Output Corridor Impulsive Control of First-order Continuous System with Non-local Attractivity Analysis

Abstract

This paper addresses the design of an impulsive controller for a continuous scalar time-invariant linear plant that constitutes the simplest conceivable model of chemical kinetics. The model is ubiquitous in process control as well as pharmacometrics and readily generalizes to systems of Wiener structure. Given the impulsive nature of the feedback, the control problem formulation is particularly suited to discrete dosing applications in engineering and medicine, where both doses and inter-dose intervals are manipulated. Since the feedback controller acts at discrete time instants and employs both amplitude and frequency modulation, whereas the plant is continuous, the closed-loop system exhibits hybrid dynamics featuring complex nonlinear phenomena. The problem of confining the plant output to a predefined corridor of values is considered. The method at the heart of the proposed approach is to design a stable periodic solution, called a 1-cycle, whose one-dimensional orbit coincides with the predefined corridor. Conditions ensuring local and global attractivity of the 1-cycle are established. As a numerical illustration of the proposed approach, the problem of intravenous paracetamol dosing is considered.

Figures (5)

• Figure F1: Open-loop dosing of paracetamol. Single IV bolus dose (in red) vs initial ($\lambda_0=2000mg$) and five subsequent IV bolus doses ($\lambda_k=1000mg, k=1,\dots,5$) (in blue). The recommended output value corridor $[y_{\min}^*,y_{\max}^*]$ is marked by red dashed lines. The black vertical lines correspond to the dose administration times, with inter-dose intervals of six hours.
• Figure F2: $Q^\prime(x)$ calculated according to \ref{['eq:Jacobian_fb']} for Case 1--Case 3.
• Figure F3: Closed-loop dosing of paracetamol, Case 1. After initial IV bolus dose ($\lambda_0=2000mg$), the pulse-modulated feedback drives the drug concentration and medication effect (in blue) into desired corridor. Upper plot: the recommended output value corridor $[y_{\min}^*,y_{\max}^*]$ is marked by red dashed lines. Lower plot: the recommended concentration corridor $[x_{\min}^*,x_{\max}^*]$ is marked by red dashed lines.
• Figure F4: Closed-loop dosing of paracetamol, Case 2. After initial IV bolus dose ($\lambda_0=2000mg$), the pulse-modulated feedback drives the drug concentration and medication effect (in blue) into desired corridor. Upper plot: the recommended output value corridor $[y_{\min}^*,y_{\max}^*]$ is marked by red dashed lines. Lower plot: the recommended concentration corridor $[x_{\min}^*,x_{\max}^*]$ is marked by red dashed lines.
• Figure F5: Closed-loop dosing of paracetamol, Case 3. After initial IV bolus dose ($\lambda_0=2000mg$), the pulse-modulated feedback drives the drug concentration and medication effect (in blue) into desired corridor. Upper plot: the recommended output value corridor $[y_{\min}^*,y_{\max}^*]$ is marked by red dashed lines. Lower plot: the recommended concentration corridor $[x_{\min}^*,x_{\max}^*]$ is marked by red dashed lines.

Theorems & Definitions (16)

• Proposition 1: ZCM12a
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1
• proof
• Proposition 4
• proof
• Proposition 5