A population model‐based linear‐quadratic Gaussian compensator for the control of intravenously infused alcohol studies and withdrawal symptom prophylaxis using transdermal sensing

Abstract

An output feedback linear‐quadratic Gaussian compensator (combined controller and state estimator) for the regulation of intravenous‐infused alcohol studies and treatment using a noninvasive transdermal alcohol biosensor is developed. The design is based on a population model involving an abstract semilinear parabolic hybrid reaction‐diffusion system involving coupled partial and ordinary differential equations with random parameters known only up to their distributions. The scheme developed is based on a weak formulation of the model equations in an appropriately constructed Gelfand triple of Bochner spaces wherein the unknown random parameters are treated as additional spatial variables. Implementation relies on a Galerkin‐based approximation and convergence theory and an abstract formulation involving linear semigroups of operators. The model is fit and validated using laboratory collected human subject data and the method of moments. The results of numerical simulations of controlled intravenous alcohol infusion are presented and discussed.

Optimal control in therapeutics and epidemiology

“…Finally, in Reference 6, focus is on the control of intravenously infused alcohol. Rather different techniques allow the authors to build feedback controls for a complex PDE‐ODE system, which incorporates parameter uncertainty.…”

Real-time recursive estimation of, and uncertainty quantification for, breath alcohol concentration via LQ tracking control-based inverse filtering of transdermal alcohol biosensor signals

“…We then briefly review the LQR control theory in Hilbert space and its finite dimensional approximation theory and convergence results. These ideas are presented in greater detail in [63,64]. We also discuss how the population model can be used to quantify the uncertainty in our eBrAC estimates and the modifications we make to the LQR theory so that it tracks a test TAC signal and so that it regularizes the eBrAC signal.…”

Blood and breath alcohol concentration from transdermal alcohol biosensor data: estimation and uncertainty quantification via forward and inverse filtering for a covariate-dependent, physics-informed, hidden Markov model*

“…Since the only observed and observable quantities in (12) are BrAC input, u, and TAC output, y, with the state variables x and w remaining hidden, elementary changes of variable (see, for example [58]), allow for the transformation of the system (12) into an equivalent system involving only four parameters which we shall denote by the vector q = [q 1 , q 2 , q 3 , q 4 ] in the positive orthant of R 4 . Moreover, since the observed output y is time-sampled (with sampling time τ dictated by the sensor hardware) and if we assume the input u is zero-order hold, using standard techniques form the theory of linear semigroups for abstract parabolic systems [37,53] together with spline based Galerkin approximations (for which there exists a rigorous convergence theory [18]), the system (12) can be equivalently written as a single input-single output convolution system as…”

“…While some of the these physical parameters can be found in the literature (typically determined empirically in-vitro in the laboratory), we instead estimated the q i 's statistically by first fitting the model (13) to data from individual drinking episodes [10,43] and then averaging those estimates over the entire population of episodes to obtain a crude PDE-based population model. Without too much additional effort, it is possible to obtain a more sophisticated population model (see, for example [1,18,50,51], and [58]), but since this study is still at the proof-of-concept stage, we did not do this.…”

“…However, we have developed more sophisticated PDE-based population models where the parameters are assumed to be random with their distributions fit to their values over the entire population. We have then shown how these random PDEs can be transformed into an equivalent weak form wherein the random parameters are now treated as additional spatial variables (see, for example [48][49][50]58]). We intend to investigate how these more sophisticated population models perform when introduced into our HMM model.…”