What Are the Basics of Deriving Control Transfer Functions?

[gl0ck]

Hello,

I got the following diagram, shown below, and I have to derive its transfer function. I think I have a general misunderstanding about the transfer functions. What I think it is, is: output/input basically. As input is the whole block of things that affect the output.

This is the system that I have.

Homework Equations

After doing some thinking how to start this I came out with the following:
$$v_m ( t ) - R_m i_m ( t ) - L_m { \frac {\ di_m (t) } { dt}} -k_m ω _m ( t ) = 0$$
It was said that we could ignore the inductance as the motor resistance is much bigger so I came up with as solving for i_m(t)
$$i_m (t) = {\frac {v_m (t) - k_m ω_m (t) }{R_m}}$$
I am given also a expression of the motor shaft equation:
$$J_{eq} ω_m (t) = τ_m (t)$$ , given that the J_eq is the total moment inertia acting on the motor. Also the torque is given by $$τ_m = k_m i_m (t)$$

I also know that the J_eq = J_m + J_h + J_d
With Jm and Jh given in a separate table.
I've found J_d to be $$J_d = {\frac {1}{2}} m r^2$$

The Attempt at a Solution

So far I've only seen transfer function of an electrical circuit like RC and RL circuits. As I am studying electrical engineering I am kind of confused how I am supposed to derive the transfer function with all this torque and angular velocity. I am asked to derive the transfer function of input voltage and the angular velocity.
I can see that I can substitute what I've found for i_m into the torque equation, but don't know how then I can turn that into the (s) domain.

Any guidance is much appreciated !
Thanks

 

[donpacino]

I can see that I can substitute what I've found for I am into the torque equation, but don't know how then I can turn that into the (s) domain.

How do you go from the time domain to the s domain in a purely electrical problem??

also I might be wrong but it looks like there is an error with the equation you were given
w_m is angular velocity.

J_eqω_m(t)=τm(t) should be J_eqω_m'(t)=τ_m(t) given that friction is 0.

also here is a handy guide that derives motor speed / armature voltage.
http://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed&section=SystemModelingnote: you may be thinking of transfer functions in an odd way. Below is something to maybe help you understand?

I think I have a general misunderstanding about the transfer functions. What I think it is, is: output/input basically.

lets take a current to voltage circuit. your transfer function needs to have the units of voltage/current.

therefore...
voltage (your output) = current (your input) * voltage/current (filter)

lets look at an airplane control yoke.
you could say turning right on the yoke rolls the plane the the right, the higher the angle, the greater the roll.

so let's say our transfer function is P(Roll Rate)/angle

well yes, but there is more to it. you see moving the yoke might just turn a knob on a potentiometer.

so really we have

p=voltage/angle (the potentiometer transfer function) * P/voltage (the roll rate transfer function)

but no, because that has to go through some com protocol and acuator gain we call X

so

p = voltage/angle (potentiometer gain) * X/voltage (voltage to actuator gain) * P/X (roll rate gain)

we can continue this process for a while.

The essence of understanding transfer functions like this is to make sure the units work out. so if you want to get a transfer function of w/voltage (angluar velocity per volt), its really just a combination of different smaller transfer functions. break it down if you have to.

in the end, it is no different from a 5 stage voltage filter ( volt/volt * volt/volt * current/volt * volt/current *volt/volt)
at the end of the day, that is a volt/volt filter
understanding the units is the key to understanding controls!

 

[Hesch]

I don't know if you are familiar with Laplace?

You start from the left with:

1) Summation block: +Vm, -e_b, output = Verr.

2) Find I_m, multiplying Verr with 1/(R_m + sL_m), output = I_m

3) Find motor torque from I_m

4) Find dω/dt from torque, inertia

5) Find ω by integration

6) Find e_b and feed back to summation block 1)

Now use Mason's rule to calculate the transfer function of the system.

 

[gl0ck]

Thank you both for the answers. donpacino, I looked at the link it helped find the transfer function in my case. What I found more helpful actually was the comment below. I also found the book Modern Control Systems quite helpful understanding some basics.

 

[zoltrix]

hello

I asked about the same question in an other thread
suppose the DC motor drives a slider-crank mechanism
the transfer function of the mechanism block might be :

G(s) = X(s)/T(s)

where
X(s) : position of the slider
T(s) : Torque at the crank

I wonder whether it makes even sense to speak of "Transfer function" even though the load is not linear ?

I own an "automation and control " book with a few mechanical examples
All of them assume that the moment of inertia referred to the shaft of the DC motor is constant but in general it is not constant

 

[zoltrix]

It is what I meant
control functions assume that the system is linear
if so, what shall you do in case of non linear but very common mechanisms such as a slider-crank or a four bar linkage ?

