We focus on Taylor Series Methods (TSM) and Automatic Differentiation (AD) for the numerical solution of Ordinary Differential Equations (ODE) characterized by a vector field given by a finite composition of elementary and standard functions. We show that computational advantages are achieved if a kind of pre-processing said Exact Quadratization (EQ) is applied to the ODE before applying the TSM and the AD. In particular, when the ODE function is given by a formal polynomial (i.e. with real powers) of n variables and m monomials, the computational complexity required by our EQ based method for the calculation of the k-th order Taylor coefficient is O(k) whereas by using the existing AD methods it amounts to O(k2).

In this paper, the stabilization problem of nonlinear time-delay systems via quantized sampled-data event-based controllers is investigated. Fully nonlinear (i.e., possibly non-affine in the control) systems affected by state delays are studied. Sufficient conditions are provided for the existence of a suitably fast sampling and of an accurate quantization of the input/output channels such that the digital implementation of the continuous-time controller at hand, updated through a proposed event-triggered mechanism, ensures the semi-global practical stability property, with arbitrarily small final target ball of the origin, of the related closed-loop system. A spline approximation methodology is used in order to cope with the problem of the possible non-availability in the buffer of suitable past values of the system variables needed for the digital implementation of the controller. The stabilization in the sample-and-hold sense theory is used as a tool to prove the results. In the theory here developed, the case of non-uniform quantization of the input/output channels and the case of aperiodic sampling are both included. The proposed theoretical results are validated through an application concerning the plasma glucose regulation problem in type-2 diabetic patients.

When a subject is at rest and meals have not been eaten for a relatively long time (e.g. during the night), presumably near-constant, zero-order glucose production occurs in the liver. Glucose elimination from the bloodstream may be proportional to glycemia, with an apparently first-order, linear elimination rate. Besides glycemia itself, unobserved factors (insulinemia, other hormones) may exert second and higher order effects. Random events (sleep pattern variations, hormonal cycles) may also affect glycemia. The time-course of transcutaneously, continuously measured glycemia (CGM) thus reflects the superposition of different orders of control, together with random system error. The problem may be formalized as a fractional random walk, or fractional Brownian motion. In the present work, the order of this fractional stochastic process is estimated on night-time CGM data from one subject.

Reinforcement learning (RL) is a wellestablished framework for the computation of optimal control policies maximizing the expected reward collected along the evolution of Markov decision processes. In this letter, we extend the RL framework to non-deterministic finite transition systems (FTSs), whose solutions are non-unique but not endowed with a probability measure. We show how to dynamically build RL controllers (possibly learning the FTS model just from experience) maximizing the best-case and worst-case return obtained from a trajectory (run) of the model, assuming full-state information. The framework is successfully applied to the case in which the considered transition system is obtained as a finite approximation of a continuous system, also called a symbolic model. Numerical results on the classical mountain car benchmark highlight the potential of the proposed approach.

A novel integrated dynamic model, the Integrated Fish Model (INTFISH), incorporating mercury (Hg) dynamics at non-steady state in marine organisms, is presented and is applied to the benthic food web in a polluted area. The integrated Fish model represents the dynamics of inorganic mercury (HgII) and methyl-mercury (MeHg) in a real marine ecosystem including environmental (seawater and sediments) and biota compartments. Mercury concentration in fish is estimated using the INTFISH model coupled, in real-time, with results from i) the seawater and sediments modules computed using the HR3DHG model, ii) a dedicated Phytoplankton model and iii) six modules for Hg fluxes within the invertebrate compartment, incorporating the main organisms included in fish diet preferences, whose variations during the whole life cycle are also taken into account to verify the sensitivity of the integrated model to the core set of parameters. The simulated total mercury concentrations (HgTOT) in specimens of red mullet (Mullus barbatus), selected as target species for the Fish model, are in excellent agreement with field observations reported from the investigated area. The intrinsic modularity of the model offers the opportunity to extend simulations to other fish species (which are part of the diet of human populations of interest) and predict Hg concentration in food. A natural extension of the model will allow to evaluate the health risks related to human consumption of contaminated fish.

In this article, we provide an analysis of noise propagation in a stochastic minimal model of chemical self-replication, where a given species can duplicate itself normally. A feedback from the end product on the source, acting as an inhibitor transcription factor, is considered. Stochasticity involves the intrinsic noise affecting gene expression, which is assumed to happen in bursts. The use of a stochastic approach is a novelty within such a framework. The investigation involves the role of the feedback: how it impacts noise attenuation with respect to different modeling choices of stochastic transcription, and with respect to different strengths of the feedback action. The quantification of noise propagation is measured by means of the so called metabolic noise, that is, the coefficient of variation of the end product. Computations are carried out numerically, according to the stochastic simulation algorithm (SSA) properly adapted for the proposed stochastic hybrid systems, as well as analytically: the latter has been achieved by exploiting the linear approximation of the nonlinear terms involved, since otherwise there are no closed loop solutions for the first- and second-order moments. In such a way, noise propagation may be linked to the model parameters, with the SSA aiming at validating the approximated formulas. Results confirm the noise reduction paradigm with feedback.

