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Nonnegative and Compartmental Dynamical Systems

Nonnegative and Compartmental Dynamical Systems

Wassim M. Haddad
VijaySekhar Chellaboina
Qing Hui
Copyright Date: 2010
Pages: 624
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  • Book Info
    Nonnegative and Compartmental Dynamical Systems
    Book Description:

    This comprehensive book provides the first unified framework for stability and dissipativity analysis and control design for nonnegative and compartmental dynamical systems, which play a key role in a wide range of fields, including engineering, thermal sciences, biology, ecology, economics, genetics, chemistry, medicine, and sociology. Using the highest standards of exposition and rigor, the authors explain these systems and advance the state of the art in their analysis and active control design.

    Nonnegative and Compartmental Dynamical Systemspresents the most complete treatment available of system solution properties, Lyapunov stability analysis, dissipativity theory, and optimal and adaptive control for these systems, addressing continuous-time, discrete-time, and hybrid nonnegative system theory. This book is an indispensable resource for applied mathematicians, dynamical systems theorists, control theorists, and engineers, as well as for researchers and graduate students who want to understand the behavior of nonnegative and compartmental dynamical systems that arise in areas such as biomedicine, demographics, epidemiology, pharmacology, telecommunications, transportation, thermodynamics, networks, heat transfer, and power systems.

    eISBN: 978-1-4008-3224-8
    Subjects: Mathematics, Technology

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xiv)
  3. Preface
    (pp. xv-xviii)
    Wassim M. Haddad
  4. Chapter One Introduction
    (pp. 1-6)

    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 [155, 171, 172, 235]. 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 [19, 287], metabolic systems [50], epidemic dynamics [155, 157], biochemical reactions [57, 171], endocrine systems [50], and lipoprotein kinetics [171]...

  5. Chapter Two Stability Theory for Nonnegative Dynamical Systems
    (pp. 7-88)

    Even though numerous results focusing on compartmental systems have been developed in the literature (see [4, 29, 88, 100, 155, 158, 209, 211,220, 259] and the numerous references therein), the development of nonnegative dynamical systems theory has received far less attention. In this chapter, we develop several basic mathematical results on stability of linear and nonlinear nonnegative dynamical systems. In addition, usinglinearLyapunov functions, we develop necessary and sufficient conditions for Lyapunov stability, semistability, that is, system trajectory convergence to Lyapunov stable equilibrium points [31, 47], and asymptotic stability for linear nonnegative dynamical systems. The consideration of a linear...

  6. Chapter Three Stability Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay
    (pp. 89-110)

    As discussed in Chapter 1, nonnegative and compartmental models play a key role in understanding many processes in biological and medical sciences [4,100,113,155,158,259]. Such models are composed of homogeneous interconnected subsystems (or compartments) which exchange variable nonnegative quantities of material with conservation laws describing transfer, accumulation, and outflows between compartments and the environment. A key physical limitation of such systems is that transfer between compartments is not instantaneous, and realistic models for capturing the dynamics of such systems should account for material, energy, or information in transit between compartments [106,155,210]. Hence, to accurately describe the evolution of compartmental systems, it...

  7. Chapter Four Nonoscillation and Monotonicity of Solutions of Nonnegative Dynamical Systems
    (pp. 111-142)

    While compartmental systems have wide applicability in biology and medicine, their use in the specific field of pharmacokinetics [95, 297] is particularly noteworthy. The goal of pharmacokinetic analysis often is to characterize the kinetics of drug disposition in terms of the parameters of a compartmental model. This is accomplished by postulating a model, collecting experimental data (typically drug concentrations in blood as a function of time), and then using statistical analysis to estimate parameter values which best describe the data. Differences between the experimental data and those predicted by the model are attributed to measurement noise. Because the ultimate disposition...

  8. Chapter Five Dissipativity Theory for Nonnegative Dynamical Systems
    (pp. 143-196)

    In control engineering, dissipativity theory provides a fundamental framework for the analysis and control design of dynamical systems using an input, state, and output system description based on system-energy-related considerations [112]. The notion of energy here refers to abstract energy notions for which a physical system energy interpretation is not necessary. The dissipation hypothesis on dynamical systems results in a fundamental constraint on their dynamic behavior, wherein a dissipative dynamical system can deliver only a fraction of its energy to its surroundings and can store only a fraction of the work done to it. Many of the great landmarks of...

