Simple and efficient moving horizon estimation based on the fast gradient method

By now many results with respect to the fast and efficient implementation of model predictive control exist. However, for moving horizon estimation, only a few results are available. We present a simple solution algorithm tailored to moving horizon estimation of linear, discrete time systems. In a first step the problem is reformulated such that only the states remain as optimization variables, i.e. process and measurement noise are eliminated from the optimization problem. This reformulation enables the use of the fast gradient method, which has recently received a lot of attention for the solution of model predictive control problems. In contrast to the model predictive control case, the Hessian matrix is time- varying in moving horizon estimation, due to the time-varying nature of the arrival cost. Therefore, we outline a tailored method to compute online the lower and upper eigenvalues of the Hessian matrix required by the here considered fast gradient method. In addition, we discuss stopping criteria and various implementation details. An example illustrates the efficiency of the proposed algorithm.