1 of 23

��Chapter 17�Stability and Robustness Margins

2 of 23

General Feedback Loop

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

2

3 of 23

Motivation

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

3

4 of 23

Motivation

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

4

The purpose of this chapter is to understand how phase influences stability

5 of 23

Definition: Crossover Frequency

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

5

6 of 23

Stability and Phase

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

6

7 of 23

Stability and Phase

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

7

Unstable Condition:

Phase = -180 at crossover

8 of 23

Phase and Gain Margin

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

8

9 of 23

Phase and Gain Margin in the Complex Plane

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

9

10 of 23

Closed Loop System and PM/GM

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

10

The {\bf open-loop} transfer function is

\[

P(s)C(s)

\]

�

The {\bf closed-loop} transfer function for tracking is

\[

\frac{P(s)C(s)}{1+P(s)C(s)}

\]

�

\noindent {\bf Observations:}

\begin{align*}

|P(j\omega)C(j\omega)|\gg 1 &\implies \left|\frac{P(j\omega)C(j\omega)}{1+P(j\omega)C(j\omega)}\right|\approx 1 \\

|P(j\omega)C(j\omega)|\ll 1 &\implies \left|\frac{P(j\omega)C(j\omega)}{1+P(j\omega)C(j\omega)}\right|\approx |P(j\omega)C(j\omega)| \\

|P(j\omega_{co})C(j\omega_{co})|=1 \text{~and PM-large} &\implies \left|\frac{P(j\omega_{co})C(j\omega_{co})}{1+P(j\omega_{co})C(j\omega_{co})}\right| < 1 \\

|P(j\omega_{co})C(j\omega_{co})|=1 \text{~and PM-small} &\implies \left|\frac{P(j\omega_{co})C(j\omega_{co})}{1+P(j\omega_{co})C(j\omega_{co})}\right| > 1 \\

|P(j\omega_{co})C(j\omega_{co})|=1 \text{~and PM}\approx 60^\circ &\implies \left|\frac{P(j\omega_{co})C(j\omega_{co})}{1+P(j\omega_{co})C(j\omega_{co})}\right| = -3 dB

\end{align*}

�

11 of 23

Closed Loop System and PM/GM

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

11

12 of 23

Closed Loop System and PM/GM

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

12

13 of 23

Closed Loop System and PM/GM

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

13

14 of 23

Closed Loop System and PM/GM

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

14

15 of 23

Closed Loop System and PM/GM

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

15

16 of 23

Bode Phase-Gain Relationship

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

16

17 of 23

Bode Phase-Gain Relationship

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

17

18 of 23

Ideal Loopshape

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

18

19 of 23

Design Study A

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

19

https://github.com/randybeard/controlbook_public/blob/master/_A_arm/python/hw17.py

20 of 23

Design Study B

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

20

Pendulum inner loop with PD control

https://github.com/randybeard/controlbook_public/blob/master/_B_pendulum/python/hw17.py

21 of 23

Design Study B

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

21

Pendulum outer loop with PID control

https://github.com/randybeard/controlbook_public/blob/master/_B_pendulum/python/hw17.py

22 of 23

Design Study C

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

22

Satellite inner loop with PD control

https://github.com/randybeard/controlbook_public/blob/master/_C_satellite/python/hw17.py

23 of 23

Design Study C

Introduction to Feedback Control, Beard, McLain, Peterson, Killpack

23

Satellite outer loop with PID control

https://github.com/randybeard/controlbook_public/blob/master/_C_satellite/python/hw17.py