Proportional-Integral-Derivative Control

The PID controller is a classic controller where we combine proportional feedback and derivative feedback with an integral feedback term. This results in feedback of the form:

We can write:

$$U=\left(K_P+K_{D}s+\frac{K_I}{s}\right)E$$

where $E=R-Y$ is the error between the reference input and the system output.

We have:

$$\begin{array}{rl}Y& =(U+W)\cdot \frac{1}{s^2-1}\\ (s^2-1)Y& =U+W\\ (s^2-1)Y& =\left(K_P+K_{D}s+\frac{K_I}{s}\right)(R-Y)+W\\ \left[s^2-1+K_P+K_{D}s+\frac{K_I}{s}\right]Y& =\left(K_P+K_{D}s+\frac{K_I}{s}\right)R+W\\ [s^3-s+K_{P}s+K_D{s}^2+K_I]Y& =(K_{P}s+K_D{s}^2+K_I)R+Ws\end{array}$$

Thus, we can write the transfer function as:

$$Y=\frac{K_D{s}^2+K_{P}s+K_I}{s^3+K_D{s}^2+(K_P-1)s+K_I}R+\frac{s}{s^3+K_D{s}^2+(K_P-1)+K_I}W$$

Stability

Based on the transfer function, recall that a necessary (but not sufficient) condition is that the coefficients of the polynomial need to be positive, which means we need:

• $K_D>0$
• $K_P>1$
• $K_I>0$

We also need $K_D(K_P-1)>K_I$ (Routh-Hurwitz Criterion for 3rd-order).

Note that we can assign coefficients arbitrarily by choosing $K_P,K_I,K_D$.

Reference Tracking

The DC gain of the transfer function between the reference input $R$ and output $Y$ can be calculated as

$$\text{DC gain}(R\to Y)=\frac{K_D{s}^2+K_{P}s+K_I}{s^3+(K_P-1)s+K_D{s}^2+K_I}{|}_{s=0}=1$$

So, with the addition of integral feedback we remove the limitation of PD control and achieve perfect tracking.

Disturbance Rejection

The DC gain of the transfer function between the disturbance input $W$ and output $Y$ can be calculated as

$$\text{DC gain}(W\to Y)=\frac{s}{s^3+(K_P-1)s+K_D{s}^2+K_I}{|}_{s=0}=0$$

So, integral gain also gives complete attenuation of constant disturbances!