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Solvability of The Output Corridor Control Problem by Pulse-Modulated Feedback

Alexander Medvedev, Anton V. Proskurnikov

TL;DR

This work addresses keeping the stationary output of a positive LTI SISO plant within a predefined corridor using pulse-modulated feedback. It reduces the closed-loop dynamics to a discrete-time map and proves that, for a class of third-order positive models, a 1-cycle (one impulse per period) always exists and can be designed to meet an arbitrary output corridor via a unique period $T^*$ and impulse weight $\lambda^*$. A constructive design equation $\Psi(T)=\frac{z_{\max}(T)}{z_{\max}(T)-z_{\min}(T)}$ with explicit expressions for $z_{\max}, z_{\min}$ enables exact computation of $(T^*,\lambda^*)$ and tight corridor bounds. The method is applied to a population of patient-specific PKPD Wiener models, showing that feasibility hinges on the Hill slope $\gamma$ of the PD nonlinearity, with low $\gamma$ commonly causing infeasibility, thus providing a practical tool for safety-focused model feasibility assessment in drug dosing scenarios.

Abstract

The problem of maintaining the output of a positive time-invariant single-input single-output system within a predefined corridor of values is treated. For third-order plants possessing a certain structure, it is proven that the problem is always solvable under stationary conditions by means of pulse-modulated feedback. The obtained result is utilized to assess the feasibility of patient-specific pharmacokinetic-pharmacodynamic models with respect to patient safety. A population of Wiener models capturing the dynamics of a neuromuscular blockade agent is studied to investigate whether or not they can be driven into the desired output corridor by clinically acceptable sequential drug doses (boluses). It is demonstrated that low values of a parameter in the nonlinear pharmacodynamic part lie behind the detected model infeasibility.

Solvability of The Output Corridor Control Problem by Pulse-Modulated Feedback

TL;DR

This work addresses keeping the stationary output of a positive LTI SISO plant within a predefined corridor using pulse-modulated feedback. It reduces the closed-loop dynamics to a discrete-time map and proves that, for a class of third-order positive models, a 1-cycle (one impulse per period) always exists and can be designed to meet an arbitrary output corridor via a unique period and impulse weight . A constructive design equation with explicit expressions for enables exact computation of and tight corridor bounds. The method is applied to a population of patient-specific PKPD Wiener models, showing that feasibility hinges on the Hill slope of the PD nonlinearity, with low commonly causing infeasibility, thus providing a practical tool for safety-focused model feasibility assessment in drug dosing scenarios.

Abstract

The problem of maintaining the output of a positive time-invariant single-input single-output system within a predefined corridor of values is treated. For third-order plants possessing a certain structure, it is proven that the problem is always solvable under stationary conditions by means of pulse-modulated feedback. The obtained result is utilized to assess the feasibility of patient-specific pharmacokinetic-pharmacodynamic models with respect to patient safety. A population of Wiener models capturing the dynamics of a neuromuscular blockade agent is studied to investigate whether or not they can be driven into the desired output corridor by clinically acceptable sequential drug doses (boluses). It is demonstrated that low values of a parameter in the nonlinear pharmacodynamic part lie behind the detected model infeasibility.
Paper Structure (10 sections, 1 theorem, 39 equations, 4 figures)

This paper contains 10 sections, 1 theorem, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Background
  4. Main Result
  5. Model feasibility analysis
  6. Model cohort
  7. Feasibility analysis
  8. Proofs of The Main Results
  9. The Existence and Technical Properties of $\tau_1,\tau_2$
  10. Proof of Theorem \ref{['th:solvability']}

Key Result

Lemma 1

Let $\alpha_1,\ldots,\alpha_n\in\mathbb{R}$ be pairwise distinct and let $\sum_{i=1}^n|c_i|>0$, where $c_1,\ldots,c_n\in\mathbb{R}$. Then the equation has no more than $n-1$ real solutions.

Figures (4)

  • Figure 1: Numerator and denominator of the function $\Psi(T)$ in design equation \ref{['eq:design_T']}. It can be seen that $\lim_{T\to\infty}\Psi(T)=1$, $\lim_{T\to 0}\Psi(T)=\infty$
  • Figure 2: The model parameter pairs in the dataset satisfy $\alpha_{\min}= 0.0270\le \alpha \le 0.0524=\alpha_{\max}$, $\gamma_{\min}=1.4030\le\gamma\le 5.5619=\gamma_{\max}$. The extreme parameter values are indicated by the Patient Identification Number (PIN). PIN 26 features two extreme values and corresponds to $(\alpha_{\max},\gamma_{\min})$.
  • Figure 3: Calculated values of $\lambda^*$ and $T^*$ across the cohort of patient models. The dashed lines show the highest clinically feasible values: $\lambda_{\max}=600~\mathrm{\mu g/kg}$, $T_{\max}=45~\mathrm{min}$. The cases exhibiting the extreme values of $\alpha$ and $\gamma$ (in red) are marked by PIN; cf. Fig. \ref{['fig:alpha_gamma']}.
  • Figure 4: The model parameter pairs $(\alpha,\gamma)$ in the dataset. Infeasible models are marked with filled circles. Models with $\lambda^*<\lambda_{\max}$ and $T^*<T_{\max}$ are plotted in blue, and models with $\lambda^*>\lambda_{\max}$ and $T^*>T_{\max}$ are plotted in cyan.

Theorems & Definitions (1)

  • Lemma 1