Approximating the Real Structured Stability Radius with Frobenius-Norm Bounded Perturbations

Abstract

We propose a fast method to approximate the real stability radius of a linear dynamical system with output feedback, where the perturbations are restricted to be real valued and bounded with respect to the Frobenius norm. Our work builds on a number of scalable algorithms that have been proposed in recent years, ranging from methods that approximate the complex or real pseudospectral abscissa and radius of large sparse matrices (and generalizations of these methods for pseudospectra to spectral value sets) to algorithms for approximating the complex stability radius (the reciprocal of the H∞ norm). Although our algorithm is guaranteed to find only upper bounds to the real stability radius, it seems quite effective in practice. As far as we know, this is the first algorithm that addresses the Frobenius-norm version of this problem. Because the cost mainly consists of computing the eigenvalue with maximal real part for continuous-time systems (or modulus for discrete-time systems) of a sequence of matrices, our algorithm remains very efficient for large-scale systems provided that the system matrices are sparse.

Faster and More Accurate Computation of the $\mathcal{H}_\infty$ Norm via Optimization

“…If the optimization method is initialized at ωj , then even if ω opt , a computed solution to (19) holds, provided that ωj does not happen to be a stationary point of g c (ω). Furthermore, (19) can only have more than one maximizer when the current estimate γ of the H ∞ norm is so low that there are multiple peaks above level-set interval I j .…”

“…If the optimization method is initialized at ωj , then even if ω opt , a computed solution to (19) holds, provided that ωj does not happen to be a stationary point of g c (ω). Furthermore, (19) can only have more than one maximizer when the current estimate γ of the H ∞ norm is so low that there are multiple peaks above level-set interval I j . Consequently, as the algorithm converges, computed maximizers of (19) will be assured to be globally optimal over I j and in the limit, over all frequencies along the entire imaginary axis.…”

“…Furthermore, (19) can only have more than one maximizer when the current estimate γ of the H ∞ norm is so low that there are multiple peaks above level-set interval I j . Consequently, as the algorithm converges, computed maximizers of (19) will be assured to be globally optimal over I j and in the limit, over all frequencies along the entire imaginary axis. By setting tight tolerances for the optimization code, maximizers of (19) can also be computed to full precision with little to no penalty, due to the superlinear or quadratic rate of convergence we can expect from the secant method or Newton's method, respectively.…”

“…Consequently, as the algorithm converges, computed maximizers of (19) will be assured to be globally optimal over I j and in the limit, over all frequencies along the entire imaginary axis. By setting tight tolerances for the optimization code, maximizers of (19) can also be computed to full precision with little to no penalty, due to the superlinear or quadratic rate of convergence we can expect from the secant method or Newton's method, respectively. If the computed optimizer of ( 19) also happens to be a global maximizer of g c (ω), for all ω ∈ R, then the H ∞ norm has indeed been computed to full precision, but the algorithm must still verify this by computing the imaginary eigenvalues of (M γ , N ) just one more time.…”

On the real stability radius of sparse systems

“…In [15] and [16], lower bounds on the real, F -norm SR were provided for the unstructured and structured cases, respectively. Recently, a number of works have appeared that use iterative algorithms to approximate the 2-norm/F -norm real SR [17,18,19,20,21]. Typically, these algorithms use two levels of iterations.…”