Observability

If the row rank of the observability matrix, defined as

${\displaystyle {\mathcal {O}}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{n-1}\end{bmatrix}}}$

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

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

${\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}.}$

The unobservable subspace ${\displaystyle N}$ of the linear system is the kernel of the linear map ${\displaystyle G}$ given by[3]

${\displaystyle {\begin{aligned}G\colon \mathbb {R} ^{n}&\rightarrow {\mathcal {C}}(\mathbb {R}$ ;\mathbb {R} ^{n})\\x(0)&\mapsto Ce^{At}x(0)\\\end{aligned}}}

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 [3]

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

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

The following properties for the unobservable subspace are valid:[3]

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

A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.[4]

Detectability conditions are important in the context of sensor networks.[5][6]

Nonlinear observers

sliding mode and cubic observers[7] can be applied for state estimation of time invariant linear systems, if the system is observable and fulfills some additional conditions.

The system is observable in [${\displaystyle t_{0}}$,${\displaystyle t_{1}}$] if and only if there exists an interval [${\displaystyle t_{0}}$,${\displaystyle t_{1}}$] 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 ${\displaystyle {\bar {t}}\in [t_{0},t_{1}]}$ and a positive integer k such that[9]

${\displaystyle rank{\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&:&\\&N_{k}({\bar {t}})&\end{bmatrix}}=n,}$

where ${\displaystyle N_{0}(t):=C(t)}$ and ${\displaystyle N_{i}(t)}$ is defined recursively as

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

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

${\displaystyle A(t)={\begin{bmatrix}t&1&0\\0&t^{3}&0\\0&0&t^{2}\end{bmatrix}}}$, ${\displaystyle C(t)={\begin{bmatrix}1&0&1\end{bmatrix}}.}$

Then ${\displaystyle {\begin{bmatrix}N_{0}(0)\\N_{1}(0)\\N_{2}(0)\end{bmatrix}}={\begin{bmatrix}1&0&1\\0&1&0\\1&0&0\end{bmatrix}}}$ , and since this matrix has rank = 3, the system is observable on every nontrivial interval of ${\displaystyle \mathbb {R} }$.