Observabilité

{\displaystyle n </span referrerpolicy=

variables d'état (voir l'espace d'état pour plus de détails sur les systèmes MIMO ) données par

{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t) <semantics><mrow><mstyle><mrow><mrow><mover><mrow><mi>x</mi></mrow><mo>˙</mo></mover></mrow></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mrow><mi>UN</mi></mrow><mrow><mi>x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mrow><mi>B</mi></mrow><mrow><mi>vous</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}</annotation></semantics></math></span><img src=

{\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t) <semantics><mrow><mstyle><mrow><mi>y</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mrow><mi>C</mi></mrow><mrow><mi>x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mrow><mi>D</mi></mrow><mrow><mi>vous</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)}</annotation></semantics></math></span><img src=

Matrice d'observabilité

Si et seulement si le rang des colonnes de la matrice d'observabilité , défini comme

{\displaystyle {\mathcal {O}}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{n-1}\end{bmatrix <semantics><mrow><mstyle><mrow><mrow><mi>O</mi></mrow></mrow><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mi>C</mi></mtd></mtr><mtr><mtd><mi>C</mi><mi>UN</mi></mtd></mtr><mtr><mtd><mi>C</mi><msup><mi>UN</mi><mrow><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mi>C</mi><msup><mi>UN</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mstyle></mrow><annotation>{\displaystyle {\mathcal {O}}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{n-1}\end{bmatrix}}}</annotation></semantics></math></span><img src=

est égal à ${\displaystyle n <semantics><mrow><mstyle><mi>n</mi></mstyle></mrow><annotation>{\displaystyle n}</annotation></semantics></math><img src=$ ${\displaystyle n <semantics><mrow><mstyle><mi>n</mi></mstyle></mrow><annotation>{\displaystyle n}</annotation></semantics></math><img src=$ ${\displaystyle n <semantics><mrow><mstyle><mi>n</mi></mstyle></mrow><annotation>{\displaystyle n}</annotation></semantics></math><img src=$ ${\displaystyle y <semantics><mrow><mstyle><mi>y</mi></mstyle></mrow><annotation>{\displaystyle y}</annotation></semantics></math><img src=$

Concepts connexes

Indice d'observabilité

L' indice d'observabilité ${\displaystyle v <semantics><mrow><mstyle><mi>v</mi></mstyle></mrow><annotation>{\displaystyle v}</annotation></semantics></math><img src=$ ${\displaystyle { ext{rank}}{({\mathcal {O}}_{v})}={ ext{rank}}{({\mathcal {O}}_{v+1}) <semantics><mrow><mstyle><mrow><mtext>rang</mtext></mrow><mrow><mo>(</mo><msub><mrow><mrow><mi>O</mi></mrow></mrow><mrow><mi>v</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mtext>rang</mtext></mrow><mrow><mo>(</mo><msub><mrow><mrow><mi>O</mi></mrow></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow></mstyle></mrow><annotation>{\displaystyle { ext{rank}}{({\mathcal {O}}_{v})}={ ext{rank}}{({\mathcal {O}}_{v+1})}}</annotation></semantics></math><img src=$

{\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}. <semantics><mrow><mstyle><msub><mrow><mrow><mi>O</mi></mrow></mrow><mrow><mi>v</mi></mrow></msub><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mi>C</mi></mtd></mtr><mtr><mtd><mi>C</mi><mi>UN</mi></mtd></mtr><mtr><mtd><mi>C</mi><msup><mi>UN</mi><mrow><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mi>C</mi><msup><mi>UN</mi><mrow><mi>v</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>.</mo></mstyle></mrow><annotation>{\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}.}</annotation></semantics></math></span><img src=

Sous-espace inobservable

Le sous-espace inobservable ${\displaystyle N <semantics><mrow><mstyle><mi>N</mi></mstyle></mrow><annotation>{\displaystyle N}</annotation></semantics></math><img src=$ ${\displaystyle G <semantics><mrow><mstyle><mi>G</mi></mstyle></mrow><annotation>{\displaystyle G}</annotation></semantics></math><img src=$

