We will consider the following feedback interconnection :
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:
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
then the feedback interconnection is stable.
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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