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Observability

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</style><div role="note" class="hatnote navigation-not-searchable">For the concept in quantum mechanics, see <a href="Observable" title="Observable">Observable</a>. For the concept in software, see <a href="Observability_(software)" title="Observability (software)">Observability (software)</a>.</div>

Observability is a measure of how well internal states of a <a href="System" title="System">system</a> can be inferred from knowledge of its external outputs.

In <a href="Control_theory" title="Control theory">control theory</a>, the observability and <a href="Controllability" title="Controllability">controllability</a> of a linear system are mathematical <a href="Duality_(mathematics)" title="Duality (mathematics)">duals</a>.

The concept of observability was introduced by the Hungarian-American engineer <a href="Rudolf_E._K%C3%A1lm%C3%A1n" title="Rudolf E. Kálmán">Rudolf E. Kálmán</a> for linear dynamic systems.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a> A dynamical system designed to estimate the state of a system from measurements of the outputs is called a <a href="State_observer" title="State observer">state observer</a> for that system, such as <a href="Kalman_filter" title="Kalman filter">Kalman filters</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Consider a physical system modeled in <a href="State-space_representation" title="State-space representation">state-space representation</a>. A system is said to be observable if, for every possible evolution of <a href="State_space_representation" class="mw-redirect" title="State space representation">state and control vectors</a>, the current state can be estimated using only the information from outputs (physically, this generally corresponds to information obtained by <a href="Sensor" title="Sensor">sensors</a>). 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.

<div class="mw-heading mw-heading2"><h2 id="Linear_time-invariant_systems">Linear time-invariant systems</h2></div>

For <a href="LTI_system_theory" class="mw-redirect" title="LTI system theory">time-invariant linear systems</a> in the state space representation, there are convenient tests to check whether a system is observable. Consider a <a href="Single-Input_and_Single-Output" class="mw-redirect" title="Single-Input and Single-Output">SISO</a> system with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> state variables (see <a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">state space</a> for details about <a href="MIMO" title="MIMO">MIMO</a> systems) given by

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}</annotation>

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</math><img src="./07c210cd2848bf62ea5cae319e5e4618b7630e55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.113ex; height:2.843ex;" alt="{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)}</annotation>

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</math><img src="./b0ee0e1b752d253f51bfcde3e5d9739d971288d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.174ex; height:2.843ex;" alt="{\displaystyle \mathbf {y} (t)=\mathbf {C} \mathbf {x} (t)+\mathbf {D} \mathbf {u} (t)}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Observability_matrix">Observability matrix</h3></div>

If and only if the column <a href="Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> of the observability matrix, defined as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{n-1}\end{bmatrix}}}</annotation>

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</math><img src="./6b7b58b11e6df0355fa3ba74a0358fe79addcb79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:15.629ex; height:17.176ex;" alt="{\displaystyle {\mathcal {O}}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{n-1}\end{bmatrix}}}" loading="lazy"></dd></dl>

is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, then the system is observable. The rationale for this test is that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> columns are linearly independent, then each of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> state variables is viewable through linear combinations of the output variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Related_concepts">Related concepts</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Observability_index">Observability index</h4></div>

The observability index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> of a linear time-invariant discrete system is the smallest natural number for which the following is satisfied: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{rank}}{({\mathcal {O}}_{v})}={\text{rank}}{({\mathcal {O}}_{v+1})}}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{rank}}{({\mathcal {O}}_{v})}={\text{rank}}{({\mathcal {O}}_{v+1})}}</annotation>

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</math><img src="./45c3194e3afee43a4092a5a84069a30f88fd71be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.765ex; height:2.843ex;" alt="{\displaystyle {\text{rank}}{({\mathcal {O}}_{v})}={\text{rank}}{({\mathcal {O}}_{v+1})}}" loading="lazy">, where

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}.}</annotation>

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</math><img src="./66b49ee4eb2e2f7c85b7fde578105cfb5b371056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:17.117ex; height:17.176ex;" alt="{\displaystyle {\mathcal {O}}_{v}={\begin{bmatrix}C\\CA\\CA^{2}\\\vdots \\CA^{v-1}\end{bmatrix}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Unobservable_subspace">Unobservable subspace</h4></div>

