Disturbance in a proportional control system

[Jason-Li]

Homework Statement

The proportional control system of FIGURE 3(a) has an input, Q1, of 10 units. The uncontrolled input, Q2, has a value of 50 units, prior to a step change down to 40 units. The result of this disturbance upon the output, Qo, is shown in FIGURE 3(b).

Calculate the change in offset in the output produced by the step change.

Relevant Equations

Control Systems Equations

Hello, I found an old thread related to this question but it doesn't look correct to me so was looking for some guidance.

I have equated Qo = (Q2+Q1*G) / (1+H*G)

I did this by rearranging Qo=Q2+G(Q1-Q0*H)

I then went on to the calculations and found before step disturbance:
Qo= (50+10*9)/(1+0.1*9) = 73.68
And After:
Qo=(40+10*9)/(1+0.1*9) = 68.42
Giving an offset of 5.26

My issue is that if you look at figure 3b the Qo - time graph 73.68 and 68.42 don't seem to line up to where the disturbance starts and where it resettles at?

 

[DaveE]

Your calculation looks good to me. You may be trying to get more detail from the plot than was intended by the instructor. For example, why didn't he label "0" or the graduations on that plot if he wanted you to measure it. Actually I don't really see any point in including that plot for this question.

 

[alan123hk]

I believe the equations and calculation results you derived are correct. I use LTspice to simulate this control system, and it seems that the simulation results also match your calculation result.

 

[DaveE]

Also, for future reference, notice that you really didn't need to include the Θ₁ term in your calculation. It is a constant input operated on by constant gains to effect the output. So, for any question that addresses perturbations in the output from other sources, you can ignore the constant inputs. They will cancel when you subtract "before" from "after". It isn't a big deal for this problem, but later it will help you focus on the important parts of feedback systems like this.

This is also a good example of the single most powerful analysis trick for linear systems: superposition. The output for multiple input linear systems can be found by adding up the individual contributions from each input with the other inputs set to zero. You'll learn more about this later in your studies.

 

[Jason-Li]

DaveE & alan123hk thank you very much for the input that helps greatly!

alan123hk I will check out that LTspice and have a play about.

DaveE I see that makes sense as I am doing a difference with constants apparent. Perfect thank you both very much!

 

[Jason-Li]

Hi DaveE & alan123hk got two more parts of the question that I am a little unsure of.

(b) Draw a modified block diagram to show how the offset could be minimised by the inclusion of another control action. Also, show by means of a sketch how the modification might be expected to affect the output response.
(c) Show, by drawing a modified block diagram, how the magnitude of the disturbance could be minimised by the inclusion of a third type of control action.

So for (b) I have done the following Block diagram, but for the sketch should it be like the first one or more like the second one with higher frequency of oscillation, (I will draw them neater once I know which is better!)

and for (c) I have drawn the following block diagram, our learning materials present no way of representing the differentiator like the integrator so I have just drawn a block and wrote differentiator in it... Also I need to add an addition sign where the differentiator enters the comparator.

If you could cast your eye over this it would be greatly appreciated! Thanks again for all the help.

 

[DaveE]

You'll want to reconsider what data the differentiator uses to modify the output. You have drawn it so that that section changes the output based on only the output.

As far as drawing style, differentiation is the inverse of integration. So, you could redraw that part as you did at the top but with "K_d" and "d/dt". This isn't a substantial difference IMO; "Differentiator" works too, we know what you mean.

 

[Jason-Li]

You'll want to reconsider what data the differentiator uses to modify the output. You have drawn it so that that section changes the output based on only the output.

As far as drawing style, differentiation is the inverse of integration. So, you could redraw that part as you did at the top but with "K_d" and "d/dt". This isn't a substantial difference IMO; "Differentiator" works too, we know what you mean.

Yeah I have seen examples online that use the same node as the integrator like below:

However in my learning materials this is shown? Doesn't look correct...

Also does the graph sketches I drew look okay?

 

[DaveE]

Yes, your top drawing is the standard way of doing this. But it's important to really understand why this is better than your first attempt. What do you really want your controller to do? Minimize error and respond to the input. Think of a cruise control in your car. Suppose you are going 55mph and then ask for 60 mph, isn't that mostly the same as going 60mph and then asking for 65mph? It's the 5mph error that is the most important thing to correct for. Your controller probably doesn't really need to know exactly what speed it's going if it can reliably respond correctly to the error.

- The differentiator's job is to respond to quick changes in the error and rapidly adjust the output. This is often the least important term and is frequently left out. It is often the most difficult to use correctly.
- The integrator's job is to only respond to the errors that are present for a long time and slowly adjust the output to eventually get it to the right place. This term is essential IF you want your system to actually get to zero error, and is really easy to use in practice.
- The proportional term is to deal with the in-between response, not too fast, not too slow. It turns out it is important when you start to deal with the stability and frequency response characteristics of these systems.

