In this brief, we present a set of techniques for finding a cost function to the time-invariant linear quadratic regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient linear matrix inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. In addition, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrix K from the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.

中文翻译：

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LQR反问题的解及其在生物系统分析中的应用

在本文中，我们介绍了一套在连续时间和离散时间情况下为时不变线性二次调节器（LQR）问题寻找成本函数的技术。我们的方法基于对LQR反问题的解决方案，它可以表示为：给定的控制器K是否描述了时不变LQR问题的解决方案；如果是，那么权重Q和R会产生K作为最优解吗？ ？我们研究此问题的动机是分析生物系统中的运动目标。我们首先描述一种有效的线性矩阵不等式（LMI）方法，用于在加权矩阵Q和R都不知道的情况下确定此反LQR问题的一般情况的解决方案。我们的第一个基于LMI的配方在可行时提供了独特的解决方案。此外，我们提出了一种基于梯度的方法，最小二乘最小化方法，可以在LMI不可行的情况下近似求解。该新方法在实践中非常有用，因为根据嘈杂的实验数据估计的增益矩阵K可能会受到估计误差的干扰，这可能会导致LMI的不可行。我们还提供了一个LMI最小化问题，以使用提出的梯度下降算法找到一个很好的最小化初始点。然后，我们提供了一组示例来说明如何将我们的方法应用于几种不同类型的问题。重要的结果是，将该技术应用于坐在移动机器人上的人体姿势控制。结果表明，我们可以恢复成本函数，这可能会对人的运动控制目标提供有用的见解。