how to extract data from sys=ss(A,B,C,D) that is , inputs and outputs

%State-space representation of the system

A= [-85 250 63];

B= [11 0; 0 11; ];

C=[0 0 1 0 0];

D=[0 0;0 0];

[Paul]

Hi welesh,

Please clarify the question. What exactly is the goal for either the open loop system or the closed loop system (or both)?

[Paul]

Which data?

Answer 1

[Sam Chak]

Hi @welesh beyene

A = [-300 0 0 0 0 0; 0 -300 0 0 0 0; 0 0 0 0 1 0; 0 0 0 0 0 1; 10.7206 -9.5648 10.7206 -9.5648 0 0; -9.5648 10.7206 -9.5648 10.7206 0 0];
B = [300 0; 0 300; 0 0; 0 0; 0 0; 0 0];              
C = [0 0 1 0 0 0; 0 0 0 1 0 0]; 
D = [0 0; 0 0]; 
Q = diag([0.1, 0.1, 0.1, 0.1, 1, 1]);
R = diag([10, 10]);
K = lqr(A, B, Q, R)

K = 2×6

0.0269 -0.0147 2.0369 -0.0147 1.2296 0.6755 -0.0147 0.0269 -0.0147 2.0369 0.6755 1.2296

sysc = ss(A-B*K,B,C,D)

sysc =
 
  A = 
           x1      x2      x3      x4      x5      x6
   x1  -308.1     4.4  -611.1     4.4  -368.9  -202.7
   x2     4.4  -308.1     4.4  -611.1  -202.7  -368.9
   x3       0       0       0       0       1       0
   x4       0       0       0       0       0       1
   x5   10.72  -9.565   10.72  -9.565       0       0
   x6  -9.565   10.72  -9.565   10.72       0       0
 
  B = 
        u1   u2
   x1  300    0
   x2    0  300
   x3    0    0
   x4    0    0
   x5    0    0
   x6    0    0
 
  C = 
       x1  x2  x3  x4  x5  x6
   y1   0   0   1   0   0   0
   y2   0   0   0   1   0   0
 
  D = 
       u1  u2
   y1   0   0
   y2   0   0
 
Continuous-time state-space model.

You can probably extract the outputs {} caused by the inputs {} like this:

[y, t] = step(sysc, 10);
for j = 1:4
    subplot(2,2,j)
    plot(t, y(:,j)), grid on, xlabel('t'), ylim([-0.3 1.1])
end

[Sam Chak]

In that case, I'd advise you to use the ode45() function to solve the dynamics and generate the data.

If you have access to Simulink, then you can also put your State-space model in the State-space Block, and insert the Matrix K into the Gain Block, provided you understand how to build block diagrams.

You also need to understand what you want to train. If you obtain only the unit step responses data, then I think the trained system only "smart enough" to produce the unit step responses data that converge at around 5 seconds.

Answer 2

[Sam Chak]

Hi @welesh beyene,

Let's try this simple state-space model and generate the data using ode45() function (click on the link):

If the input is designed as , the feedback closed-loop system becomes:

I'm referring to some examples in anfis() documentation. If you find the demo and MATLAB code helpful, please consider voting 👍 the Answer.

[t, y] = ode45(@system, [0 10], [1; 0]);
plot(t, y), grid on, xlabel('t'), ylabel('\bf{y}(t)'), legend('y_1', 'y_2')

% organizing the data
out1 = y(:,1);              % position signal
out2 = y(:,2);              % velocity signal
in_u = - y(:,1) - 2*y(:,2); % force signal
FISin  = out1;              % input to the FIS that we want to train
FISout = in_u;              % output to the FIS that we want to train
data1 = [FISin FISout];

% setting up the ANFIS
genOpt = genfisOptions('GridPartition');
genOpt.NumMembershipFunctions = 3;
genOpt.InputMembershipFunctionType = 'gaussmf';
inFIS = genfis(data1(:,1), data1(:,2), genOpt);
opt = anfisOptions('InitialFIS', inFIS, 'EpochNumber', 60);

fis = anfis(data1, opt);

ANFIS info:
	Number of nodes: 16
	Number of linear parameters: 6
	Number of nonlinear parameters: 6
	Total number of parameters: 12
	Number of training data pairs: 73
	Number of checking data pairs: 0
	Number of fuzzy rules: 3


Start training ANFIS ...

