Observability

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

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

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

Observability matrix

If and only if the column rank of the observability matrix, defined as

${\displaystyle \mathcal{O}=\left[\begin{array}{c}C\\ CA\\ C{A}^2\\ ⋮\\ C{A}^{n-1}\end{array}\right]}$

is equal to ${\displaystyle n}$, then the system is observable. The rationale for this test is that if ${\displaystyle n}$ columns are linearly independent, then each of the ${\displaystyle n}$ state variables is viewable through linear combinations of the output variables ${\displaystyle y}$.

Related concepts

Observability index

The observability index ${\displaystyle v}$ of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied: ${\displaystyle \text{rank}({\mathcal{O}}_v)=\text{rank}({\mathcal{O}}_{v+1})}$, where

${\displaystyle {\mathcal{O}}_v=\left[\begin{array}{c}C\\ CA\\ C{A}^2\\ ⋮\\ C{A}^{v-1}\end{array}\right].}$

Unobservable subspace

The unobservable subspace ${\displaystyle N}$ of the linear system is the kernel of the linear map ${\displaystyle G}$ given by

${\displaystyle \begin{array}{cc}G:{\mathbb{R}}^n& \to \mathcal{C}(\mathbb{R};{\mathbb{R}}^n)\\ x(0)& \mapsto C{e}^{At}x(0)\end{array}}$

where ${\displaystyle \mathcal{C}(\mathbb{R};{\mathbb{R}}^n)}$ is the set of continuous functions from ${\displaystyle \mathbb{R}}$ to ${\displaystyle {\mathbb{R}}^n}$. ${\displaystyle N}$ can also be written as

${\displaystyle N=\bigcap _{k=0}^{n-1}\ker (C{A}^k)=\ker \mathcal{O}}$

Since the system is observable if and only if ${\displaystyle \mathrm{rank}(\mathcal{O})=n}$, the system is observable if and only if ${\displaystyle N}$ is the zero subspace.

The following properties for the unobservable subspace are valid:

• ${\displaystyle N\subset Ke(C)}$
• ${\displaystyle A(N)\subset N}$
• ${\displaystyle N=\bigcup \{S\subset R^n\mid S\subset Ke(C),A(S)\subset N\}}$

Detectability

A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.

Detectability conditions are important in the context of sensor networks.

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

${\displaystyle M(t_0,t_1)={\int }_{t_0}^{t_1}\phi (t,t_0{)}^{T}C(t{)}^{T}C(t)\phi (t,t_0)dt}$

where ${\displaystyle \phi }$ 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 ${\displaystyle t_1\ge t_0}$
• ${\displaystyle M(t_0,t_1)}$ 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

${\displaystyle M(t_0,t_1)=M(t_0,t)+\phi (t,t_0{)}^{T}M(t,t_1)\phi (t,t_0)}$

Observability matrix generalization

The system is observable in ${\displaystyle [t_0,t_1]}$ if and only if there exists an interval ${\displaystyle [t_0,t_1]}$ in ${\displaystyle \mathbb{R}}$ such that the matrix ${\displaystyle M(t_0,t_1)}$ is nonsingular.

If ${\displaystyle A(t),C(t)}$ are analytic, then the system is observable in the interval [${\displaystyle t_0}$,${\displaystyle t_1}$] if there exists ${\displaystyle \overline{t}\in [t_0,t_1]}$ and a positive integer k such that

${\displaystyle \mathrm{rank}\left[\begin{array}{ccc}& N_0(\overline{t})& \\ & N_1(\overline{t})& \\ & ⋮& \\ & N_k(\overline{t})& \end{array}\right]=n,}$

where ${\displaystyle N_0(t):=C(t)}$ and ${\displaystyle N_i(t)}$ is defined recursively as

${\displaystyle N_{i+1}(t):=N_i(t)A(t)+\frac{\mathrm{d}}{\mathrm{d}t}{N}_i(t),i=0,\dots ,k-1}$

Example

Consider a system varying analytically in ${\displaystyle (-\mathrm{\infty },\mathrm{\infty })}$ and matrices

${\displaystyle A(t)=\left[\begin{array}{ccc}t& 1& 0\\ 0& t^3& 0\\ 0& 0& t^2\end{array}\right],C(t)=\left[\begin{array}{ccc}1& 0& 1\end{array}\right].}$

Then ${\displaystyle \left[\begin{array}{c}{N}_0(0)\\ N_1(0)\\ N_2(0)\end{array}\right]=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]}$ , and since this matrix has rank = 3, the system is observable on every nontrivial interval of ${\displaystyle \mathbb{R}}$.

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

${\displaystyle d{\mathcal{O}}_s(x_0)=\mathrm{span}(d{h}_1(x_0),\dots ,d{h}_p(x_0),d{L}_{v_i}{L}_{v_{i-1}},\dots ,L_{v_1}{h}_j(x_0)),j\in p,k=1,2,\dots .}$

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar, Kou, Elliot and Tarn, and Singh.

There also exist an observability criteria for nonlinear time-varying systems.

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 ${\displaystyle {\mathbb{R}}^n}$. 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.