The complexity of the glucose-insulin system makes the glucose control problem a hard task to accomplish. In this context, a decentralized approach can be of help, through the exploitation of contracts theory, which allows to formalize the fulfillment of safety/invariance specifications over a system in terms of set of assumptions and guarantees over the composing subsystems.We here take a compositional modelbased approach considering, as a first attempt, simplified scalar glucose and insulin subsystems. Assumptions and guarantees sets are piecewise-constant time-varying intervals, computed at sampling times, on the basis of the glucose measurements, so they are not completely known a priori. Updating the intervals may lead to temporary violation of the contracts, according to their classical definition, until the system reaches the new target set. By exploiting the property of monotonicity of the involved subsystems, we define a minimum-time reachability problem, which is solved in closed form to minimize the worst-case contract time violation, and such that the insulin subsystem is steered to a controlled invariant set (reachand- stay specification). Simulations performed in a non-ideal scenario confirm the potential of the proposed approach.

This note investigates how noise propagates in cascades of metabolic transformations. Motivation stems from recent single cell experiments that have shown that noise generated in gene expression and enzymes fluctuations propagates to growth rate through metabolic fluxes. A stochastic approach based on Continuous-Time Markov Chains (CTMC) is exploited to model all reactions, with a special interest in the substrate production, assumed to happen in bursts. Different noise features are dealt with, including correlation of intermediate players, noise impact on the end-product and the role of a feedback from the end-product that may control the substrate production. Most results are given in terms of analytical solutions of the CTMC, in some cases exploiting linear approximations; in all these cases, the findings are validated via Monte Carlo stochastic simulations. The proposed results highlight how substrate production in bursts, cascade length and distance among species affect fluctuations and correlations, with the feedback possibly playing a crucial role in favor of noise propagation.

Reinforcement learning, thanks to the observation-action approach, represents a useful control tool, in particular when the dynamics are characterized by strong non-linearity and complexity. In this sense, it has a natural application in the biological systems field where the complexity of the dynamics makes the automatic control particularly challenging. This paper presents a combined application of neural networks and reinforcement learning, in the so-called field of deep reinforcement learning, for the glucose regulation problem in patients with diabetes mellitus. The glucose control problem is solved through the Deep Deterministic Policy Gradient (DDPG) and the Soft Actor-Critic (SAC) algorithms, where the environment exploited for the agent's interactions is represented by a glucose model that is completely unknown to agents. Preliminary results show that the DDPG and SAC agents can suitably control the glucose dynamics, making the proposed approach promising for further investigations. The comparison between the two agents shows a better behaviour of SAC algorithm.

In this paper, the plasma glucose regulation problem for Type 2 diabetic patients by subcutaneous insulin infusion is studied. A nonlinear time-delay model of the glucose-insulin regulatory system, which takes into account the subcutaneous infusion of insulin, is exploited for the design of a quantized sampled-data static state feedback glucose controller. Quantization in both input/output channels is considered. Spline interpolation methodologies are used in order to obtain an approximation of suitable needed past values of the system state that cannot be always available from the buffer. The stabilization in the sample-and-hold sense is used in order to prove that the proposed digital glucose regulator ensures the semi-global practical stability of the related closed-loop tracking error system, with arbitrarily small final target ball. The proposed digital glucose regulator is validated through simulations.

Attitude control systems for ground vehicles have been an important topic in automotive research for decades, and have been extensively studied by resorting to classical continuous-time nonlinear design. Although this approach can incorporate saturation constraints and actuator dynamics in the design, the computed control laws are often approximated and applied within digital environments in absence of formal performance guarantees. In this letter, we present a quantized sampled-data approach to the vehicle attitude control problem. Starting from classical nonlinear design achieving tracking of prescribed trajectories in continuous time (emulation approach), we derive conditions for preserving the practical stability of the error dynamics by means of quantized sampled-data event-based controllers. Simulations performed in an non-ideal setting confirm the potential of the approach.

This paper deals with the hierarchical real-time control of unmanned aerial vehicles (UAVs) with rule-based strategy for mission time and energetic references generator based on optimal control theory. The objective of this work is to design a control which ensures an energy consumption close to the optimality, and easily implementable thanks to its low computational cost. The first part of the work deals with the extraction of simple and immediate rules for the determination of the optimal mission time, and the generation of 'energetic trajectories' from the analysis of the optimal control strategy results, to minimize the consumption over a high heterogeneous amount of simulations. Then, a hierarchical real-time controller is proposed to track desired energetic trajectories, identified as optimal. The results provided are validated through numerical experiments and compared in terms of energy performance with the optimal solution.

Diabetes is a widespread disease characterized by chronic hyperglycemia so that diabetic individuals usually require the administration of exogenous insulin for survival. As a consequence, in the context of the so-called artificial pancreas, many glucose control methods have been presented in the last few years. In this work, we focus on type-2 diabetes and propose a novel model-based glucose control technique based on the use of symbolic models, which are finite approximations of complex dynamical systems. This framework allows taking into account nonlinearities and delays in the dynamics, uncertainties, and input bounds, as well as nonidealities coming from the interaction between physical plant and digital environment. The methodology is extensively validated over a virtual patient model, broadly accepted as a substitute to animal trials in the preclinical testing of closed-loop glucose control strategies. The results show the effectiveness and the robustness of the approach.

