Model Predictive Control

The Kalman Filter (KF) is a recursive state estimator proposed by Rudolf E. Kálmán in 1960. It operates in two steps: prediction (propagating state and covariance through the dynamics model) and correction (updating with a new measurement, weighted by the Kalman gain matrix). It is optimal in the MMSE sense for linear systems with Gaussian noise. In robotics and control, the KF is the foundation of localization, object tracking, sensor fusion (IMU + GPS + LiDAR), fault detection, and state prediction for MPC. The Extended Kalman Filter (EKF) linearises the nonlinear model at the operating point; the Unscented Kalman Filter (UKF) propagates deterministic sigma points instead; the Particle Filter uses Monte Carlo approximation for arbitrary distributions. The Kalman Filter is a building block of virtually every modern FDI and FTC system — the residuals it generates (differences between predicted and observed output) are the primary signal for fault detection.

Fault Detection and Isolation (FDI), also called Fault Detection and Diagnosis (FDD), is a discipline of control engineering concerned with automatic detection, isolation, and characterisation of faults in dynamic systems. FDI comprises three phases: (1) detection — determining that a fault has occurred (by residual threshold crossing); (2) isolation — identifying the fault location (which sensor/actuator/component); (3) identification — characterising the fault (type, magnitude, time profile). Model-based methods include state observers (Luenberger, KF), parity equations, observer banks, Dedicated Observer Scheme (DOS), and Generalised Observer Scheme (GOS). Data-driven methods use ML classifiers (SVM, neural networks, autoencoders) trained on fault data. FDI is a prerequisite for Active FTC — without reliable diagnosis, controller reconfiguration cannot happen in time. Key FDI metrics: False Alarm Rate (FAR), Missed Detection Rate (MDR), detection time T_d, and isolation time T_i. Standards: IEC 61511 (SIS), ISO 13849 (machinery), DO-178C (aviation).

Fault-Tolerant Control (FTC) is a branch of control theory concerned with systems capable of preserving stability and acceptable control performance in the presence of faults. It is divided into two main classes: Passive FTC (PFTC), where the controller is designed to be robust against a predefined fault set without explicitly detecting it, and Active FTC (AFTC), where a Fault Detection and Diagnosis (FDD) module detects and isolates the fault and a reconfiguration mechanism then alters the controller structure or parameters. In robotics, FTC is applied to redundant manipulators, multirotor drones, humanoid robots, and autonomous vehicles — anywhere a single actuator or sensor failure must not result in loss of control. Typical methods include Control Allocation with redistribution onto healthy actuators, Multiple Model Adaptive Control, Sliding Mode Control, and Model Predictive Control with an updated faulty-plant model. FTC is tightly coupled with FDI (Fault Detection and Isolation) — without reliable diagnosis, the controller cannot be reconfigured in time. Key performance metrics include detection time, reconfiguration time, the bound on performance degradation, and the share of faults the system can handle. Standards such as ISO 26262 (automotive), DO-178C/DO-254 (aviation), and IEC 61508 (industry) mandate the use of FTC techniques in safety-critical applications.