Observability

A system is observable if, for any possible sequence of control inputs, the current internal state vector can be uniquely determined from a finite history of the system's outputs (measurements). Formally, a linear time-invariant system is observable if and only if its observability matrix has full column rank. For nonlinear systems, the concept is extended via local observability and observability analysis using Lie derivatives.

Observability is the dual concept to controllability in linear systems theory. While controllability asks whether any state can be reached with appropriate control inputs, observability asks whether any state can be identified from the outputs. This duality is formalized in the Kalman decomposition, which separates a system's state-space into four parts: controllable and observable, controllable but not observable, observable but not controllable, and neither.

An unobservable state is a component of the system's internal state that produces no effect on the measured outputs. Unobservable states form an unobservable subspace. In sensor fusion, this manifests as degenerate motions—movements the robot can make that are invisible to the sensor suite. A classic example is a robot with only a monocular camera moving along the camera's optical axis; the scale of translation is unobservable without additional sensors like an IMU.

Observability directly dictates the performance and stability of state estimation algorithms like the Kalman Filter and its variants. If the system is unobservable, the estimator's covariance matrix for the unobservable states will not converge—their uncertainty remains unbounded. Engineers perform observability analysis during sensor suite design to ensure all critical degrees of freedom (e.g., position, orientation, velocity, bias states) are observable. For example, a Visual-Inertial Odometry (VIO) system's observability of scale depends on accelerometer excitation.

In physical systems, observability is not a binary property but a continuum influenced by:

• Sensor Geometry: The placement and type of sensors (e.g., baseline between stereo cameras).
• Excitation Motion: The specific trajectory of the robot (e.g., a car driving straight provides poor observability of lateral dynamics).
• Environmental Features: The richness of the perceived environment (e.g., a textureless wall degrades visual observability). Poor observability leads to filter divergence, drift, and unreliable state estimates, compromising navigation and control.

Engineers use specific methods to analyze observability:

• Linear Systems: Checking the rank of the observability matrix.
• Nonlinear Systems (VIO, SLAM): Employing Lie algebra and computing the observability Gramian to test for local weak observability.
• Empirical Analysis: Examining the Fisher Information Matrix (FIM) or the Hessian in bundle adjustment; eigenvalues near zero indicate poorly observable directions.
• Simulation: Injecting noise into unobservable directions in simulation to verify that estimator uncertainty grows appropriately.

State estimation is the core algorithmic process of inferring the internal, often unmeasured, variables (the state) of a dynamic system from a sequence of noisy sensor measurements. The state typically includes quantities like position, orientation, velocity, and bias terms.

• It is the practical application of the theoretical property of observability.
• If a system is observable, a state estimator (like a Kalman filter) can be designed to converge to the true state.
• Common algorithms include the Kalman Filter, Particle Filter, and optimization-based smoothers.

A Kalman filter is an optimal recursive algorithm for state estimation in linear dynamic systems with Gaussian noise. It operates in a two-step predict-update cycle:

• Predict: Projects the current state and its uncertainty (covariance) forward using the system's process model.
• Update: Corrects the prediction using a new sensor measurement and the measurement model, weighting the correction based on the relative certainty of the prediction and the measurement.

Its optimality is contingent on the system's observability; unobservable states cannot be corrected by the measurements.

These two mathematical models are the essential components of any state estimation framework and directly determine observability.

• Process Model: Describes how the system's state evolves over time (e.g., x_{k+1} = f(x_k, u_k) + w_k). It defines the dynamics.
• Measurement Model: Describes how sensor observations are generated from the true state (e.g., z_k = h(x_k) + v_k). It defines what the sensors can see.

The combination of f() and h() defines which state variables are observable. A state is unobservable if changes in it produce no change in the predicted measurements.

Controllability is the dual concept to observability in linear systems theory. It determines whether a system's state can be driven to any desired value using appropriate control inputs over a finite time.

• Observability: Can you deduce the internal state from outputs?
• Controllability: Can you influence the internal state with inputs?

For a linear time-invariant system, the Kalman rank test can check observability, and a similar Popov-Belevitch-Hautus (PBH) test checks controllability. In optimal control (e.g., Linear Quadratic Regulator - LQR), controllability is a prerequisite.

Sensor fusion is the engineering practice of algorithmically combining data from multiple, often heterogeneous, sensors to produce a state estimate that is more accurate, complete, and reliable than one from a single source.

• It relies on the principle that different sensors have complementary observability properties. For example, a camera may observe position well but not velocity, while an IMU observes velocity but drifts in position.
• Architectures are classified as loosely-coupled (fusing high-level state estimates) or tightly-coupled (fusing raw sensor data), with the latter generally providing better observability and accuracy.

For the vast majority of robotic systems (which are nonlinear), nonlinear observability analysis is required. Unlike the linear case, observability can depend on the specific trajectory and inputs.

• Analysis often uses Lie derivatives to construct an observability matrix and check its rank locally.
• A classic example: a robot with a monocular camera. Its scale (absolute distance) is unobservable from images alone, as doubling the robot's size and halving its distance from all landmarks produces identical images. This is resolved by fusing with an IMU (observes scale through acceleration) or using stereo vision.
• Algorithms like the Unscented Kalman Filter (UKF) and Error-State Kalman Filter (ESKF) are designed for nonlinear observability problems.