checkstability = truefalse

True means check whether the system is stable; if it is not stable, an error occurs. False means skip the check. The default is true.

output = C or O or list of these names

Specifies the returned values. If equal to C, then the controllability grammian is returned. If equal to O, then the observability grammian is returned. If a list of these names, then the output is a list/sequence (see returnlist) with each C replaced with the controllability grammian and each O replaced with the observability grammian. The default is [C,O].

returnlist = truefalse

True means return a list; false means return an expression sequence. The default is false.

The Grammians command computes the grammians of sys, a state-space system.

Depending on the value of the output option, either the controllability grammian, the observability grammian, or both, is computed.

For a grammian to exist, the system must be stable. For a continuous-time system, all eigenvalues, $\mathrm{\lambda }$, of A must lie in the open left-half plane: $\mathrm{\Re }\left(\mathrm{\lambda }\right)<0$. For a discrete-time system, all eigenvalues, $\mathrm{\lambda }$, of A must lie in the open unit-circle: $\left|\mathrm{\lambda }\right|<1$. If sys is not stable, an error occurs, unless the option checkstability is false.

A grammian is the positive-definite matrix X that solves the appropriate Lyapunov equation (see LyapunovSolve).

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\&colon;$

$\mathrm{aSys}≔\mathrm{StateSpace}\left(⟨⟨-5\&comma;3⟩|⟨3\&comma;-4⟩⟩\&comma;⟨⟨2\&comma;3⟩⟩\&comma;⟨⟨1\&comma;0⟩|⟨0\&comma;1⟩⟩\&comma;⟨⟨0\&comma;0⟩⟩\right)\&colon;$

$\mathrm{Cg}≔\mathrm{Grammians}\left(\mathrm{aSys}\&comma;\mathrm{output}=C\right)$

$\mathrm{Cg}≔\left[\begin{array}{cc}1.68181818181818& 2.13636363636364\\ 2.13636363636364& 2.72727272727273\end{array}\right]$

$\mathbf{use}\mathrm{aSys}\mathbf{in}\mathrm{`.`}\left(a\&comma;\mathrm{Cg}\right)\&plus;\mathrm{`.`}\left(\mathrm{Cg}\&comma;a\right)\&equals;-\mathrm{`.`}\left(b\&comma;b^{\mathrm{\%T}}\right)\mathbf{end\; use}$

$\left[\begin{array}{cc}\mathrm{-4.00000000000000}& \mathrm{-6.00000000000000}\\ \mathrm{-6.00000000000000}& \mathrm{-9.00000000000001}\end{array}\right]=\left[\begin{array}{cc}\mathrm{-4}& \mathrm{-6}\\ \mathrm{-6}& \mathrm{-9}\end{array}\right]$