Proportional Feedback

In feedback control systems, proportional feedback is a simple method to correct the error between a reference input and system output.

• $R$ is the reference input
• $E$ is the error, calculated as $E=R-Y$
• $K_P$ is the proportional gain
• $U$ is the control input to the plant, given by $U=K_{P}E$
• The plant (system to be controlled) has a transfer function $\frac{1}{s^2-1}$
• $Y$ is the system output.

We want to find a value of $K_P$ is that stabilizes the closed-loop system. The closed-loop transfer function $\frac{Y}{R}$ is defined as:

$$\frac{Y}{R}=\frac{K_P\cdot \frac{1}{s^2-1}}{1+K_P\cdot \frac{1}{s^2-1}}=\frac{K_P}{s^2-1+K_P}$$

To check the system’s stability, we can examine the characteristic polynomial in the denominator:

$$s^2+(K_P-1)=0$$

Note that we have a coefficient of zero for the $s$ term. For a system to be stable, a necessary condition (but not sufficient) is that all power of $s$ must have nonzero positive coefficients. Hence, the system cannot be stabilized by adjusting $K_P$; the system is not stable for any value of $K_P$.