[Paper Review] Structured Singular Value of a Repeated Complex Full-Block Uncertainty

Talha Mushtaq, Diganta Bhattacharjee+2 more|arXiv (Cornell University)|Nov 11, 2022

Probabilistic and Robust Engineering Design4 citations

TL;DR

This paper proposes efficient algorithms to compute less conservative upper and lower bounds for the structured singular value (μ) in systems with repeated complex full-block uncertainties—common in fluid dynamics. Using a gradient-based method of centers for the upper bound and a generalized power iteration for the lower bound, the method improves computational efficiency and reveals more accurate stability margins compared to approximating repeated blocks as non-repeated ones.

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.

Motivation & Objective

• Address the lack of efficient methods to compute structured singular value (μ) bounds for systems with repeated complex full-block uncertainties, which are common in fluid dynamics and convective systems.
• Reduce conservatism in μ bounds that arises when repeated full-block structures are approximated as non-repeated ones, as done in standard tools like MATLAB's Robust Control Toolbox.
• Develop computationally efficient algorithms that leverage gradient information rather than Hessian computation to improve scalability for large-dimensional systems.
• Demonstrate that correctly modeling the repeated block structure is essential for accurate physical interpretation of instability mechanisms in systems like plane Couette flow.

Proposed method

• Formulate the upper bound on μ as a semi-definite program (SDP) solved via the method of centers, using only gradient information to enhance computational efficiency.
• Generalize the power iteration method of Packard et al. to compute a lower bound on μ for repeated complex full-block uncertainties, extending existing approaches for scalars and non-repeated blocks.
• Implement the upper bound algorithm using an interior-point method with gradient-only updates, avoiding costly Hessian computations.
• Apply the proposed algorithms to a 4×4 state-space model of incompressible fluid flow and a synthetic academic example to validate performance and accuracy.
• Compare results with standard methods that replace repeated full-blocks with non-repeated ones, showing reduced conservatism.
• Use the structured singular value to analyze input-output gain across temporal frequencies, revealing physical instability characteristics.

Experimental results

Research questions

• RQ1 How can upper and lower bounds on the structured singular value (μ) be efficiently computed for systems with repeated complex full-block uncertainties?
• RQ2 What is the impact of approximating repeated complex full-blocks as non-repeated blocks on μ bounds and their physical interpretation in fluid flow systems?
• RQ3 Can gradient-based optimization methods improve computational efficiency in μ computation without sacrificing accuracy?
• RQ4 How does the proposed method reveal more accurate temporal behavior and instability mechanisms compared to existing approaches that ignore the repeated block structure?
• RQ5 What is the gap between the true μ and the convex upper bound (D-scaling) in the case of a single repeated complex full-block?

Key findings

• The proposed upper bound computation using gradient-based method of centers achieves higher computational efficiency than Hessian-based methods, enabling scalability to large systems.
• The generalized power iteration method successfully computes a tighter lower bound on μ for repeated complex full-blocks, outperforming methods that treat them as non-repeated blocks.
• At ω = 1.896, the μ lower bound from the non-repeated approximation was approximately 1.7 times larger than the true bound from the repeated structure, demonstrating significant conservatism.
• The global peak of the μ upper bound occurred at ω > 0 in the non-repeated case but at ω < 0 in the repeated case, indicating that ignoring the repeated structure can lead to incorrect conclusions about temporal instability behavior.
• The proposed algorithms yield less conservative bounds than existing methods, improving stability-margin estimates and enabling more accurate physical interpretation of instability mechanisms in convective systems.
• The results on the plane Couette flow model confirm that modeling the repeated block structure is essential for capturing the true input-output gain and instability characteristics of fluid flows.