How can i get the control effort/output of the PID?

sys = (2.293*s + 9.172)/(s^2 + 10.29*s + 25.17);

Kp = 5; Ki = 50; Kd = 0.5;

[Jon]

You have only given the numerator and denominator of a transfer function, maybe this is your plant, and assigned some values to some variables that are named suggestively as PID controller parameters.

In order to find the output of a PID controller you would need to make a PID controller, (perhaps as a transfer function parameterized by the Kp, Ki, Kd values you have assigned), and then connect it appropriately in a closed loop with your plant.

You will also need to drive your system with some sort of input signal, or perhaps set it to some initial condition and let it settle to steady state.

Once you have got that far you can further look into how to get particular values of interest out.

[Jon]

Also, as a suggestion for impoved style in submitting questions, it is better to have the subject of your question be the general topic. ie. "How to get output from a PID controller", and then put your specific, detailed question within the body of your question.

[Rupali Gawde]

Thanks for the explanation. I have performed all that you mentioned. I have have got stuck with the control effort part.

[Jon]

I'm not really sure exactly where you are stuck. If you could post the code you have tried so far, and give some more details on exactly what problem you are having (error message, post those too) or more conceptual (try to explain what you tried and why it didn't work) I or others may be able to give you some more ideas.

By if you do post your code, if you use the Code button on the MATLAB answers toolbar it will format it nicely for you.

[Jon]

One thought, in case this helps. Assuming you are modeling , analyzing and simulating this using transfer functions in MATLAB you may have to do a little block diagram algebra first to get the transfer functions of interest.

Let's say you have defined appropriate transfer functions for your PID controller and lets call that C, along with your plant G, and perhaps some transfer function in the feedback path H. Then with some block diagram algebra you could find that the relationship between the reference input r, and the controller output u would be given by u = Cr/(1 +CGH). So the closed loop transfer function would be C/(1 + CHG) Please check my math I did this quickly.You could then make this closed loop transfer function between the reference and the output u in MATLAB for example (assuming you have already defined the transfer functions C,G and H

Tclr = C/(1+C*H*G)

And then analyze and simulate this. Note the step response of this transfer function would give you the control effort that result from a step change in the reference (setpoint) value.

Similarly if you were looking for the response of the controller output to a disturbance at the plant output, some block diagram algebra would give the appropriate transfer function Tcld = -CH/(1+CHG) Please check this too. And you could similarly define this in MATLAB using

Tcld = -C*H/(1 + C*H*G)

again the output for example step(Tcld) would give you the response of the control output a step change in the disturbance value.

[Rupali Gawde]

Find code in attachment. Please help and check the last section "Control Effort". Please help correct the code if needed. I need to find the output from all the 4 controllers -ZN-PID, GA-PID, PSO-PID, ABC-PID, TLBO-PID.

Thanks.

[Jon]

Now I see what you are doing I will post my response as an "Answer". Hope this helps. If it answers your question you can accept it. Otherwise please follow up with further comments to the "Answer".

Answer 1

[Jon]

So I see you are on the right track, you are trying to define the appropriate closed loop transfer functions but using MATLAB's feedback function instead of doing the block diagram algebra by hand. That seems like a good approach. I think you just need to look more at the details of how you are applying the feedback function.

I see you want the closed loop output of the controller to a setpoint change. In this case the forward path would be the PID controller's transfer function, but the feedback path would be the rest of the elements in the closed loop, in this case, the plant or as you call it sys. Note in your case the closed loop always consists of two elements the controller and the plant, but what part of it is the forward path and what part is the feedback path depends upon what output you are interested in. When you are interested in the plant output as the "output" then the controller and the system are in the forward path with unity feedback, that is 1 in the feedback path. When you want the controller output as the "output" then the forward path is just the controller, and the feedback path is the system.

So, for example for your y_zn calculation you should use

y_zn = lsim(feedback(zn_cont,sys),sp,t);

similar for the others.

