The following lemma is taken from references firstly.
Lemma 1 [1-2] Given matrices $Q=Q^{T} , F, M$ and $N$ of appropriate dimensions, then $$Q+MFN+N^{T}F^{T}M^{T}<0$$ for all $F$ satisfying $F^{T}F\leq I$ , if and only if there exists some scalar $\epsilon$ such that $$Q+\epsilon MM^{T}+\frac{1}{\epsilon}N^{T}N<0 \quad(1)$$ The superscripts $T$ stands for the matrix transpose. For a square matrix $X$ and $Y$, $X<Y$ represents $Y-X$ is a symmetric positive definite matrix.
Now, we consider the following output equation $$y(t)=Cx(t)+Dd(t)$$ where $x\in \mathbb{R}^{n}, y\in {{\mathbb{R}}^{p}},d\in {{\mathbb{R}}^{m}}$ represent the state vector, measurement output vector and disturbance vector respectively. Furthermore, we assume that partial state of $x$ are cannot measurable. Therefore, $x$ is unknown in the output equation, and only $y$ is known in the output equation. $C, D$ are constants matrices of appropriate dimension. Without loss of generality, we can assume $C$ is full row rank, i.e., $\text{rank}(C)=p$. For matrix $C$, the inequality $C^{T}C\leq I$ may be not true.
The question is stated as following.
If we replace $F$ with $C$ , can we still get a similar inequality as (1) for inequality $$Q+MCN+N^{T}C^{T}M^{T}<0\quad $$
In the new inequality, $M$ and $N$ can be separated as $ MM^{T}$ and $N^{T}N$ .
For this question, first of all, if we can turn $C$ into a matrix $E$ which satisfies $E^{T}E\leq I$ by left transformation (we can only employ left transformation because of the output equation, where $C$ is in the left side of $x$), then the question is solved. Now, the question is whether there exists such a left transformation and how can we fulfill the transformation?
Secondly, if we cannot find out such left transformation, can we get a similar inequality as (1) by other methods?
Thanks very much for your attention.
References
[1]Wang Y, Xie L, de Souza C E. Robust control of a class of uncertain nonlinear systems. Systems & Control Letters, 1992, 19(2): 139-149.
[2]Xie L. Output feedback H∞ control of systems with parameter uncertainty. International Journal of Control, 1996, 63(4): 741-750.
