Fig. 3

Visualization of the impact of DC and lack thereof on the output using time-varying step-like changes in the reference input r(t) and disturbance d(t) in different combinations. Gaussian noise was added to the constant value of r(t) and d(t) to demonstrate that the system output remains identical independent of the value of s also for complex signals. y(t) and z(t) show the comparison of the step response when s is 0.25 and 1.5. As we have started y(t) and z(t) with a distance from the equilibrium point, it takes time to converge to the stable situation resulting in having some difference between but then the output y(t) remained identical–the difference (green dashed line) equals zero. A hallmark of the system is \(\mathbb {P}\)-invariant with regard to s