Stability and Dissipativity Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay

  • Wassim M. Haddad
  • VijaySekhar Chellaboina
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 38)


Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences and typically involve the exchange of nonnegative quantities between subsystems or compartments wherein each compartment is assumed to be kinetically homogeneous. However, in many engineering and life science systems, transfers between compartments are not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. Including some information of past system states in the system model leads to infinite-dimensional delay nonnegative dynamical systems. In this chapter we present necessary and sufficient conditions for stability of nonnegative and companmental dynamical systems with time delay. Specifically, asymptotic stability conditions for linear and nonlinear as well as continuous-time and discrete-time nonnegative dynamical systems with time delay are established using linear Lyapunov-Krasovskii functionals. Furthermore, we develop new notions of dissipativity theory for nonnegative dynamical systems with time delay using linear storage functionals with linear supply rates. These results are then used to develop general stability criteria for feedback interconnections of nonnegative dynamical systems with time delay.


Time Delay Time Delay System Supply Rate Compartmental System Compartmental Matrix 
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  1. 1.
    L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications. New York, NY: John Wiley & Sons, [2000].Google Scholar
  2. 2.
    W. M. Haddad, V. Chellaboina, and E. August, “Stability and dissipativity theory for nonnegative dynamical systems: A thermodynamic framework for biological and physiological systems,“ in Proc. IEEE Conf. Dec. Contr., (Orlando, FL), pp. 442–458, [2001].Google Scholar
  3. 3.
    D. S. Bernstein and D. C. Hyland, “Compartmental modeling and second-moment analysis of state space systems,” SIAM J. Matrix Anal. Appl., vol. 14, pp. 880–901, [1993].MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    W. Sandberg, “On the mathematical foundations of compartmental analysis in biology, medicine and ecology,” IEEE Trans. Circuits and Systems, vol. 25, pp. 273–279, [1978].MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    J. A. Jacquez, Compartment Analysis in Biology and Medicine, 2nd ed. Ann Arbor University of Michigan Press, [1985].Google Scholar
  6. 6.
    J. K. Hale and S. M. VerduynLunel, Introduction to Functional Differential Equations. New York Springer-Verlag, [1993].zbMATHGoogle Scholar
  7. 7.
    S. I. Nicutescu, Delay Effects on Stability: A Robust Control Approach. New York Springer, [2001].Google Scholar
  8. 8.
    J. C. Willems. “Dissipative dynamical systems part I: General theory,” Arch. Rational Mech. Anal.” vol. 45, pp. 321–351, [1972].MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    N. N. Krasovskii, Stability of Motion. Stanford: Stanford University Press. [1963].Google Scholar
  10. 10.
    S. A. Campbell and J. Belar, “Multiple-delayed differential equations as models for biological control systems,” in Proc. World Math. Conf. pp. 3110–3117, [1993].Google Scholar
  11. 11.
    K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, [1992].Google Scholar
  12. 12.
    Y. Kuang, Delay Differential Equotions with Applications in Population Dynamics. Baston Academic Press, [1993].Google Scholar
  13. 13.
    C. W. Marcus and R. M. Westervelt, “Stability of analog neuml networks with delay,” Phys. Rev., vol. 34, pp. 347–359, [1989].MathSciNetGoogle Scholar
  14. 14.
    W. M. Haddad, T. Hayakawa, and J. M. Bailey, “Nonlinear adaptive control for intensive care unit sedation and opemting room hypnosis,” in Proc. Amer. Contr. Conf., (Denver, CO), pp. 1808–1813, [June 2003].Google Scholar
  15. 15.
    W. M. Haddad. V. Chellaboina, and T. Rajpurohit, “Dissipalivity theory for nonnegative and compartmental dynamical systems with time delay,” in Proc. Amer. Contr. Conf., (Denver, CO), pp. 857–862, [June 2003].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Wassim M. Haddad
    • 1
  • VijaySekhar Chellaboina
    • 2
  1. 1.School of Aerospace Engineering, Georgia Institute of TechnologyAtlanta
  2. 2.Mechanical and Aerospace Engineering, University of MissouriColumbia

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