# Stability of an open loop controller

• icesalmon
In summary, the conversation discusses the controller and plant equations for an open loop system and how the transfer function can be expressed as a ratio of polynomials. It is mentioned that the roots of the characteristic equation of this transfer function cannot have any roots in the right half plane (RHP). The conversation also touches on the idea that attempting to cancel unstable roots of the plant using the controller may not be effective due to modeling uncertainties. The conversation concludes by stating that the only way to stabilize the system is to remove the pole from the RHP.
icesalmon
Given the following Controller equation Gol(s) and Plant equation Dol(s) for an open loop system the transfer function can be expressed as a ratio of polynomials where:
Gol(s) = b(s)/a(s)and Dol = c(s)/d(s).

For the open loop system the transfer function Tol = Gol(s)Dol(s) = b(s)c(s)/a(s)d(s), the roots of the characteristic equation (the denominator) of this transfer function cannot have any roots in the RHP.

What I'm confused about is that my notes say "An attempt to cancel unstable roots of a(s) of the plant by using c(s) of the controller will be useless. Although, cancelled, physically the unstable pole still remains. The slightest modelling uncertainty will cause the output to diverge"

This doesn't make sense to me as it seems to be saying that the mathematical modelling of the physical plant doesn't actually fully impact what happens physically. How is this possible? I would think that if you design a controller based off of an equation that counteracts the instability of the other controller, how can this same thing not happen when you physically build the thing?

Averagesupernova
icesalmon said:
The slightest modelling uncertainty will cause the output to diverge"
icesalmon said:
I would think that if you design a controller based off of an equation that counteracts the instability of the other controller,
It is saying that in real life your counteraction can not be perfect, the modeling uncertainty becomes a fatal flaw.

icesalmon, berkeman and Tom.G
This makes more sense to me now, thank you!

A pole on the RHP is a vertical asymptote. It represents an exponentially increasing wave.
Placing a zero in the exact same position is more difficult than eliminating the pole from the RHP.
The only sure way to stabilise the system is to remove the pole from the RHP.

icesalmon

## 1. What is an open loop controller?

An open loop controller is a type of control system in which the output is not affected by the input. It does not use feedback to make adjustments and relies on a predetermined set of instructions to control the system.

## 2. How does an open loop controller work?

An open loop controller works by taking an input signal, processing it through a predetermined set of instructions, and producing an output without any feedback or correction. The output is solely based on the input and the programmed instructions.

## 3. What is the stability of an open loop controller?

The stability of an open loop controller refers to its ability to maintain a steady and predictable output despite changes in the input or external disturbances. A stable open loop controller will produce a consistent output, while an unstable one may produce erratic or unpredictable results.

## 4. How is the stability of an open loop controller determined?

The stability of an open loop controller is determined by analyzing its transfer function, which describes the relationship between the input and output signals. A stable transfer function will have all of its poles (roots of the denominator) in the left half of the complex plane, while an unstable one will have at least one pole in the right half.

## 5. What are some factors that can affect the stability of an open loop controller?

Some factors that can affect the stability of an open loop controller include changes in the system parameters, external disturbances, and nonlinearities in the system. Additionally, the choice of control algorithm and the accuracy of the measurements can also impact the stability of the controller.

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