A Deterministic Least Squares Approach for Simultaneous Input and State Estimation

In this article, we consider a deterministic estimation problem to find the input and state of a linear dynamical system, which minimize a weighted integral squared error between the resulting output and the measured output. A completion of squares approach is used to find the unique optimum in terms of the solution of a Riccati differential equation. The optimal estimate is obtained from a two-stage procedure that is reminiscent of the Kalman filter. The first stage is an end-of-interval estimator for the finite horizon, which may be solved in real time as the horizon length increases. The second stage computes the unique optimum over a fixed horizon by a backward integration over the horizon. A related tracking problem is solved in an analogous manner. Making use of the solution to both the estimation and tracking problems, a constrained estimation problem is solved which shows that the Riccati equation solution has a least squares interpretation that is analogous to the meaning of the covariance matrix in stochastic filtering. This article shows that the estimation and tracking problems considered here include the Kalman filter and the linear quadratic regulator as special cases. The infinite horizon case is also considered for both the estimation and tracking problems. Stability and convergence conditions are provided and the optimal solutions are shown to take the form of left inverses of the original system.