Compartmental Dynamical Systems Research Articles

Representing and Understanding the Carbon Cycle Using the Theory of Compartmental Dynamical Systems

AbstractModels representing exchange of carbon between the atmosphere and the terrestrial biosphere include a large variety of processes and mechanisms, and have increased in complexity in the last decades. These models are no exception of the simulation versus understanding conundrum previously articulated for models of the physical climate, which states that increasing detail in process representation in models, and the simulations they produce, hinders understanding of holistic system behavior. However, recent theoretical progress on the mathematical representation of the carbon cycle in ecosystems may help to provide a general framework for the qualitative understanding of models without compromising detail in process representation. Here we (1) briefly review recent ideas on the theory of transient dynamics of the terrestrial carbon cycle and its matrix representation, pointing out issues of interpretation, (2) show that these ideas can be further generalized in the mathematical concept of nonautonomous compartmental systems, and (3) provide thoughts on how this framework can be used to address a new set of questions in carbon cycle science.

Adaptive disturbance rejection control for compartmental systems with application to intraoperative anesthesia influenced by hemorrhage and hemodilution effects

AbstractCompartmental system models involve dynamic states whose values are nonnegative. These models are widespread in biological and physiological sciences and play a key role in understanding these processes. In this paper, we develop a direct adaptive disturbance rejection control framework for compartmental dynamical systems with exogenous bounded disturbances. The proposed framework is Lyapunov based and guarantees partial asymptotic stability of the closed‐loop system, that is, asymptotic stability with respect to part of the closed‐loop system states associated with the plant dynamics. The remainder of the states associated with the adaptive controller gains are shown to be Lyapunov stable. In the case of bounded energy ℒ︁2 disturbances, the proposed approach guarantees a nonexpansivity constraint on the closed‐loop input–output map between the plant disturbances and performance variables. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for surgery in the face of continuing hemorrhage and hemodilution is provided. Copyright © 2008 John Wiley & Sons, Ltd.

Adaptive control for nonlinear compartmental dynamical systems with applications to clinical pharmacology

There are significant potential clinical applications of adaptive control for pharmacology in general, and anesthesia and critical care unit medicine in particular. Specifically, monitoring and controlling the levels of consciousness in surgery are of particular importance. Nonnegative and compartmental models provide a broad framework for biological and physiological systems, including clinical pharmacology, and are well suited for developing models for closed-loop control of drug administration. In this paper, we develop a direct adaptive control framework for nonlinear uncertain nonnegative and compartmental systems with nonnegative control inputs. The proposed framework is Lyapunov-based and guarantees partial asymptotic set-point regulation, that is, asymptotic set-point regulation with respect to part of the closed-loop system states associated with the plant. In addition, the adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of consciousness for noncardiac surgery is provided to demonstrate implementation of the proposed approach.

Adaptive Control of Mammillary Drug Delivery Systems with Actuator Amplitude Constraints and System Time Delays

In this paper, we develop a direct adaptive control framework for uncertain linear compartmental dynamical systems with actuator amplitude constraints and system time delays. The specific focus of the paper is on compartmental pharmacokinetic models and their applications to drug delivery systems. In particular, we develop a Lyapunov–Krasovskii-based direct adaptive control framework for guaranteeing set-point regulation of the closed-loop system in the nonnegative orthant in the presence of unknown system time delays and control amplitude constraints. The framework additionally guarantees nonnegativity of the control signal. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for noncardiac surgery is provided to demonstrate the framework on a drug delivery mammillary model for general anesthesia involving system time delays and control infusion rate constraints.

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 note, we develop 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.

Adaptive control for nonlinear nonnegative dynamical systems

Nonnegative and compartmental models are widespread in engineering systems and life sciences and play a key role in the understanding of these systems. In this paper, we develop a direct adaptive control framework for nonlinear uncertain nonnegative and compartmental dynamical systems. The proposed framework is Lyapunov-based and guarantees partial asymptotic set-point regulation; that is, asymptotic set-point regulation with respect to part of the closed-loop system states associated with the plant. In addition, the adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space.

On monotonicity of solutions of discrete-time nonnegative and compartmental dynamical systems

Nonnegative and compartmental dynamical system models are widespread in biological, physiological, and pharmacological sciences. Since the state variables of these systems are typically masses or concentrations of a physical process, it is of interest to determine necessary and sufficient conditions under which the system states possess monotonic solutions. In this paper, we present necessary and sufficient conditions for identifying discrete-time nonnegative and compartmental dynamical systems that only admit monotonic solutions.

Stability and dissipativity theory for discrete-time non-negative and compartmental dynamical systems

Non-negative and compartmental dynamical systems are derived from mass and energy balance considerations that involve dynamic states whose values are non-negative. These models are widespread in engineering, biomedicine and ecology. In this paper we develop several results on stability, dissipativity and stability of feedback interconnections of discrete-time linear and non-linear non-negative dynamical systems. Specifically, using linear Lyapunov functions we first develop necessary and sufficient conditions for Lyapunov stability and asymptotic stability for non-negative systems. In addition, using linear and non-linear storage functions with linear supply rates we develop new notions of dissipativity theory for non-negative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of non-negative dynamical systems.

Stability theory for nonnegative and compartmental dynamical systems with time delay

Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations and involve the exchange of nonnegative quantities between subsystems or compartments. These models are widespread in biological and physical sciences and play a key role in understanding these processes. A key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments. In this paper, we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay. Specifically, asymptotic stability conditions for linear and nonlinear nonnegative dynamical systems with time delay are established using linear Lyapunov–Krasovskii functionals.

Adaptive control for non‐negative and compartmental dynamical systems with applications to general anesthesia

AbstractNon‐negative and compartmental dynamical system models are composed of homogeneous interconnected subsystems or compartments which exchange variable non‐negative quantities of material with conservation laws describing transfer, accumulation, and elimination between the compartments and the environment. These models are widespread in biological and physiological sciences and play a key role in understanding these processes. In this paper, we develop a direct adaptive control framework for linear uncertain non‐negative and compartmental systems. The proposed framework is Lyapunov‐based and guarantees partial asymptotic set‐point regulation; that is, asymptotic set‐point stability with respect to part of the closed‐loop system states associated with the plant. In addition, the adaptive controller guarantees that the physical system states remain in the non‐negative orthant of the state space. Finally, a numerical example involving the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for non‐cardiac surgery is provided to demonstrate the efficacy of the proposed approach. Copyright © 2003 John Wiley & Sons, Ltd.

Hybrid nonnegative and computational dynamical systems

Nonnegative and Compartmental dynamical 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 typically possess hierarchical (and possibly hybrid) structures and are remarkably effective in capturing the phenomenological features of many biological and physiological dynamical systems. In this paper we develop several results on stability and dissipativity of hybrid nonnegative and Compartmental dynamical systems. Specifically, using linear Lyapunov functions we develop sufficient conditions for Lyapunov and asymptotic stability for hybrid nonnegative dynamical systems. In addition, using linear and nonlinear storage functions with linear hybrid supply rates we develop new notions of dissipativity theory for hybrid nonnegative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of hybrid nonnegative dynamical systems.