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