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

Manuel De La Sen

doi:10.3390/ijerph16152689

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.

Highlights

Studies to design controllers to improve the basic properties of dynamic systems is very important in theoretical studies and in many industrial applications as well as in the study of biological systems and epidemic models under appropriate controls
This paper has developed a formalism for hyperstability and asymptotic hyperstability of controlled dynamic systems whose feed-forward part excluding potential controls consist of the additive contributions of a known linear dynamics, an unknown one and unknown nonlinear disturbances under wide classes of controllers which satisfy a Popov’s-type inequality
The known linear part is given by a positive real transfer function, the unknown dynamics is assumed stable but it is unknown except some “a priori” knowledge of its resonance peak, that is, the maximum gain in the frequency domain

Summary

Introduction

Studies to design controllers to improve the basic properties of dynamic systems is very important in theoretical studies and in many industrial applications as well as in the study of biological systems and epidemic models under appropriate controls. Note that Theorem 1 establishes that the zero-state input-output energy on a nonzero finite time interval [0, t] is positive under Assumption A1–A4 if the input is nonzero on some subinterval of nonzero measure. The following result relies on weakening Assumption A4 if the input converges to zero and it is bounded on [0, ∞) concluding in the non-negativity of the input-output energy on any time interval [t1 , t2 ] of nonzero measure. Lim sup t→∞|y(t)|(|cT0 b0 | ρ 0 +|e cTe b| Kρe +(k1 +d)) 1−k m−ku e (iii) y(t) is uniformly bounded on R0+ for any given finite initial conditions if d = k1 = k3 = 0 and e(s) are strictly stable transfer functions satisfying: k2 < 1 for any control u ∈ L2 if G(s) = G0 (s) + d and G cTe b Kρe Mu y(t) ≤ 1−k. Note from the above concepts that a hypestable system is a passive system

Hyperstability and Asymptotic Hyperstability of the Closed-Loop System

Examples on Epidemic Models Subject to Vaccination and Treatment Controls

Conclusions