${\displaystyle {\begin{aligned}G\colon \mathbb {R} ^{n}& ightarrow {\mathcal {C}}(\mathbb {R} ;\mathbb {R} ^{n})\\x(0)&\mapsto Ce^{At}x(0)\end{aligned <semantics><mrow><mstyle><mrow><mtable><mtr><mtd><mi>G</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mtd><mtd><mi></mi><mo>→</mo><mrow><mrow><mi>C</mi></mrow></mrow><mo>(</mo><mrow><mi>R</mi></mrow><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mtd></mtr><mtr><mtd><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo></mtd><mtd><mi></mi><mo>↦</mo><mi>C</mi><msup><mi>e</mi><mrow><mi>UN</mi><mi>t</mi></mrow></msup><mi>x</mi><mo>(</mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable></mrow></mstyle></mrow><annotation>{\displaystyle {\begin{aligned}G\colon \mathbb {R} ^{n}& ightarrow {\mathcal {C}}(\mathbb {R}</annotation></semantics></math>;\mathbb {R} ^{n})\\x(0)&\mapsto Ce^{At}x(0)\end{aligned}}} <img src=$

où ${\displaystyle {\mathcal {C}}(\mathbb {R} ;\mathbb {R} ^{n}) <semantics><mrow><mstyle><mrow><mrow><mi>C</mi></mrow></mrow><mo>(</mo><mrow><mi>R</mi></mrow><mo>;</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mstyle></mrow><annotation>{\displaystyle {\mathcal {C}}(\mathbb {R}</annotation></semantics></math>;\mathbb {R} ^{n})} <img src=$ ${\displaystyle \mathbb {R} <semantics><mrow><mstyle><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle \mathbb {R} ^{n <semantics><mrow><mstyle><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbb {R} ^{n}}</annotation></semantics></math><img src=$ ${\displaystyle N <semantics><mrow><mstyle><mi>N</mi></mstyle></mrow><annotation>{\displaystyle N}</annotation></semantics></math><img src=$

{\displaystyle N=\bigcap _{k=0}^{n-1}\ker(CA^{k})=\ker {\mathcal {O <semantics><mrow><mstyle><mi>N</mi><mo>=</mo><munderover><mo>⋂</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>ker</mi><mo>⁡</mo><mo>(</mo><mi>C</mi><msup><mi>UN</mi><mrow><mi>k</mi></mrow></msup><mo>)</mo><mo>=</mo><mi>ker</mi><mo>⁡</mo><mrow><mrow><mi>O</mi></mrow></mrow></mstyle></mrow><annotation>{ displaystyle N = bigcap _ {k = 0} ^ {n-1} ker (CA ^ {k}) = ker { mathcal {O}}}</annotation></semantics></math></span><img src=

Puisque le système est observable si et seulement si ${\displaystyle \operatorname {rank} ({\mathcal {O}})=n <semantics><mrow><mstyle><mi>rang</mi><mo>⁡</mo><mo>(</mo><mrow><mrow><mi>O</mi></mrow></mrow><mo>)</mo><mo>=</mo><mi>n</mi></mstyle></mrow><annotation>{\displaystyle \operatorname {rang} ({\mathcal {O}})=n}</annotation></semantics></math><img src=$ ${\displaystyle N <semantics><mrow><mstyle><mi>N</mi></mstyle></mrow><annotation>{\displaystyle N}</annotation></semantics></math><img src=$

Les propriétés suivantes du sous-espace non observable sont valides :

${\displaystyle N\subset Ke(C) <semantics><mrow><mstyle><mi>N</mi><mo>⊂</mo><mi>K</mi><mi>e</mi><mo>(</mo><mi>C</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle N\subset Ke(C)}</annotation></semantics></math><img src=$
${\displaystyle A(N)\subset N <semantics><mrow><mstyle><mi>UN</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>⊂</mo><mi>N</mi></mstyle></mrow><annotation>{\displaystyle A(N)\subset N}</annotation></semantics></math><img src=$
${\displaystyle N=\bigcup \{S\subset R^{n}\mid S\subset Ke(C),A(S)\subset N\ <semantics><mrow><mstyle><mi>N</mi><mo>=</mo><mo>⋃</mo><mo>{</mo><mi>S</mi><mo>⊂</mo><msup><mi>R</mi><mrow><mi>n</mi></mrow></msup><mo>∣</mo><mi>S</mi><mo>⊂</mo><mi>K</mi><mi>e</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>,</mo><mi>UN</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>⊂</mo><mi>N</mi><mo>}</mo></mstyle></mrow><annotation>{\displaystyle N=\bigcup \{S\subset R^{n}\mid S\subset Ke(C),A(S)\subset N\}}</annotation></semantics></math><img src=$

Détectabilité

La notion de détectabilité est légèrement moins précise que celle d'observabilité . Un système est détectable si tous les états non observables sont stables.

Les conditions de détectabilité sont importantes dans le contexte des réseaux de capteurs .

