Unity Feedback Systems

In a unity feedback system, is a control system where the output is fed back directly to be compared with the input without any additional processing or modification — meaning the feedback transfer function is 1.

Given a plant $P(s)$ and controller $P(s)$, the closed-loop transfer function is

$$T(s)=\frac{C(s)P(s)}{1+C(s)P(s)}$$

The closed-loop characteristic polynomial is the denominator set to zero:

$$1+C(s)P(s)=0$$

In the case of a unity feedback system, we have $C(s)=1$.

In this structure, instead of taking the information of the reference and the measurement independently, the controller is driven by the difference between the reference and the measurement. Such a structure is simpler than the 2DOF structure since the controller is a SISO system and is actually a special case of the 2DOF structure by setting

$$\mathbf{C}\left[\begin{array}{c}r\\ -y\end{array}\right]=C(r-y)$$

Consider the unity feedback system with

$$P(s)=\frac{10}{(s+1)(s+2)}$$

Let $C(s)$ be a proportional-integral (PI) controller, i.e.,

$$C(s)=K_P+\frac{K_I}{s}$$

Find the ranges of $K_P$ and $K_I$ so that the closed-loop system is stable.

Since $C(s)$ is proper and $P(s)$ is strictly proper, we only need to check the stability of the closed-loop characteristic polynomial:

$$d(s)=s^3+3s^2+(2+10K_P)s+10K_I$$

Construct the Routh table:

By the Routh criterion, we need

$$\frac{3(2+10K_P)-10K_I}{3}>0,10K_I>0⟺3+15K_P-5K_I>0,K_I>0$$

which gives us a stability region:

Unity vs. General Feedback

In general, a closed-loop transfer function has the form

$$T(s)=\frac{G(s)}{1+G(s)H(s)}$$

where $G(s)$ is the forward path transfer function and $H(s)$ is the feedback path. We also called $G(s)$ the open-loop transfer function.

In the unity feedback special case, we have $H(s)=1$, leading to

$$T(s)=\frac{G(s)}{1+G(s)}$$

In the above, we broke $G(s)$ into two parts,

$$T(s)=\frac{C(s)P(s)}{1+C(s)P(s)}$$

where $C(s)$ is the controller and $G(s)$ is the plant.

Relevant Examples