Stability and Dissipativity Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay
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.
KeywordsTime Delay Time Delay System Supply Rate Compartmental System Compartmental Matrix
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- 1.L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications. New York, NY: John Wiley & Sons, .Google Scholar
- 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, .Google Scholar
- 5.J. A. Jacquez, Compartment Analysis in Biology and Medicine, 2nd ed. Ann Arbor University of Michigan Press, .Google Scholar
- 7.S. I. Nicutescu, Delay Effects on Stability: A Robust Control Approach. New York Springer, .Google Scholar
- 9.N. N. Krasovskii, Stability of Motion. Stanford: Stanford University Press. .Google Scholar
- 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, .Google Scholar
- 11.K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics. Kluwer Academic Publishers, .Google Scholar
- 12.Y. Kuang, Delay Differential Equotions with Applications in Population Dynamics. Baston Academic Press, .Google Scholar
- 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.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