PI controller and its effect on time response

A PI (proportional–integral) controller produces an output proportional to the error plus the integral of the error:

$$u(t)=K_pe(t)+K_i\int_0^te(t)\,dt,\qquad G_c(s)=K_p+\frac{K_i}{s}=\frac{K_p\left(s+\frac{K_i}{K_p}\right)}{s}$$

It adds a pole at the origin and a zero at $s=-K_i/K_p$ to the open-loop transfer function.

          +-->[ Kp ]----+
 e(t) ----|             (+)--> u(t)
          +-->[Ki/s]----+

Effect on time response

Consider a plant $G(s)=\dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ with PI control. The open loop becomes

$$G_c(s)G(s)=\frac{\omega_n^2(K_ps+K_i)}{s^2(s+2\zeta\omega_n)}$$

1. System type increases by one (type 1 to type 2). Step error stays zero and the ramp steady-state error becomes zero ($K_v=\infty$). This is the main benefit: steady-state accuracy improves.
2. Order increases (2nd to 3rd order). The characteristic equation $s^3+2\zeta\omega_ns^2+K_p\omega_n^2s+K_i\omega_n^2=0$ is stable only if $2\zeta\omega_nK_p>K_i$; too large $K_i$ causes instability.
3. Transient response: the integrator adds phase lag, so damping decreases. Overshoot increases and settling time becomes longer; rise time usually decreases a little.
4. The PI controller acts as a low-pass filter (like a lag compensator), so it reduces high-frequency noise.

Example: a speed-control loop with only P control settles with a small offset; adding the integral term removes the offset, but the speed may overshoot more before settling.