Stereoscopic spectroscopy

In control theory, the controllability Gramian is a Gramian used to determine whether or not a linear system is controllable. For the time-invariant linear system

$\dot{x}=Ax+Bu$

if all eigenvalues of $A$ have negative real part, then the unique solution $W_c$ of the Lyapunov equation

$A{W}_c+W_c{A}^T=-B{B}^T$

is positive definite if and only if the pair $(A,B)$ is controllable. $W_c$ is known as the controllability Gramian and can also be expressed as

$W_c=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{A\tau }B{B}^T{e}^{A^T\tau }d\tau$

A related matrix used for determining controllability is

$W_c(t)=\underset{0}{\overset{t}{\int }}{e}^{A\tau }B{B}^T{e}^{A^T\tau }d\tau =\underset{0}{\overset{t}{\int }}{e}^{A(t-\tau )}B{B}^T{e}^{A^T(t-\tau )}d\tau$

The pair $(A,B)$ is controllable if and only if the matrix $W_c(t)$ is nonsingular, for any $t>0$. A physical interpretation of the controllability Gramian is that if the input to the system is white gaussian noise, then $W_c$ is the covariance of the state.

Linear time-variant state space models of form

$\dot{x}(t)=A(t)x(t)+B(t)u(t)$,

$y(t)=C(t)x(t)+D(t)u(t)$

are controllable in an interval $[t_0,t_1]$ if and only if the rows of the matrix product $\mathrm{\Phi }(t_0,\tau )B(\tau )$ where $\mathrm{\Phi }$ is the state transition matrix are linearly independent. The Gramian is used to prove the linear independency of $\mathrm{\Phi }(t_0,\tau )B(\tau )$. To have linear independency Gramian matrix $W_c$ have to be nonsingular, i.e., invertible.

$W_c(t)=\underset{t_0}{\overset{t}{\int }}\mathrm{\Phi }(t_0,\tau )B(\tau )B^T(\tau ){\mathrm{\Phi }}^T(t_0,\tau )d\tau$

See also

• Observability Gramian
• Gramian matrix
• Minimum energy control

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Mathematica function to compute the controllability Gramian

Template:Applied-math-stub