  9. Chapter Six Hybrid Nonnegative and Compartmental Dynamical Systems
    (pp. 197-222)

    Complex biological and physiological systems typically possess a multiechelon hierarchical hybrid structure characterized by continuous-time dynamics in the lower-level units and logical decision-making units at the higher levels of the hierarchy. The logical decision-making units serve to coordinate and reconcile the (sometimes competing) actions of the lower-level units. Due to their multiechelon hierarchical structure, hybrid dynamical systems are capable of simultaneously exhibiting continuous-time dynamics, discretetime dynamics, logic commands, discrete events, and resetting events. Hence, hybrid dynamical systems involve aninteractingcountable collection of dynamical systems wherein control actions are not independent of one another and yet not all control actions...

  10. Chapter Seven System Thermodynamics, Irreversibility, and Time Asymmetry
    (pp. 223-262)

    Energy is a concept that underlies our understanding of all physical phenomena and is a measure of the ability of a dynamical system to produce changes (motion) in its own system state as well as changes in the system states of its surroundings. Thermodynamics is a physical branch of science that deals with laws governing energy flow from one body to another and energy transformations from one form to another. These energy flow laws are captured by the fundamental principles known as the first and second laws of thermodynamics. The first law of thermodynamics gives a precise formulation of the...

  11. Chapter Eight Finite-Time Thermodynamics
    (pp. 263-280)

    In Chapter 7, we used a compartmental dynamical systems perspective to provide a system-theoretic approach to thermodynamics. Specifically, using a state space formulation, we developed a nonlinear compartmental dynamical system model characterized by energy conservation laws that is consistent with basic thermodynamic principles. In the case where the compartmental system is isolated we showed that the dynamical system asymptotically evolves toward a state of energy equipartition. However, in physical systems, energy and temperature equipartition is achieved in finite time rather than merely asymptotically.

    In this chapter, we merge the theories of semistability and finite-time stability developed in [32–34] to...

  12. Chapter Nine Modeling and Analysis of Mass-Action Kinetics
    (pp. 281-314)

    Mass-action kinetics are used in chemistry and chemical engineering to describe the dynamics of systems of chemical reactions, that is, reaction networks [281]. These models are a special form of compartmental systems, which involve mass- and energy-balance relations [29, 158]. Aside from their role in chemical engineering applications, mass-action kinetics have numerous analytical properties that are of inherent interest from a dynamical systems perspective. For example, mass-action kinetics give rise to systems of differential equations having polynomial nonlinearities. Polynomial systems are notorious for their intricate analytical properties even in lowdimensional cases [159, 179, 180, 265]. Because of physical considerations, however,...

  13. Chapter Ten Semistability and State Equipartition of Nonnegative Dynamical Systems
    (pp. 315-342)

    In Chapter 7, we developed a compartmental dynamical systems approach to thermodynamics. Specifically, each compartment represented the energy content of the different parts of the dynamical system, and different compartments interacted by exchanging heat. A key assumption in Chapter 7 was that intercompartmental energy flows between connected compartments were bidirectional. However, in many applications of thermal and fluid sciences the assumption of bidirectional energy flow or fluid flow between compartments can be limiting. In addition, transfers between compartments are not instantaneous and realistic models for capturing the dynamics of fluid and thermal systems should account for material or energy in...

  14. Chapter Eleven Robustness of Nonnegative Dynamical Systems
    (pp. 343-358)

    Even though numerous analysis results for nonnegative and compartmental dynamical systems have been developed over the last several years in the literature, robustness properties of these systems have been largely ignored. Robustness here refers to sensitivity of the system stability in the face of model uncertainty. In this chapter, we build on the results of Chapters 8 and 10 to examine the robustness of several nonnegative and compartmental systems of a specified structure. In particular, we develop sufficient conditions for robust stability of compartmental systems involving higher-order perturbation terms that scale in a consistent fashion with respect to a scaling...

  15. Chapter Twelve Modeling and Control for Clinical Pharmacology
    (pp. 359-378)

    Control technology impacts modern medicine through robotic surgery, electrophysiological systems (pacemakers and automatic implantable de-fibrillators), life support (ventilators, artificial hearts), and image-guided therapy and surgery. An additional area of medicine suited for applications of control is clinical pharmacology, in which mathematical modeling plays a prominent role [155, 305]. Although numerous drugs are available for treating disease, proper dosing is often imprecise, resulting in increased costs, morbidity, and mortality. In this chapter, we discuss potential applications of control technology to clinical pharmacology, specifically the control of drug dosing [223].

    We begin by considering how dosage guidelines are developed. Drug development begins...

