Observability is a measure of how well internal states of a system can be inferred from knowledge of its external outputs.

In control theory, the observability and controllability of a linear system are mathematical duals.

The concept of observability was introduced by the Hungarian-American engineer Rudolf E. Kálmán for linear dynamic systems.^[1][2] A dynamical system designed to estimate the state of a system from measurements of the outputs is called a state observer or simply an observer for that system.

Definition

Consider a physical system modeled in state-space representation. A system is said to be observable if, for every possible evolution of state and control vectors, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by sensors). In other words, one can determine the behavior of the entire system from the system's outputs. On the other hand, if the system is not observable, there are state trajectories that are not distinguishable by only measuring the outputs.

Linear time-invariant systems

For time-invariant linear systems in the state space representation, there are convenient tests to check whether a system is observable. Consider a SISO system with ${\displaystyle n}$ state variables (see state space for details about MIMO systems) given by

${\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}$

${\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)}$

Linear time-varying systems

Consider the continuous linear time-variant system

${\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\,}$

${\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t).\,}$

Suppose that the matrices ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ are given as well as inputs and outputs ${\displaystyle u}$ and ${\displaystyle y}$ for all ${\displaystyle t\in [t_{0},t_{1}];}$ then it is possible to determine ${\displaystyle x(t_{0})}$ to within an additive constant vector which lies in the null space of ${\displaystyle M(t_{0},t_{1})}$ defined by

${\displaystyle M(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}\varphi (t,t_{0})^{T}C(t)^{T}C(t)\varphi (t,t_{0})\,dt}$

where ${\displaystyle \varphi }$ is the state-transition matrix.

It is possible to determine a unique ${\displaystyle x(t_{0})}$ if ${\displaystyle M(t_{0},t_{1})}$ is nonsingular. In fact, it is not possible to distinguish the initial state for ${\displaystyle x_{1}}$ from that of ${\displaystyle x_{2}}$ if ${\displaystyle x_{1}-x_{2}}$ is in the null space of ${\displaystyle M(t_{0},t_{1})}$.

Note that the matrix ${\displaystyle M}$ defined as above has the following properties:

• ${\displaystyle M(t_{0},t_{1})}$ is symmetric
• ${\displaystyle M(t_{0},t_{1})}$ is positive semidefinite for ${\displaystyle t_{1}\geq t_{0}}$
• ${\displaystyle M(t_{0},t_{1})}$ satisfies the linear matrix differential equation

${\displaystyle {\frac {d}{dt}}M(t,t_{1})=-A(t)^{T}M(t,t_{1})-M(t,t_{1})A(t)-C(t)^{T}C(t),\;M(t_{1},t_{1})=0}$

• ${\displaystyle M(t_{0},t_{1})}$ satisfies the equation

${\displaystyle M(t_{0},t_{1})=M(t_{0},t)+\varphi (t,t_{0})^{T}M(t,t_{1})\varphi (t,t_{0})}$^[7]

Nonlinear systems

Given the system ${\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j}}$, ${\displaystyle y_{i}=h_{i}(x),i\in p}$. Where ${\displaystyle x\in \mathbb {R} ^{n}}$ the state vector, ${\displaystyle u\in \mathbb {R} ^{m}}$ the input vector and ${\displaystyle y\in \mathbb {R} ^{p}}$ the output vector. ${\displaystyle f,g,h}$ are to be smooth vector fields.

Define the observation space ${\displaystyle {\mathcal {O}}_{s}}$ to be the space containing all repeated Lie derivatives, then the system is observable in ${\displaystyle x_{0}}$ if and only if ${\displaystyle \dim(d{\mathcal {O}}_{s}(x_{0}))=n}$, where

${\displaystyle d{\mathcal {O}}_{s}(x_{0})=\operatorname {span} (dh_{1}(x_{0}),\ldots ,dh_{p}(x_{0}),dL_{v_{i}}L_{v_{i-1}},\ldots ,L_{v_{1}}h_{j}(x_{0})),\ j\in p,k=1,2,\ldots .}$^[9]

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,^[10] Kou, Elliot and Tarn,^[11] and Singh.^[12]

There also exist an observability criteria for nonlinear time-varying systems.^[13]

Static systems and general topological spaces

Observability may also be characterized for steady state systems (systems typically defined in terms of algebraic equations and inequalities), or more generally, for sets in ${\displaystyle \mathbb {R} ^{n}}$.^[14][15] Just as observability criteria are used to predict the behavior of Kalman filters or other observers in the dynamic system case, observability criteria for sets in ${\displaystyle \mathbb {R} ^{n}}$ are used to predict the behavior of data reconciliation and other static estimators. In the nonlinear case, observability can be characterized for individual variables, and also for local estimator behavior rather than just global behavior.

See also

• Controllability
• Hautus lemma
• Identifiability
• State observer
• State space (controls)

References

External links

• "Observability". PlanetMath.
• MATLAB function for checking observability of a system
• Mathematica function for checking observability of a system