Measures to track the reference input

A permanent mismatch between the reference and the output is a steady-state error $e_{ss}$. It can be removed by using a closed-loop system with integral action (PI or PID controller), i.e. by raising the type number of the loop.

Justification

For unity feedback, $E(s) = \dfrac{R(s)}{1+G(s)}$ and

$$e_{ss} = \lim_{s\to0} sE(s)$$

For a unit step, $e_{ss} = \dfrac{1}{1+K_p}$, where the position error constant is $K_p = \lim_{s\to0}G(s)$.

• Type-0 system, e.g. $G(s) = \dfrac{K}{\tau s+1}$: $K_p = K$, so $e_{ss} = \dfrac{1}{1+K} \ne 0$. The mismatch is permanent.
• Raising $K$ reduces the error (e.g. $K = 99$ gives 1%) but never makes it zero, and high gain can make higher-order systems oscillatory or unstable.
• Add an integrator (PI controller $G_c(s) = K_c + K_i/s$): the loop becomes type 1 and the position constant $K_p \to \infty$, so

$$e_{ss} = \frac{1}{1+\infty} = 0$$

The integrator keeps changing its output as long as any error exists, so the output is forced to follow the reference exactly.

Other measures

• If the system is open loop, convert it to closed loop with a sensor so that the error is measured and corrected.
• For ramp references, use a type-2 loop (double integration) or a lag compensator to raise $K_v$.
• Calibrate the sensor and feedback element: an error in $H$ appears directly in the output, since $C/R \approx 1/H$.