From where comes the pole and zero in real worl...??

[koolguy]

Hi again,

You posted a question about the state equations so that's why i made the example like that.

But if we talk about matrix D can we use it here?

 

[MrAl]

Hi,


Yes, i'll get to that next. The 'D' comes in in the other part of the state equation as you probably know. In a block diagram which is very informative you'll see D directly between the input and the output, which means that the D gain multiples the input and adds that result directly to the output.

Here's a quick block diagram to show how simple this is:

SIGNAL FLOW BLOCK DIAGRAM

   +---------->-----[ D ]-------------------------------->--+
   |                                                        |
   |                                                        |
IN o--->--[Other stuff with feedback and integrators]---->--o OUT

So the output is:
OUT=IN*OtherStuff+IN*D

so if D=2 we would get:
OUT=IN*OtherStuff+IN*2

 

[koolguy]

so if D=2 we would get:

AS D = 2 but i have seen many problem where D is zero and in some problem in form of matrix, why?

 

[MrAl]

D is only going to be present when it is required to describe the system adequately. Why is D=0 a problem for you?

 

[koolguy]

Because many time in problem D is taken zero i was just confused.

 

[koolguy]

OK, If we want to know Transfer function of the MIMO system the state vaiable are used then converted to freq. domain from time domain..
By C(s)/R(s)= B*[SI-A]-1*B

but there are other property like Φ(t)...etc what the use of them?

 

[MrAl]

Hi,

You can solve the time domain state equations using the transition matrix by:
e^At

Is that what you mean?

Your questions are always a little vague so it's hard to know what you are asking almost all the time.

 

[koolguy]

OK, i know it will be confusing..................
just changing the subject for a while.

If we talk about PID controller.......
what can we do of it?? as i was searching it in you tube there comes an obstacle robot which was working on PID controller. When P was used it was not having good artificial intelligence as it was moved to PID it was more stable.
https://www.youtube.com/watch?v=ALVo4aJpcF0

But i don't understand the main purpose or to use it in real...

 

[MrAl]

Hello again,

That's a cute robot
Adding a controller usually gives us smoother action because it's measuring the error and giving the system the ability to update itself at a fast rate and so correct the errors before they get too large.

Here is the generalized transfer function for the PID controller:
Out=In*(s*K3+K2/s+K1+P(s))/(F(s)*(s*K3+K2/s+K1+P(s))+1)

where
P(s) is the plant,
F(s) is the feedback sensor,
K1 is the proportional gain,
K2 is the integral gain,
K3 is the derivative gain.

That equation can be simplified but then we loose some generality. As it is, we can construct any kind of controller we want like PI, PD, ID, PD, or leave it as it is for PID.
For example, to make a PI controller, eliminate all the terms that contain K3. If we did that we would get:
Out=In*(K2/s+K1+P(s))/(F(s)*(K2/s+K1+P(s))+1)

and then this can be simplified and analyzed.

We can look at an example if you like.

 

[koolguy]

Here is the generalized transfer function for the PID controller:
Out=In*(s*K3+K2/s+K1+P(s))/(F(s)*(s*K3+K2/s+K1+P(s))+1)

where
P(s) is the plant,
F(s) is the feedback sensor,
K1 is the proportional gain,
K2 is the integral gain,
K3 is the derivative gain.

THe equation seems to be long but i will preffer to know differnt in these controllers P,PI,PD,PID,etc.......
what they are meant for??

 

[MrAl]

For example, the integral controller is meant to reduce the steady state error. The steady state error SSE is the difference between the output signal and the input signal after a relatively long time has passed (or a time period defined by the design requirements of the system).

For example, say we have a simple voltage regulator. We have 5v input (reference) and we have 4.50v output after 1 second. The steady state error after 1 second is therefore -0.50 volts. If we add an integrator, we might reduce that error to -0.05v which would be much better.
Many times the process integrates anyway so we end up with some kind of inherent integration, so all we have to do is increase the gain (and of course check stability).

There are other kinds of feedback systems too but it would take a lot of time and space to show everything here.

 

[koolguy]

If we add an integrator, we might reduce that error to -0.05v which would be much better.
Many times the process integrates anyway so we end up with some kind of inherent integration, so all we have to do is increase the gain (and of course check stability).

How does it really work??
We can use diff. amplifiers with reff. of 5V to stabilize the o/p using feed back like that so, what does integrator circuit really do??

 

[MrAl]

Hi again,

The integrator circuit really does increase the steady state accuracy. With a perfect integrator you'd get 1.000 out for 1.000 in. With the integrator only with nothing else, you'd get the perfect accuracy but the response would be ringing until it stabilized. This may not be a problem in some systems but with the added P and D we can get a much nicer output.

