Is it possible for the state variables to regulate to a set point instead of the output?

Given the state variables being measurable (no need for an observer), instead of the output tracking the setpoint, can we make the state variables track the setpoint instead? If it can't track the setpoint at steady state, can it be close enough to the setpoint? What is the control structure that best suits this scenario?

Will appreciate your answer on this.

Answer 1

[Birdman]

Consider the following state space representation:

x_dot=Ax+Bu;%x_dot is the derivative of x wrt time.

If we take the Laplace transform of this expression, we will obtain

X(s)      B
____  =  ____
U(s)     sI-A

Taking into consideration that this is a transfer function, where U(s) is input and X(s) is output, you can imagine the state variables as an output and make it track a reference signal. It can track the reference without steady state if you include an integrator in controller between reference and system. Therefore, it is possible to make state variables track a reference signal. Just by rewriting the expressions, you can obtain the necessary transfer functions. Hope this helps.

[Mr. NailGuy]

Thanks for the answer! I'll try this one out.

[Mr. NailGuy]

Hi cvklpstunc, I tried it but my B is 4x2 while my (sI-A) is 4x4 so I would not be able to get a proper transfer function relating the two.

[Birdman]

Send A and B matrices.

[Mr. NailGuy]

Here are my matrices. You could take a look on it.

A =

   -7.2487   -0.0399   -5.1500    3.5846
   -4.5740    4.5018   -4.3662   -1.5683
    3.7698   16.1212  -15.6103    4.4936
   -9.8978    8.3742   -4.4331   -6.4318

B =

    0.1564    0.0319
    0.0173   -0.0200
    4.4939    2.3363
   -1.4269   -0.2730

[Birdman]

Sorry, I just missed a part. Use the following command:

s=tf('s');
Gs=inv(s*eye(4)-A)*B;

You will obtain 8 transfer functions. There you have it.

[Mr. NailGuy]

Hi cvklpstunc looks good. So does the B/(s*eye(4)-A) still holds for this? Will (s*eye(4)-A) be on the left side when you inverse it? :-) Also how can I implement this on Simulink with 4 references? Should I use a continuous time transfer function block?

Thanks

[Birdman]

Well the first thing I said will not apply since there are two inputs. (s*eye(4)-A) will be on left side exactly :)

Yes you can implement it by using transfer function block.

Note that since you are going to control each state variable separately, you will have eight controllers, eight reference signal and so on.

[Mr. NailGuy]

Thanks cvklpstunc, I would try it and hope that it will work. Tho, I still don't have an idea regarding how to make the controller and put the 4 reference input and use a continuous time T.F. block with the resulting Gs function :)

[Birdman]

Use the following notation:

num1=Gs(1,1).Numerator{1};
den1=Gs(1,1).Denominator{1};

and the same for the rest.

[Mr. NailGuy]

Hi cvklpstunc, would try it and thanks for the help. The problem is that the system is a MIMO system where I might need to decouple it too to control individually so it would be quite tough to do this but I will try. Thanks

[Mr. NailGuy]

Hi cvklpstunc, the only problem with this solution is that it I only have 4 references to input to the system. Is there any other solution like using the state feedback controller and solve for gain K? I almost make it work, only that I don't know how get my K. I have troubles in the concepts. Thanks