Is the MATLAB command lsim good for linearized state space models?

Question

A state space model is on the form: dx = Ax + Bu y = Cx + Du

Nonlinear state space models which is linearized is on this form:

dΔx = AΔx + BΔu
Δy =  CΔx + DΔu

Where:

Δx = [x1 - x10; x2 - x20; x3 - x30; .... ; xn - xn0]
Δu = [u1 - u10; u2 - u20; u3 - u30; .... ; um - um0]

The x10, x20, x30, xn0, u10, u20, u30, um0 is constants/initial values for the linearization.

So! The question is about the MATLAB command "lsim":

lsim (sys, u, t, x0)

In this case, sys is the A, B, C, D matrices from the lineraized state space model. u is the insignal vector, t is the time vector. But x0....can i say that x0 is x10, x20, x30, .... ,xn0?

Can I also say that u = u - u0 , and u0 is , u10 u10, u20, u30,... , um0 ?

Example:

u = linspace(5, 5, 100); % insignal 5
t = linspace(0, 100, 100); % 100 seconds
u0 = [0.2; -1.2; -3];
u = u - u0; % 
x0 = [-2; 2; -1]
lsim (sys, u, t, x0)

This will results:

Δx = [x1 - 2; x2 + 2; x3 - 1]
Δu = [u1 + 0.2; u2 - 1.2; u3 - 3]

Am I right?

Is there any reason that make you think you may not be right? The answer for the question "is this function specifically designed for this task good at this task?" is relatively clear I think, so what is the real question? — Ander Biguri, May 14 '17 at 18:37
The question is i can simulate my linearized state space model as I shown above. — euraad, May 14 '17 at 21:11
The approach is generally: try it, if it works, then nice! If it does not work, then you come and ask about it — Ander Biguri, May 15 '17 at 07:11

score 0 · Accepted Answer · answered May 14 '17 at 18:53

There is a readily available Simulink S-Function dsfunc.m which is the simulation of a State Space model, and which can be easily adapted for included linearized State Space matrices, changing at every time step.

Just build a simulink file dsfunc.slx and run it with the sim command sim('dsfunc') command.

The good part is the S-Function deals properly with the state, and you just adapt the State Space model each time, keeping the state from one iteration to another.

Also you can do it equally easily with a single for loop. Linearizing constant are included as shown.

function [t,u,x,y]=example2
% Start
dt=0.001;
N=1000;
n=2;
r=1;
t=(0:N-1)'*dt;
u=rand(N,r);
% First
[A,B,C,D,x0,y0]=ABCD(t(1),zeros(n,1),u(1,:)');
x(1,:)=x0';
y(1,:)=(C*x(1,:)'+D*u(1,:)'+y0)';
% Cycle
for i=2:N
    [A,B,C,D,x0,y0]=ABCD(t(i),x(i-1,:)',u(i-1,:)');
    x(i,:)=(A*x(i-1,:)'+B*u(i-1,:)'+x0)';
    y(i,:)=(C*x(i,:)'+D*u(i,:)'+y0)';
end
plot(t,[u x y]);legend({'u' 'x1' 'x2' 'y'});


function [A,B,C,D,x0,y0]=ABCD(t,x,u)
A=[0.1 -0.9;0 0.1];
B=0.5*ones(2,1);
C=[1 1];
x0=[0];y0=[0];

Yeah i though you wanted to linearize at several points. Happens you only need a linear model to go. That is ok :\. — Brethlosze, May 15 '17 at 02:39

Is the MATLAB command lsim good for linearized state space models?

1 Answers1