Structured feedback optimization for Metzler dynamics

James Anderson, Riley Murray

Abstract:

We propose a method to compute convergent lower bounds for the state-feedback controller design problem
\begin{equation*}
\inf\limits_{F}\ \{L(F): A+BF\ \mathbf{is\ Metzler}\ \&\ \mathbf{stable}\}
\end{equation*}
where $L$ is a convex loss function of $F$. The theory behind the approach is simple: relying only on an extension of Perron-Frobenius to Metzler matrices, and popular discrete optimization techniques. The method itself has two tuning parameters (which enable faster recovery of solutions, with possible introduction of optimality gaps) and is practical for systems with a non-trivial state dimension. A convergence result with respect to the optimal solution is derived, and a direct heuristic algorithm based on linear programming is given. We explain how projecting $A$ onto the set of stable Metzler matrices is essentially the hardest of these problems, and we focus our numerical examples on precisely this case.