The unobservable subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> of the linear system is the kernel of the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> given by<a href="#cite_note-:1-3">[3]</a><blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./74abb61436a5748231e40778f603c9a3092bedc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.333ex; height:6.509ex;" alt="{\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}}}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}(\mathbb {R} ;\mathbb {R} ^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}(\mathbb {R} ;\mathbb {R} ^{n})}</annotation>

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</math><img src="./09be72dd6758691d553798bebc512f201b37dc07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.657ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}(\mathbb {R} ;\mathbb {R} ^{n})}" loading="lazy"> is the set of continuous functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> can also be written as <a href="#cite_note-:1-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle N=\bigcap _{k=0}^{n-1}\ker(CA^{k})=\ker {\mathcal {O}}}</annotation>

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</math><img src="./feab4d19441e78e8b06eb1a217466ff80e04fe44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.722ex; height:7.509ex;" alt="{\displaystyle N=\bigcap _{k=0}^{n-1}\ker(CA^{k})=\ker {\mathcal {O}}}" loading="lazy"></dd></dl>

Since the system is observable if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rank} ({\mathcal {O}})=n}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {rank} ({\mathcal {O}})=n}</annotation>

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</math><img src="./82009b62372d65a639723ba24561294354c7bdbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.746ex; height:2.843ex;" alt="{\displaystyle \operatorname {rank} ({\mathcal {O}})=n}" loading="lazy">, the system is observable if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is the zero subspace.

The following properties for the unobservable subspace are valid:<a href="#cite_note-:1-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle N\subset Ke(C)}</annotation>

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</math><img src="./9e16c3cc95ccd3a72960812b74bd1536b11bf809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.887ex; height:2.843ex;" alt="{\displaystyle N\subset Ke(C)}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle A(N)\subset N}</annotation>

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</math><img src="./7e260a8f792c013feddf6612021761f1f6a1d877.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.778ex; height:2.843ex;" alt="{\displaystyle A(N)\subset N}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle N=\bigcup \{S\subset R^{n}\mid S\subset Ke(C),A(S)\subset N\}}</annotation>

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</math><img src="./a2a1ee661145ca1bf7dc4ad7211c54c969ca6868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:42.156ex; height:3.843ex;" alt="{\displaystyle N=\bigcup \{S\subset R^{n}\mid S\subset Ke(C),A(S)\subset N\}}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Detectability">Detectability</h4></div>

A slightly weaker notion than observability is detectability. A system is detectable if all the unobservable states are stable.<a href="#cite_note-4">[4]</a>

Detectability conditions are important in the context of <a href="Sensor_Networks" class="mw-redirect" title="Sensor Networks">sensor networks</a>.<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="Linear_time-varying_systems">Linear time-varying systems</h2></div>

Consider the <a href="Continuous_function" title="Continuous function">continuous</a> <a href="Linear" class="mw-redirect" title="Linear">linear</a> <a href="Time-variant_system" title="Time-variant system">time-variant system</a>

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<annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\,}</annotation>

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</math><img src="./850cec2c9e8365764a20bdfb1f4cc711e9fd4c31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.385ex; height:2.843ex;" alt="{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)\,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t).\,}</annotation>

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Suppose that the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in [t_{0},t_{1}];}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in [t_{0},t_{1}];}</annotation>

</semantics>

</math><img src="./c4746264c5543ef673683cc11e9e98d1250312fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.442ex; height:2.843ex;" alt="{\displaystyle t\in [t_{0},t_{1}];}" loading="lazy"> then it is possible to determine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t_{0})}</annotation>

</semantics>

</math><img src="./30dccfced64263036f1ce61e61cb0219d1cf48de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle x(t_{0})}" loading="lazy"> to within an additive constant vector which lies in the <a href="Null_space" class="mw-redirect" title="Null space">null space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t_{0},t_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> defined by

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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><img src="./3e4990ef9cd75261f8a2e82ce0bbb22a3108767c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.926ex; height:6.509ex;" alt="{\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}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is the <a href="State-transition_matrix" title="State-transition matrix">state-transition matrix</a>.

