Parametrical Non-Complex Tests to Evaluate Partial Decentralized Linear-Output Feedback Control Stabilization Conditions from Their Centralized Stabilization Counterparts

Abstract

. This paper formulates sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization. The provided tests are simple to evaluate, while they are based on the quantification of the sufficiently smallness of the parametrical error norms between the control, output, interconnection and open-loop system dynamics matrices and the corresponding control gains in the decentralized case related to the centralized counterpart. The tolerance amounts of the various parametrical matrix errors are described by the maximum allowed tolerance upper-bound of a small positive real parameter that upper-bounds the various parametrical error norms. Such a tolerance is quantified by considering the first or second powers of such a small parameter. The results are seen to be directly extendable to quantify the allowed parametrical errors that guarantee the closed-loop linear output-feedback stabilization of a current system related to its nominal counterpart. Furthermore, several simulated examples are also discussed.

Special Issue on Mathematical Modeling Using Differential Equations and Network Theory

“…In [5], the authors formulate sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization.…”

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

“…See, for instance, Reference [15]. Assume that the basic reproduction number [5,6], is less than unity so that the disease-free equilibrium point x e = (S e , 0 , 0 , N(0) − S e ) T 0 is globally asymptotically stable, [3,5,6]. Since x(t) → x e as t → ∞ , it turns out that, for t * = +∞ , no other targeted state x * ( x e ) can be prefixed as objective for any given initial state x(0) even for the current linearized version.…”

“…Usually, real dynamic systems are neither time-invariant nor linear in their whole operation rank since there are usually saturation and dead-zone type nonlinearities at the input, saturated behaviors in the state and output variables and sometimes nonlinear dynamics. See, for instance [1][2][3] and some references therein. However, very relevant information about their properties is often obtained from the knowledge of their equilibrium points, or their equilibrium steady-state oscillations, and the Jacobian matrices which describe the linearized trajectory solutions around such point for small deviations of linearity.…”

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“…Such models, because of their structure, become very appropriate to study the equilibrium points, the oscillatory behaviors, the illness permanence and the vaccination and treatment controls. See, for instance, [18][19][20][21][22][23][24] and some of the references therein. More recently, entropy-based models have been proposed for epidemic models.…”