Determine the scalar system-level PI gain region

Determine the scalar system-level robust PI gain region for the first-order non-affine single-input single-output uncertainty class, given that the common-quadratic sufficient region and the pointwise linear necessary region bracket but do not determine the exact region.

Background

For first-order non-affine systems with one controlled coordinate, the paper proves an exact PI gain-region characterization only for dimensions m≥2. The sufficient common-quadratic condition is the inequality k_i+Lk_p<bk_p^2 with positive gains, while the admissible scalar linear system yields the necessary condition bk_p>L.

Because pointwise asymptotic regulation is weaker than the uniform exponential regulation required in the system-level definition, the scalar linear outer region does not resolve the scalar problem. The paper explicitly leaves the scalar system-level PI region undetermined between these bounds.