It is possible to determine a unique <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t_{0})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t_{0})}</annotation>

</semantics>

</math><img src="./30dccfced64263036f1ce61e61cb0219d1cf48de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:2.843ex;" alt="{\displaystyle x(t_{0})}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t_{0},t_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> is <a href="Algebraic_curve#Singularities" title="Algebraic curve">nonsingular</a>. In fact, it is not possible to distinguish the initial state for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation>

</semantics>

</math><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"> from that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

</semantics>

</math><img src="./d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}-x_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}-x_{2}}</annotation>

</semantics>

</math><img src="./e8738200f3dcb9d8a7bbdf29aa44218a554ac3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.608ex; height:2.343ex;" alt="{\displaystyle x_{1}-x_{2}}" loading="lazy"> is in the null space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t_{0},t_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy">.

Note that the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> defined as above has the following properties:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> is <a href="Symmetric_matrix" title="Symmetric matrix">symmetric</a></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> is <a href="Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">positive semidefinite</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}\geq t_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{1}\geq t_{0}}</annotation>

</semantics>

</math><img src="./d8ceb42464df69817cb989e7a3de11c126be34fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.886ex; height:2.343ex;" alt="{\displaystyle t_{1}\geq t_{0}}" loading="lazy"></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> satisfies the linear <a href="Matrix_differential_equation" title="Matrix differential equation">matrix differential equation</a></li></ul>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

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<msup>

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<msub>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\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><img src="./10228bd6238b01c13c963114abbb0fb8c350503c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:72.942ex; height:5.509ex;" alt="{\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}" loading="lazy"></dd></dl></dd></dl>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> satisfies the equation</li></ul>

<msub>

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</msub>

<msub>

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</msub>

<msub>

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</msub>

<msub>

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</msub>

<msup>

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<msub>

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<annotation encoding="application/x-tex">{\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><img src="./bb4cf6ddddf94fc364742b42a0010399c8a52d2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.633ex; height:3.176ex;" alt="{\displaystyle M(t_{0},t_{1})=M(t_{0},t)+\varphi (t,t_{0})^{T}M(t,t_{1})\varphi (t,t_{0})}" loading="lazy"><a href="#cite_note-7">[7]</a></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Observability_matrix_generalization">Observability matrix generalization</h3></div>

The system is observable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [t_{0},t_{1}]}">

<msub>

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</msub>

<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle [t_{0},t_{1}]}</annotation>

</semantics>

</math><img src="./ffe2ab6560fe2acf9a63ad878ad482164b79012d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.115ex; height:2.843ex;" alt="{\displaystyle [t_{0},t_{1}]}" loading="lazy"> if and only if there exists an interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [t_{0},t_{1}]}">

<msub>

</mrow>

</msub>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [t_{0},t_{1}]}</annotation>

</semantics>

</math><img src="./ffe2ab6560fe2acf9a63ad878ad482164b79012d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.115ex; height:2.843ex;" alt="{\displaystyle [t_{0},t_{1}]}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> such that the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t_{0},t_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(t_{0},t_{1})}</annotation>

</semantics>

</math><img src="./e04ae154a1a781a5a0a92e5ae7ffe3d3249c8593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.073ex; height:2.843ex;" alt="{\displaystyle M(t_{0},t_{1})}" loading="lazy"> is nonsingular.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(t),C(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(t),C(t)}</annotation>

</semantics>

</math><img src="./f807d6bf038b612dc78abd2d7ce9729dfb2766c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.841ex; height:2.843ex;" alt="{\displaystyle A(t),C(t)}" loading="lazy"> are analytic, then the system is observable in the interval [<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation>

</semantics>

</math><img src="./02d3006c4190b1939b04d9b9bb21006fb4e6fa4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}" loading="lazy">,<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{1}}</annotation>

</semantics>

</math><img src="./cb0768c0bd659f2f84fb5ef9f4b74f336123d915.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{1}}" loading="lazy">] if there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {t}}\in [t_{0},t_{1}]}">

<mover>

</mover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {t}}\in [t_{0},t_{1}]}</annotation>

</semantics>

</math><img src="./ae5d1a6d7fbd649936e22c4225538f8d64f5cf88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.312ex; height:2.843ex;" alt="{\displaystyle {\bar {t}}\in [t_{0},t_{1}]}" loading="lazy"> and a positive integer k such that<a href="#cite_note-:0-8">[8]</a>

