# PID Control: Controller of the type $\dfrac{1}{s+k}$

In PID control an integrator action is represented by $\dfrac{1}{s}$

Does anyone know what would be the effect of using $\dfrac{1}{s+k}$ instead of $\dfrac{1}{s}$, where $k$ is a constant? What would this controller be called?

Expanding on Ben's answer, let's presume we construct a PI with the integral replaced with a single pole low-pass filter. The TF for our controller is $$C(s) = K_p + \frac{K_f}{s+k} = \frac{K_ps+kK_p+K_f}{s+k},$$ so you see you've essentially turned your PI into a Lead-Lag compensator with pole/zero combo $$p = -k,\ \ z = -\left(k+\frac{K_f}{K_p}\right).$$ The main difference between PID controllers versus Lead-lag compensators is whether or not you want to regulate the time (PID) or frequency (lead-lag) domain responses.

• interesting, do you know the general form of HPF and if they are similar? – Bajie Jun 7 '16 at 4:21

"Single-pole lowpass filter", typically.

The difference is that the lowpass filter has finite gain (1/k) at low frequencies, while the integrator theoretically has infinite gain at DC.

One reason for using an integrator in the forward path of a closed-loop system (if integration is not present, inherently) is that it gives unity 'DC gain' or 'steady-state gain'. That is, the controller's I term forces the system steady-state error signal to zero for a step input. And, of equal importance, the I term also gives complete rejection of steady-state disturbances. A lead-lag controller or a simple first order lag, $\large\frac{1}{s+k}$, has neither of these attributes.

For example, a velocity control loop, where the controlled device is a DC motor, does not have an inherent integration (apart from the trivial zero-friction/pure inertial load case). So the controller would require integral action in order to ensure that there is no offset between required and actual velocities for a steady input command, and that any external constant disturbance will be reduced to zero.

However, the same motor in a position control loop would not require controller integral action in order to provide those desirable features, as there is an inherent pure integration built-in to the motor: displacement is the integral of velocity.