1002/asjc.v22.3

Issue’s Table of Contents

© 2018 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd.

Publisher

John Wiley & Sons, Inc.

United States

Publication History

Published: 28 May 2020

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 28 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

Mondal RDey J(2023)Fractional order internal model control of non‐minimum phase and time‐delayed plantsAsian Journal of Control10.1002/asjc.301525:4(3208-3229)Online publication date: 2-Jul-2023
https://dl.acm.org/doi/10.1002/asjc.3015
Gu DLiu QLiu Y(2022)Parametric Design of Functional Interval Observer for Time-Delay Systems with Additive DisturbancesCircuits, Systems, and Signal Processing10.1007/s00034-021-01906-341:5(2614-2635)Online publication date: 1-May-2022
https://dl.acm.org/doi/10.1007/s00034-021-01906-3
Messaoud R(2020)Reduced nonlinear unknown inputs observer using mean value theorem and patternsearch algorithmAutomatica (Journal of IFAC)10.1016/j.automatica.2019.108708112:COnline publication date: 1-Feb-2020
https://dl.acm.org/doi/10.1016/j.automatica.2019.108708
Cong Huong DThanh Huynh VTrinh H(2020)Interval functional observers for time‐delay systems with additive disturbancesInternational Journal of Adaptive Control and Signal Processing10.1002/acs.314934:9(1281-1293)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.1002/acs.3149

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents