Priess M Cody, Conway Richard, Choi Jongeun, Popovich John M, Radcliffe Clark
Michigan State University Dept. of Mechanical Engineering and the MSU Center for Orthopedic Research (MSUCOR), East Lansing, MI 48824.
Western Digital Inc.
IEEE Trans Control Syst Technol. 2015 Mar;23(2):770-777. doi: 10.1109/TCST.2014.2343935. Epub 2014 Aug 19.
In this paper, 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 describe the solution to a time-invariant LQR problem, and if so, what weights and produce 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 and are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. Additionally, 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 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.
在本文中，我们提出了一套技术，用于在连续时间和离散时间情况下找到时不变线性二次调节器（LQR）问题的代价函数。我们的方法基于逆LQR问题的解，该问题可以表述为：给定的控制器是否描述了时不变LQR问题的解，如果是，哪些权重矩阵 (Q) 和 (R) 会产生该控制器作为最优解？我们研究这个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，用于在权重矩阵 (Q) 和 (R) 均未知的情况下确定该逆LQR问题一般情形的解。我们基于LMI的第一种公式化表述在可行时提供唯一解。此外，我们提出一种基于梯度的最小二乘最小化方法，当LMI不可行时，该方法可用于近似求解。这种新方法在实际中非常有用，因为从噪声实验数据估计的增益矩阵 (\hat{K}) 可能会受到估计误差的干扰，这可能导致LMI不可行。我们还提供了一个LMI最小化问题，以使用所提出的梯度下降算法找到最小化的良好初始点。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。一个重要的结果是该技术在人类坐在移动机器人上时的姿势控制中的应用。结果表明，我们可以恢复一个代价函数，这可能为人类运动控制目标提供有用的见解。
