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Scalable computation of viability kernels and a viability-theoretic approach to guaranteeing safety for closed-loop medical devices Maidens, John Norman

Abstract

As closed-loop controllers become increasingly prevalent in medical technology, increasing emphasis is being placed on ensuring that such systems operate in a safe manner. In our approach to guaranteeing the safe operation of a physiologic closed-loop control system, we wish to provide a mathematical guarantee that, despite limited control authority, the system’s state can be confined to a region designated as safe. The largest subset of the safe region for which there exists an admissible control input that keeps the state within the safe region is known as the viability kernel, or maximal controlled invariant set. Many methods are known for computing viability kernels in low-dimensional systems, but these existing methods rely on gridding the state space and hence their time complexity increases exponentially with the state dimension. In this thesis we describe a new connection between reachability and viability theory that enables us to approximate the viability kernel using Lagrangian methods which scale well with the state dimension. We present four new viability kernel approximation algorithms using polytope-, ellipsoid- and support vector-based set representations and we compare their performances in terms of accuracy and scalability with the state dimension. Using the support vector and ellipsoidal techniques, we are able to accurately approximate the viability kernel for systems of much larger state dimension than was previously feasible using existing Eulerian methods. We also present a viability theoretic solution to the problem of determining when a physiologic closed-loop control system should initiate a fallback mode of operation. The viability-based method allows impending safety violations to be detected in advance, allowing the fallback mode to be initiated earlier than using a naive approach. Our new approach to fallback mode initiation is examined in two sample contexts: the closed-loop control of carbon dioxide partial pressure under mechanical ventilation, and the control of the concentration of the anaesthetic drug Propofol using a paediatric model of Propofol pharmacokinetics.

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