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.     

Neural network adaptive control for nonlinear nonnegative dynamical systems

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. In this paper, we develop a full-state feedback neural adaptive control framework for adaptive set-point regulation of nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains. In addition, the neural adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state-space for nonnegative initial conditions.     

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.     

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.     

A dissipative dynamical systems approach to stability analysis of time delay systems

In this paper the concepts of dissipativity and the exponential dissipativity are used to provide sufficient conditions for guaranteeing asymptotic stability of a time delay dynamical system. Specifically, representing a time delay dynamical system as a negative feedback interconnection of a finite‐dimensional linear dynamical system and an infinite‐dimensional time delay operator, we show that the time delay operator is dissipative with respect to a quadratic supply rate and with a storage functional involving an integral term identical to the integral term appearing in standard Lyapunov–Krasovskii functionals. Finally, using stability of feedback interconnection results for dissipative systems, we develop sufficient conditions for asymptotic stability of time delay dynamical systems. The overall approach provides a dissipativity theoretic interpretation of Lyapunov–Krasovskii functionals for asymptotically stable dynamical systems with arbitrary time delay. Copyright © 2004 John Wiley & Sons, Ltd.     

Observer-based control of a class of time-delay nonlinear systems having triangular structure

Time-delay systems constitute a special class of dynamical systems that are frequently present in many fields of engineering. It has been shown in the literature that the existence of a stabilizing observer-based controller is related to delay-dependent conditions that are generally satisfied for a small time delay. Motivating works towards reducing the conservatism of the results are among the on-going research topics especially when partial-state measurements are imposed. This paper investigates the problem of observer-based stabilization of a class of time-delay nonlinear systems written in triangular form. First, we show that a delay nonlinear observer is globally convergent under the global Lipschitz condition of the system nonlinearity. Then, it is shown that a parameterized linear feedback that uses the observer states can stabilize the system whatever the size of the delay. An illustrative example is provided to approve the theoretical results.     

Linear and nonlinear stabilizing continuous controllers ofuncertain dynamical systems including state delay

The problem of robust stabilization of a class of time-varying dynamical systems with disturbance, uncertain parameters, and a bounded time-varying state delay is considered. Two classes of stabilizing continuous controllers are proposed. The proposed controllers can guarantee the existence of the solution to the dynamical system in the usual sense and can also be directly implemented in the practical control problems. Moreover, since the proposed controllers are completely independent of the time delay (which is only assumed to be any nonnegative bounded and continuous function), the results developed in this note are applicable to a class of dynamical systems with uncertain time delays. Finally, a numerical example is given to demonstrate the validity of the results     

多模式网络化控制系统稳定性分析

网络传输引起的时滞在网络化控制系统中经常出现．针对存在多个运动模式的网络化控制系统,建立了时滞混杂自动机模型,分析了系统平衡点的稳定性,给出了平衡点保持稳定的充分条件在于构造的与离散状态相关的Lyapunov泛函在离散状态的切换时刻是非增的．当时滞混杂自动机中连续性子系统为线性时不变时滞系统时,可以利用线性矩阵不等式方法来寻找系统的公共Lyapunov泛函,并求解使系统平衡点保持稳定的时滞上界．最后,给出了一个例子说明了分析结果．     

Passivity-based neural network adaptive output feedback control for nonlinear nonnegative dynamical systems

The potential clinical applications of adaptive neural network control for pharmacology in general, and anesthesia and critical care unit medicine in particular, are clearly apparent. Specifically, monitoring and controlling the depth of anesthesia in surgery is 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 neural adaptive output feedback control framework for adaptive set-point regulation of nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains. The approach is applicable to nonlinear nonnegative systems with unmodeled dynamics of unknown dimension and guarantees that the physical system states remain in the nonnegative orthant of the state-space for nonnegative initial conditions. Finally, a numerical example involving the infusion of the anesthetic drug midazolam for maintaining a desired constant level of depth of anesthesia for noncardiac surgery is provided to demonstrate the efficacy of the proposed approach.     

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.     

A gap metric based multiple model approach for nonlinear switched systems

Many applications in chemical engineering often exhibit a switching character due to the presence of discrete modes in the course of their operation. First principles models of such systems constructed using process simulators are far too complex for use in online applications, especially in model-based control. For such systems, numerous control-relevant modeling approaches have been reported in the literature such as mixed logic dynamical (MLD) models [1] and piece wise affine (PWA) [2] models among others. These models describe the evolution of states in each discrete mode using linear equations. Fewer control-relevant models have been reported that address the nonlinear behavior of switched systems. To model nonlinear hybrid systems, Nandola and Bhartiya [3] proposed a multiple linear model approach wherein multiple linear models are used to describe the dynamic behavior in each mode of the hybrid system. However, no guidelines were provided to select the number of models necessary in each mode and their region of validity. In this work, we address these lacunae by presenting a systematic multiple model approach to describe nonlinear switched systems. The method involves a trajectory based linearization and employs a model bank with a set of local linear models for each discrete operational mode. The model bank is generated by linearizing the first principles model across a carefully designed trajectory based on accuracy of multi-step ahead predictions. The numerous models thus obtained are clustered using the gap metric as the distance measure and representative models are selected. The selected linear models are aggregated using Bayesian or Fuzzy approaches to obtain the global model for the nonlinear switched system. A simulation case study of spherical two-tank system and an experimental case study of a benchmark problem consisting of three tanks are used to validate the proposed modeling strategy.     

