Noise sensitivity transfer function

Question

Consider we have a second order system (plant) described by the following transfer function:

$$ P(s) = \frac{b_0}{s^2+a_1*s+a_2} $$

which is controller by a PD-controller:

$$ C(s) = K_p+K_ds $$

In order to study the impact that the external noise has to the system, I have come across the so called noise sensitivity function which is described by the below equation:

$$ N(s) = \frac{C(s)}{1+P(s)C(s)} $$

However, deriving $\ N(s) $ produces an improper transfer function with degree of numerator being $\ 3 $ and of denominator being $\ 2 $. This results, for example, to the imcapability of obtaining a step response in order to study the behaviour. Is the definition of the noise sensitivity function wrong or am I missing something ?

Answer 1

This is because your controller is also improper. A common way to correct this is to add a low-pass filter to the derivative (effectively making it a high-pass filter). It can be noted that the PD controller could then also be seen as a lead-lag filter.

However, if you are only interested in the step response you could also use that a step is the integral of an impulse. So you could add an integrator to the system, which raises the order of the denominator, and use impulse instead of step on the causal modified system.