Speaker
Description
In this talk, we consider computing the worst-case (highest) $\mathcal{H}_\infty$ norm of a either a continuous-time or discrete-time linear time-invariant parametric system, where the state-space matrices all depend on a single real-valued scalar parameter $\mathsf{p}$ on a domain $\mathcal{P}$ consisting of a finite number of intervals. On each interval in $\mathcal{P}$, we assume that $\mathsf{p}$ may vary nonlinearly and that each state-space matrix is differentiable with respect to $\mathsf{p}$ almost everywhere, but we do permit that the parameter may be discontinuous at a finite number of points inside each interval of $\mathcal{P}$. Using the framework of interpolation-based globality certificates, which were first devised as a fast way to compute Kreiss constants and the distance to uncontrollability, we present a new optimization-with-restarts algorithm that, under reasonable assumptions, solves the underlying nonconvex optimization with global convergence and thus computes the worst-case $\mathcal{H}_\infty$ norm to arbitrary accuracy. Our algorithm does $\mathcal{O}(kn^3)$ work, where $n$ is the dimension of the state vector and $k$ is the total number of function evaluations needed to globally approximate our certificate functions over the parameter domain $\mathcal{P}$ using, say, Chebfun. In practice, $k$ is not strongly correlated with $n$ and typically only a handful of optimization restarts are needed before a global optimizer attaining the value of the worst-case $\mathcal{H}_\infty$ norm is found. Moreover, the overall cost of our algorithm is almost entirely due to the cost of approximating the final certificate function, which asserts that the final maximizer found is in indeed a global maximizer. Experiments show that our new algorithm is both significantly faster and more reliable than other reasonable approaches for computing the worst-case $\mathcal{H}_\infty$ norm.