- #1

- 611

- 116

- Homework Statement
- Using the small gain theorem, derive the condition for multiplicative inverse uncertainty

- Relevant Equations
- Block diagram

Hi,

I have a question that I am quite confused about. Please note this is at the undergraduate level.

and the fact that the system is connected in feedback with controller [itex] K(s) [/itex], use the small gain theorem to derive the condition for the closed loop system to be stable: [tex] ||W(s) G(s) \cdot \frac{1}{1 + K(s)G(s)} ||_{\infty} < 1 [/tex]

My problems are as follows:

1. I am not 100% confident about what the method is to do this

2. I am not sure how to draw this as a block diagram

For q1, does the following method sound correct?

- Draw the block diagram

- Relate the output [itex] Y(s) [/itex] to the system [itex] G(s) [/itex], controller [itex] K(s) [/itex], and uncertainties (I am not quite sure how to properly do this step)

- Then use the small gain theorem to get the above expression

For q2, I have looked around on the internet, but the examples are ones where there isn't a [itex] G(s) [/itex] term in the denominator. I am especially confused about how to include an inverse in a block diagram. Does the following look like the correct start?

Thanks in advance for any help.

I have a question that I am quite confused about. Please note this is at the undergraduate level.

**Question:**Given the transfer function with inverse multiplicative uncertainty [tex] \bar G (s) = \frac{G(s)}{1+\Delta \cdot W(s) \cdot G(s)} [/tex]and the fact that the system is connected in feedback with controller [itex] K(s) [/itex], use the small gain theorem to derive the condition for the closed loop system to be stable: [tex] ||W(s) G(s) \cdot \frac{1}{1 + K(s)G(s)} ||_{\infty} < 1 [/tex]

**Attempt:**My problems are as follows:

1. I am not 100% confident about what the method is to do this

2. I am not sure how to draw this as a block diagram

For q1, does the following method sound correct?

- Draw the block diagram

- Relate the output [itex] Y(s) [/itex] to the system [itex] G(s) [/itex], controller [itex] K(s) [/itex], and uncertainties (I am not quite sure how to properly do this step)

- Then use the small gain theorem to get the above expression

For q2, I have looked around on the internet, but the examples are ones where there isn't a [itex] G(s) [/itex] term in the denominator. I am especially confused about how to include an inverse in a block diagram. Does the following look like the correct start?

Thanks in advance for any help.