Controller recommended to reduce steady-state error

To reduce (or eliminate) steady-state error, an integral-type controller, usually a PI controller (or PID), is recommended. In frequency-domain design, the equivalent is a lag compensator.

Why integral action

The steady-state error of a unity feedback system depends on the system type and the error constants:

$$e_{ss}(\text{step})=\frac{1}{1+K_{pos}},\qquad e_{ss}(\text{ramp})=\frac{1}{K_v}$$

where $K_{pos}=\lim_{s\to0}G(s)$ is the position error constant and $K_v=\lim_{s\to0}sG(s)$ the velocity error constant.

A PI controller

$$G_c(s)=K_p+\frac{K_i}{s}=\frac{K_p(s+K_i/K_p)}{s}$$

adds a pole at the origin, so the system type increases by one:

Physically, the integral keeps growing while any error remains, so the controller output keeps changing until the error is exactly zero.

The zero at $-K_i/K_p$ is placed close to the origin so that the extra phase lag (and its effect on the transient response and stability) stays small.

Alternatives

• Increasing proportional gain reduces the error but cannot remove it, and makes the response oscillatory.
• Lag compensator $\frac{1+Ts}{1+\beta Ts}$ raises the low-frequency gain by $\beta$ ($K_v$ multiplied by β) without changing the transient much.

Example: $G(s)=\frac{10}{s+2}$ with unity-gain P control has $K_{pos}=5$ and step error $\frac{1}{1+5}=0.167$. With PI control the open loop has a pole at the origin and the step error becomes zero.