Many incentive design problems must contend with information asymmetries due to non-observation of efficiency (adverse selection) or non-observation of effort (moral hazard). And although a growing body of literature considers incentive design in control systems, the problem of designing incentives for control systems under information asymmetries has been less well-studied. This paper considers a model of moral hazard within control systems. In our model, the control system is described by an (affine) linear time-invariant (LTI) system with process noise. There is an agent who gets to choose (from between two choices) a linear state-feedback controller to apply to the LTI system, with one of the state-feedback controllers having a higher quadratic cost on the control inputs than the other. Our goal is to design a payment scheme that incentivizes the agent to choose the state-feedback controller that minimizes a quadratic cost on system states plus the time-discounted payment amount, subject to the understanding that the agent bears the control cost while being risk-averse with respect to their time-discounted payment. We formulate the problem as a constrained optimization, and prove that for a payment given after a fixed (but optimizable) time horizon the optimal payment scheme chooses the payment amount using a likelihood ratio hypothesis test. We numerically demonstrate our results by applying the derived optimal payment scheme to two examples: load frequency control (LFC) in power systems and wellness interventions for body weight loss.

翻译：许多激励设计问题必须应对由效率不可观测（逆向选择）或努力不可观测（道德风险）导致的信息不对称。尽管越来越多的文献关注控制系统中的激励设计，但在信息不对称条件下为控制系统设计激励的问题仍未得到充分研究。本文研究了一个控制系统中存在道德风险的模型。在该模型中，控制系统由一个带有过程噪声的（仿射）线性时不变（LTI）系统描述。存在一个智能体，其可以从两种选择中选取一个线性状态反馈控制器应用于该LTI系统，其中一个状态反馈控制器在控制输入上的二次成本高于另一个。我们的目标是设计一种支付方案，激励智能体选择能够最小化系统状态二次成本与时间折现支付金额之和的状态反馈控制器，同时需考虑智能体承担控制成本且对其时间折现支付具有风险厌恶特征。我们将该问题建模为约束优化问题，并证明：当支付在固定（但可优化）的时间范围后执行时，最优支付方案通过似然比假设检验确定支付金额。我们通过将所推导的最优支付方案应用于两个示例（电力系统的负荷频率控制（LFC）和体重减轻的健康干预）来数值验证结果。