One of the systems I want to control includes disturbances in the input matrix. How can I convert this state space equation into a transfer function?

I want to control one system includes disturbances in the input matrix. I tried to write the function to control it from Matlab simulink but it didn't work correctly. How can I convert this state space equation into a transfer function. Could you share a source where I can study on it? Thank you very much.

Answer 1

[Paul]

Combine u and d into a single input vector. Then the "B" matrix is composed of the Bbar and Ebar matrices concatenated together as shown here:

xdot = Abar * xbar + Bbar * ubar + Ebar *dbar

xdot = Abar * xbar + [Bbar , Ebar] * [ubar ; dbar]

Now all the Control System Toolbox functions can be used, like

sys = ss(Abar, [Bbar , Ebar], Cbar, 0)

sys will, of course, be a multi-input system, three inputs in this case based on the dimensions of Bbar and Ebar.

[veysel burçak]

If you have time, can you write and send the matlab codes for me to get the transfer function. Thank you very much.

Answer 2

[Sam Chak]

Hi @veysel burçak

Sounds interesting. I wonder how you will design the controller for the system subject to the mismatched disturbance.

Presuming that is the same as , and the state-space is a Multi Input, Single Output system then the transfer functions from the three inputs to the output (2nd state, ) can be obtained

% No change (from you)
A = 1e-3*[-4.46566  4.45684;
           5.15227 -5.15227]

A = 2×2

-0.0045 0.0045 0.0052 -0.0052

B = [3.16315e-3; 
     0]

B = 2×1

0.0032 0

E = [2.92761e-3 0; 
     0 -2.0318e-8]

E = 2×2

0.0029 0 0 -0.0000

% Input Matrix Concatenation method according to Paul
Abar = A;
Bbar = [B E]

Bbar = 2×3

0.0032 0.0029 0 0 0 -0.0000

Cbar = [0 1];   % second state is the output
Dbar = 0;

sys = ss(Abar, Bbar, Cbar, Dbar)

sys =
 
  A = 
              x1         x2
   x1  -0.004466   0.004457
   x2   0.005152  -0.005152
 
  B = 
               u1          u2          u3
   x1    0.003163    0.002928           0
   x2           0           0  -2.032e-08
 
  C = 
       x1  x2
   y1   0   1
 
  D = 
       u1  u2  u3
   y1   0   0   0
 
Continuous-time state-space model.

% Converting the state-space model to transfer functions
G = tf(sys)

G =
 
  From input 1 to output:
            1.63e-05
  ----------------------------
  s^2 + 0.009618 s + 4.544e-08
 
  From input 2 to output:
           1.508e-05
  ----------------------------
  s^2 + 0.009618 s + 4.544e-08
 
  From input 3 to output:
    -2.032e-08 s - 9.073e-11
  ----------------------------
  s^2 + 0.009618 s + 4.544e-08
 
Continuous-time transfer function.

Primarily you should use the 1st transfer function to design the controller for the disturbance-free response. After that, run simulations on the 2nd and 3rd TFs to check its disturbance rejection capability. Then, fine-tune or improve the controller, or even modify the controller structure (if needed).

[veysel burçak]

Thank you for your answer. It's a system that was controlled with PID before. I want to try it with my own controller.

For parameters:MODELING AND MODEL IDENTIFICATION OF A PRESSURIZER AT THE PAKS NUCLEAR POWER PLANT

Control with PID:Fault Detection in Nuclear Systems Using Sliding Mode Observers

Answer 3

[veysel burçak]

Thank you very much for your answers, your explanations are very understandable.After I finish my work, I will post a video on the subject. @Paul @Sam Chak

[Sam Chak]

You're most welcome! That sounds like a great idea! I'm looking forward to watching your video on the subject once you finish your work. If you have any more questions or need further assistance, feel free to reach out. Good luck with your project!

[veysel burçak]

I designed the same system with these transfer functions using the PID parameters in the paper. But I couldn't get the same graph, can you check?

First of all, I need to get the same graphic as this paper for comparison. I will also have checked the correctness of the transfer function. Thank you.

[Sam Chak]

Hi @veysel burçak,

Suggest you update the new info in your question. I get almost a similar response from the simulation below. Maybe due the effects of the unaccounted disturbances because they are not provided.

% Run this on Workspace
A = 1e-3*[-4.46566  4.45684;
           5.15227 -5.15227]
B = [3.16315e-3; 
     0];
E = [2.92761e-3 0; 
     0 -2.0318e-8];
% For State-space block
Abar = A;
Bbar = [B E]
Cbar = [1 0];   % first state is the output (Temperature)
Dbar = [0 0 0];

[veysel burçak]

Really thank you so much. You made me so happy.

[veysel burçak]

Can you check my matlab simulink result should increase from 300 to 350 but the temperature value is going down. Ksc and Lambda values are my variables. Do you have any suggestions for optimization?@Sam Chak

Answer 4

[veysel burçak]

I tried to write a matlab function before, but I couldn't finish it. I am new to using Matlab simulink. Can you help me get the above graph with space state function block or transfer function? Thank you very much.