Append a free integrator to a Transfer Function

[Alw]

Homework Statement

PI Design

The transfer functions represent the dynamics of a typical commercial turbofan engine between fuel flow rate $$\Delta$$WF in pps and fan speed increment $$\Delta$$N1 in rpm and between fuel flow rate and high-pressure turbine outlet temperature increment $$\Delta$$T in degrees Rankine.

"Since zero steady-state error is desired, an integral control approach will be used. To simplify the process, append a free integrator to Eqs. 1 and 2 and regard the resulting transfer functions as design plants. Keep in mind that the integrator has already been included when designing your controllers."

Homework Equations

(1) $$\Delta$$N1(s) / ($$\Delta$$WF(s)) = 247.37(s+8.906) / ((s+2.855)(s+5.421))

(2) $$\Delta$$T(s) / ($$\Delta$$WF(s)) = 167.995(s+4.79)(s+2.075) / ((s+2.855)(s+5.421))

The Attempt at a Solution

At this point i really don't even know what a "free integrator" is, and searching my prof's notes and the internet hasn't really given me anything to work with. I'm hoping this is something of a brain fart, because this is only the first part of a very long assignment. Any help would be appreciated.

 

[KataKoniK]

Hello,

Thank you for your post. I understand your confusion and will do my best to clarify the concept of a "free integrator" and how it relates to the transfer functions provided.

A free integrator, also known as an "ideal integrator," is a mathematical tool used in control systems to eliminate steady-state error. It essentially performs the mathematical operation of integration, which is the summing of small changes over time, without introducing any delay or error. In a control system, this means that the output will continue to increase or decrease until it reaches the desired value, without any deviation or error.

In the context of this forum post, the suggestion is to append a free integrator to the transfer functions provided in equations (1) and (2). This means that the transfer functions should be modified to include a free integrator, resulting in a new transfer function that represents the dynamics of the system with the integral control approach.

To better understand this concept, let's take a look at equation (1) as an example:

\DeltaN1(s) / (\DeltaWF(s)) = 247.37(s+8.906) / ((s+2.855)(s+5.421))

To append a free integrator to this transfer function, we simply add an "s" term to the numerator, which represents the integration operation. This results in the following transfer function:

\DeltaN1(s) / (\DeltaWF(s)) = 247.37(s+8.906) / ((s+2.855)(s+5.421)(s))

This new transfer function can now be regarded as the design plant for the integral control approach.

I hope this explanation helps to clarify the concept of a free integrator and how it relates to the provided transfer functions. Please let me know if you have any further questions or need clarification on any other aspects of this assignment.

 

[Mene]

I would suggest that a "free integrator" refers to an integrator that is not constrained by any specific parameters or conditions. In other words, it is a pure integrator that can freely integrate any input without any limitations.

In the context of the transfer functions provided, appending a free integrator would mean adding an additional term in the form of 1/s to the equations. This would result in the following transfer functions:

(1) \DeltaN1(s) / (\DeltaWF(s)) = 247.37(s+8.906) / ((s+2.855)(s+5.421)(s))

(2) \DeltaT(s) / (\DeltaWF(s)) = 167.995(s+4.79)(s+2.075) / ((s+2.855)(s+5.421)(s))

The purpose of this free integrator is to ensure that there is zero steady-state error in the control system. It will continuously integrate the error between the desired output and the actual output, thus ensuring that the system reaches the desired output without any error.

However, it is important to note that the controllers designed for these transfer functions should already include an integrator. This means that the free integrator should not be included in the design process, but rather should be considered as a part of the design plant.

In summary, appending a free integrator to the transfer functions means adding an additional term of 1/s to the equations in order to eliminate steady-state error in the control system.