Solving Problem: LMIs and block matrices

Question

I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find conditions that would satisfy

\begin{equation} \mathcal{W} \triangleq \dot{V}(\xi) + \vert \vert \xi \vert \vert^2 - \mu \vert \vert \eta \vert \vert^2 \leq 0 \end{equation} where $V(\xi) = \xi^T \hspace{1mm} \mathbb{Q} \hspace{1mm} \xi $ is the Lyapunov function.

Substituting for the state dynamics (assuming I have taken the derivative properly), $\dot{\xi}$, we get \begin{align} \mathcal{W} &= 2 \hspace{0.1cm} \xi^T \mathbb{Q} \left[\begin{pmatrix} \mathbb{A}_{\overline{K}} & \overline{B} \overline{K} \tilde{B}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix} \hspace{0.1cm} \xi + \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix} \hspace{0.1cm} \eta \right] + \vert \vert \xi \vert \vert^2 - \mu \vert \vert \eta \vert \vert^2\\ &= \left(2 \hspace{0.1cm} \xi^T \hspace{0.1cm} \left[\mathbb{Q} \left\{ \begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix} + \begin{pmatrix} \mathcal{O} & \overline{B} \overline{K} \tilde{B}\\ \mathcal{O} & \mathcal{O} \end{pmatrix} \right\} \right] \hspace{0.1cm} \xi + \xi^T \hspace{0.1cm} \xi \right)+ 2 \hspace{0.1cm} \xi^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix} \hspace{0.1cm} \eta - \mu \hspace{0.1cm} \eta^T \hspace{0.1cm} \eta \end{align}

The final form that the paper skips to and that I am trying to attain is:

\begin{equation} \mathcal{W} = \begin{pmatrix} \xi\\ \eta \end{pmatrix}^T \hspace{0.1cm} \Pi \hspace{0.1cm} \begin{pmatrix} \xi\\ \eta \end{pmatrix} \end{equation}

with $\Pi = \begin{pmatrix} \begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix}^T \mathbb{Q} + \mathbb{Q} \begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix} + \mathbb{I}_{2n} & \mathbb{Q} \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix}\\ \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix}^T \mathbb{Q}^T & -\mu \mathbb{I}_{q} \end{pmatrix} + \begin{pmatrix} \mathbb{Z}^{-1} \overline{B} \overline{K}\\ \mathcal{O}\\ \mathcal{O} \end{pmatrix} \begin{pmatrix} \mathcal{O} & \tilde{B} & \mathcal{O} \end{pmatrix} + \begin{pmatrix} \mathcal{O}\\ \tilde{B}^T\\ \mathcal{O} \end{pmatrix} \begin{pmatrix} \left(\overline{B} \overline{K}\right)^T \mathbb{Z}^{-1} & \mathcal{O} & \mathcal{O} \end{pmatrix}$

If you expand further, you'll see that the following needs to be true for them to be equal:

1. $\mathbb{Q} \begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix} + \mathbb{Q} \begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix}$ is the same as $\begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix}^T \mathbb{Q} + \mathbb{Q} \begin{pmatrix} \mathbb{A}_{\overline{K}} & \mathcal{O}\\ \mathcal{O} & \mathbb{A}_{L} \end{pmatrix}$

2. Given that $\mathbb{Q} = \begin{pmatrix} \mathbb{Z}^{-1} & \mathcal{O}\\ \mathcal{O} & \mathbb{P} \end{pmatrix}$, $\mathbb{P}$ is a positive-definite matrix with matrices $\mathbb{Z}$ and $\mathbb{Q}$ being symmetric, $2 \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix} \mathcal{O} & \overline{B} \overline{K} \tilde{B}\\ \mathcal{O} & \mathcal{O} \end{pmatrix}$ is the same as $\begin{pmatrix} \mathcal{O} & \mathbb{Z}^{-1} \hspace{1mm} \overline{B} \overline{K} \hspace{1mm} \tilde{B}\\ \tilde{B} \hspace{1mm} \left(\overline{B} \overline{K}\right)^T \mathbb{Z}^{-1} & \mathcal{O} \end{pmatrix}$

3. $2 \hspace{0.1cm} \xi^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix} \eta$ is the same as $\eta \hspace{0.1cm} \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix}^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \xi + \xi^T \hspace{0.1cm} \mathbb{Q} \hspace{0.1cm} \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix} \eta$

The curious thing is the above three relations hold (as far as I know) only when the matrices are symmetric which in turn means the submatrices in the block matrices are symmetric but knowing the structure of these matrices, they are not. I think there is something silly I am doing and can't figure it out.

P.S. I tried to be as succinct as I could have been but I can understand if I am missing some info. Please let me know.

Answer 1

I assume $\mathcal{W}$ is a scalar. In such case it can also be written as

$$ \mathcal{W} = \frac{1}{2}\left(\mathcal{W}+\mathcal{W}^\top\right). $$

This is common practice when formulating a LMI, since it has the advantage that when you factor out $\begin{bmatrix} \zeta^\top & \mu^\top\end{bmatrix}^\top$ the $\Pi$ matrix is symmetric. Writing your initial inequality $\mathcal{W} \leq 0$ in terms of $\Pi=\Pi^\top$ allows you to also write it as the LMI $\Pi\preceq 0$.

For example for state feedback of a LTI system you have

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

and you want to find

$$ u=K\,x, \\ V(x)=x^\top P\,x, \\ P=P^\top\succ0, \\ \dot{V}(x)<0\ \forall\,x\neq0. $$

Where $\dot{V}(x)$ can be written as

$$ \dot{V}(x)=2\,x^\top P\,(A+B\,K)\,x, $$

but more commonly is written as

$$ \dot{V}(x)=x^\top\left(P\,(A+B\,K) + (A+B\,K)^\top P\right)\,x. $$