1 	 0.0559898
2 	 0.0548279
3 	 0.0537759
4 	 0.0528129
Step size increases to 0.011000 after epoch 5.
5 	 0.0519196
6 	 0.0510792
7 	 0.0502002
8 	 0.0493556
Step size increases to 0.012100 after epoch 9.
9 	 0.0485363
10 	 0.0477374
11 	 0.0468815
12 	 0.0460528
Step size increases to 0.013310 after epoch 13.
13 	 0.045258
14 	 0.0445051
15 	 0.0437341
16 	 0.0430297
Step size increases to 0.014641 after epoch 17.
17 	 0.0423943
18 	 0.0418257
19 	 0.041271
20 	 0.0407799
Step size increases to 0.016105 after epoch 21.
21 	 0.0403412
22 	 0.0399454
23 	 0.0395517
24 	 0.0391966
Step size increases to 0.017716 after epoch 25.
25 	 0.0388743
26 	 0.0385768
27 	 0.0382683
28 	 0.037974
Step size increases to 0.019487 after epoch 29.
29 	 0.0376911
30 	 0.0374207
31 	 0.0371399
32 	 0.0368765
Step size increases to 0.021436 after epoch 33.
33 	 0.0366245
34 	 0.036376
35 	 0.0360996
36 	 0.0358142
Step size increases to 0.023579 after epoch 37.
37 	 0.0355162
38 	 0.0352057
39 	 0.0348567
40 	 0.0345119
Step size increases to 0.025937 after epoch 41.
41 	 0.0341598
42 	 0.033809
43 	 0.0333839
44 	 0.0335596
45 	 0.0329597
46 	 0.0332002
Step size decreases to 0.023344 after epoch 47.
47 	 0.0323918
48 	 0.0326266
49 	 0.0317615
50 	 0.0315423
51 	 0.0312428
Step size increases to 0.025678 after epoch 52.
52 	 0.03111
53 	 0.0302846
54 	 0.0299535
55 	 0.0288002
Step size increases to 0.028246 after epoch 56.
56 	 0.0276869
57 	 0.0260923
58 	 0.0241319
59 	 0.0224591
Step size increases to 0.031070 after epoch 60.
60 	 0.0212941

Designated epoch number reached. ANFIS training completed at epoch 60.

Minimal training RMSE = 0.0212941

% plotting the results
u = data1(:,1);
anfisOutput = evalfis(fis, u);
plot(u, data1(:,2), '*r', u, anfisOutput,'.b')
grid on, xlabel('y_1'), ylabel('u'),
legend('Training Data', 'ANFIS Output', 'Location', 'best')

function dydt = system(t, y)
    u    = - y(1) - 2*y(2); % input
    dydt = [y(2);           % state #1 differential equation
            u];             % state #2 differential equation
end

[Sam Chak]

I don't know what exactly you want to train because you just mentioned that there are two inputs and two outputs. Moreover, what is the scientific motivation behind the training activity?

You have designed the LQR to make the system to behave as desired.

Do you think that training something would improve the performance? It would be helpful you can provide a little background.

I strongly advise you to write/provide the mathematical equation for the inputs and . You are going need them in your ode45() code to generate the data points for the inputs and outputs.

This is just my example to generate the input u based on the equation that I defined:

[t, y] = ode45(@system, [0 10], [1; 0]);
out1 = y(:,1);              % generate output 1
out2 = y(:,2);              % generate output 2
in_u = - y(:,1) - 2*y(:,2); % generate input

% System dynamics
function dydt = system(t, y)
    u    = - y(1) - 2*y(2); % input
    dydt = [y(2);           % state #1 differential equation
            u];             % state #2 differential equation
end

Since you have two inputs, I expect there are two equations.