System identification approach for inverse optimal control of finite-horizon linear quadratic regulators

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.