Asymptotic Stability Properties Research Articles

Input Mapping Design for Batch-to-Batch Optimization With Limited Memory

This brief discusses data-driven design techniques for batch-to-batch optimization problems and proposes a new input-mapping-based online uncertainty compensation method for optimization-based iterative learning control (ILC) with limited memory. Since process uncertainties are generally inevitable, we collect historical data that incorporates past inputs and outputs to provide more insight into plant uncertain dynamics, resulting in a more accurate optimization model for optimal ILC solution. Instead of learning from a single step, our proposed method maintains a memory of the latest executed steps and updates the ILC solution using a linear combination of the memory and quadratic programming. A rigorous theoretical analysis shows that such a design provides robust benefits as well as asymptotic and monotonic stability properties under mild conditions. Finally, we demonstrate our design through an illustrative numerical example.

Global asymptotic behavior of a discrete system of difference equations with delays

In the present paper, we mainly investigate the qualitative behavior of the solutions of a discrete system of difference equations xn+1 = ? + Pmi =1 xn?i yn , yn+1 = ? + Pmi =1 yn?i xn , n ? N where ?, ? ? (0,?), m ? Z+, x?i and y?i are non-negative real numbers for i ? {0, 1, ...,m}. Namely, we discuss the boundedness character and the asymptotic stability properties of steady states of the mentioned system. Finally, for this system, we give a rate of convergence result which has an important place in the discrete dynamical systems. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.

The random two-sector RSS model: On discounted optimal growth without Ramsey-Euler conditions

This paper shows that the introduction of uncertainty in the two-sector model due to Robinson-Solow-Srinivasan (RSS) fully subdues the veritable plethora of the results that have been obtained the theory of deterministic optimal growth. Rather than an “anything goes” theorem that admits optimal cyclical and chaotic trajectories for the discrete-time deterministic version, we present results on the existence, uniqueness, asymptotic stability and comparative-static properties of the steady state measure. We relate the basic intuition of our result to global games, and note that the properties of value and policy functions we identify rely on “supermodularity” and “increasing-differences property” of Veinott-Topkis-Milgrom-Shannon. While of interest in themselves, our results highlight a methodological advance in developing the theory of optimal growth without Ramsey-Euler conditions.

Receding-Horizon Control of Constrained Switched Systems with Neural Networks as Parametric Function Approximators

This work studies receding-horizon control of discrete-time switched linear systems subject to polytopic constraints for the continuous states and inputs. The objective is to approximate the optimal receding-horizon control strategy for cases in which the online computation is intractable due to the necessity of solving mixed-integer quadratic programs in each discrete time instant. The proposed approach builds upon an approximated optimal finite-horizon control law in closed-loop form with guaranteed constraint satisfaction. The paper derives the properties of recursive feasibility and asymptotic stability for the proposed approach. A numerical example is provided for illustration and evaluation of the approach.

Dynamical analysis of a generalized hepatitis B epidemic model and its dynamically consistent discrete model

The aim of this work is to study qualitative dynamical properties of a generalized hepatitis B epidemic model and its dynamically consistent discrete model. Positivity, boundedness, the basic reproduction number and asymptotic stability properties of the model are analyzed rigorously. By the Lyapunov stability theory and the Poincare–Bendixson theorem in combination with the Bendixson–Dulac criterion, we show that a disease-free equilibrium point is globally asymptotically stable if the basic reproduction number R0≤1 and a disease-endemic equilibrium point is globally asymptotically stable whenever R0>1. Next, we apply the Mickens’ methodology to propose a dynamically consistent nonstandard finite difference (NSFD) scheme for the continuous model. By rigorous mathematical analysis, it is proved that the constructed NSFD scheme preserves essential mathematical features of the continuous model for all finite step sizes. Finally, numerical experiments are conducted to illustrate the theoretical findings and to demonstrate advantages of the NSFD scheme over standard ones. The obtained results in this work not only improve but also generalize some existing recognized works.

