Uncertainty that leads to instability

I was studying multivariable feedback control and robust stability criterions so started to read documentation of robstub function.

But there is something i couldnt understand in the first example(Robust Stability Margin of Closed-Loop System).

Basically meaning of bounds that are greater than 1 is that there is no any combination of uncertanties that will lead to instability of the system.

But than again,by substituting worst case uncertainty to the nominal plant,we see that system is unstable by examining its poles. How is it possible? Im confused.

Answer 1

[Ashutosh Bajpai]

In the example you mentioned, the robstub function is used to compute the robust stability margin of a closed-loop system with uncertain parameters. The robust stability margin is a measure of how much the uncertain parameters can vary before the closed-loop system becomes unstable.

If the bounds on the uncertain parameters are greater than 1, it means that no combination of these uncertain parameters will lead to instability of the closed-loop system. This means that the robust stability margin is infinite, indicating that the closed-loop system is robustly stable.

However, even though the closed-loop system is robustly stable, it is still possible for the nominal plant (the plant without any uncertainty) to be unstable. This is because the robstub function only considers the worst-case uncertainty, and the worst-case uncertainty may not be the same as the actual uncertainty in the system.

For example, consider a closed-loop system with uncertain parameters that have bounds greater than 1. If the actual uncertainty in the system is such that the nominal plant is stable, then the closed-loop system will be stable as well. However, if the actual uncertainty is such that the nominal plant is unstable, then the robstub function will still report that the robust stability margin is infinite, even though the closed-loop system is actually unstable.

In other words, the robstub function only provides a conservative estimate of the robust stability margin, and it does not guarantee that the closed-loop system will always be stable. It is still possible for the closed-loop system to be unstable if the actual uncertainty in the system is different from the worst-case uncertainty considered by the robstub function.

I hope this helps clarify the behavior of the robstub function.