We will consider the following feedback interconnection
:
![{\displaystyle {\begin{cases}v=Gu+f,\\u=\Delta (v)+r\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca68040d5e8e0443d420933e57a4278b0fcc9fd3)
where
and
are exogeneous inputs.
and
are two casual operators.
Let
be a measurable Hermitian-valued function,
and
be a bounded casual operator.
such that
Then the feedback interconnection of
and
is stable.
is a linear time-invariant system with the state space realization:
![{\displaystyle {\begin{cases}{\dot {x}}=Ax+Bu\\y=Cx+Du\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf0810441837df88428962288a570996cc2e7bf)
where
is the state.
Any
can be factorized as
where
and
. Denote the state space realization of
by
.
A state space realization for the system
is
If there exists a matrix
such that
![{\displaystyle {\begin{bmatrix}{\hat {A}}^{T}P+P{\hat {A}}&P{\hat {B}}\\{\hat {B}}^{T}P&0\end{bmatrix}}+{\begin{bmatrix}{\hat {C}}^{T}\\{\hat {D}}^{T}\end{bmatrix}}M{\begin{bmatrix}{\hat {C}}&{\hat {D}}\end{bmatrix}}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/858ea0132fd737c09ef427f57c51cc31ee0f5b70)
then the feedback interconnection
is stable.
A. Megretski and A. Rantzer, "System analysis via integral quadratic constraints," in IEEE Transactions on Automatic Control, vol. 42, no. 6, pp. 819-830, June 1997, doi: 10.1109/9.587335
P. Seiler, "Stability Analysis With Dissipation Inequalities and Integral Quadratic Constraints," in IEEE Transactions on Automatic Control, vol. 60, no. 6, pp. 1704-1709, June 2015, doi: 10.1109/TAC.2014.2361004