Moving horizon estimation is an optimization approach that uses a series of measurements observed over time, containing noise and other inaccuracies, and produces estimates of unknown variables or parameters. Unlike deterministic approaches, MHE requires an iterative approach that relies on linear programming or nonlinear programming solvers to find a solution. MHE reduces to the Kalman filter under certain simplifying conditions. A critical evaluation of the extended Kalman filter and MHE found improved performance of MHE with the only cost of improvement being the increased computational expense. Because of the computational expense, MHE has generally been applied to systems where there are greater computational resources and moderate to slow system dynamics. However, in the literature there are some methods to accelerate this method.
Overview
The application of MHE is generally to estimate measured or unmeasured states of dynamical systems. Initial conditions and parameters within a model are adjusted by MHE to align measured and predicted values. MHE is based on a finite horizon optimization of a process model and measurements. At time the current process state is sampled and a minimizing strategy is computed for a relatively short time horizon in the past:. Specifically, an online or on-the-fly calculation is used to explore state trajectories that find an objective-minimizing strategy until time. Only the last step of the estimation strategy is used, then the process state is sampled again and the calculations are repeated starting from the time-shifted states, yielding a new state path and predicted parameters. The estimation horizon keeps being shifted forward and for this reason the technique is called moving horizon estimation. Although this approach is not optimal, in practice it has given very good results when compared with the Kalman filter and other estimation strategies.
Principles of MHE
Moving horizon estimation is a multivariable estimation algorithm that uses:
an optimization cost function J over the estimation horizon,
to calculate the optimum states and parameters. The optimization estimation function is given by: without violating state or parameter constraints With: = i -th model predicted variable = i -th measured variable = i -th estimated parameter = weighting coefficient reflecting the relative importance of measured values = weighting coefficient reflecting the relative importance of prior model predictions = weighting coefficient penalizing relative big changes in Moving horizon estimation uses a sliding time window. At each sampling time the window moves one step forward. It estimates the states in the window by analyzing the measured output sequence and uses the last estimated state out of the window, as the prior knowledge.
