It is inconvenient and often difficult to consider systems, or environments, that are not completely known. However, with the proliferation of autonomous systems, where the success or survival of the system is coupled to the ability to make decisions with incomplete information, it is becoming increasingly important. This work introduces an optimal control algorithm in which the Koopman operator is used to solve for the probabilistically optimal input in the presence of initial condition and/or parametric uncertainty. The proposed approach is minimally restrictive; applicable to nonlinear systems and non-Gaussian uncertainty. Additionally, it offers unique computational advantages over alternative uncertainty quantification techniques, such as Monte Carlo methods, providing a practical method to compute a probabilistically optimal input. In the context of the airdrop problem, these inputs are the optimal package release point, aircraft run-in, and transition altitude. Given an objective–or cost–function defined over the drop zone and a joint probability density function (PDF) quantifying the uncertainty in the system parameters and initial state, the objective function is pulled back to the drop altitude using the Koopman operator, and an expected value is computed with the joint PDF. With the control inputs embedded in the initial joint PDF and the system dynamics, the problem is treated as probabilistic from the beginning, a fundamental shift from the current state-of-the-art. Simulation examples are presented highlighting performance of the algorithm in real-world scenarios. Results compare favorably with those achieved through deterministic methods.
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Probabilistic methods for decision making in precision airdrop
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