Observability Gramian
In control theory, we may need to find out whether or not a system such as x ˙ ( t ) = A x ( t ) + B u ( t ) y ( t ) = C x ( t ) + D u ( t ) {\displaystyle {\begin{array}{c}{\dot {\boldsymbol {x}}}(t){\boldsymbol {=Ax}}(t)+{\boldsymbol {Bu}}(t)\\{\boldsymbol {y}}(t)={\boldsymbol {Cx}}(t)+{\boldsymbol {Du}}(t)\end{array}}} is observable, where A {\displaystyle {\boldsymbol {A}}} , B {\displaystyle {\boldsymbol {B}}} , C {\displaystyle {\boldsymbol {C}}} and D {\displaystyle {\boldsymbol {D}}} are, respectively, n × n {\displaystyle n\times n} , n × p {\displaystyle n\times p} , q × n {\displaystyle q\times n} and q × p {\displaystyle q\times p} matrices. One of the many ways one can achieve such goal is by the use of the Observability Gramian.