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It's well-known that a control system with integral control can track a constant reference signal (command) with zero steady-state error.

What is needed to generate a controller that can track a ramping reference signal (i.e. one that changes at a constant rate) with zero steady-state error?

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Consider a controller $C(s)$ for plant $G(s)$ and feedback $H(s)$.

The closed loop is

$$\frac{Y(s)}{U(s)}=\frac{CG}{1+CGH}$$

The ramp is $U(s)=\frac1{s^2}$

Thus the error is

$$E=Y(s)-\frac{1}{s^2}=(\frac{CG}{1+CGH}-1)\frac1{s^2}$$

According to the final value theorem,

$$\lim_{t\to \infty} e(t)=\lim_{s\to 0} sE(s)=\lim_{s\to 0} (\frac{CG-1-CGH}{1+CGH} \times \frac1{s})=0$$

For a known case, you can find out how many $s$ in the denominator of $C$ will reduce the steady state error to zero.

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  • $\begingroup$ Feedforward might also be a solution (so that the feedback controller only has to attenuate disturbances). $\endgroup$
    – fibonatic
    Jan 19 '18 at 12:23
  • $\begingroup$ @fibonatic, True, this can include $H=0$. $\endgroup$
    – Arash
    Jan 20 '18 at 3:19

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