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   The modeling of nonlinear dynamics based on Koopman operator theory, originally applicable only to autonomous systems with no control, is extended to nonautonomous control system without approximation of the input matrix. Prevailing methods using a least square estimate of the input matrix may result in an erroneous input matrix, misinforming the controller. Here, a new method for constructing a Koopman model that yields the exact input matrix is presented. A set of state variables are introduced so that the control inputs are linearly involved in the dynamics of actuators. With these variables, a lifted linear model with the exact input matrix, called a Control-Coherent Koopman Model, is constructed by superposing control input terms, which are linear in local actuator dynamics, to the Koopman operator of the associated autonomous nonlinear system. As an example, the proposed method is applied to multi degree-of-freedom robotic arms, which are controlled with Model Predictive Control (MPC). It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate input matrix does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct input matrix, even performing better than the bilinear formulation of the Koopman operator. 

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3. Indecision Modeling

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4. Indecision Modeling

   McElfresh, Duncan C; Chan, Lok; Doyle, Kenzie; Sinnott-Armstrong, Walter; Conitzer, Vincent; Schaich Borg, Jana; Dickerson, John P (April 2021, Proceedings of the AAAI Conference on Artificial Intelligence)

   AI systems are often used to make or contribute to important decisions in a growing range of applications, including criminal justice, hiring, and medicine. Since these decisions impact human lives, it is important that the AI systems act in ways which align with human values. Techniques for preference modeling and social choice help researchers learn and aggregate peoples' preferences, which are used to guide AI behavior; thus, it is imperative that these learned preferences are accurate. These techniques often assume that people are willing to express strict preferences over alternatives; which is not true in practice. People are often indecisive, and especially so when their decision has moral implications. The philosophy and psychology literature shows that indecision is a measurable and nuanced behavior---and that there are several different reasons people are indecisive. This complicates the task of both learning and aggregating preferences, since most of the relevant literature makes restrictive assumptions on the meaning of indecision. We begin to close this gap by formalizing several mathematical indecision models based on theories from philosophy, psychology, and economics; these models can be used to describe (indecisive) agent decisions, both when they are allowed to express indecision and when they are not. We test these models using data collected from an online survey where participants choose how to (hypothetically) allocate organs to patients waiting for a transplant. 

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