Moving-Horizon Estimation
MHE estimates states by solving a constrained optimization over a sliding window of recent measurements, the estimation dual of predictive control.
Estimation as optimization
Moving-horizon estimation reconstructs the current state by fitting the model to a window of the most recent measurements. At each step it solves an optimization that minimizes the weighted mismatch between predicted and measured outputs over the window, plus a penalty tying the window's start to prior information. Only the estimate at the end of the window is used; then the window slides forward. It is the estimation counterpart of receding-horizon control.
The arrival cost
Discarding data older than the window would lose information, so MHE summarizes it in an arrival cost, a penalty on the state at the window's start that approximates all past measurements. A common choice updates the arrival cost from a Kalman-filter-like covariance recursion. A poor arrival-cost approximation degrades performance, so it is treated carefully.
Why use it over a Kalman filter
The extended Kalman filter is cheaper but handles constraints and nonlinearity poorly. MHE can enforce hard constraints on the states, such as nonnegativity of a concentration or physical bounds, directly in the optimization, and it handles nonlinear models more accurately because it re-linearizes over the window rather than at a single point. The cost is solving an optimization each step.
- Fit the model to a sliding window of measurements
- Arrival cost summarizes discarded past data
- Enforces physical constraints on state estimates
- More accurate than EKF for constrained nonlinear systems
MHE pairs naturally with nonlinear MPC: both solve constrained optimizations over horizons, share model and solver infrastructure, and together form a constrained output-feedback scheme. Stability of the combined estimator-controller loop can be established under detectability and observability assumptions.
For a design-stage plant where some states are unmeasured and physically bounded, MHE would produce constraint-respecting estimates for the controller, evaluated in simulation. It is the modern optimization-based observer.