[Jon]

Seem to be some further technical issues with this as now the closed loop transfer function is non-causal (numerator is higher order than denominator) and lsim fails. I have to think about what is going on here a little more

[Jon]

The problem is your non-causal representation of the derivative term in your pid controller. Note that the controller output response to a step would involve differentiating a step change, which would give an infinite output. MATLAB doesn't like this and it is quite non-physical. Instead I suggest you approximate the derivative term in the PID control instead of just using s instead you could use s/(tau*s + 1) where if tau is sufficiently small it will give a good approximation but still be causal.

[Jon]

I see you also in some parts of your code use MATLAB's PID controller function to make the PID transfer function. In this case you need to add the additional final argument to specify the time constant for the derivative approximation. You can type doc pid on the command line to get documentation on how MATLAB makes PID controllers.

One other comment. Looking quickly through your code, it looks like the only difference between many of your controllers are the control gains.

As a suggestin for the future, rather than copying and pasting all of the code defining each controller it would be much better to make a for loop to loop through the gains. Just have the code to define the controllers appear once inside of the loop.Copying and pasting is very error prone, when you try to maintain the code, as if you find an error in one of the copy and pastes you have to make sure you fix it in all of the places it appears.

[Jon]

I hope you were you able to follow this and get the results you wanted. If this answered your question please mark it as answered so that others that may have a similar question will know that an answer is available. Otherwise, if you still need more help please let me know.

[Rupali Gawde]

Thanks Jon.

I also need help finding the 1-Norm, 2-Norm and ∞-Norm for the tuning methods (ZN, PSO,ABC and TLBO). Can you help me provide the Matlab syntax code for the same?

[Jon]

Hi I think you should be able to use the norm command for this. On the command line you can type doc norm to bring up the detailed documentation. In particular you will find

n = norm(v,p)

where v is the vector of values, for example the error signal you want to compute the norm of and

p = 1 gives the 1-norm, p = 2, gives the 2-Norm (Euclidean Norm), and p = inf gives the infinity norm (maximum absolute value)

[Rupali Gawde]

Hi,

Can you please confirm if the below code is correct:

norm1 = norm((sp - y_zn'),1)

norm2 = norm((sp - y_zn'),2)

normi = norm((sp - y_zn'),inf)

Also, the values for the norms are in attachment. They look incorrect to me.

Thanks.

[Jon]

I think that these expressions will give the 1, 2, and infinity norm respectively of the argument you provide (sp - y_zn'). Be sure that both sp and y_zn' are both either row vectors or column vectors. Current versions of MATLAB allow you to subtract a column vector from a row vector, or a row vector from a column vector but this gives a matrix result, not what you want. To be sure they are both columns it is safest to subtract (sp(:) - y_zn(:)). I think from your earlier code that sp is a row and so is y_zn' so I don't think this is a problem for you, but you should check to be sure, as otherwise you will get the norm of a matrix rather than the vector which you wanted.

Otherwise I think it is a matter of how you are intrepreting the results and how you are using these norms. In some cases it may be more appropriate to normalize by the number of elements, for example the root mean square error is closely related to the 2-norm but includes a further normalization by the number of elements

[Rupali Gawde]

Let me look into this.

Also, can you please help me with the Simulink Block method (for ZN, PSO, ABC, TLBO) for the same problem.

Attached is what i have worked. I need with the coding and the correct simulink model.

Thanks.

[Jon]

The function linmod is used to linearize a non-linear Simulink model. You don't need to use this because your model is already linear. There is no further information you can get out of running this in Simulink that you haven't already gotten using lsim. What exacly is the problem you are trying solve by using Simulink?

[Rupali Gawde]

I need to verify the time response parameters that i got by lsim with the block simulink model method.

[Jon]

Why, you will get the same answer as with lsim.

[Jon]

Is it for an academic assignment just to get experience using Simulink? If so then what problem are you having using Simulink? If you don't have much experience using Simulink I would recommend first completing the Simulink on-ramp training. It will get you going very quickly https://www.mathworks.com/learn/tutorials/simulink-onramp.html