We examine Lagrangian techniques for computing underapproximations of finite-time horizon, stochastic reachavoid level-sets for discrete-time, nonlinear systems. We use the concept of reachability of a target tube in the control literature to define robust reach-avoid sets which are parameterized by the target set, safe set, and the set in which the disturbance is drawn from. We unify two existing Lagrangian approaches to compute these sets and establish that there exists an optimal control policy of the robust reach-avoid sets which is a Markov policy. Based on these results, we characterize the subset of the disturbance space whose corresponding robust reachavoid set for the given target and safe set is a guaranteed underapproximation of the stochastic reach-avoid level-set of interest. The proposed approach dramatically improves the computational efficiency for obtaining an underapproximation of stochastic reach-avoid level-sets when compared to the traditional approaches based on gridding. Our method, while conservative, does not rely on a grid, implying scalability as permitted by the known computational geometry constraints. We demonstrate the method on two examples: a simple twodimensional integrator, and a space vehicle rendezvous-docking problem.
A real-time, feedback-capable, variable spectrum lighting system was recently installed at the University of New Mexico Hospital to facilitate biomedical research on the health impacts of lighting. The system consists of variable spectrum troffers, color sensors, occupancy sensors, and computing and communication infrastructure, and is the only such clinical facility in the US. The clinical environment posed special challenges for installation as well as for ongoing maintenance and operations. Pilot studies are currently underway to evaluate the effectiveness of the system to regulate circadian phase in subjects with delayed sleep-wake phase disorder.
Objective: Lighting is a strong synchronizer for circadian rhythms, which in turn drives a wide range of biological functions. The objective of our work is a) to construct a clinical in-patient testbed with smartİ lighting, and b) evaluate its feasibility for use in future clinical studies. Methods: A feedback capable, variable spectrum lighting system was installed at the University of New Mexico Hospital. The system consists of variable spectrum lighting troffers, color sensors, occupancy sensors, and computing and communication infrastructure. We conducted a pilot study to demonstrate proof of principle, that 1) this new technology is capable of providing continuous lighting and sensing in an active clinical environment, 2) subject recruitment and retention is feasible for round-the-clock, multi-day studies, and 3) current techniques for circadian regulation can be deployed in this unique testbed. Unlike light box studies, only troffer-based lighting was used, and both lighting intensity and spectral content were varied. Results: The hardware and software functioned seamlessly to gather biometric data and provide the desired lighting. Salivary samples that measure dim-light melatonin onset showed phase advancement for all three subjects. Conclusion: We executed a five-day circadian rhythm study that varied intensity, spectrum, and timing of lighting as proof-of-concept or future clinical studies with troffer-based, variable spectrum lighting. Clinical Impact: The ability to perform circadian rhythm experiments in more realistic environments that do not overly constrain the subject is important for translating lighting research into practice, as well as for further research on the health impacts of lighting.INDEX TERMS Lighting control, system implementation, circadian rhythm, biomedical engineering.
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