In this article we introduce the use of recently developed min/max-plus techniques in order to solve the optimal attitude estimation problem in filtering for nonlinear systems on the special orthogonal (SO(3)) group. This work helps obtain computatio
nally efficient methods for the synthesis of deterministic filters for nonlinear systems -- i.e. optimal filters which estimate the state using a related optimal control problem. The technique indicated herein is validated using a set of optimal attitude estimation example problems on SO(3).
A robust (deterministic) filtering approach to the problem of optimal sensor selection is considered herein. For a given system with several sensors, at each time step the output of one of the sensors must be chosen in order to obtain the best state
estimate. We reformulate this problem in an optimal control framework which can then be solved using dynamic programming. In order to tackle the numerical computation of the solution in an efficient manner, we exploit the preservation of the min-plus structure of the optimal cost function when acted upon by the dynamic programming operator. This technique yields a grid free numerical approach to the problem. Simulations on an example problem serve to highlight the efficacy of this generalizable approach to robust multi-sensor state estimation.
The design of deterministic filters can be cast as a problem of minimizing an associated cost function for an optimal control problem. Employing the min-plus linearity property of the dynamic programming operator (associated with the control problem)
results in a computationally feasible approach (while avoiding linearization of the system dynamics/output). This article describes the salient features of this approach and a specific form of pruning/projection, based on clustering, which serves to facilitate the numerical efficiency of these methods.
This article approaches deterministic filtering via an application of the min-plus linearity of the corresponding dynamic programming operator. This filter design method yields a set-valued state estimator for discrete-time nonlinear systems (nonline
ar dynamics and output functions). The energy bounds in the process and the measurement disturbances are modeled using a sum quadratic constraint. The filtering problem is recast into an optimal control problem in the form of a Hamilton-Jacobi-Bellman (HJB) equation, the solution to which is obtained by employing the min-plus linearity property of the dynamic programming operator. This approach enables the solution to the HJB equation and the design of the filter without recourse to linearization of the system dynamics/ output equation.
In this article we explore a modification in the problem of controlling the rotation of a two level quantum system from an initial state to a final state in minimum time. Specifically we consider the case where the qubit is being weakly monitored --
albeit with an assumption that both the measurement strength as well as the angular velocity are assumed to be control signals. This modification alters the dynamics significantly and enables the exploitation of the measurement backaction to assist in achieving the control objective. The proposed method yields a significant speedup in achieving the desired state transfer compared to previous approaches. These results are demonstrated via numerical solutions for an example problem on a single qubit.
The relationship between efficient quantum gate synthesis and control theory has been a topic of interest in the quantum control literature. Motivated by this work, we describe in the present article how the dynamic programming technique from optimal
control may be used for the optimal synthesis of quantum circuits. We demonstrate simulation results on an example system on SU(2), to obtain plots related to the gate complexity and sample paths for different logic gates.
