How do I make a plot with all three states [x1 x2 x3]' in SImulink?

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DB 2020-11-26

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评论： DB 2020-12-4

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I have made a model on people [x1 x2 x3]' influencing each other by influence matrix A.

How can I plot the changes in the states over k (discrete) and time (continuous)?

The initial value is x0 which has values for

x = [x1 x2 x3]'

with

x0 = [12 7.57 5]'

And matrix A is a 3x3 matrix with unstable eigenvalues.

My discrete model looks like this:

And my continuous model ike this:

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Paul 2020-12-2

编辑：Paul 2020-12-2

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I still don't think the simulations are set up correctly based on the description of the problem. The initial state should not be modeled as a constant input. Note that in the continuous case, the initial states are all zero on the plot. The feedback loop around around the discrete time state space model is suspect as well, and I don't understand why a state space model is used for the discrete time case when only a delay is needed (analgous to the integrator in the continuous case).

Do you really need to do this in Simulink? The problem you've described can be easily implemented in Matlab alone. The Control System Toolbox can also be used.

Let's try to first get a handle on the problem you're trying to solve. Let's focus on the continuous case first. As I understand it, you want to solve the differential equation

xdot(t) = A*x(t), x(0)= x0 = [12 7.57 5].'

where A is a constant 3 x 3 matrix. To simplify this discussion, let's assume a diagonal A

A = diag([-1 -2 -3])

Because of the form of A, we know the solution to the differential equation

x1(t)=x0(1)*exp(-1*t)
x2(t)=x0(2)*exp(-2*t)
x3(t)=x0(3)*exp(-3*t)

We can compare this closed-form solution to our computed solution, which we develop as follows. In general, the solution to the given differential equation is

x(t)=e^(A*t)*x0

where e^(A*t) is the matrix exponential.

Let's put this all together and compare the computed solution to the close form solution.

x0 = [12 7.57 5].';
t = 0:.01:5;
A = diag([-1 -2 -3]);
x = nan(numel(t),3);
for ii = 1:numel(t), x(ii,:) = (expm(A*t(ii))*x0).'; end
xclosed = [x0(1)*exp(-1*t(:)) x0(2)*exp(-2*t(:)) x0(3)*exp(-3*t(:))];
plot(t,x,t(1:10:end),xclosed(1:10:end,:),'o'),grid

As must be the case, the solution starts from the initial conditions and decays to zero.

You can also get the same solution using the Control System Toolbox.

sys=ss(A,zeros(3,1),eye(3),zeros(3,1));  % Use eye(3) for the C-matrix so that each state is an output
y = initial(sys,x0,t);
plot(t,y,t(1:10:end),xclosed(1:10:end,:),'o'),grid

Either of these methods can be used with your particular A-matrix (which would be helpful to post if you can).

Is this the type of solution you're expecting? If it isn't, you'll need to provide more clarity on your problem. If it is, then the next step would be to use a similar approach for the discrete time case, and we can look at Simulink implementations if that's really needed.

Paul 2020-12-3

I'm not sure how much more (if any) help I can offer. I'm used to doing the opposite, i.e., starting with a continuous time model (differential equation) and then coming up with a discrete time approximation (difference equation). You seem to want to go the other way. One could come up with a differential equation that is approximated by the difference equation, but the outputs of those equations in response to the initial condition will not look like the plots you've posted.

The plot you posted shows a group of three (?) red lines and two blue lines. As best I can understand based on the problem description, the blue lines cannot be initial condition responses from discrete and continuous models where one model is the approximation of the other. One can come up with a model with two outputs that have the same appearance as the blue curves, but I wouldn't call that model a discrete model with a continuous approximation.

Sorry I couldn't be of more help.

DB 2020-12-4

@Paul Thank you very much for your help. I do acknowledge I have a vague problem and I'll work on it further. I think your help has made me understand it a whole lot better.

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