Observabilité fonctionnelle

L'observabilité fonctionnelle est une propriété qui étend la notion classique d'observabilité aux cas où l'observabilité complète de l'état n'est ni possible ni requise (en raison du manque de signaux de mesure ou de l'emplacement des capteurs). Plutôt que d'exiger une reconstruction complète de l'état, l'observabilité fonctionnelle établit la condition sous laquelle une fonction linéaire peut être considérée comme observable. ${\displaystyle \mathbf {z} (t)=\mathbf {F} \mathbf {x} (t) <semantics><mrow><mstyle><mrow><mi>z</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mrow><mi>F</mi></mrow><mrow><mi>x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {z} (t)=\mathbf {F} \mathbf {x} (t)}</annotation></semantics></math><img src=$ ${\displaystyle r imes n <semantics><mrow><mstyle><mi>r</mi><mo>×</mo><mi>n</mi></mstyle></mrow><annotation>{\displaystyle r imes n}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {F} <semantics><mrow><mstyle><mrow><mi>F</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {F} }</annotation></semantics></math><img src=$ ${\displaystyle r\leq n <semantics><mrow><mstyle><mi>r</mi><mo>≤</mo><mi>n</mi></mstyle></mrow><annotation>{\displaystyle r\leq n}</annotation></semantics></math><img src=$

{\displaystyle \operatorname {rank} {\begin{bmatrix}{\mathcal {O}}\\\mathbf {F} \end{bmatrix}}=\operatorname {rank} {\mathcal {O}}. <semantics><mrow><mstyle><mi>rang</mi><mo>⁡</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mrow><mrow><mi>O</mi></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mi>F</mi></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mi>rang</mi><mo>⁡</mo><mrow><mrow><mi>O</mi></mrow></mrow><mo>.</mo></mstyle></mrow><annotation>{\displaystyle \operatorname {rank} {\begin{bmatrix}{\mathcal {O}}\\\mathbf {F} \end{bmatrix}}=\operatorname {rank} {\mathcal {O}}.}</annotation></semantics></math></span><img src=

L'observabilité fonctionnelle est un concept important car elle détermine la condition nécessaire et suffisante sous laquelle un observateur fonctionnel (également appelé observateur de Darouach ) peut être conçu pour estimer asymptotiquement ${\displaystyle \mathbf {z} (t) <semantics><mrow><mstyle><mrow><mi>z</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {z} (t)}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {z} (t) <semantics><mrow><mstyle><mrow><mi>z</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {z} (t)}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {x} (t) <semantics><mrow><mstyle><mrow><mi>x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {x} (t)}</annotation></semantics></math><img src=$

Systèmes linéaires variant dans le temps

Considérons le système linéaire continu variant dans le temps

{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\, <semantics><mrow><mstyle><mrow><mrow><mover><mrow><mi>x</mi></mrow><mo>˙</mo></mover></mrow></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>UN</mi><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mi>x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>B</mi><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mi>vous</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mspace></mspace></mstyle></mrow><annotation>{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\,}</annotation></semantics></math></span><img src=

{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t).\, <semantics><mrow><mstyle><mrow><mi>y</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mi>x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>.</mo><mspace></mspace></mstyle></mrow><annotation>{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t).\,}</annotation></semantics></math></span><img src=

Supposons que les matrices ${\displaystyle A <semantics><mrow><mstyle><mi>UN</mi></mstyle></mrow><annotation>{\displaystyle A}</annotation></semantics></math><img src=$ ${\displaystyle B <semantics><mrow><mstyle><mi>B</mi></mstyle></mrow><annotation>{\displaystyle B}</annotation></semantics></math><img src=$ ${\displaystyle C <semantics><mrow><mstyle><mi>C</mi></mstyle></mrow><annotation>{\displaystyle C}</annotation></semantics></math><img src=$ ${\displaystyle u <semantics><mrow><mstyle><mi>vous</mi></mstyle></mrow><annotation>{\displaystyle u}</annotation></semantics></math><img src=$ ${\displaystyle y <semantics><mrow><mstyle><mi>y</mi></mstyle></mrow><annotation>{\displaystyle y}</annotation></semantics></math><img src=$ ${\displaystyle t\in [t_{0},t_{1}]; <semantics><mrow><mstyle><mi>t</mi><mo>∈</mo><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo><mo>;</mo></mstyle></mrow><annotation>{\displaystyle t\in [t_{0},t_{1}];}</annotation></semantics></math><img src=$ ${\displaystyle x(t_{0}) <semantics><mrow><mstyle><mi>x</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle x(t_{0})}</annotation></semantics></math><img src=$ ${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$