Stability of multi-dimensional switched systems with an application to open multi-agent systems

A multi-dimensional switched system or multi-mode multi-dimensional (M3D) system extends the classic switched system by allowing different subsystem dimensions. The stability problem of the M3D system, whose state transitions at switching instants can be discontinuous due to the dimension-varying feature, is studied. The discontinuous state transition is formulated by an affine map that captures both the dimension variations and the state impulses, with no extra constraint imposed. In the presence of unstable subsystems, the general criteria featuring a series of Lyapunov-like conditions for the practical and asymptotic stability properties of the M3D system are provided under the proposed slow/fast transition-dependent average dwell time framework. Then, by considering linear subsystems, we propose a class of parametric multiple Lyapunov functions to verify the obtained Lyapunov-like stability conditions and explicitly reveal a connection between the practical/asymptotic stability property and the non-vanishing/vanishing property of the impulsive effects in the state transition process. Further, the obtained stability results for the M3D system are applied to the consensus problem of the open multi-agent system (MAS), whose network topology can be switching and size-varying due to the migrations of agents. It shows that through a proper transformation, the seeking of the (practical) consensus performance of the open MAS with disconnected digraphs boils down to that of the (practical) stability property of an M3D system with unstable subsystems.

Construction of a Lyapunov function for a linear large-scale periodic system with possibly unstable subsystems

This article proposes an approach to construct a Lyapunov function for a linear large-scale periodic system. In this case, in contrast to various variants of small-gain stability conditions for large-scale systems, the presence of the asymptotic stability property of independent subsystems is not assumed. To analyze the asymptotic stability of a large-scale system, the direct Lyapunov method is used in combination with the discretization method and identities of the commutator calculus. The main results are illustrated by means of examples.

A Hamiltonian control approach for electric microgrids with dynamic power flow solution

In this paper, a control scheme for islanded electric microgrids is proposed. Exhibiting a dynamical structure and based on passivity, cascaded systems, and input-to-state stability arguments, the asymptotic stability properties of the closed-loop system are formally established to guarantee that voltage regulation and a power network balance are achieved. In contrast to other approaches, the controller design considers the differential–algebraic structure of the system obtained by the explicit inclusion of the network’s dynamic, the existence of both a grid-forming and grid-following nodes, and the highly nonlinear structure of the power balance equations.

A nonlinear optimal control method against the spreading of epidemics

To define a vaccination policy and antiviral treatment against the spreading of viral infections a nonlinear optimal (H-infinity) control approach is proposed. Actually, because of the scarcity of the resources for treating infectious diseases in terms of vaccines, antiviral drugs and other medical facilities, there is need to implement optimal control against the epidemics deployment. In this approach, the state-space model of the epidemics dynamics undergoes first approximate linearization around a temporary operating point which is recomputed at each time-step of the control method. The linearization is based on Taylor series expansion and on the computation of the associated Jacobian matrices. Next, an optimal (H-infinity) feedback controller is developed for the approximately linearized model of the epidemics. To compute the controller’s feedback gains an algebraic Riccati equation is solved at each iteration of the control algorithm. Furthermore, the global asymptotic stability properties of the control scheme are proven through Lyapunov stability analysis. This paper’s results confirm that optimal control of the infectious disease dynamics allows for eliminating its spreading while also keeping moderate the consumption of the related medication, that is vaccines and antiviral drugs.

Стійкість систем векторного керування напругою асинхронного генератора

The paper presents an analysis of the stability properties of typical structures of induction generators-based vector control systems with linear proportional-integral currents and DC-link voltage controllers. The study is based on the mathematical model of an induction machine taking into account the magnetization curve. It provides the modification of the typical voltage-flux vector control algorithm by considering the saturation of the magnetic system. The stability proof problem of an induction generator-based generation system is that its model is nonlinear and nonminimal-phase, and that the dynamics of the DC-link voltage is nonlinear even for a constant flux linkage and speed due to the presence of nonlinear components which are proportional to active losses. The conditions which provide consideration of the reduced-order system dynamics are formulated and properties of local asymptotic stability of the generation system with typical vector control algorithms are proven based on the singular perturbation theory. It is shown that the local asymptotic stability is ensured when the regulation processes of the DC-link voltage and the torque-forming stator current component are decoupled, which is achieved due to the proposed special adjustment of the voltage and current controllers. A modified vector control algorithm of induction generator is studied in simulation and experimentally. At the first stage, the system dynamics is investigated when the flux subsystem is in equilibrium point. As a result, the possibility of the reduced-order system consideration for the design and analysis of the voltage regulation subsystem was confirmed. At the second stage, the dynamic behavior of the voltage control loop for different settings of the controllers was experimentally investigated. It is shown that the configuration of the voltage control algorithm proposed in the work provides the timescale separation of the current and DC-link voltage regulation processes, as well as their quasi-decoupling.

