On the Design of Hyperstable Feedback Controllers for a Class of Parameterized Nonlinearities. Two Application Examples for Controlling Epidemic Models

Abstract

This paper studies the hyperstability and the asymptotic hyperstability of a single-input single-output controlled dynamic system whose feed-forward input-output dynamics is nonlinear and eventually time-varying consisting of a linear nominal part, a linear incremental perturbed part and a nonlinear and eventually time-varying one. The nominal linear part is described by a positive real transfer function while the linear perturbation is defined by a stable transfer function. The nonlinear and time-varying disturbance is, in general, unstructured but it is upper-bounded by the combination of three additive absolute terms depending on the input, output and input-output product, respectively. The non-linear time-varying feedback controller is any member belonging to a general class which satisfies an integral Popov’s-type inequality. This problem statement allows the study of the conditions guaranteeing the robust stability properties under a variety of the controllers designed for the controlled system and controller disturbances. In this way, set of robust hyperstability and asymptotic hyperstability of the closed-loop system are given based on the fact that the input-output energy of the feed-forward controlled system is positive and bounded for all time and any given initial conditions and controls satisfying Popov’s inequality. The importance of those hyperstability and asymptotic hyperstability properties rely on the fact that they are related to global closed-loop stability, or respectively, global closed-loop asymptotic stability of the same uncontrolled feed-forward dynamics subject to a great number of controllers under the only condition that that they satisfy such a Popov’s-type inequality. It is well-known the relevance of vaccination and treatment controls for Public Health Management at the levels of prevention and healing. Therefore, two application examples concerning the linearization of known epidemic models and their appropriate vaccination and/or treatment controls on the susceptible and infectious, respectively, are discussed in detail with the main objective in mind of being able of achieving a fast convergence of the state- trajectory solutions to the disease- free equilibrium points under a wide class of control laws under deviations of the equilibrium amounts of such populations.

CIRD-F: Spread and Influence of COVID-19 in China

“…Therefore, we need a capital asset pricing model with a dummy variable to solve this problem. Traditional capital asset pricing model (CAPM) mainly studies the relationship between expected return of assets in the securities market and systematic risk [4] . Therefore, it shows how a certain capital asset is influenced in the background of a certain market.…”

“…Furthermore, there are few related researches on the influence of official policies. Most existing analyses focus on the impacts of the epidemic itself [3][4][5] . Also, some researches confuse the influence of official policies and the influence of the epidemic after the policies were published.…”

On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models

“…Such terms deviate the solution trajectory from the linear behavior about the equilibrium points. See, for instance, [5][6][7][8][9][10][11][12][13][14] and also [16][17][18][19][20][21] and some of the references therein. Example 1.…”

“…Some of the relevant properties of positivity and stability of epidemic models are already qualitatively reflected in their linearized versions around their equilibrium points. See, for instance, [5,6] and references therein. The background literature on epidemic models is very abundant, including the use of either vaccination or treatment controls as well as combinations of both types of controls.…”

“…The background literature on epidemic models is very abundant, including the use of either vaccination or treatment controls as well as combinations of both types of controls. See, for instance, [5][6][7][8][9][10][11][12][13][14] and the references therein. Such controls may reduce the value of the basic reproduction number related to the case of absence of controls, so the average number of contagions per each primary infectious case, and they are also able to change the components of the equilibrium points, that is the numbers of each subpopulation at the equilibrium, and the rates the convergence to such equilibrium points.…”