# Norm bound in simultaneous stability to semidefinite program

In the context of robust control, I remember hearing that the two following problems are equivalent.

1. Find $$P \succ 0$$, such that $$A P + P A^{\top} \prec 0$$ for all $$A \in \mathscr{A}$$ where $$\mathscr{A} = \left\{ A+B \Delta C:\|\Delta\|_{\text{op}} \leqslant 1 \right\}$$

2. Find $$P\succ0$$ such that $$\left[\begin{array}{cc}A P+P A^{\top} & P B \\ B P & -C^{\top} C\end{array}\right] \preceq 0$$

There is another case where we require $$\mathscr{A} = \left\{ A+B \Delta C:\|\Delta\|_{\text{op}}\leq 1, \Delta \text{ is a diagonal matrix} \right\}$$ also known as diagonal norm bound.

Question. Are there any sources describing why these two formulations are equivalent, and also more information about their relation to the diagonal norm bound?

Despite searching on the internet for some time, I did not find anything relevant. This is also not in Boyd et al$$^\color{magenta}{\dagger}$$. I am pretty sure that this is an old result.

$$\color{magenta}{\dagger}$$ Stephen Boyd, Laurent El Ghaoui, Eric Feron, Venkataramanan Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Studies in Applied and Numerical Mathematics, SIAM, 1994.

• Is the operator norm the spectral norm? Commented Dec 28, 2023 at 10:34
• @RodrigodeAzevedo Thank you for editing my post and including the source. This presumably is from robust control, but the only thing I know is that it is related to simultaneous stability. I would assume that the operator norm means the spectral norm, but I'm just exactly transcribing down what I heard. Commented Dec 28, 2023 at 17:47
• Are you acquainted with Parrilo's doctoral dissertation? The list of references might be useful. Commented Dec 28, 2023 at 18:05
• Have you taken a look at chapter 5 of Boyd et al$^\color{magenta}{\dagger}$? Or at Boyd & Yang's Structured and simultaneous Lyapunov functions for system stability problems? Commented Dec 28, 2023 at 18:19
• @RodrigodeAzevedo Thank you for the resource. I think it might be really in the first few pages of Chapter 5 of Boyd et al. However, this book is so dense, and I cannot understand many of his derivations. For instance his norm bound LDI and diagonal norm bound LDI seems to skip a lot of details. Commented Jan 4 at 2:35

The well-known result is the following:

• There exists a matrix $$P\succ0$$ such that $$(A+B\Delta C)^TP+P(A+B\Delta C)\prec0$$ for all $$||\Delta||\le 1$$.
• There exist a matrix $$P\succ0$$ and a scalar $$\epsilon>0$$ such that $$\begin{bmatrix} A^TP+PA+\epsilon C^TC & PB\\ B^TP & -\epsilon I \end{bmatrix}\prec0.$$

It was initially proven in https://www.sciencedirect.com/science/article/pii/0167691187901022 by Petersen and is often referred to as "Petersen's lemma". The original proof is a bit convoluted and easier proofs exist, such as the one based on the S-procedure, which can be found in Briat's book on LPV systems (among others) together with many extensions of the result to structured uncertainties.