Mathematical study of transmission dynamics of SARS-CoV-2 with waning immunity

<abstract><p>The aim of this work is to provide a new mathematical model that studies transmission dynamics of Coronavirus disease 2019 (COVID-19) caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). The model captures the dynamics of the disease taking into consideration some measures and is represented by a system of nonlinear ordinary differential equations including seven classes, which are susceptible class (S), exposed class (E), asymptomatic infected class (A), severely infected class (V), hospitalized class (H), hospitalized class but in ICU (C) and recovered class (R). We prove positivity and boundedness of solutions, compute the basic reproduction number, and investigate asymptotic stability properties of the proposed model. As a consequence, dynamical properties of the model are established fully and some mitigation and prevention measures of COVID-19 outbreaks are also suggested. Furthermore, the model is fitted to COVID-19 confirmed cases in South Africa during the Omicron wave from November 27, 2021 to January 20, 2022 which helped determine the model parameters value for our numerical simulation. A set of numerical experiments using real data is conducted to support and illustrate the theoretical findings. Numerical simulation results show that fast waning of infection-induced immunity can increase the occurrence of outbreaks.</p></abstract>

Accelerated Continuous-Time Approximate Dynamic Programming via Data-Assisted Hybrid Control

We introduce a new closed-loop architecture for the online solution of approximate optimal control problems in the context of continuous-time systems. Specifically, we introduce the first algorithm that incorporates dynamic momentum in actor-critic structures to control continuous-time dynamic plants with an affine structure in the input. By incorporating dynamic momentum in our algorithm, we are able to accelerate the convergence properties of the closed-loop system, achieving superior transient performance compared to traditional gradient-descent based techniques. In addition, by leveraging the existence of past recorded data with sufficiently rich information properties, we dispense with the persistence of excitation condition traditionally imposed on the regressors of the critic and the actor. Given that our continuous-time momentum-based dynamics also incorporate periodic discrete-time resets that emulate restarting techniques used in the machine learning literature, we leverage tools from hybrid dynamical systems theory to establish asymptotic stability properties for the closed-loop system. We illustrate our results with a numerical example.

Stability Analysis Using Generalized Sup-Delay Inequalities

We provide a new generalization of a recent trajectory-based approach for proving global asymptotic stability properties, using new sup-delay inequalities. Our generalization eliminates the usual contractivity requirement from previous sup-delay inequality approaches. We show how the requirements of our generalization hold under less restrictive Halanay's conditions than those that were previously reported. While the usual Halanay's inequality requires the decay rate to exceed the gain, our less restrictive conditions allow the gain to exceed the decay rate. We apply our work to systems with switching delays and to a continuous-discrete system.

On stability with respect to the part of variables of a non-autonomous system in a cylindrical phase space

Abstract. The stability problem of a system of differential equations with a right-hand side periodic with respect to the phase (angular) coordinates is considered. It is convenient to consider such systems in a cylindrical phase space which allows a more complete qualitative analysis of their solutions. The authors propose to investigate the dynamic properties of solutions of a non-autonomous system with angular coordinates by constructing its topological dynamics in such a space. The corresponding quasi-invariance property of the positive limit set of the system’s bounded solution is derived. The stability problem with respect to part of the variables is investigated basing of the vector Lyapunov function with the comparison principle and also basing on the constructed topological dynamics. Theorem like a quasi-invariance principle is proved on the basis of a vector Lyapunov function for the class of systems under consideration. Two theorems on the asymptotic stability of the zero solution with respect to part of the variables (to be more precise, non-angular coordinates) are proved. The novelty of these theorems lies in the requirement only for the stability of the comparison system, in contrast to the classical results with the condition of the corresponding asymptotic stability property. The results obtained in this paper make it possible to expand the usage of the direct Lyapunov method in solving a number of applied problems.

ON SPECIFIC ASYMPTOTIC STABILITY SOLUTIONS OF A LINEAR HOMOGENEOUS WELTERER INTEGRO-DIFFERENTIAL EQUATION OF THE FOURTH ORDER

Specific sufficient conditions for the asymptotic stability of a linear homogeneous fourthorder integro-differential equation of the Volterra type are established in the case when all nonzero solutions of the corresponding fourth-order differential equation do not have the property of asymptotic stability of the solutions. In this paper, we obtain estimates on the semiaxis of the solution and the derivative up to the third order.

Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form Δ∗β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(0,1) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

On a Discrete SEIR Epidemic Model with Exposed Infectivity, Feedback Vaccination and Partial Delayed Re-Susceptibility

A new discrete Susceptible-Exposed-Infectious-Recovered (SEIR) epidemic model is proposed, and its properties of non-negativity and (both local and global) asymptotic stability of the solution sequence vector on the first orthant of the state-space are discussed. The calculation of the disease-free and the endemic equilibrium points is also performed. The model has the following main characteristics: (a) the exposed subpopulation is infective, as it is the infectious one, but their respective transmission rates may be distinct; (b) a feedback vaccination control law on the Susceptible is incorporated; and (c) the model is subject to delayed partial re-susceptibility in the sense that a partial immunity loss in the recovered individuals happens after a certain delay. In this way, a portion of formerly recovered individuals along a range of previous samples is incorporated again to the susceptible subpopulation. The rate of loss of partial immunity of the considered range of previous samples may be, in general, distinct for the various samples. It is found that the endemic equilibrium point is not reachable in the transmission rate range of values, which makes the disease-free one to be globally asymptotically stable. The critical transmission rate which confers to only one of the equilibrium points the property of being asymptotically stable (respectively below or beyond its value) is linked to the unity basic reproduction number and makes both equilibrium points to be coincident. In parallel, the endemic equilibrium point is reachable and globally asymptotically stable in the range for which the disease-free equilibrium point is unstable. It is also discussed the relevance of both the vaccination effort and the re-susceptibility level in the modification of the disease-free equilibrium point compared to its reached component values in their absence. The influences of the limit control gain and equilibrium re-susceptibility level in the reached endemic state are also explicitly made viewable for their interpretation from the endemic equilibrium components. Some simulation examples are tested and discussed by using disease parameterizations of COVID-19.

Mathematical assessment of the impact of cohort vaccination on pneumococcal carriage and serotype replacement

ABSTRACT Although pneumococcal vaccines are quite effective in reducing disease burden, factors such as imperfect vaccine efficacy and serotype replacement present an important challenge against realizing direct and herd protection benefits of the vaccines. In this study, a novel mathematical model is designed and used to describe the dynamics of two Streptococcus pneumoniae (SP) serotypes, in response to the introduction of a cohort vaccination program which targets one of the two serotypes. The model is fitted to a pediatric SP carriage prevalence data from Atlanta, GA. The model, which is rigorously analysed to investigate the existence and asymptotic stability properties of the associated equilibria (in addition to exploring conditions for competitive exclusion), is simulated to assess the impact of vaccination under different levels of serotype-specific competition and illustrate the phenomenon of serotype replacement. The calibrated model is used to forecast the carriage prevalence in the pediatric cohort over 30 years.

On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

In this paper, we study the nonnegativity and stability properties of the solutions of a newly proposed extended SEIR epidemic model, the so-called SE(Is)(Ih)AR epidemic model which might be of potential interest in the characterization and control of the COVID-19 pandemic evolution. The proposed model incorporates both asymptomatic infectious and hospitalized infectious subpopulations to the standard infectious subpopulation of the classical SEIR model. In parallel, it also incorporates feedback vaccination and antiviral treatment controls. The exposed subpopulation has three different transitions to the three kinds of infectious subpopulations under eventually different proportionality parameters. The existence of a unique disease-free equilibrium point and a unique endemic one is proved together with the calculation of their explicit components. Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. The reproduction numbers in the presence of one or both controls is linked to the control-free reproduction number to emphasize that such a number decreases with the control gains. We also prove that, depending on the value of the basic reproduction number, only one of them is a global asymptotic attractor and that the solution has no limit cycles.

Study and modelling of droop-controlled islanded mesh microgrids

The active and reactive power sharing of distributed generation sources (DGs) connected to isolated microgrids with a single point of common coupling (mono-PCC) to which the loads are also connected has already been the subject of several studies. A high penetration rate of DGs based on renewable energies has as a logical consequence the development and implementation of mesh and more complex multi- PCC microgrids. In this paper, a developed droop control method for synchronization and power sharing between different DGs connected to a mesh islanded multi-PCC microgrid with many distributed generation sources (DGs) and different type of loads (including active load (CPL)) randomly connected to different PCCs is applied. Then, a state model of the entire mesh microgrid is developed integrating the generators with their controllers, power lines, droop algorithms and dynamic loads. This model is then used to study the asymptotic stability and robustness properties of the system. The simulation results confirm the effectiveness of the applied strategies for the synchronization of the different DGs to the microgrid while ensuring an efficient active and reactive power sharing. also, they confirm the validity of the developed state space model of the system.