02.1 Basics of Closed Loop

What is an Open-Loop Plant?

An open-loop plant $P(s)$ is simply the transfer function of the system you want to control, without any feedback. It describes how the system naturally responds to inputs.

In your case: \(P(s) = \frac{\omega^2}{s^2 + 2\zeta\omega s + \omega^2}\)

This is a standard second-order system (like a mass-spring-damper), where:

• $\omega$ is the natural frequency
• $\zeta$ is the damping ratio

Unity Negative Feedback Configuration

Here’s the standard block diagram:

        +        +------+       +------+
r(s) --->(sum)-->| C(s) |------>| P(s) |----+----> y(s)
        - ^      +------+       +------+    |
          |                              |
          +------------------------------+

Where:

• $r(s)$ = reference input
• $y(s)$ = output
• $C(s)$ = compensator (what we’re solving for)
• $P(s)$ = plant

Deriving the Closed-Loop Transfer Function

The loop transfer function is: $L(s) = C(s) \cdot P(s)$

For unity negative feedback, the closed-loop transfer function is:

Why? Let’s derive it from the block diagram:

1. At the summing junction: $e(s) = r(s) - y(s)$
2. Through the compensator and plant: $y(s) = C(s) \cdot P(s) \cdot e(s)$
3. Substituting: $y(s) = C(s)P(s)[r(s) - y(s)]$
4. Expanding: $y(s) = C(s)P(s)r(s) - C(s)P(s)y(s)$
5. Collecting $y(s)$ terms: $y(s)[1 + C(s)P(s)] = C(s)P(s)r(s)$
6. Therefore: $\boxed{T(s) = \frac{y(s)}{r(s)} = \frac{C(s)P(s)}{1 + C(s)P(s)}}$

The reason it is called unity negative feedback is because we are subtracting the output from the reference. For instance, at step 3:

$e(s)=r(s)-1\times y(s)$, where gain = 1. That is the “unity” part, and the subtraction is the “negative feedback” part.