 

[rude man]

Hello,

I got the following diagram, shown below, and I have to derive its transfer function. I think I have a general misunderstanding about the transfer functions. What I think it is, is: output/input basically. As input is the whole block of things that affect the output.

This is the system that I have.
View attachment 186906

Homework Equations

I am given also a expression of the motor shaft equation:
Jeqωm(t)=τm(t) , given that the J_eq is the total moment inertia acting on the motor.

Whoever gave you that equation owes you an apology. It's dimensionally incorrect. Otherwise I think your approach is good.
First, you have two k_m, they are not at all the same constant, so come up with a new one relating current to torque, say k_t so that torque = k_t times current..

OK, it's been years since I worked on this forum and my Latex is rusty so let me give you some verbal leads:

1. Change that equation to torque = J dw/dt where w is your lower-case omega.
2. Use your other two equations relating motor current to torque and dw/dt. Laplace-transform all three equations, then you have
W(s) = F(s)V(s),
W - transformed w
V = t ransformed v_m
F will contain K_m, k_t, R_m, J_equiv, R_m and of course s.

You could also find the transfer function relating dw/dt to v_m if you wish. That would relate applied input voltage to acceleration instead of velocity.

If you have a finite load, the torque-current equation must be modified. If the load toque is constant or linear in one or more derivatives of your dependent variable then you just solve that modified ODE. Otherwise you are stuck with a non-linear system. Cf. hint of post 6.

 

[DaveE]

what shall you do in case of non linear but very common mechanisms such as a slider-crank or a four bar linkage ?

Work harder with less satisfying results. It's nearly impossible to give any generic description of non-linear systems. They could be practically anything.

Steve Strogatz has a nice set of non-linear systems lectures available through MIT. You may want to look at those.

 

[zoltrix]

thanks DaveE

do you mean that it is impossible to control position and velocity of the slider of a slider-crank mechanisms ,
i.e to achieve the wanted trajectory by mean of a PID controller and DC motor ?

 

[DaveE]

thanks DaveE

do you mean that it is impossible to control position and velocity of the slider of a slider-crank mechanisms ,
i.e to achieve the wanted trajectory by mean of a PID controller and DC motor ?

No. Its just harder to analyze nonlinear systems. Lots of useful math tools don't work. Answers are more sensitive to specific conditions.

 

[zoltrix]

thanks again

suppose there is a flywheel
moment of inertia at the shaft = Constant + variable
does it make sense to consider a linear system + noise ?

 

[DaveE]

thanks again

suppose there is a flywheel
moment of inertia at the shaft = Constant + variable
does it make sense to consider a linear system + noise ?

Sorry, I don't understand your flywheel system. Plus I'm an EE, so not great with mechanics, and I don't feel like guessing.

Noise is different from nonlinear systems because it is modelled as random, uncorrelated with the system state. You would normally use different tools/models for these two things.

You're not going to get a simple answer from us about nonlinear systems. They just always (OK, almost always) depend too much on the details.

One very common technique is to use a linear approximation around the operating point. This often works, but not always. Basically, you'll need to study control systems a bit to deal with these systems.

 

[rude man]

Yes it's still linear. Your J now becomes J1 + J2(t). Substitute that in your ODE and solve for w(t) or dw/dt (you could theoretically solve for angle position also but it would be impractical I think due to the uncertainties in your km and kt.).
The reason your system would still be linear is as I said that the dependent variable w or dw/dt appear only to the 1st power.
If you had a specific function in mind for J1(t) you could put the equation into wolfram alpha for an easy solution.

 

[zoltrix]

Hi DaveE and rude man

the motor constant Km can be evaluted, and kt too, being a related parameter

take the glock's diagram which you can find, as example, in any automation and control books

the point is , in my opinion, that the inertia Jd at the DC shaft is alwayes assumed to be a constant value while in most mechanisms it is not

however as you said yourself you can write

Jd = J1 + j2(x,w)

the constant inertia J1 depends on the flywheel and ,in part ,on the mechanism itself
j1 can be easly calculated

the variable inertia J2 depends on the current geometry of the mechanism (x) and on the current velocity (w)
also J2 value can be evaluated but it is a quite complicated non linear function since contains trigonometric functions

if the flywheel is correctly sized I think it is reasonable to assume that J1 >> J2

I wonder whether the glock's diagram is still valid provide you add a block for a sort of noise to take into account the non linearity of the system

I agree that noise is , in general, a random signal while J2 is a analytical function but I suppose that you should be able to treat it same as a "standard noise" since its key parameters (max,min,sigma etc) are known

on the other hand I doubt that a linear approximation around the operating point could be of help in this case
my goal is to obtain an about constant velocity of the slider over a reasonable long path
let's say, for example, about 25 % or even less of the stroke of the slider cycle