The inverse optimal control for finite-horizon discrete-time linear quadratic regulators is investigated in this paper, which is to estimate the parameters in the objective function using noisy measurements of partial optimal states only. By the Pontryagin’s minimum principle, the concerned inverse optimal control problem is recast as the identification of a parameterized causal-and-anticausal mixed system excited by boundary conditions. Sufficient identifiability conditions for the unknown parameters are provided in terms of the system model itself, rather than relying on the exact values of optimal states or control inputs. In addition, an elegant algebraic solution is provided for the concerned identification problem that is inherently a challenging optimization problem with trilinear equality constraints, and it can recover the true parameters (up to a scalar ambiguity) in the absence of measurement noise or can consistently identify the parameters (up to a scalar ambiguity) in the presence of white measurement noise. The presented algebraic solution relies on recursive matrix calculations so that its computational burden is much less than directly solving a high-dimensional non-convex optimization problem as done in many existing works. The effectiveness of the proposed method as well as its noise sensitivity issue is illustrated by simulation examples.

本文研究了有限水平离散时间线性二次调节器的逆最优控制，其目的是仅通过对部分最优状态的噪声测量来估计目标函数中的参数。根据庞特里亚金极小原理，将相关的逆最优控制问题重铸为对边界条件激发的参数化因果和反混合系统的识别。对于未知参数，有足够的可识别性条件是根据系统模型本身提供的，而不是依赖于最佳状态或控制输入的确切值。此外，还为相关的识别问题提供了一种优雅的代数解决方案，该问题本质上是具有三线性等式约束的具有挑战性的优化问题，并且它可以在没有测量噪声的情况下恢复真实参数（最高标量模糊度），或者可以在存在白色测量噪声的情况下一致地识别参数（最高标量模糊度）。所提出的代数解决方案依赖于递归矩阵计算，因此其计算负担远不如许多现有工作中那样直接解决高维非凸优化问题。仿真算例说明了所提方法的有效性及其噪声敏感性问题。所提出的代数解决方案依赖于递归矩阵计算，因此其计算负担远不如许多现有工作中那样直接解决高维非凸优化问题。仿真算例说明了所提方法的有效性及其噪声敏感性问题。所提出的代数解决方案依赖于递归矩阵计算，因此其计算负担远不如许多现有工作中那样直接解决高维非凸优化问题。仿真算例说明了所提方法的有效性及其噪声敏感性问题。