<mrow>

<mtr>

<mtd>

<msub>

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<mover>

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<mtr>

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<msub>

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<mover>

</mover>

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</mtr>

<mtr>

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</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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<mover>

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</mtd>

</mtr>

</mtable>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {rank} {\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&\vdots &\\&N_{k}({\bar {t}})&\end{bmatrix}}=n,}</annotation>

</semantics>

</math><img src="./4f8c05ec743a267301b9f1b174147f7d6703f01e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:24.739ex; height:14.176ex;" alt="{\displaystyle \operatorname {rank} {\begin{bmatrix}&N_{0}({\bar {t}})&\\&N_{1}({\bar {t}})&\\&\vdots &\\&N_{k}({\bar {t}})&\end{bmatrix}}=n,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{0}(t):=C(t)}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle N_{0}(t):=C(t)}</annotation>

</semantics>

</math><img src="./a5d69179bb51b100eea09aec080cad1b4812c3a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.73ex; height:2.843ex;" alt="{\displaystyle N_{0}(t):=C(t)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{i}(t)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle N_{i}(t)}</annotation>

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</math><img src="./fc942a79abed0c383484b50c27db49750e5c392a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.315ex; height:2.843ex;" alt="{\displaystyle N_{i}(t)}" loading="lazy"> is defined recursively as

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<annotation encoding="application/x-tex">{\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><img src="./2d832a77b583982a2ce326ebb47cd79dd24edb76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:49.061ex; height:5.509ex;" alt="{\displaystyle N_{i+1}(t):=N_{i}(t)A(t)+{\frac {\mathrm {d} }{\mathrm {d} t}}N_{i}(t),\ i=0,\ldots ,k-1}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div>

Consider a system varying analytically in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-\infty ,\infty )}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle (-\infty ,\infty )}</annotation>

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</math><img src="./0c8c11c44279888c9e395eeb5f45d121348ae10a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.299ex; height:2.843ex;" alt="{\displaystyle (-\infty ,\infty )}" loading="lazy"> and matrices<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}.}">

<mrow>

<mtr>

<mtd>

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<annotation encoding="application/x-tex">{\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>

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</math><img src="./189e6ebfcbfb5d3e54a050807d3dcddad41bad65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:40.697ex; height:9.509ex;" alt="{\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}}.}" loading="lazy"> </blockquote>Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\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}}}">

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<annotation encoding="application/x-tex">{\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>

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</math><img src="./d9c222aaf3c016147c5fcfdaea855f5b168483ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:24.827ex; height:9.843ex;" alt="{\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}}}" loading="lazy"> , and since this matrix has rank = 3, the system is observable on every nontrivial interval of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

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</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Nonlinear_systems">Nonlinear systems</h2></div>

Given the system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j}}</annotation>

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</math><img src="./202139e918dd31d31bc474f59e57a78e8af4a718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.825ex; height:7.176ex;" alt="{\displaystyle {\dot {x}}=f(x)+\sum _{j=1}^{m}g_{j}(x)u_{j}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}=h_{i}(x),i\in p}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}=h_{i}(x),i\in p}</annotation>

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</math><img src="./dbce14940c92e6dc7fac251c77decdfc43e48545.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.161ex; height:2.843ex;" alt="{\displaystyle y_{i}=h_{i}(x),i\in p}" loading="lazy">. Where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} ^{n}}</annotation>

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</math><img src="./c520ee2cb6ccf8a93c89a8c58a8378796bd52e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle x\in \mathbb {R} ^{n}}" loading="lazy"> the state vector, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in \mathbb {R} ^{m}}">

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</math><img src="./64ebec3a2ba3f7f2ae46dcdc3c9b890a0232f552.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.523ex; height:2.343ex;" alt="{\displaystyle u\in \mathbb {R} ^{m}}" loading="lazy"> the input vector and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {R} ^{p}}">

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<annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {R} ^{p}}</annotation>

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</math><img src="./fb62f1d39c2775088d317eae4da44e4d46ce7f83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.733ex; height:2.676ex;" alt="{\displaystyle y\in \mathbb {R} ^{p}}" loading="lazy"> the output vector. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g,h}">

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</math><img src="./40bac5cdee81bb0c899b395c37f6ab3e3a8a0193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.801ex; height:2.509ex;" alt="{\displaystyle f,g,h}" loading="lazy"> are to be smooth vector fields.

