Moving Horizon Estimation (MHE)

Unlike the Kalman Filter, which uses all past data, MHE solves an optimization problem over a sliding window of the most recent N measurements. This window, the estimation horizon, moves forward in time with each new measurement. The core optimization minimizes the discrepancy between model predictions and measurements within this window, plus a prior weighting term that incorporates information from before the horizon. This structure provides a natural mechanism for data forgetting, preventing outdated information from corrupting the current estimate.

A primary advantage of MHE is its ability to enforce hard constraints on state and parameter estimates. These constraints encode physical realities and prior knowledge that a pure statistical filter cannot guarantee.

• Physical Limits: Enforce that a tank level must be between 0% and 100%, or a concentration must be non-negative.
• Model-Based Knowledge: Constrain parameters like reaction rates to known feasible ranges.
• Feasibility Guarantees: The optimization ensures the estimated state trajectory is consistent with both the measurements and the system's physical operating envelope, leading to more realistic and reliable estimates than unconstrained methods.

MHE formulates estimation as a Nonlinear Programming (NLP) problem, allowing it to directly incorporate nonlinear dynamic models. This is critical for complex systems like chemical reactors, robotics, and aerospace vehicles where dynamics are inherently nonlinear.

The optimization framework does not require linearization for the core algorithm (though solvers may use it internally). It seeks the state trajectory that best fits the nonlinear model to the data. This makes MHE more accurate than the Extended Kalman Filter (EKF) for highly nonlinear systems, as it avoids the linearization errors that can accumulate in the EKF's recursive update.

The arrival cost is a critical term in the MHE objective function that summarizes the information from measurements prior to the estimation horizon. It prevents the loss of historical data when the window slides.

• Function: It penalizes estimates at the beginning of the horizon that are inconsistent with past data.
• Approximation: Often approximated using the covariance from a Kalman Filter or an Extended Kalman Filter (EKF), effectively "bridging" the recursive filtering world with the optimization-based MHE framework.
• Stability: A well-designed arrival cost is essential for guaranteeing the stability of the MHE, ensuring estimates do not diverge.

MHE is the natural dual or inverse problem of Model Predictive Control (MPC). This duality creates a powerful, symmetric framework for advanced control:

• MPC (Forward Problem): Uses a model to predict future states and optimize future control inputs.
• MHE (Inverse Problem): Uses a model and past measurements to reconstruct past states and estimate current states.

In practice, they are often used in tandem: MHE provides the accurate initial state estimate required for MPC's prediction. This MPC-MHE cascade is the standard architecture for high-performance constrained control of complex systems, such as autonomous vehicles and chemical processes.

The main challenge of MHE is its computational cost. At each time step, it must solve a (potentially large) Nonlinear Programming problem within the system's sampling period.

• Solvers: Relies on efficient NLP solvers like IPOPT or Sequential Quadratic Programming (SQP) methods.
• Optimization: Techniques like direct multiple shooting or collocation discretize the problem for numerical stability.
• Real-Time: Advances in solver speed and warm-starting (using the previous solution as an initial guess) have made real-time MHE feasible for many industrial applications with sampling times on the order of seconds. For very fast systems, Explicit MHE or approximate methods may be required.

Model Predictive Control (MPC) is the dual problem to MHE. It is an advanced control method that uses an internal dynamic model to predict future system behavior over a finite horizon. At each time step, it solves an online optimization problem to determine a sequence of optimal control inputs, applying only the first input before re-solving with new measurements.

• Core Principle: Receding horizon control.
• Mathematical Foundation: Optimal Control Problem (OCP).
• Key Relationship to MHE: MPC requires an accurate initial state for its predictions, which is precisely the output provided by MHE. Together, they form an MHE-MPC cascade, the gold standard for constrained, nonlinear control.

The Kalman Filter (KF) is the foundational recursive algorithm for state estimation in linear dynamic systems with Gaussian noise. It provides the optimal (minimum mean-square error) state estimate by fusing predictions from a model with incoming measurements.

• Key Limitation: Designed for unconstrained, linear systems.
• Contrast with MHE: MHE generalizes the KF by explicitly handling nonlinear dynamics, hard constraints on states, and using a finite window of past data. For linear, unconstrained problems, the KF and a properly tuned MHE yield identical results, but MHE provides a unified framework for more complex scenarios.

The Extended Kalman Filter (EKF) is the de facto standard for nonlinear state estimation. It linearizes the system's nonlinear dynamics and measurement models around the current state estimate at each time step, then applies the standard Kalman Filter equations.

• Primary Use Case: Real-time estimation for moderately nonlinear systems.
• Contrast with MHE: The EKF's linearization can introduce significant errors for highly nonlinear systems or when estimates approach constraints. MHE solves a full nonlinear optimization over a data window, providing more accurate estimates for constrained, strongly nonlinear problems, albeit at a higher computational cost.

An Optimal Control Problem (OCP) is the core mathematical formulation solved by both MPC and MHE. It is defined by:

• A cost function (or objective function) to be minimized.

• A dynamic model describing the system's evolution.

• A set of constraints on states and inputs.

• A specified time horizon.

• In MPC: The OCP is solved forward in time to find optimal future controls.

• In MHE: The OCP is solved over a recent past horizon to find the most likely sequence of past states that explains the observed measurements, given the model and constraints. MHE is essentially an OCP for estimation.

Full Information Estimation (FIE) is the theoretical ideal where the optimization for state estimation considers the entire history of measurements from time zero to the current time. It provides the best possible estimate but grows computationally unbounded over time.

• Key Relationship to MHE: MHE is a practical approximation of FIE. Instead of using all past data, MHE uses a fixed-length moving window of the most recent measurements. This imposes a finite, manageable problem size while retaining most of the performance benefits of FIE, especially for systems where older data becomes less informative.

The Arrival Cost is a critical term in the MHE cost function that summarizes the information from data outside the estimation window. It penalizes the estimated state at the beginning of the window based on all prior measurements, ensuring the finite-horizon MHE problem approximates the full-history FIE problem.

• Function: Prevents the loss of historical information as the window slides.
• Common Approximation: Often approximated using the covariance matrix from a standard Kalman Filter or EKF running in parallel. The accurate design and approximation of the arrival cost is essential for MHE stability and performance.