Determine the scalar system-level PID gain region

Determine the scalar system-level robust PID gain region for the second-order non-affine single-input single-output uncertainty class, given that the common-quadratic region and the linear-subclass region provide strict inner and outer bounds but do not determine the exact region.

Background

For second-order non-affine systems with one controlled coordinate, the paper exactly characterizes two auxiliary gain regions: the common-quadratic PID region, described by a four-vertex LMI condition, and the linear-subclass PID region, described by scalar Routh–Hurwitz inequalities. Unlike the multivariable case, the paper proves that these regions are strictly separated.

The system-level robust PID region lies between these two endpoints. The authors explicitly state that the endpoints bracket the scalar region without determining it, leaving unresolved which gains guarantee uniform exponential regulation for every admissible scalar nonlinear system. Resolving this problem would close the endpoint gap and yield a complete scalar PID gain-region characterization.