First Order Control System

In summary, the control system tries to keep the water in a tank at a certain height by feedng in more water when the height is low and less water when the height is high. There is also an additional flow in from a separate pipe that is regulated. The question is to find the transfer function, time constant and steady state gain.
  • #1
pobatso
17
0
Hi all, got a Control question here, and I'm struggling with what I assume is a simple algebraic step. Thanks in advance!

Homework Statement


A closed loop control system governs the level of water in a tank (H(s)) to meet a target height (Hi(S)). The flow of water into the tank is controlled by a transducer that feeds the current level of the tank into a differencing junction that works out the error (H(s)-Hi(s)). The flow rate of water pumped in is proportional to this error, with gain K.

The flow out of the tank is also constrained by a linearized flow restrictor, with flow out equal to the height/constant (Qd=H(s)/R).

There is also an additional flow into the tank from a separate pipe, with flow rate Qd.

The question is to find the transfer function, time constant and steady state gain. I've attached a diagram.

Homework Equations



The Attempt at a Solution



So far I've gotten as far as the governing equation:
Qi + Qd - Qo = A.dH(t)/Dt
Laplace: Qi(s) + Qd(s) - Qo(s) = A.s.H(s)

Where Qi = Flow in
Qd = Additional disturbance flow
Qo = Flow out
A = XSection area of tank

Using the information about the individual components this goes to:

K.Hi(s) - K.H(s) + Qd(s) - H(s)/R = A.s.H(s)

The correct way to describe transfer function (Checked with answer booklet):

H(s)=(R.K.Hi + R.Qd(s)) / (R.k +R.A.s + 1)

But I can't get the hang of expressing it in a way that would allow me to get the specific time function, ie I can't arrange it into form H(s)/Hi(s)=u/(1+Ts) where u is the SS Gain and T is the time constant.

Any help with this step would be fab!

Regards
 

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  • #2
You have the right equation, so why can't you form H/Hi?

Is you problem the fact that you have a transform of the form a/(bs + c + 1)? Surely you know how to change that to the form d/(es+1)? High school algebra! :-)
 
  • #3
Yes mate, that's the one. I know that I'm being a bit of a moron with this one, but its just left me. Watching youtube vids as well to help bring it back, but I think just having one example from someone else with one of these that is related to this topic area as well would be a great help, even if it is stuff I once covered in what feels like a very long time ago :)
 
  • #4
pobatso said:
Yes mate, that's the one. I know that I'm being a bit of a moron with this one, but its just left me. Watching youtube vids as well to help bring it back, but I think just having one example from someone else with one of these that is related to this topic area as well would be a great help, even if it is stuff I once covered in what feels like a very long time ago :)

Righto! Let's take the general expression a/(bs+c+1). Now, divide numerator and denominator by c+1. What do you get?
 
  • #5
Think that's it! Know its pretty stupid, but I can't get any sense out of the division. Not a good sign to get to 2nd year Uni without learning algebraic division :/ Thanks again!
 
  • #6
pobatso said:
Think that's it! Know its pretty stupid, but I can't get any sense out of the division. Not a good sign to get to 2nd year Uni without learning algebraic division :/ Thanks again!

How about a/(bs + c+1) = d/(es+1) where

d = a/(c+1)
e = b/(c+1)

?
 
  • #7
Yeah, I can't do it. Have been trying for a bit now, wasting far too much time on something this simple!
 
  • #8
Have gotten something like:

(R^2(K^2.Hi + K.Qd(s)) - R(K.Hi + Qd(s))) / (R^2(A.K.s + K^2) - R.A.s -1
 
  • #9
pobatso said:
Using the information about the individual components this goes to:

K.Hi(s) - K.H(s) + Qd(s) - H(s)/R = A.s.H(s)

Move all the H(s) terms to the left-hand side of this equation (which you yourself correctly derived). Then form H(s)/Hi(s). You must realize that there are really two transfer functions: H(s)/Hi(s) and H(s)/Qd(s). The problem ask you for the former only, so disregard the latter.

Then reduce the denominator of H(s)/Hi(s) to es+1 as I've shown you.

Don't give up!
 

What is a First Order Control System?

A First Order Control System is a type of control system in which the output of the system depends on the input and the past behavior of the system. It consists of a feedback loop that compares the output of the system with the desired output and makes adjustments accordingly.

What are the components of a First Order Control System?

The components of a First Order Control System include a plant, a controller, and a feedback loop. The plant is the system being controlled, the controller is responsible for comparing the output with the desired output, and the feedback loop is used to make adjustments to the system.

What are the advantages of using a First Order Control System?

Some advantages of using a First Order Control System include simplicity, low cost, and easy implementation. It is also suitable for systems with simple dynamics and can be easily modified for different control objectives.

What are the limitations of a First Order Control System?

One of the limitations of a First Order Control System is its inability to accurately control systems with complex dynamics. It also has a slow response time and is not suitable for systems that require precise control.

How is stability achieved in a First Order Control System?

Stability in a First Order Control System is achieved by ensuring that the feedback loop is properly designed and the controller is able to accurately adjust the system according to the desired output. It is also important to choose the correct parameters for the controller and the plant to prevent oscillations or instability.

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