Decay rate estimations for linear quadratic optimal regulators

Let $u(t)=-Fx(t)$ be the optimal control of the open-loop system $x'(t)=Ax(t)+Bu(t)$ in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system $x'(t)=(A-BF)x(t)$. Main attention is given to the case of a skew-Hermitian matrix $A$. Given an operator $A$, for a class of cases, we find a matrix $B$ that provides an almost optimal decay rate. We show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems $(A, B_j)$, $j=1, \dots, N$, and illustrate this on an example of a concrete mechanical system. At the end of the article, we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem.Comment: 25 pages, 1 figur

Instability in clinical risk stratification models using deep learning

While it has been well known in the ML community that deep learning models suffer from instability, the consequences for healthcare deployments are under characterised. We study the stability of different model architectures trained on electronic health records, using a set of outpatient prediction tasks as a case study. We show that repeated training runs of the same deep learning model on the same training data can result in significantly different outcomes at a patient level even though global performance metrics remain stable. We propose two stability metrics for measuring the effect of randomness of model training, as well as mitigation strategies for improving model stability.Comment: Accepted for publication in Machine Learning for Health (ML4H) 202