  16. Chapter Thirteen Optimal Fixed-Structure Control for Nonnegative Systems
    (pp. 379-404)

    In this chapter, we develop optimal output feedback controllers for set-point regulation of linear nonnegative and compartmental dynamical systems. In particular, we extend the optimal fixed-structure control framework [28, 30] to develop optimal output feedback controllers that guarantee that the trajectories of the closed-loop plant system states remain in the nonnegative orthant of the state space for nonnegative initial conditions. The proposed optimal fixed-structure control framework is aconstrainedoptimal control methodology that does not seek to optimize a performance measure perse, but rather seeks to optimize performance within a class of fixedstructure controllers satisfying internal controller constraints that guarantee...

  17. Chapter Fourteen H₂ Suboptimal Control for Nonnegative Dynamical Systems Using Linear Matrix Inequalities
    (pp. 405-424)

    Even though nonnegative systems are often encountered in numerous application areas, nonnegative orthant stabilizability and holdability has received little attention in the literature. Notable exceptions include [20, 193]. In addition, fixed-structure control for linear nonnegative dynamical systems, and adaptive and neuroadaptive control of nonnegative and compartmental systems have been recently developed in [120, 122, 132, 133, 228]. In this chapter, we use linear matrix inequalities (LMIs) to developH₂(sub) optimal estimators and controllers for nonnegative dynamical systems. Linear matrix inequalities provide a powerful design framework for linear control problems [39,72,236,260]. Since LMIs lead to convex or quasiconvex optimization problems,...

  18. Chapter Fifteen Adaptive Control for Nonnegative Systems
    (pp. 425-490)

    One of the fundamental problems in feedback control design is the ability of the control system to guarantee robust stability and robust performance with respect to system uncertainties in the design model. To this end, adaptive control along with robust control theory have been developed to address the problem of system uncertainty in control-system design. The fundamental differences between adaptive control design and robust control design can be traced to the modeling and treatment of system uncertainties as well as the controller architecture structures. In particular, adaptive control [10,153,227] is based on constant linearly parameterized system uncertainty models of a...

  19. Chapter Sixteen Adaptive Disturbance Rejection Control for Compartmental Systems
    (pp. 491-522)

    In Chapter 15 a direct adaptive control framework for linear and nonlinear nonnegative and compartmental systems was developed. The framework in Chapter 15 is Lyapunov based and guarantees partial asymptotic setpoint regulation, that is, asymptotic set-point stability with respect to the closed-loop system states associated with the plant. In addition, the adaptive controllers in Chapter 15 guarantee that the physical system states remain in the nonnegative orthant of the state space. In this chapter, we extend the results of Section 15.2 to develop a direct adaptive control framework for adaptive stabilization and disturbance rejection for compartmental dynamical systems with exogenous...

  20. Chapter Seventeen Limit Cycle Stability Analysis and Control for Respiratory Compartmental Models
    (pp. 523-552)

    Acute respiratory failure due to infection, trauma, and major surgery is one of the most common problems encountered in intensive care units, and mechanical ventilation is the mainstay of supportive therapy for such patients. Numerous mathematical models of respiratory function have been developed in the hope of better understanding pulmonary function and the process of mechanical ventilation [17, 46, 77, 214, 299]. However, the models that have been presented in the medical and scientific literature have typically assumed homogeneous lung function. For example, in analogy to a simple electrical circuit, the most common model has assumed that the lungs can...

  21. Chapter Eighteen Identification of Stable Nonnegative and Compartmental Systems
    (pp. 553-570)

    As discussed throughout this monograph, nonnegative and compartmental systems are essential in capturing the phenomenological behavior of biological and physiological systems, and their use in the specific field of pharmacokinetics is particularly noteworthy. The goal of pharmacokinetic analysis often is to characterize the kinetics of drug disposition in terms of the parameters of a compartmental model. This is accomplished by postulating a model, collecting experimental data (typically drug concentrations in blood as a function of time), and then using statistical analysis to estimate the parameter values that best describe the data. There are numerous sources of noise in the data,...

  22. Chapter Nineteen Conclusion
    (pp. 571-572)

    In this monograph, we have developed a dynamical systems and control theory framework for nonnegative and compartmental systems. These systems are conceptually simple and yet remarkably effective in describing the essential features of a wide range of dynamical systems involving compartments containing variable nonnegative quantities of a particular substance coupled by connectors governed by intercompartmental flow laws. Such models include biomedical, demographic, epidemic, ecological, economic, pharmacological, telecommunications, transportation, power, heat transfer, fluid, structural vibration, network, and thermodynamic systems. In addition, suboptimal, optimal, and adaptive feedback control architectures for nonnegative and compartmental dynamical systems were developed. It was shown that these...

  23. Bibliography
    (pp. 573-598)
  24. Index
    (pp. 599-605)