On the convexity of QFT bounds and its relation to automatic loop-shaping

A difficult problem in quantitative feedback theory (QFT) is the design of a nominal-loop function. A promising direction for QFT loop-shaping involves convex optimization. However, the underlying QFT bounds are often non-convex sets requiring the bounds be approximated by convex sets to allow for convex optimization. We propose an automatic loop-shaping technique via linear programming. Specifically, we transform the open-loop QFT bounds into closed-loop QFT bounds to reduce design conservation due to approximation. We also present a sufficiency condition for convexity of the closed-loop QFT bounds.