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

In adaptive control of uncertain dynamical systems, it is well known that the presence of actuator and/or unmodeled dynamics in feedback loops can yield to unstable closed‐loop system trajectories. Motivated by this standpoint, this paper focuses on the analysis and synthesis of multiple adaptive architectures for control of uncertain dynamical systems with both actuator and unmodeled dynamics. Specifically, we first analyze model reference adaptive control architectures with standard, hedging‐based, and expanded reference models for this class of uncertain dynamical systems and develop sufficient stability conditions. We then synthesize a robustifying term for the latter architecture and analytically show that this term can allow for a relaxed sufficient stability condition. The proposed theoretical treatments involve Lyapunov stability theory, linear matrix inequalities, and matrix mathematics. Finally, we compare the resulting sufficient stability conditions of the considered adaptive control architectures on a benchmark mechanical system subject to actuator and unmodeled dynamics.     

Neural network adaptive output feedback control for intensive care unit sedation and intraoperative anesthesia.

The potential applications of neural adaptive control for pharmacology, in general, and anesthesia and critical care unit medicine, in particular, are clearly apparent. Specifically, monitoring and controlling the depth of anesthesia in surgery is 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 neural adaptive output feedback control framework for nonlinear uncertain nonnegative and compartmental systems with nonnegative control inputs. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals. In addition, the neural adaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space. Finally, the proposed approach is used to control the infusion of the anesthetic drug propofol for maintaining a desired constant level of depth of anesthesia for noncardiac surgery.     

A model of angiogenesis by hybrid systems with delay on the piecewise constant part

An important approach towards understanding the cancer dynamics is the modeling of angiogenesis process. There have been several attempts to model this process. Among them angiogenesis models with time delays, caused by the physical distance between the tumor and the vessel, are the most realistic ones. Recent studies have suggested that those delays can cause oscillatory behavior in the angiogenesis process. In this work we employed piecewise linear hybrid systems with delay on the piecewise constant part. Our approach is based on piecewise linearization of the system behavior where the delays occur at threshold crossings and state transitions. Piecewise linear systems with a single threshold for each variable are useful in approximating and modeling the dynamical systems especially when the model might need to be calibrated by the observations. Therefore, we used piecewise linear systems where the delays are introduced in piecewise constant part of the equations. Our approach allows tractable approximation of the angiogenesis process with possible advances of incorporating more variables, involving the effect of some possible external inputs, and possible adjustment or correction of parameters by observations.     

Robust H∞ Filtering and Deconvolution for Continuous Time Delay Systems Based on Game‐Theoretic Approach

Many dynamical systems involve not only process and measurement noise signals but also parameter uncertainty and unknown input signals. This paper aims to estimate the state and unknown input for linear continuous time‐varying systems subject to time delay in state, norm‐bounded parameter uncertainty, and a known input. Such a problem is reformulated into a two‐player differential game whose saddle point solution gives rise to one sufficient solvable condition for the estimation problem. The possible optimal estimators are obtained by solving the two coupled Riccati differential equations. We demonstrate, through two examples, how the proposed estimator is valid for estimating state and unknown input.     

Adaptive Control for the Systems Preceded by Hysteresis

Hysteresis hinders the effectiveness of smart materials in sensors and actuators. It is a challenging task to control the systems with hysteresis. This note discusses the adaptive control for discrete time linear dynamical systems preceded with hysteresis described by the Prandtl-Ishlinskii model. The time delay and the order of the linear dynamical system are assumed to be known. The contribution of the note is the fusion of the hysteresis model with adaptive control techniques without constructing the inverse hysteresis nonlinearity. Only the parameters (which are generated from the parameters of the linear system and the density function of the hysteresis) directly needed in the formulation of the controller are adaptively estimated online. The proposed control law ensures the global stability of the closed-loop system, and the output tracking error can be controlled to be as small as required by choosing the design parameters. Simulation results show the effectiveness of the proposed algorithm.     

Solvability of static output-feedback stabilization for LTI positive systems

This paper addresses the stabilization problem of positive linear systems which have nonnegative states whenever the initial conditions are nonnegative. The synthesis of static output-feedback controllers that ensure the positivity and asymptotic stability of the closed-loop system is investigated. It is shown that this important problem is completely solved for single-input and single-output positive systems. The proposed approach can be applied to multi-input positive systems with controllers having one rank gains. All the provided conditions are necessary and sufficient and can be solved in terms of Linear Programming.     

Stable inversion of LPTV systems with application in position-dependent and non-equidistantly sampled systems

Many control applications, including feedforward and learning control, involve the inverse of a dynamical system. For nonminimum-phase systems, the response of the inverse system is unbounded. For linear time-invariant (LTI), nonminimum-phase systems, a bounded, noncausal inverse response can be obtained through an exponential dichotomy. For generic linear time-varying (LTV) systems, such a dichotomy does not exist in general. The aim of this paper is to develop an inversion approach for an important class of LTV systems, namely linear periodically time-varying (LPTV) systems, which occur in, e.g. position-dependent systems with periodic tasks and non-equidistantly sampled systems. The proposed methodology exploits the periodicity to determine a bounded inverse for general LPTV systems. Conditions for existence are provided. The method is successfully demonstrated in several application cases, including position-dependent and non-equidistantly sampled systems.