Define the observation space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{s}}">

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</math><img src="./7dd70366667e3650837b2242c95f70f38d720782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.853ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{s}}" loading="lazy"> to be the space containing all repeated <a href="Lie_derivative" title="Lie derivative">Lie derivatives</a>, then the system is observable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(d{\mathcal {O}}_{s}(x_{0}))=n}">

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<annotation encoding="application/x-tex">{\displaystyle \dim(d{\mathcal {O}}_{s}(x_{0}))=n}</annotation>

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</math><img src="./d279f624f41f81d4ae942de64330898eff4782aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.44ex; height:2.843ex;" alt="{\displaystyle \dim(d{\mathcal {O}}_{s}(x_{0}))=n}" loading="lazy">, where

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<annotation encoding="application/x-tex">{\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>

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</math><img src="./6342cd9b7a9ea9e0dde946699a325ffd92a84017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:84.192ex; height:3.009ex;" alt="{\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 .}" loading="lazy"><a href="#cite_note-9">[9]</a></dd></dl>

Early criteria for observability in nonlinear dynamic systems were discovered by Griffith and Kumar,<a href="#cite_note-10">[10]</a> Kou, Elliot and Tarn,<a href="#cite_note-11">[11]</a> and Singh.<a href="#cite_note-12">[12]</a>

There also exist an observability criteria for nonlinear time-varying systems.<a href="#cite_note-13">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Static_systems_and_general_topological_spaces">Static systems and general topological spaces</h2></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy">.<a href="#cite_note-14">[14]</a><a href="#cite_note-15">[15]</a> Just as observability criteria are used to predict the behavior of <a href="Kalman_filter" title="Kalman filter">Kalman filters</a> or other observers in the dynamic system case, observability criteria for sets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> are used to predict the behavior of <a href="Data_validation_and_reconciliation" title="Data validation and reconciliation">data reconciliation</a> 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.

<ul><li><a href="Controllability" title="Controllability">Controllability</a></li>

<li><a href="Hautus_lemma" title="Hautus lemma">Hautus lemma</a></li>

<li><a href="Structural_identifiability" title="Structural identifiability">Identifiability</a></li>

<li><a href="State_observer" title="State observer">State observer</a></li>

<li><a href="State_space_(controls)" class="mw-redirect" title="State space (controls)">State space (controls)</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFMartinelli2022" class="citation journal cs1">Martinelli, Agostino (2022). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/9790057">"Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems"</a>. IEEE Transactions on Automatic Control. 67 (9): 5002–5008. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAC.2022.3180771">10.1109/TAC.2022.3180771</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9286">0018-9286</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:251957578">251957578</a>.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFStanleyMah1981" class="citation journal cs1">Stanley, G. M.; Mah, R. S. H. (1981). <a rel="nofollow" class="external text" href="https://gregstanleyandassociates.com/CES-1981a-ObservabilityRedundancy.pdf">"Observability and redundancy in process data estimation"</a> (PDF). Chemical Engineering Science. 36 (2): 259–272. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981ChEnS..36..259S">1981ChEnS..36..259S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0009-2509%2881%2985004-X">10.1016/0009-2509(81)85004-X</a>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFStanleyMah1981" class="citation journal cs1">Stanley, G.M.; Mah, R.S.H. (1981). <a rel="nofollow" class="external text" href="https://gregstanleyandassociates.com/CES-1981b-ObservabilityRedundancyProcessNetworks.pdf">"Observability and redundancy classification in process networks"</a> (PDF). Chemical Engineering Science. 36 (12): 1941–1954. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0009-2509%2881%2980034-6">10.1016/0009-2509(81)80034-6</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/Observability">"Observability"</a>. <a href="PlanetMath" title="PlanetMath">PlanetMath</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.mathworks.com/help/toolbox/control/ref/obsv.html">MATLAB function for checking observability of a system</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120219025315/http://www.mathworks.com/help/toolbox/control/ref/obsv.html">Archived</a> 2012-02-19 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="http://reference.wolfram.com/mathematica/ref/ObservableModelQ.html">Mathematica function for checking observability of a system</a></li></ul></div><div><div class="zim-footer">

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