{\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 <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mi>φ</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><msup><mo>)</mo><mrow><mi>T</mi></mrow></msup><mi>C</mi><mo>(</mo><mi>t</mi><msup><mo>)</mo><mrow><mi>T</mi></mrow></msup><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>φ</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo><mspace></mspace><mi>d</mi><mi>t</mi></mstyle></mrow><annotation>{\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}</annotation></semantics></math></span><img src=

où ${\displaystyle \varphi <semantics><mrow><mstyle><mi>φ</mi></mstyle></mrow><annotation>{\displaystyle \varphi }</annotation></semantics></math><img src=$

Il est possible de déterminer une unique ${\displaystyle x(t_{0}) <semantics><mrow><mstyle><mi>x</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle x(t_{0})}</annotation></semantics></math><img src=$ ${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$ ${\displaystyle x_{1 <semantics><mrow><mstyle><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle x_{1}}</annotation></semantics></math><img src=$ ${\displaystyle x_{2 <semantics><mrow><mstyle><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle x_{2}}</annotation></semantics></math><img src=$ ${\displaystyle x_{1}-x_{2 <semantics><mrow><mstyle><msub><mi>x</mi><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mrow><mn>2</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle x_{1}-x_{2}}</annotation></semantics></math><img src=$ ${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$

Notez que la matrice ${\displaystyle M <semantics><mrow><mstyle><mi>M</mi></mstyle></mrow><annotation>{\displaystyle M}</annotation></semantics></math><img src=$

${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$
${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$ ${\displaystyle t_{1}\geq t_{0 <semantics><mrow><mstyle><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle t_{1}\geq t_{0}}</annotation></semantics></math><img src=$
${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$

{\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 <semantics><mrow><mstyle><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>−</mo><mi>UN</mi><mo>(</mo><mi>t</mi><msup><mo>)</mo><mrow><mi>T</mi></mrow></msup><mi>M</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>M</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>UN</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>C</mi><mo>(</mo><mi>t</mi><msup><mo>)</mo><mrow><mi>T</mi></mrow></msup><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation>{\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}</annotation></semantics></math></span><img src=

${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$

{\displaystyle M(t_{0},t_{1})=M(t_{0},t)+\varphi (t,t_{0})^{T}M(t,t_{1})\varphi (t,t_{0}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>φ</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><msup><mo>)</mo><mrow><mi>T</mi></mrow></msup><mi>M</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>φ</mi><mo>(</mo><mi>t</mi><mo>,</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})=M(t_{0},t)+\varphi (t,t_{0})^{T}M(t,t_{1})\varphi (t,t_{0})}</annotation></semantics></math></span><img src=

généralisation de la matrice d'observabilité

Le système est observable dans ${\displaystyle [t_{0},t_{1}] <semantics><mrow><mstyle><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo></mstyle></mrow><annotation>{\displaystyle [t_{0},t_{1}]}</annotation></semantics></math><img src=$ ${\displaystyle [t_{0},t_{1}] <semantics><mrow><mstyle><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo></mstyle></mrow><annotation>{\displaystyle [t_{0},t_{1}]}</annotation></semantics></math><img src=$ ${\displaystyle \mathbb {R} <semantics><mrow><mstyle><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle M(t_{0},t_{1}) <semantics><mrow><mstyle><mi>M</mi><mo>(</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>)</mo></mstyle></mrow><annotation>{\displaystyle M(t_{0},t_{1})}</annotation></semantics></math><img src=$

Si ${\displaystyle A(t),C(t) <semantics><mrow><mstyle><mi>UN</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle A(t),C(t)}</annotation></semantics></math><img src=$ ${\displaystyle t_{0 <semantics><mrow><mstyle><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle t_{0}}</annotation></semantics></math><img src=$ ${\displaystyle t_{1 <semantics><mrow><mstyle><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle t_{1}}</annotation></semantics></math><img src=$ ${\displaystyle {\bar {t}}\in [t_{0},t_{1}] <semantics><mrow><mstyle><mrow><mrow><mover><mi>t</mi><mo>¯</mo></mover></mrow></mrow><mo>∈</mo><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo></mstyle></mrow><annotation>{\displaystyle {\bar {t}}\in [t_{0},t_{1}]}</annotation></semantics></math><img src=$ k tel que

