Investigation of the solutions optimal with respect to the degree of stability at proportional-plus-integral control

The solutions optimal with respect to the degree of stability are examined for PI control of dynamic systems governed by transfer functions with a zero power polynomial in the numerator. The paper shows that there exist 7 cases of maximum degree of stability attainment in control systems. Sufficient optimality conditions are formulated for each case. The hodographs of closed-loop control systems meeting either only necessary or only sufficient optimality conditions are built. The shape of optimal systems' hodographs demonstrate the geometric sense of the solutions obtained.