Barrier function nonlinear optimization for optimal Decompression of divers

Abstract

Abstract-This paper is based on a comprehensive dynamic mathematical model (Copernicus) of vascular bubble formation and growth during and after decompression from a dive. The model describes the underlying relationship between Venous Gas Emboli (VGE) and risk of severe Decompression Sickness (DCS). By using the Copernicus model the diving decompression problem can be formulated as a nonlinear optimal control problem, where the objective is to minimize the total ascend time subject to constraints on the maximum number of bubbles in the pulmonary artery (also referred to as the decompression stress). A recent study reveals that the optimal solution can be obtained by solving the optimization problem with some equality constraints. Inspired by which, a simpler approach using barrier function is proposed in this paper, through which we achieve a more efficient and robust numerical implementation. The paper also studies the effect of ascent profile parameterization.

Stabilization of Nonlinear Control-Affine Systems With Multiple State Constraints

“…The controller was presented in a closed-form making it unnecessary to solve the QP online. On the other hand, barrier functions have also been applied in solving nonlinear optimal control problems subject to the trajectory, input, state, or output constraints [38]- [40]. With the introduction of optimization skills, closed-loop systems' performances can be further improved.…”

Optimal Decompression Through Multi-parametric Nonlinear Programming

“…In addition, this parameterization tends to give a "flat minimum" with associated numerical challenges. This motivated another parameterization method used in Gutvik et al [2009b] and studied in Feng et al [2009] The idea is to first define a curve function with two parameters and (illustrated in Figure 2). For a given and , the curve can be quantized and generate the stepwise formulation and also a total ascend time according to stopping depths ( 1 , 2 , .…”

“…Obviously this parameterization and quantization method leads to a simplified optimization problem but may also introduces some conservatism, that is, the total ascending time ( ) may be longer and/or the stress threshold may be exceeded. The benefit and performance loss of this parameterization method was studied by simulations in Feng et al [2009].…”

“…Using barrier functions as in Feng et al [2009] (4) can be considered an mp-NLP, which means that the optimal solution should be found for a given range of the parameter values. We seek an explicit approximate representation of the solution as a function of these parameters.…”

“…For safety reasons the convergence and correctness of the numerical optimization result needs to be verified. These are considered prohibitive for practical implementation (Feng et al [2009]). However, the approximate explicit solution of nonlinear constrained optimization using multi-parametric nonlinear programming (mp-NLP) (Grancharova et al [2007], Johansen [2004]) is expected to be very well suited.…”