Skip to main content
SpringerLink
Log in

Stability Analysis and Constrained Control of a Class of Fuzzy Positive Systems with Delays Using Linear Copositive Lyapunov Functional

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

This paper deals with the stability of nonlinear continuous-time positive systems with delays represented by the Takagi–Sugeno (T-S) fuzzy model. A simpler sufficient condition of stability based on linear copositive Lyapunov functional (LCLF) is derived which is not relevant to the magnitude of delays. Based on the result of stability, the problem of controller design via the so-called parallel distributed compensation (PDC) scheme is solved. The control is under a positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. Constrained positive control is also considered, further requiring that the trajectory of the closed-loop system is bounded by a prescribed boundary if the initial condition is bounded by the same boundary. The stability results are formulated as linear programs (LPs) and linear matrix inequalities (LMIs), and the control laws can be obtained by solving a set of bilinear matrix inequalities (BMIs). A numerical example and a real plant are studied to demonstrate the efficiency of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Abbreviations

A⪰0:

All entries of matrix A are nonnegative

A⪯0:

All entries of matrix A are nonpositive

A≻0:

All entries of matrix A are positive

A≺0:

All entries of matrix A are negative

A T :

The transpose of matrix A

\(\boldsymbol{R}^{n}_{+}\) :

n-dimensional positive vector space

R n :

n-dimensional nonnegative vector space

R n :

n-dimensional real vector space

R n×m :

The set of all real matrices of (n×m)-dimension

N :

The set of the natural numbers

M :

the set of Metaler matrices whose off diagonal entries are nonnegative

References

  1. L. Benvenuti, L. Farina, Positive and compartmental systems. IEEE Trans. Autom. Control 47(2), 70–373 (2002)

    Article  MathSciNet  Google Scholar 

  2. A. Benzaouia, A. Hmamed, A.EL. Hajjaji, Stabilization of controlled positive discrete-time T-S fuzzy systems by state feedback control. Int. J. Adapt. Control Signal Process. 24, 1091–1106 (2010)

    Article  MATH  Google Scholar 

  3. A. Benzaouia, D. Mehdi, A.El. Hajjaji, M. Nachidi, Relaxed stabilization of controlled positive discrete-time T-S fuzzy systems, in 18th Mediterranean Conference on Control Automation Congress Palace Hotel (2010), pp. 23–25

    Google Scholar 

  4. A. Berman, M. Neumann, R.J. Stern, Nonnegative Matrices in Dynamic Systems (Wiley, New York, 1989)

    MATH  Google Scholar 

  5. H. Dong, Z. Wang, W.C.Ho. Daniel, H. Gao, Robust H ∞ fuzzy output-feedback control with multiple probabilistic delays and multiple missing measurements. IEEE Trans. Fuzzy Syst. 18(4), 712–725 (2010)

    Article  Google Scholar 

  6. L. Farina, S. Rinaldi, Positive Linear Systems (Wiley Interscience Series, New York, 2000)

    Book  MATH  Google Scholar 

  7. G. Feng, A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst. 14(5), 676–697 (2006)

    Article  Google Scholar 

  8. G. Feng, C.L. Chen, D. Sun, Y. Zhou, H ∞ controller synthesis of fuzzy dynamic systems based on piecewise Lyapunov functions and bilinear matrix inequalities. IEEE Trans. Fuzzy Syst. 13(1), 94–103 (2005)

    Article  Google Scholar 

  9. K.C. Goh, L. Turan, M.G. Safonov, G.P. Papavassilopoulos, J.H. Ly, Biaffine matrix inequality properties and computational methods, in Proc. Amer. Control Conf, Baltimore, MD (1994), pp. 850–855

    Google Scholar 

  10. X. Guan, C. Chen, P. Shi, On robust stability for uncertain time-delay systems: a polyhedral Lyapunov–Krasovskii approach. Circuits Syst. Signal Process. 24(1), 1–18 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. W.M. Haddad, V. Chellaboina, Stability theory for nonnegative and compartmental dynamical systems with time delay. Syst. Control Lett. 51(5), 355–361 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Hinrichsen, N.K. Son, P.H.A. Ngoc, Stability radii of higher order positive difference systems. Syst. Control Lett. 49(5), 377–388 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Hu, Z. Wang, H. Gao, A delay fractioning approach to robust sliding mode control for discrete-time stochastic systems with randomly occurring non-linearities. IMA J. Math. Control Inf. 28(3), 345–363 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Kaczorek, Locally positive nonlinear systems. Int. J. Appl. Math. Comput. Sci. 13(4), 505–509 (2003)

    MathSciNet  MATH  Google Scholar 

  15. X. Liu, C. Dang, Stability analysis of positive switched linear systems with delays. IEEE Trans. Autom. Control 56(7), 1684–1690 (2011)

    Article  MathSciNet  Google Scholar 

  16. X. Liu, L. Wang, W. Yu, S. Zhong, Constrained control of positive discrete-time systems with delays. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 55(2), 193–197 (2008)

    Article  Google Scholar 

  17. X. Liu, W. Yu, L. Wang, Stability analysis for continuous-time positive systems with time-varying delays. IEEE Trans. Autom. Control 55(4), 1024–1028 (2010)

    Article  MathSciNet  Google Scholar 

  18. B. Nagy, M. Matolcsi, A lower bound on the dimension of positive realizations. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50(6), 782–784 (2003)

    Article  MathSciNet  Google Scholar 

  19. M.A. Rami, F. Tadeo, Controller synthesis for positive linear systems with bounded controls. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 54(2), 151–155 (2007)

    Article  Google Scholar 

  20. T. Takagi, M. Sugeno, Fuzzy identification of systems and its application to modeling and control. IEEE Trans. Syst. Man Cybern. 15(11), 116–132 (1985)

    MATH  Google Scholar 

  21. H.B. Zhang, C.Y. Dang, Piecewise H ∞ controller design of uncertain discrete-time fuzzy systems with time delays. IEEE Trans. Fuzzy Syst. 16(6), 1649–1655 (2008)

    Article  Google Scholar 

Download references

Acknowledgements

The work described in this paper was partially supported partly by a grant from the National Natural Science Foundation of China (Grant No. 60904004), partly by a grant from the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2009J020), and partly by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No.: CityU 112809], the grant from Chengdu Administration of Science and Technology (10GGYB184GX-023).

Author information

Authors and Affiliations

  1. Centre for Nonlinear and Complex Systems, School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, P. R. China

    Yanbing Mao & Hongbin Zhang

  2. Department of Systems Engineering & Engineering Management, City University of Hong Kong, Kowloon, Hong Kong

    Chuangyin Dang

Authors
  1. Yanbing Mao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongbin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chuangyin Dang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanbing Mao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mao, Y., Zhang, H. & Dang, C. Stability Analysis and Constrained Control of a Class of Fuzzy Positive Systems with Delays Using Linear Copositive Lyapunov Functional. Circuits Syst Signal Process 31, 1863–1875 (2012). https://doi.org/10.1007/s00034-012-9401-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-012-9401-6

Keywords

Search

Navigation