State Estimation (Observer)

Observability is a fundamental system property that determines whether the internal states can be uniquely inferred from the system's outputs over a finite time interval. A system is observable if, by observing the outputs, one can deduce the initial state, and therefore all future states. This is a prerequisite for effective state estimation. For linear systems, observability is mathematically verified using the Observability Matrix or the Popov-Belevitch-Hautus (PBH) test. Without observability, no estimator can converge to the true state.

A primary function of a state estimator is to separate the true system signal from measurement noise and process disturbances. Optimal estimators, like the Kalman Filter, are designed to minimize the mean squared error of the state estimate by statistically modeling the noise. They use a prediction-correction cycle:

• Prediction: Projects the state forward using the system model.
• Correction (Update): Fuses the prediction with the new measurement, weighted by the Kalman Gain. This gain is computed to optimally balance trust in the model's prediction versus the new sensor reading, effectively providing a low-pass filter for the state.

A state estimator must be asymptotically stable, meaning its estimation error converges to zero over time, regardless of the initial guess. For a Luenberger Observer (for deterministic linear systems), stability is ensured by placing the poles of the observer's error dynamics in the left-half of the complex plane (for continuous time). For stochastic estimators like the Kalman Filter, convergence is guaranteed under conditions of detectability and stabilizability. The speed of convergence is dictated by the chosen observer poles or the filter's inherent dynamics, allowing designers to trade off between fast response and noise sensitivity.

The accuracy of a state estimator is intrinsically tied to the fidelity of the internal model it uses. This model, typically a set of differential or difference equations, describes the expected system dynamics. Model mismatch—differences between the estimator's internal model and the true plant—is a primary source of estimation error and can lead to bias or divergence. Advanced techniques like Adaptive Observers or Moving Horizon Estimation (MHE) can partially compensate for model errors by simultaneously estimating states and key model parameters online.

Different estimators are suited to different system classes and noise assumptions:

• Luenberger Observer: For deterministic, linear time-invariant (LTI) systems.
• Kalman Filter (KF): The optimal linear estimator for systems with Gaussian white noise.
• Extended Kalman Filter (EKF): A nonlinear variant that linearizes the model around the current estimate.
• Unscented Kalman Filter (UKF): Uses a deterministic sampling approach to better handle nonlinearities without linearization.
• Moving Horizon Estimation (MHE): Solves a constrained optimization over a sliding window of past data, explicitly handling constraints and nonlinear models, acting as the dual to Model Predictive Control (MPC).

In Model Predictive Control (MPC), the state estimator provides the initial condition for the prediction horizon. The MPC algorithm solves an optimization problem starting from this estimated state. Therefore, the performance of the MPC controller is directly limited by the accuracy of the state estimate. An inaccurate initial state can lead to suboptimal or even infeasible control actions. This tight coupling is why Moving Horizon Estimation (MHE) is often paired with MPC, forming a unified estimation-and-control framework where both components use the same system model and constraint sets.

The Kalman Filter is an optimal recursive algorithm for estimating the state of a linear dynamic system from a series of noisy measurements. It operates in a two-step predict-update cycle:

• Predict: Projects the current state estimate forward using the system model and accounts for process noise.
• Update: Corrects the prediction with a new measurement, weighting the correction based on the estimated uncertainty (covariance). It is the foundational observer for Linear MPC, providing the minimum mean-square error estimate under Gaussian noise assumptions.

The Extended Kalman Filter (EKF) is the de facto standard for state estimation in Nonlinear MPC (NMPC). It linearizes the nonlinear system dynamics and measurement models around the current state estimate at each time step.

• Process: The Jacobian matrices of the nonlinear functions are computed to create a local linear approximation.
• Application: Enables the use of the Kalman Filter framework for nonlinear systems, though it is only optimal for mildly nonlinear problems.
• Use Case: Critical for robotics where models for orientation (e.g., using quaternions) or complex dynamics are inherently nonlinear.

Moving Horizon Estimation (MHE) is the dual problem to MPC. Instead of using a recursive filter, it solves an online optimization problem over a finite window of past measurements to estimate the current state.

• Mechanism: Minimizes a cost function that penalizes estimation error and model mismatch, explicitly handling state constraints and nonlinear models.
• Advantage: Provides more accurate estimates for constrained or highly nonlinear systems compared to EKF, at the cost of higher computational load.
• Relation to MPC: Often paired with NMPC in a unified estimation-and-control architecture for maximum performance.

Observability is a fundamental system property that determines whether the internal state of a system can be uniquely reconstructed from its output measurements over a finite time interval.

• Mathematical Test: For linear systems, a system is observable if the observability matrix has full column rank.
• Implication for MPC: If a system is unobservable, a state estimator cannot converge to the true state, rendering the MPC predictions invalid. Observability analysis is a prerequisite for observer design.
• Detectability: A slightly weaker condition where only the unstable states need to be observable, which is sufficient for stabilizing controllers.

A Luenberger Observer is a deterministic state estimator for linear systems designed via pole placement. It uses a copy of the system dynamics plus a correction term driven by the output estimation error.

• Design: The observer gain matrix is chosen to place the observer poles (eigenvalues) faster than the controller poles, ensuring the estimation error decays rapidly.
• Relation to Kalman Filter: For a linear time-invariant system with no noise, the steady-state Kalman Filter reduces to a Luenberger Observer with a specific gain.
• Use Case: Commonly taught as the foundational concept for state feedback control when full state measurement is unavailable.

Sensor Fusion is the process of combining data from multiple, often heterogeneous, sensors to produce a state estimate that is more accurate, complete, and reliable than from any single sensor.

• Methods: Algorithms like the Kalman Filter are naturally multi-sensor. Complementary Filtering is a simpler, frequency-domain approach often used in robotics (e.g., fusing gyroscope and accelerometer data for attitude).
• Role in Embodied Systems: Essential for robotics, where data from LiDAR, cameras, IMUs, and wheel encoders must be fused to form a coherent estimate of pose, velocity, and environment.
• Challenge: Requires careful modeling of each sensor's noise characteristics and time synchronization.