Moving Horizon Estimation (MHE)

The natural counterpart to MHE, forming a complete dual estimation-control framework. While MHE solves the optimal state estimation problem over a past horizon, MPC solves the optimal control problem over a future prediction horizon.

• Both share the same dynamic model and optimization solver infrastructure
• MHE provides the initial state for MPC's prediction at each time step
• The receding horizon principle is identical: the window slides forward with each sample
• Together they enable constraint-aware, optimal closed-loop performance for non-linear processes

The recursive predecessor to MHE for unconstrained linear systems. The Kalman filter provides the optimal state estimate by minimizing mean squared error analytically, without solving an online optimization problem.

• Computationally lightweight: O(n³) per step vs. MHE's optimization
• MHE reduces to the Kalman filter for linear, unconstrained problems with infinite horizon
• Kalman filters cannot enforce physical constraints like non-negative concentrations or valve saturation limits
• The Extended Kalman Filter (EKF) handles non-linearity via linearization but lacks MHE's robustness to poor initial guesses

The foundational discipline that builds the dynamic process models required by MHE. Without an accurate state-space representation, MHE cannot reconcile measurements with predictions.

• Identifies model structure and parameters from input-output data
• Techniques include subspace identification, prediction error methods, and regularized regression
• The model's quality directly determines MHE's estimation accuracy
• Grey-box modeling combines first-principles physics with data-driven parameter estimation for MHE applications

A critical implementation detail that summarizes all measurement data before the estimation window into a single penalty term. This prevents information loss when the horizon truncates.

• Without a proper arrival cost, MHE behaves like a finite-memory estimator and may diverge
• The arrival cost is typically approximated using an Extended Kalman Filter covariance update
• Poor arrival cost approximation is the most common cause of MHE performance degradation
• Advanced formulations use particle filters or moving horizon versions of the unscented transform

The computational engine that makes real-time MHE feasible. Each MHE iteration requires solving a constrained nonlinear optimization problem within the control sample time.

• Interior-point methods and sequential quadratic programming (SQP) are standard approaches
• Solvers like IPOPT, SNOPT, and acados are optimized for the structured sparsity of MHE problems
• Real-time iteration schemes sacrifice optimality for speed by performing only one Newton step per sample
• Hardware acceleration via GPUs and FPGAs enables MHE for millisecond-scale processes

The mathematical prerequisite that determines whether MHE can uniquely reconstruct the internal state from available measurements. An unobservable system will produce ambiguous or divergent estimates.

• Linear observability is checked via the rank of the observability matrix
• Nonlinear systems require Lie derivative-based observability rank conditions
• MHE can exploit moving horizons to recover practical observability even when the full system is theoretically unobservable
• Sensor placement optimization uses observability Gramians to maximize MHE estimation quality