Observability

Author: Tianjiang Shuo

Reference: Qian Yingjing (2014), "Research on Autonomous Navigation and Orbit Keeping of Spacecraft on Quasi-Periodic Orbits in Cislunar Space"

Website: https://cislunarspace.cn

Definition

Observability is a core concept in systems theory and control theory, describing the degree to which the internal states of a system can be determined through external output (measurement) information. In navigation system design, observability analysis is used to determine: given a set of sensor configurations and measurement information, whether the system states can be uniquely and accurately estimated.

For autonomous navigation systems, observability analysis is a critical step in navigation scheme design. A navigation system that fails to satisfy observability requirements cannot provide convergent state estimates, even with an optimal filtering algorithm.

Observability of Linear Systems

Definition

For a linear time-invariant system:

x˙=Ax+Bu\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}

y=Cx+Du\mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u}

its observability matrix is:

O=[CCACA2⋮CAn−1]\mathcal{O} = \begin{bmatrix} \mathbf{C} \\ \mathbf{C}\mathbf{A} \\ \mathbf{C}\mathbf{A}^2 \\ \vdots \\ \mathbf{C}\mathbf{A}^{n-1} \end{bmatrix}

where nn is the state dimension. The necessary and sufficient condition for complete observability is rank(O)=n\text{rank}(\mathcal{O}) = n.

Observability Criteria

CriterionConditionApplicable Scenario
Rank criterionrank(O)=n\text{rank}(\mathcal{O}) = nGeneral linear systems
GRAM criterionOTO\mathcal{O}^T\mathcal{O} is positive definiteContinuous systems
PBH criterionrank[sI−A,C]=n,∀s\text{rank}[s\mathbf{I}-\mathbf{A}, \mathbf{C}] = n, \forall sLinear systems

Observability of Nonlinear Systems

Locally Weak Observability

For a nonlinear system:

x˙=f(x)\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x})

y=h(x)\mathbf{y} = h(\mathbf{x})

the locally weakly observable criterion can be applied. The system is locally weakly observable in a neighborhood if and only if:

rank(∂∂xLkh(x))=n,∀k≥0,∀x\text{rank}\left(\frac{\partial}{\partial \mathbf{x}} \mathcal{L}^k h(\mathbf{x})\right) = n, \forall k \geq 0, \forall \mathbf{x}

where Lkh\mathcal{L}^k h is the kk-th Lie derivative of hh.

Degree of Observability

In practical engineering, even when a system satisfies the observability criteria, state estimation accuracy may still be poor due to insufficient information content. The degree of observability is used to quantify this "level of observability":

obs=σmin⁡σmax⁡\text{obs} = \frac{\sigma_{\min}}{\sigma_{\max}}

where σmin⁡\sigma_{\min} and σmax⁡\sigma_{\max} are the minimum and maximum singular values of the observability matrix, respectively. A degree of observability closer to 1 indicates that all state components can be estimated with similar ease; a degree closer to 0 indicates that certain state components are difficult to estimate.

Applications in Navigation System Design

Sensor Configuration Optimization

Observability analysis is used to optimize sensor configuration. When studying Sun-Earth-Moon-based autonomous navigation, Qian Yingjing (2014) compared three sensor configuration schemes through observability analysis:

  1. Scheme 1: Sun sensor + Earth sensor + Moon sensor
  2. Scheme 2: Star tracker + Sun/Earth sensor combination
  3. Scheme 3: Optical camera + image processing

The analysis results showed significant differences in the degree of observability among the different configuration schemes, requiring the optimal configuration to be selected based on mission requirements.

Sampling Strategy Optimization

Observability is closely related to observation arc length. Longer continuous observation arcs generally provide better observability but increase computational burden and response latency. In practical design, a balance must be struck between observability and system real-time performance.

Observability analysis results directly influence filter design:

  • For system state components with poor observability, prior information or constraints need to be added
  • For unobservable states, reduced-order filters or fixed-value treatment should be employed

Limitations of Observability Analysis

  1. Local nature: Locally weak observability analysis only guarantees local uniqueness; global observability is difficult to determine
  2. Model dependence: Analysis results depend on the accuracy of the system dynamics model
  3. Numerical stability: Observability matrix computation for high-dimensional systems may suffer from numerical ill-conditioning
  4. Time-varying systems: Observability analysis of time-varying systems is more complex

References

  • Hermann R, Krener A J. Nonlinear controllability and observability [J]. IEEE Transactions on Automatic Control, 1977.
  • Qian Yingjing. Research on Autonomous Navigation and Orbit Keeping of Spacecraft on Quasi-Periodic Orbits in Cislunar Space [D]. Harbin Institute of Technology, 2014.