If you look at the attachment it shows two waveforms for a motor speed control. The blue wave is for the uncompensated system where you can see not only an oscillation before it settles but also a steady state error of about 5 percent. The red wave shows the speed response after the PID controller is inserted into the feed forward path. You can see we got a much nicer response now. Not only did we get rid of lots of oscillation, we also got rid of the steady state error which would be very low now.

The waveform without the P and D added (only I) comes out with a near zero steady state error, but there is still oscillation. Adding all three and we get a nice output as shown in red.

 

[koolguy]

Hi again,

That mean PID is better than other or single controller but in case we have to design a system where we want to use it but how any method from where we can start???
one more video.

 

[MrAl]

Hi,

In the motor we have three main sections:
G1=Km/(s*L+R)
G2=1/(s*J+f)
H1=Kb

where
Km is a motor constant,
Kb is another motor constant,
R=armature resistance,
L=armature inductance,
J=rotational inertia,
f=friction.

The input is in the form of a voltage Va, and we simplify a little by leaving out the disturbance torque Td which would in real life add between G1 and G2, and we do the analysis with a step input signal that steps from 0 to Va. We also take H1=1 for simplicity here.

The transfer function of this system is:
Va*G1*G2/(G1*G2*H1+1)

The controller would insert after G1 for example, so we would end up with:
Va*G1*C*G2/(G1*C*G2*H1+1)

and note that the controller C is always multiplied by G2.
Looking again at G2 we have:
G2=1/(s*J+f)

and the rotational inertia is responsible for part of the time it takes to get the system spinning, and the friction damps the signal, but also note that if the denominator was also in the numerator it just so happens to have the same form as a PD controller (no I though) so we can start by making C=s*J+f. We can then add an integrator part so we get:
C=s*J+f+K3/s

and so now the transfer function becomes:
Va*G1*C*G2/(G1*C*G2*H1+1)

and reducing a little we get:
(Va*G1*K3)/(G1*H1*K3+s)

and with simplified H1=1 we get:
(Va*G1*K3)/(G1*K3+s)

and substitute for G1 we get:
(Km*Va*K3)/(s^2*L+s*R+Km*K3)

and here if we keep the relationship:
R^2-4*Km*K3*L > 0

we should get a nice damped response.

So you see the more we know about the system to begin with the faster we can come up with values that will work, but almost always they have to be checked in the application to make sure it works properly under any expected real world conditions.

To start cold without any knowledge of the system like he did in the video, you would set all to zero:
K1=0
K2=0
K3=0

and start from there. To optimize you might have to make measurements of the system which may or may not be simple.

The form of the controller he used in the video was digital, we have discussed an analog technique here. For the digital computer controlled technique you would be entering lines of code that makes up the controller part, and have some external means to measure the required quantities like the output. In the case of the motor speed controller the main measurement would be the shaft speed.

 

[koolguy]

If we want to design a line follower robot using PID controller ...how can we go with it??
and Thanks for writing big reply but it was difficult to understand.

 

[MrAl]

Hi,


I've included a quick diagram of a system we can look at. The sensor for this system would be two LDR's wired in series and two LEDs, with a positive reference voltage to the top LDR and a negative reference voltage to the bottom LDR so that when the car goes off the line one LDR increases in resistance and the other LDR decreases in resistance and that provides us with an error signal for the feedback.

For this discussion i'll use:
w for motor speed
O for shaft angle
Od for disturbance torque
OT for result angle

and the following values:
Km=10
L=0.1
R=0.1
J=2
f=10
Steering and Sensor=2/(s+2)
C=controller

The result angle OT is the angle of the car which is zero when the car is properly
following the line. If the car drifts to the right or left the angle changes and
appears as a disturbance angle Od. The result angle OT then changes and it is
the job of the feedback to adjust the angle of the motor shaft O so that the
result angle again goes back to zero.

The system as shown in the diagram is unstable when the controller network C is equal to 1 (which is no compensator at all) so we design a controller to stabilize the system.



Does this make sense to you so far?

 

[koolguy]

HI again,

I was just searching this for help the 1st link was god to start.....

The system as shown in the diagram is unstable when the controller network C is equal to 1 (which is no compensator at all) so we design a controller to stabilize the system.

after searching Compensator it was like controlling pressure device..., right?

Does this make sense to you so far?

Sure..!!
But i can't able to understand that equation in att. file( figure).

 

[koolguy]

HI again,

I was just searching this for help the 1st link was god to start.....

The system as shown in the diagram is unstable when the controller network C is equal to 1 (which is no compensator at all) so we design a controller to stabilize the system.

after searching Compensator it was like controlling pressure device..., right?

Does this make sense to you so far?

Sure..!!
But i can't able to understand that equation in att. file( figure).

 

[MrAl]

Hi,

Which equation is that? Inside the blocks or the whole equation for the system, or something else?