Body weight control is gaining interest since its dysregulation eventually leads to obesity and metabolic disorders. An accurate mathematical description of the behavior of physiological variables in humans after food intake may help in understanding regulation mechanisms and in finding treatments. This work proposes a multi-compartment mathematical model of food intake that accounts for glucose-insulin homeostasis and ghrelin dynamics. The model involves both food volumes and glucose amounts in the two-compartment system describing the gastro-intestinal tract. Food volumes control ghrelin dynamics, whilst glucose amounts clearly impact on the glucose-insulin system. The qualitative behavior analysis shows that the model solutions are mathematically coherent, since they stay positive and provide a unique asymptotically stable equilibrium point. Ghrelin and insulin experimental data have been exploited to fit the model on a daily horizon. The goodness of fit and the physiologically meaningful time courses of all state variables validate the efficacy of the model to capture the main features of the glucose-insulin-ghrelin interplay.

An event-based controller is here proposed for nonlinear retarded systems. Main novelty with respect to the literature is given by the following facts which here hold contemporarily: 1) control Lyapunov-Krasovskii functionals are used; 2) non-smooth feedbacks are allowed; 3) only sampled-data measures of variables in finite dimensional spaces are necessary; 4) non-uniform sampling is allowed. Practical semiglobal stability is proved, with arbitrarily small final target ball of the origin, under suitably high sampling frequency.

In this article, we deal with the problem of stabilizing a nonlinear system with state-delays by means of quantized sampled-data state feedback control laws. Quantization in the state measurements and in the input signal are simultaneously considered. Fully nonlinear (i.e., possibly nonaffine in the control) time-delay systems are studied. Sufficient conditions are provided such that suitably fast sampling and accurate quantization of the state feedback at hand yield semiglobal practical stability, with arbitrarily small final target ball of the origin. Nonlinear delay-free systems are addressed as a special case: it is shown that the above sufficient conditions, ensuring the semiglobal practical stability, are satisfied if the continuous-time static state feedback controller is a global stabilizer. The theory of stabilization in the sample-and-hold sense is used. The theoretical results are validated through an example.

Nonlinear systems with time delays have been object of intense investigation in the last decades for their ability of capturing the behavior of many natural and engineering systems, including ecological, biomedical, industrial, robotic, and networked systems. In this article, a method for the event-triggered stabilization in the sample-and-hold sense of nonlinear time-delay systems with time-varying state delays is presented. The approach exploits control Lyapunov-Razumikhin functions and a spline-based approximation of the functional state feedback, resulting in a finite-memory controller that avoids (by construction) continuous-state monitoring while ensuring also that the minimal interevent time is strictly positive. Examples of application of the illustrated technique highlight the potential of the approach taken.

In this paper, the plasma glucose regulation problem for type 2 diabetic patients is studied. A nonlinear time-delay model of the glucose-insulin regulatory system is exploited to design a quantized sampled-data static output feedback control, using only glucose measurements. It is shown that the proposed control law achieves semiglobal practical stability of the related quantized sampled-data closed-loop glucose-insulin system with arbitrary small steady-state tracking error. The controller involves past values of the glucose, which may not be available in the buffer, for instance because of non-uniform sampling. Such a drawback is overcome by means of spline interpolation. Furthermore, quantization in both input and output channels are taken into account. A pre-clinical validation, concerning the performances of the proposed glucose regulator, is carried out by means of a well-known simulator of diabetic patients, broadly accepted for testing insulin infusion therapies. The simulation results pave the way for further clinical evaluation.

The double phosphorylation/dephosphorylation cycle consists of a symmetric network of biochemical reactions of paramount importance in many intracellular mechanisms. From a network perspective, they consist of four enzymatic reactions interconnected in a specular way. The general approach to model enzymatic reactions in a deterministic fashion is by means of stiff Ordinary Differential Equations (ODEs) that are usually hard to integrate according to biologically meaningful parameter settings. Indeed, the quest for model simplification started more than one century ago with the seminal works by Michaelis and Menten, and their Quasi Steady-State Approximation methods are still matter of investigation nowadays. This work proposes an effective algorithm based on Taylor series methods that manages to overcome the problems arising in the integration of stiff ODEs, without settling for model approximations. The double phosphorylation/dephosphorylation cycle is exploited as a benchmark to validate the methodology from a numerical viewpoint.

In this paper we study the mathematical model of auxiliary (or coupled) reactions, a mechanism which describes several chemical reactions. In order to apply singular perturbation techniques, we determine an appropriate perturbation parameter ? (which is related to the kinetic constants and initial conditions of the model), the inner and outer solutions and the matched expansions of the solutions, up to the first order in ?, in the total quasi-steady-state approximation (tQSSA) framework. The contribution of these expansions can be useful for the estimation of the kinetic parameters of the reaction by means of the interpolation of experimental data with the explicit approximations of the solutions. Some numerical results are discussed, showing the high reliability of the tQSSA with respect to the standard QSSA.