This work addresses the optimal covariance control problem for stochastic discrete-time linear timevarying systems subject to chance constraints. Covariance steering is a stochastic control problem to steer the system state Gaussian distribution to another Gaussian distribution while minimizing a cost function. To the best of our knowledge, covariance steering problems have never been discussed with probabilistic chance constraints although it is a natural extension. In this work, first we show that, unlike the case with no chance constraints, the covariance steering with chance constraints problem cannot decouple the mean and covariance steering sub-problems. Then we propose an approach to solve the covariance steering with chance constraints problem by converting it to a semidefinite programming problem. The proposed algorithm is verified using two simple numerical simulations.
We consider the problem of finite-horizon optimal control of a discrete linear time-varying system subject to a stochastic disturbance and fully observable state. The initial state of the system is drawn from a known Gaussian distribution, and the final state distribution is required to reach a given target Gaussian distribution, while minimizing the expected value of the control effort. We derive the linear optimal control policy by first presenting an efficient solution for the diffusion-less case, and we then solve the case with diffusion by reformulating the system as a superposition of diffusionless systems. This reformulation leads to a simple condition for the solution, which can be effectively solved using numerical methods. We show that the resulting solution coincides with a LQG problem with particular terminal cost weight matrix. This fact provides an additional justification for using a linear in state controller. In addition, it allows an efficient iterative implementation of the controller.
SynchroWatch is a one-handed interaction technique for smartwatches that uses rhythmic correlation between a user's thumb movement and on-screen blinking controls. Our technique uses magnetic sensing to track the synchronous extension and reposition of the thumb, augmented with a passive magnetic ring. The system measures the relative changes in the magnetic field induced by the required thumb movement and uses a time-shifted correlation approach with a reference waveform for detection of synchrony. We evaluated the technique during three distraction tasks with varying degrees of hand and finger movement: active walking, browsing on a computer, and relaxing while watching online videos. Our initial offline results suggest that intentional synchronous gestures can be distinguished from other movement. A second evaluation using a live implementation of the system running on a smartwatch suggests that this technique is viable for gestures used to respond to notifications or issue commands. Finally, we present three demonstration applications that highlight the technique running in real-time on the smartwatch.
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