{\displaystyle \operatorname {rank} {\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&\vdots &\\&N_{k}({\bar {t}})&\end{bmatrix}}=n, <semantics><mrow><mstyle><mi>rang</mi><mo>⁡</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd></mtd><mtd><msub><mi>N</mi><mrow><mn>0</mn></mrow></msub><mo>(</mo><mrow><mrow><mover><mi>t</mi><mo>¯</mo></mover></mrow></mrow><mo>)</mo></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>N</mi><mrow><mn>1</mn></mrow></msub><mo>(</mo><mrow><mrow><mover><mi>t</mi><mo>¯</mo></mover></mrow></mrow><mo>)</mo></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mo>⋮</mo></mtd><mtd></mtd></mtr><mtr><mtd></mtd><mtd><msub><mi>N</mi><mrow><mi>k</mi></mrow></msub><mo>(</mo><mrow><mrow><mover><mi>t</mi><mo>¯</mo></mover></mrow></mrow><mo>)</mo></mtd><mtd></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mi>n</mi><mo>,</mo></mstyle></mrow><annotation>{\displaystyle \operatorname {rank} {\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&\vdots &\\&N_{k}({\bar {t}})&\end{bmatrix}}=n,}</annotation></semantics></math></span><img src=

où ${\displaystyle N_{0}(t):=C(t) <semantics><mrow><mstyle><msub><mi>N</mi><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>:=</mo><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle N_{0}(t):=C(t)}</annotation></semantics></math><img src=$ ${\displaystyle N_{i}(t) <semantics><mrow><mstyle><msub><mi>N</mi><mrow><mi>je</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle N_{i}(t)}</annotation></semantics></math><img src=$

{\displaystyle N_{i+1}(t):=N_{i}(t)A(t)+{\frac {\mathrm {d} }{\mathrm {d} t}}N_{i}(t),\ i=0,\ldots ,k-1 <semantics><mrow><mstyle><msub><mi>N</mi><mrow><mi>je</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>:=</mo><msub><mi>N</mi><mrow><mi>je</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mi>UN</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mrow><mfrac><mrow><mi>d</mi></mrow><mrow><mrow><mi>d</mi></mrow><mi>t</mi></mrow></mfrac></mrow><msub><mi>N</mi><mrow><mi>je</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mtext> </mtext><mi>je</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn></mstyle></mrow><annotation>{\displaystyle N_{i+1}(t):=N_{i}(t)A(t)+{\frac {\mathrm {d} }{\mathrm {d} t}}N_{i}(t),\ i=0,\ldots ,k-1}</annotation></semantics></math></span><img src=

Exemple

Considérons un système variant analytiquement en ${\displaystyle (-\infty ,\infty ) <semantics><mrow><mstyle><mo>(</mo><mo>−</mo><mi>∞</mi><mo>,</mo><mi>∞</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle (-\infty ,\infty )}</annotation></semantics></math><img src=$

${\displaystyle A(t)={\begin{bmatrix}t&1&0\\0&t^{3}&0\\0&0&t^{2}\end{bmatrix}},\,C(t)={\begin{bmatrix}1&0&1\end{bmatrix}}. <semantics><mrow><mstyle><mi>UN</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mi>t</mi></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msup><mi>t</mi><mrow><mn>3</mn></mrow></msup></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msup><mi>t</mi><mrow><mn>2</mn></mrow></msup></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>,</mo><mspace></mspace><mi>C</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>.</mo></mstyle></mrow><annotation>{\displaystyle A(t)={\begin{bmatrix}t&1&0\\0&t^{3}&0\\0&0&t^{2}\end{bmatrix}},\,C(t)={\begin{bmatrix}1&0&1\end{bmatrix}}.}</annotation></semantics></math><img src=$

Alors ${\displaystyle {\begin{bmatrix}N_{0}(0)\\N_{1}(0)\\N_{2}(0)\end{bmatrix}}={\begin{bmatrix}1&0&1\\0&1&0\\1&0&0\end{bmatrix <semantics><mrow><mstyle><mrow><mrow><mo>[</mo><mtable><mtr><mtd><msub><mi>N</mi><mrow><mn>0</mn></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo></mtd></mtr><mtr><mtd><msub><mi>N</mi><mrow><mn>1</mn></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo></mtd></mtr><mtr><mtd><msub><mi>N</mi><mrow><mn>2</mn></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mrow><mo>[</mo><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mstyle></mrow><annotation>{\displaystyle {\begin{bmatrix}N_{0}(0)\\N_{1}(0)\\N_{2}(0)\end{bmatrix}}={\begin{bmatrix}1&0&1\\0&1&0\\1&0&0\end{bmatrix}}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbb {R} <semantics><mrow><mstyle><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbb {R} }</annotation></semantics></math><img src=$