Think about when you are driving your car. You are the controller, monitoring your speedometer and trying to get your car to go the speed you want. When you step on the gas quickly anticipating a need for more power, you are acting like a differentiator (like maybe when you start to go up a steep hill). When you guess how much to push the gas part way down because you see your speed isn't right you are acting like a proportional controller. When you slowly make small adjustments to eventually get your speed perfect, you are acting like an integral controller.

However, there are lots of times that we do just feed back the output to to adjust the input in feedback systems. This was, in fact, the genesis of feedback in electronic systems. The telephone company needed amplifiers that wouldn't change their gain with perturbations like temperature, age, parts replacement, etc. The only information they needed to do that was how the amplifier was working, the inputs didn't matter. I would call this local feedback and isn't really the subject at hand, control systems. This type of feedback is undoubtable buried in the box labeled "amplifier", "G", or "H".

Sorry, I just don't understand the context of the second diagram. It's not standard, it's not a PID controller, but that doesn't mean it's wrong for whatever problem they were working on.

 

[Jason-Li]

Yes, your top drawing is the standard way of doing this. But it's important to really understand why this is better than your first attempt. What do you really want your controller to do? Minimize error and respond to the input. Think of a cruise control in your car. Suppose you are going 55mph and then ask for 60 mph, isn't that mostly the same as going 60mph and then asking for 65mph? It's the 5mph error that is the most important thing to correct for. Your controller probably doesn't really need to know exactly what speed it's going if it can reliably respond correctly to the error.

- The differentiator's job is to respond to quick changes in the error and rapidly adjust the output. This is often the least important term and is frequently left out. It is often the most difficult to use correctly.
- The integrator's job is to only respond to the errors that are present for a long time and slowly adjust the output to eventually get it to the right place. This term is essential IF you want your system to actually get to zero error, and is really easy to use in practice.
- The proportional term is to deal with the in-between response, not too fast, not too slow. It turns out it is important when you start to deal with the stability and frequency response characteristics of these systems.

Think about when you are driving your car. You are the controller, monitoring your speedometer and trying to get your car to go the speed you want. When you step on the gas quickly anticipating a need for more power, you are acting like a differentiator (like maybe when you start to go up a steep hill). When you guess how much to push the gas part way down because you see your speed isn't right you are acting like a proportional controller. When you slowly make small adjustments to eventually get your speed perfect, you are acting like an integral controller.

However, there are lots of times that we do just feed back the output to to adjust the input in feedback systems. This was, in fact, the genesis of feedback in electronic systems. The telephone company needed amplifiers that wouldn't change their gain with perturbations like temperature, age, parts replacement, etc. The only information they needed to do that was how the amplifier was working, the inputs didn't matter. I would call this local feedback and isn't really the subject at hand, control systems. This type of feedback is undoubtable buried in the box labeled "amplifier", "G", or "H".

Sorry, I just don't understand the context of the second diagram. It's not standard, it's not a PID controller, but that doesn't mean it's wrong for whatever problem they were working on.

Hi DaveE, that explanation should have been printed in my learning materials, makes more sense than anything I have read on this subject thank you!

Sorry to keep probing regarding the graph sketches, does one of them look right? I can see that the integrator would reduce the offset to negligible but would I be correct in saying that oscillation frequency would still be high?

Thanks again.

 

[DaveE]

Hi DaveE, that explanation should have been printed in my learning materials, makes more sense than anything I have read on this subject thank you!

Sorry to keep probing regarding the graph sketches, does one of them look right? I can see that the integrator would reduce the offset to negligible but would I be correct in saying that oscillation frequency would still be high?

Thanks again.

There is nothing in the problem statement or in the block diagrams that tells us anything about the dynamic response of the system. You would need much more detailed information about the gains and frequency response of each of the elements in the system to draw any conclusion. The only indication that the system has oscillatory behavior is the transient response plot, which is shown without context.

For example, the simulation that @alan123hk showed us doesn't show any oscillatory behavior. BTW he does specify the frequency characteristics of the elements of his example in the part that says "Laplace..."; you'll learn about that later if it's not familiar to you yet.

About the only thing I would say is that adding additional control methods to your system will change the dynamic response somehow. The derivative terms will tend to make a system that responds faster but is more likely to oscillate. The integral terms will probably not have much effect, unless you are way off in setting up those gains.

In practice, when people optimize feedback systems, these different gains all tend to interact a bit. So it can be hard to separate their effects. It sounds confusing in this context, but when you learn the tools to analyze system dynamics it's much clearer. The thing is we speak a slightly different language for that then this approach. For us it's all about the number and location (frequency) of poles and zeros in the various parts of the system. The whole PID approach is OK, but kind of crude, for experts. For example, in that language, I have designed several systems that you might call IPDPII controllers. That sounds complex or exotic, but it's not when you really understand the context and methods.

 

[Tromso80]

Hi Jason-Li,

How did you manage to calculate the graph for question 5(b)? I'm struggling to see how we accurately determine the graph?

 

[Jason-Li]

Hi Jason-Li,

How did you manage to calculate the graph for question 5(b)? I'm struggling to see how we accurately determine the graph?

It's just a rough sketch, if you know the sort of response the system would have, roughly sketching it is enough - no major calculations or accuracy needed