largest possible perturbation that preserves stability in a matrix

Question

We say a matrix is stable if all of its eigenvalues lie in the open left half of the complex plane. Given a stable matrix $A$ and some other (not necessarily stable) matrix B, is there any efficient way to find the "largest" interval $\Delta$ such that $0\in\Delta$ and $A+tB$ is stable $\ \forall t\in\Delta$? This is relevant to a research problem I am working on, and in the context of the problem the only assumption on $B$ is that it has at least one row of zeros

Answer 1

There are many ways, you can do that. This problem is related to the real structured singular value and this is not an easy problem in general. There are some papers dedicated to finding condition to compute it in the same way as we would do for the usual structured singular value.

I am describing here more empirical ways to approximately compute it, which are simple and good enough for some applications. Those procedures may not be acceptable depending on the problem at hand and the type of solutions that are sought.

One possible way is to use a bisection algorithm on the parameter $t$ that will compute the eigenvalues of the pencil $A+tB$ at each iteration and increase $t$ if the pencil value is Hurwitz stable and decrease it otherwise. That's not very elegant, but that works.

Another way is to view at the problem as the positive feedback interconnection of the stable transfer function

$$H(s)=(sI-A)^{-1}B$$ and the uncertain operator $\Delta = tI$. In this case, we have that the interconnection is stable for all $t\in(T_1,T_2)$ if and only if

$$\det(I-H(j\omega)t)\ne0$$ for all $\omega\ge0$ and all $t\in(T_1,T_2)$.

This implies that we need to look at all the values of $\omega$ for which at least one of the eigenvalues is real and positive, we denote those eigenvalues by $\pi_i$, and for which at least one of the eigenvalues is real and positive, we denote those eigenvalues by $-\nu_i$.

Then, the interconnection will remain stable for all $t\in(T_1,T_2)$ where $T_1=\min_i\{1/\nu_i\}$ and $T_2=\min_i\{1/\pi_i\}$.