Systèmes non linéaires

Étant donné le système ${\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j <semantics><mrow><mstyle><mrow><mrow><mover><mi>x</mi><mo>˙</mo></mover></mrow></mrow><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mi>g</mi><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><msub><mi>vous</mi><mrow><mi>j</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j}}</annotation></semantics></math><img src=$ ${\displaystyle y_{i}=h_{i}(x),i\in p <semantics><mrow><mstyle><msub><mi>y</mi><mrow><mi>je</mi></mrow></msub><mo>=</mo><msub><mi>h</mi><mrow><mi>je</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mi>je</mi><mo>∈</mo><mi>p</mi></mstyle></mrow><annotation>{\displaystyle y_{i}=h_{i}(x),i\in p}</annotation></semantics></math><img src=$ ${\displaystyle x\in \mathbb {R} ^{n <semantics><mrow><mstyle><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle x\in \mathbb {R} ^{n}}</annotation></semantics></math><img src=$ ${\displaystyle u\in \mathbb {R} ^{m <semantics><mrow><mstyle><mi>vous</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle u\in \mathbb {R} ^{m}}</annotation></semantics></math><img src=$ ${\displaystyle y\in \mathbb {R} ^{p <semantics><mrow><mstyle><mi>y</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>p</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle y\in \mathbb {R} ^{p}}</annotation></semantics></math><img src=$ ${\displaystyle f,g,h <semantics><mrow><mstyle><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi></mstyle></mrow><annotation>{\displaystyle f,g,h}</annotation></semantics></math><img src=$

Définir l'espace d'observation ${\displaystyle {\mathcal {O}}_{s <semantics><mrow><mstyle><msub><mrow><mrow><mi>O</mi></mrow></mrow><mrow><mi>s</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle {\mathcal {O}}_{s}}</annotation></semantics></math><img src=$ ${\displaystyle x_{0 <semantics><mrow><mstyle><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle x_{0}}</annotation></semantics></math><img src=$ ${\displaystyle \dim(d{\mathcal {O}}_{s}(x_{0}))=n <semantics><mrow><mstyle><mi>faible</mi><mo>⁡</mo><mo>(</mo><mi>d</mi><msub><mrow><mrow><mi>O</mi></mrow></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>=</mo><mi>n</mi></mstyle></mrow><annotation>{\displaystyle \dim(d{\mathcal {O}}_{s}(x_{0}))=n}</annotation></semantics></math><img src=$

{\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 . <semantics><mrow><mstyle><mi>d</mi><msub><mrow><mrow><mi>O</mi></mrow></mrow><mrow><mi>s</mi></mrow></msub><mo>(</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>portée</mi><mo>⁡</mo><mo>(</mo><mi>d</mi><msub><mi>h</mi><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mi>d</mi><msub><mi>h</mi><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>,</mo><mi>d</mi><msub><mi>L</mi><mrow><msub><mi>v</mi><mrow><mi>je</mi></mrow></msub></mrow></msub><msub><mi>L</mi><mrow><msub><mi>v</mi><mrow><mi>je</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>L</mi><mrow><msub><mi>v</mi><mrow><mn>1</mn></mrow></msub></mrow></msub><msub><mi>h</mi><mrow><mi>j</mi></mrow></msub><mo>(</mo><msub><mi>x</mi><mrow><mn>0</mn></mrow></msub><mo>)</mo><mo>)</mo><mo>,</mo><mtext> </mtext><mi>j</mi><mo>∈</mo><mi>p</mi><mo>,</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>.</mo></mstyle></mrow><annotation>{\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 .}</annotation></semantics></math></span><img src=

Les premiers critères d'observabilité dans les systèmes dynamiques non linéaires ont été découverts par Griffith et Kumar, Kou, Elliot et Tarn, et Singh.

Il existe également des critères d’observabilité pour les systèmes non linéaires variant dans le temps.

Systèmes statiques et espaces topologiques généraux

L'observabilité peut également être caractérisée pour les systèmes à l'état stationnaire (systèmes généralement définis en termes d'équations et d'inégalités algébriques), ou plus généralement, pour les ensembles. ${\displaystyle \mathbb {R} ^{n <semantics><mrow><mstyle><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbb {R} ^{n}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbb {R} ^{n <semantics><mrow><mstyle><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbb {R} ^{n}}</annotation></semantics></math><img src=$