Robust Stability: criterion for inverse multiplicative uncertainty

[Master1022]

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.

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

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 $Y(s)$ to the system $G(s)$, controller $K(s)$, 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 $G(s)$ 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.

 

[Valkarie]

Yes, the approach you have outlined is correct. To draw the block diagram, you should include the inverse multiplicative uncertainty (ΔWG) in the feedback loop, as shown below:[Input] -> [G(s)] -> [ΔWG] -> [K(s)G(s)] -> [Output]The small gain theorem states that for a system to be stable, the gain around any closed loop must be less than 1. In this case, the gain is given by ||W(s) G(s) \frac{1}{1 + K(s)G(s)}||_{\infty}, and so the condition for stability is ||W(s) G(s) \frac{1}{1 + K(s)G(s)}||_{\infty} < 1.