Closed Loop Control Systems

Systems

Systems can eventually be simplified so we can analyze them easily.

$x (s) \to t (s) \to y (s)$

Where we can analyze $t (s)$ with inverse Laplace transforms.

Signals

Input

Reference (generally a step or ramp): $r (t), R (s)$

Perturbations

Disturbance: $w (t), W (s)$
Sensor Noise: $v (t), V (s)$

Output

Output response: $y (t), Y (s)$

We always want $y (t)$ .

Analysis

Is the plant $G (s)$ stable?
Where are the poles/zeros? $G (s) = \frac{b ( s )}{a ( s )}$ where the poles are roots of $a (s) = 0$ and zeros are roots of $b (s) = 0$
What does the impulse response, step response, and ramp response look like?
Is the time-domain behavior acceptable?

Reviewid1((Some text))

Open Loop Topology

In theory, if we don’t like the behavior of the Plant, we can design the controller to replace the poles:

$\to Dc (s) \to G (s) \to$

This is an open loop with no feedback:

G (s) = \frac{b ( s )}{a ( s )} Dc (s) = \frac{c ( s )}{d ( s )}

The overall transfer function becomes:

T (s) = Dc (s) G (s) T (s) = \frac{c ( s )}{d ( s )} \times \frac{b ( s )}{a ( s )}

If $c (s) = a (s)$ , we can cancel the poles of the Plant and replace them with $d (s)$ . The plant must be stable for us to do this. Pole cancellation does not work on unstable plants.

Closed Loop Topology

flowchart LR
id1((Σ)) --> id2("Dc(s)")
id2 --> id3("G(s)")

If the output

Example of Closed Loop Inspection

G (s) = \frac{1}{s ^{2} + 3 s - 4} = \frac{b ( s )}{a ( s )} a (s) = s^{2} + 3 s - 4 = a_{2} s^{2} + a_{1} s + a_{0} Stable if a < 0 Do quadratic equation/factor to find poles s = - 4, s = + 1

This must be unstable because there is a pole in the Ride-hand plane/

Can the plant be made stable?

Move the unstable pole from the RHP to the LHP. How do we do this? feedback

In a PID controller, $Dc (s) = k$ , where $k$ is a constant gain.

Note that $\to k \to G (s) \to$ is forward gain, which is equivalent to $\to K G (s) \to$

The $H (s)$ is the feedback gain.

When $k = 1$ , we call it unity gain.

Notes

Explorer