Editing Observability (section)

== Linear time-varying systems ==

Consider the [[continuous function|continuous]] [[linear]] [[time-variant system]]

: <math>\dot{\mathbf{x}}(t) = A(t) \mathbf{x}(t) + B(t) \mathbf{u}(t) \, </math>
: <math>\mathbf{y}(t) = C(t) \mathbf{x}(t). \, </math>

Suppose that the matrices <math>A</math>, <math>B</math>  and <math>C</math> are given as well as inputs and outputs <math>u</math> and <math>y</math> for all <math>t \in [t_0,t_1];</math> then it is possible to determine <math>x(t_0)</math> to within an additive constant vector which lies in the [[null space]] of <math>M(t_0,t_1)</math> defined by
: <math>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</math>
where <math>\varphi</math> is the [[state-transition matrix]].

It is possible to determine a unique <math>x(t_0)</math> if <math>M(t_0,t_1)</math> is [[Algebraic curve#Singularities|nonsingular]].  In fact, it is not possible to distinguish the initial state for <math>x_1</math> from that of <math>x_2</math> if <math>x_1 - x_2</math> is in the null space of <math>M(t_0,t_1)</math>.

Note that the matrix <math>M</math> defined as above has the following properties:
* <math>M(t_0,t_1)</math> is [[symmetric matrix|symmetric]]
* <math>M(t_0,t_1)</math> is [[positive semidefinite matrix|positive semidefinite]] for <math>t_1 \geq t_0</math>
* <math>M(t_0,t_1)</math> satisfies the linear [[matrix differential equation]]
:: <math>\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</math>
* <math>M(t_0,t_1)</math> satisfies the equation
:: <math>M(t_0,t_1) = M(t_0,t) + \varphi(t,t_0)^T M(t,t_1)\varphi(t,t_0)</math><ref>{{cite book|first=Roger W.|last=Brockett|title=Finite Dimensional Linear Systems|publisher=John Wiley & Sons|year=1970|isbn=978-0-471-10585-5}}</ref>

=== Observability matrix generalization ===
The system is observable in <math>[t_0,t_1]</math> if and only if there exists an interval <math>[t_0,t_1]</math> in <math>\mathbb{R}</math> such that the matrix <math>M(t_0,t_1)</math> is nonsingular.

If <math>A(t), C(t)</math> are analytic, then the system is observable in the interval [<math>t_0</math>,<math>t_1</math>] if there exists <math>\bar{t} \in [t_0,t_1]</math> and a positive integer ''k'' such that<ref name=":0">Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.</ref>

: <math> \operatorname{rank} \begin{bmatrix}
 & N_0(\bar{t}) & \\ 
 & N_1(\bar{t}) & \\ 
 & \vdots & \\ 
 & N_{k}(\bar{t}) & 
\end{bmatrix} = n, </math>

where <math>N_0(t):=C(t)</math> and <math>N_i(t)</math> is defined recursively as

: <math>N_{i+1}(t) := N_i(t)A(t) + \frac{\mathrm{d}}{\mathrm{d} t}N_i(t),\  i = 0, \ldots, k-1 </math>

==== Example ====

Consider a system varying analytically  in <math> (-\infty,\infty) </math> and matrices<blockquote><math>A(t) = \begin{bmatrix}
t & 1 & 0\\ 
0 & t^{3} & 0\\ 
0 & 0 & t^{2} 
\end{bmatrix},\, C(t) = \begin{bmatrix}
1 & 0 & 1
\end{bmatrix}.</math> </blockquote>Then <math> \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}</math> , and since this matrix has rank = 3, the system is observable on every nontrivial interval of  <math>\mathbb{R}</math>.