Feedback System for Regulation

Regulation problem: Given plant $P$, design a 2-DOF controller $C=[C_1{C}_2]$ so that good performance in tracking and disturbance rejection is achieved.

The system can be expressed as:

$$U(s)=\left[\begin{array}{cc}{C}_1(s)& -C_2(s)\end{array}\right]\left[\begin{array}{c}R(s)\\ Y(s)\end{array}\right]$$

• The minus sign in front of $C_2$ suggests negative feedback.

The above feedback system for regulation is well posed if the feedback system for stabilization formed by $P$ and $C_2$ is well posed.

All the transfer functions are:

We now have a gang of 6, involving a gang of four formed by $P(s)$ and $C_2(s)$:

$$\frac{1}{1+P(s)C_2(s)},\frac{P(s)}{1+P(s)C(s)},\frac{C_2(s)}{1+P(s)C(s)},\frac{P(s)C_2(s)}{1+P(s)C(s)}$$

along with two additional ones involving $C_1(s)$:

$$\frac{C_1(s)}{1+P(s)C_2(s)},\frac{P(s)C_1(s)}{1+P(s)C_2(s)}$$