# Robust Stability: criterion for inverse multiplicative uncertainty

• Engineering
• Master1022
In summary: Therefore, in summary, the condition for the closed loop system to be stable when connected in feedback with controller K(s) is ||W(s) G(s) \frac{1}{1 + K(s)G(s)}||_{\infty} < 1.
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.

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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.

## What is robust stability?

Robust stability is a measure of a system's ability to maintain stability despite uncertainties or disturbances in its parameters or inputs.

## What is the criterion for inverse multiplicative uncertainty?

The criterion for inverse multiplicative uncertainty is a mathematical condition that must be satisfied for a system to be considered robustly stable. It involves analyzing the sensitivity of the system's transfer function to variations in its parameters.

## Why is robust stability important?

Robust stability is important because it ensures that a system will continue to perform correctly and reliably even in the presence of uncertainties or disturbances. This is crucial for systems that are used in critical applications, such as aerospace or medical devices.

## How is robust stability tested?

Robust stability is typically tested through mathematical analysis and simulations. This involves examining the system's transfer function and evaluating its stability margins, which indicate how much uncertainty the system can tolerate before becoming unstable.

## What are the benefits of achieving robust stability?

Achieving robust stability can lead to increased reliability and performance of a system. It can also reduce the need for frequent maintenance and adjustments, saving time and resources in the long run.

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