不可区分齐次主体的逆线性二次离散时间有限时域最优控制：一种凸优化方法
Inverse linear-quadratic discrete-time finite-horizon optimal control for indistinguishable homogeneous agents: a convex optimization approach
摘要：
逆线性二次型最优控制问题是一个系统辨识问题，其目标是根据最优轨迹的观测值恢复二次型费用函数，从而恢复闭环系统矩阵。在本文中，离散时间，有限的地平线的情况下，代理商也被假定为同质和不可区分的。后者意味着代理都具有相同的动力学和目标函数，并且观察结果是所有代理在不同时刻的“快照”，但不知道的是连续观察的“哪个代理移动到哪里”。这种缺乏链接的最佳轨迹使得问题具有挑战性。我们首先表明，这个问题是全球性的。然后，对于无噪声观测的情况下，我们表明，真正的成本矩阵，因此闭环系统矩阵，可以恢复为唯一的全局最优解的凸优化问题。接下来，对于噪声观测的情况下，我们制定了一个估计作为唯一的全局最优解的修改后的凸优化问题。此外，该估计的统计一致性。最后，所提出的方法的性能证明了一些数值例子。
英文摘要：
The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal trajectories. In this paper, the discrete-time, finite-horizon case is considered, where the agents are also assumed to be homogeneous and indistinguishable. The latter means that the agents all have the same dynamics and objective functions and the observations are in terms of "snap shots" of all agents at different time instants, but what is not known is "which agent moved where" for consecutive observations. This absence of linked optimal trajectories makes the problem challenging. We first show that this problem is globally identifiable. Then, for the case of noiseless observations, we show that the true cost matrix, and hence the closed-loop system matrices, can be recovered as the unique global optimal solution to a convex optimization problem. Next, for the case of noisy observations, we formulate an estimator as the unique global optimal solution to a modified convex optimization problem. Moreover, the statistical consistency of this estimator is shown. Finally, the performance of the proposed method is demonstrated by a number of numerical examples.
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