Über die Autorin bzw. den Autor

Wassim M. Haddad, VijaySekhar Chellaboina & Qing Hui

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Nonnegative and Compartmental Dynamical Systems

PRINCETON UNIVERSITY PRESS

Contents

Preface.....................................................................................................................................xvChapter 1. Introduction.....................................................................................................................1Chapter 2. Stability Theory for Nonnegative Dynamical Systems...............................................................................7Chapter 3. Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay.............................................89Chapter 4. Nonoscillation and Monotonicity of Solutions of Nonnegative Dynamical Systems....................................................111Chapter 5. Dissipativity Theory for Nonnegative Dynamical Systems...........................................................................143Chapter 6. Hybrid Nonnegative and Compartmental Dynamical Systems...........................................................................197Chapter 7. System Thermodynamics, Irreversibility, and Time Asymmetry.......................................................................223Chapter 8. Finite-Time Thermodynamics.......................................................................................................263Chapter 9. Modeling and Analysis of Mass-Action Kinetics....................................................................................281Chapter 10. Semistability and State Equipartition of Nonnegative Dynamical Systems..........................................................315Chapter 11. Robustness of Nonnegative Dynamical Systems.....................................................................................343Chapter 12. Modeling and Control for Clinical Pharmacology..................................................................................359Chapter 13. Optimal Fixed-Structure Control for Nonnegative Systems.........................................................................379Chapter 14. H2 Suboptimal Control for Nonnegative Dynamical Systems Using Linear Matrix Inequalities.....................405Chapter 15. Adaptive Control for Nonnegative Systems........................................................................................425Chapter 16. Adaptive Disturbance Rejection Control for Compartmental Systems................................................................491Chapter 17. Limit Cycle Stability Analysis and Control for Respiratory Compartmental Models.................................................523Chapter 18. Identification of Stable Nonnegative and Compartmental Systems..................................................................553Chapter 19. Conclusion......................................................................................................................571Bibliography................................................................................................................................573Index.......................................................................................................................................599

Chapter One

With the ever-increasing influence of mathematical modeling and engineering on biological, social, and medical sciences, it is not surprising that dynamical system theory has played a central role in the understanding of many biological, ecological, and physiological processes. With this confluence it has rapidly become apparent that mathematical modeling and dynamical system theory are the key threads that tie together these diverse disciplines. The dynamical models of many biological, pharmacological, and physiological processes such as pharmacokinetics, metabolic systems, epidemic dynamics, biochemical reactions, endocrine systems, and lipoprotein kinetics are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. Hence, it follows from physical considerations that the state trajectory of such systems remains in the nonnegative orthant of the state space for nonnegative initial conditions. Such systems are commonly referred to as nonnegative dynamical systems in the literature.

A subclass of nonnegative dynamical systems are compartmental systems. Compartmental systems involve dynamical models that are characterized by conservation laws (e.g., mass, energy, fluid, etc.) capturing the exchange of material between coupled macroscopic subsystems known as compartments. Each compartment is assumed to be kinetically homogeneous, that is, any material entering the compartment is instantaneously mixed with the material of the compartment. The range of applications of nonnegative systems and compartmental systems is not limited to biological, social, and medical systems. Their usage includes chemical reaction systems, queuing systems, large-scale systems, stochastic systems (whose state variables represent probabilities), ecological systems, economic systems, demographic systems, telecommunications systems, transportation systems, power systems, heat transfer systems, thermodynamic systems, and structural vibration systems, to cite but a few examples.

In economic systems the interaction of raw materials, finished goods, and financial resources can be modeled by compartments representing various interacting sectors in a dynamic economy. Similarly, network systems, computer networks, and telecommunications systems are all amenable to compartmental modeling with intercompartmental flow laws governed by nodal dynamics and rerouting strategies that can be controlled to minimize waiting times and optimize system throughput. Compartmental models can also be used to model the interconnecting components of power grid systems with energy flow between regional distribution points subject to control and possible failure. Road, rail, air, and space transport systems also give rise to compartmental systems with interconnections subject to failure and real-time modification.

Since the aforementioned dynamical systems have numerous input, state, and output properties related to conservation, dissipation, and transport of mass, energy, or information, nonnegative and compartmental models are conceptually simple yet remarkably effective in describing the essential phenomenological features of these dynamical systems. Furthermore, since such systems are governed by conservation laws and are comprised of homogeneous compartments which exchange variable nonnegative quantities of material via intercompartmental flow laws, these systems are completely analogous to network thermodynamic (advection-diffusion) systems with compartmental masses, energies, or information playing the role of heat and temperatures.

The goal of the present monograph is directed toward developing a general stability analysis and control design framework for nonlinear nonnegative and compartmental dynamical systems. However, as in general nonlinear systems, nonlinear nonnegative dynamical systems can exhibit a very rich dynamical behavior, such as multiple equilibria, limit cycles, bifurcations, jump resonance phenomena, and chaos, which can make general nonlinear nonnegative system analysis and control notoriously difficult. In addition, since nonnegative and compartmental dynamical systems have specialized structures, nonlinear nonnegative system stabilization has...