Structured Singular Value of a Repeated Complex Full-Block Uncertainty

TL;DR

This work targets robust stability analysis of LTI systems with a structured uncertainty consisting of repeated complex full-blocks, $\Delta_r = I_v \otimes \Delta_1$. It introduces two complementary algorithms: a gradient-based Method of Centers to compute a tight $D$-scale upper bound via a semidefinite program, and a generalized power-iteration scheme to obtain a non-conservative lower bound for $\mu$ under repeated full-blocks. The methods are implemented to rely on gradient information rather than Hessians, improving computational efficiency, and are validated on an incompressible plane Couette flow model and a simple academic example, showing reduced conservatism compared with non-repeated-block treatments. Together, these results yield tighter stability margins and better physical insight for convective systems where block repetition naturally arises, highlighting the importance of preserving the true uncertainty structure in SSV computations.

Abstract

The structured singular value (SSV), or mu, is used to assess the robust stability and performance of an uncertain linear time-invariant system. Existing algorithms compute upper and lower bounds on the SSV for structured uncertainties that contain repeated (real or complex) scalars and/or non-repeated complex full blocks. This paper presents algorithms to compute bounds on the SSV for the case of repeated complex full blocks. This specific class of uncertainty is relevant for the input output analysis of many convective systems, such as fluid flows. Specifically, we present a power iteration to compute a lower bound on SSV for the case of repeated complex full blocks. This generalizes existing power iterations for repeated complex scalar and non-repeated complex full blocks. The upper bound can be formulated as a semi-definite program (SDP), which we solve using a standard interior-point method to compute optimal scaling matrices associated with the repeated full blocks. Our implementation of the method only requires gradient information, which improves the computational efficiency of the method. Finally, we test our proposed algorithms on an example model of incompressible fluid flow. The proposed methods provide less conservative bounds as compared to prior results, which ignore the repeated full block structure.

Figures (5)

• Figure 1: The $\alpha_{\max}$ and $\beta_{\max}$ results over the wavenumber pair $(\kappa_x, \kappa_z)$ grid. The top row plots represent the upper bounds $\alpha_{\max}$ and the bottom row plots represent the lower bounds $\beta_{\max}$.
• Figure 2: The percentage difference between $\alpha_{\max}$ and $\beta_{\max}$ values over the wavenumber pair $(\kappa_x, \kappa_z)$ grid. The stopping ratio between upper and lower bounds for Algorithm \ref{['alg:moc']} was set to $1.05$, which means that all the computed upper bounds must be within $5\%$ of the lower bounds. Therefore, the majority of percentage differences in (b) are $\leq 5\%$. The only upper bounds that failed to achieve the stopping criterion are given by the red hotspots.
• Figure 3: The $\alpha$ and $\beta$ results over the temporal frequency ($\omega$) grid for $\Delta \in \mathbf{\Delta}_\mathrm{nr}$. The (a) and (b) correspond to wavenumber pairs where the gap between $\alpha_{\max}$ and $\beta_{\max}$ are the largest and smallest, respectively.
• Figure 4: The $\alpha$ and $\beta$ results over the temporal frequency ($\omega$) grid for $\Delta \in \mathbf{\Delta}_\mathrm{r}$. The (a) and (b) correspond to wavenumber pairs where the gap between $\alpha_{\max}$ and $\beta_{\max}$ are the largest and smallest, respectively.
• Figure 5: The $\alpha$ and $\beta$ results over the temporal frequency ($\omega$) grid. Although we consider $\omega \in [-10^{1.5}, 10^{1.5}]$, the results are shown for $\omega \in [-10,10]$ to better highlight the local behavior of the bounds.

Theorems & Definitions (1)

• definition 1