An (experimental) Julia package for computing the Structured Singular Value (SSV, also μ) of a (in general) complex matrix. In fact, just lower and upper bounds are computed as the computation of the exact value is known to be NP hard.
In order to explain what structured singular value is, we need to recall what classical singular value of a matrix is. In particular, we focus on the largest singular value σ₁(M), which plays the role of an operator norm for the matrix M.
Furthermore, we need to be able to specify a structure of a matrix. Loosely speaking, we just specify which elements in the matrix are zero and which a free to assume real or complex values. For some matrix Δ, we say that if it has the given structure, it belong to some set 𝚫, that is, Δ∈𝚫.
With the concept of a singular value and some way to characterize a structure of a matrix, we can define the (largest) structured singular value of a matrix M in the following way
μ(M) = 1/(min {σ₁(Δ): Δ∈𝚫, det(I-MΔ)=0}).
Clearly, if the matrix Δ has no structure, the structured singular value μ of the matrix M is equal to the the reciprocal value of the largest singular value σ₁(Δ) of some smallest (in the sense of σ₁) matrix Δ that makes the determinant of I-MΔ vanish. But this is exactly equal to the standard (largest) singular value σ₁(M). That is, in this unstructured case μ(M)=σ₁(M). In presence of some structure imposed on Δ, μ(M)≦σ₁(M).
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An anticipated use of this computation is in analysis of robustness of linear time invariant (LTI) dynamical systems. If such a system is modelled by a (matrix) transfer function M(s), structured singular value(s) are computed for a grid of frequencies ωₖ, k=1,…,N, at which the transfer function M(s) evaluates to M(jωₖ). This evaluation of the matrix M(s) at a grid of points along the imaginary axis and subsequent computation of the SSV is left for the user. In this package we only aim at computing the SSV for a given (constant) matrix.
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