error in discrete space state depeding on coefficient

Im trying to make a space-state controller and there is a problem.

I refered to here: https://ctms.engin.umich.edu/CTMS/index.php?example=MotorPosition&section=SystemModeling

There was no problem with continuous state-space open loop response based on transfer function, and discrete space-state, at first.

But i encountered with a problem that the response of discrete state-space doesnt work properly when using another values at J,b,K, etc, of motor.

Here is my code.

This works properly:

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clc;clear;

J = 0.01;

b = 0.1;

K = 0.01;

R = 1;

L = 0.5;

A = [-b/J K/J

-K/L -R/L];

B = [0

1/L];

C = [1 0];

D = 0;

sys=ss(A,B,C,D);

step(sys)

title('Continuous SS')

x=[0 0]';

u=heaviside(1);

Ts=0.1;

t=Ts:Ts:140;

for i=1:1400

x=(A*Ts+eye(2))*x+B*Ts*u;

y(i)=x(1);

end

plot(t,y)

xlim([0 3.5])

title('Discrete SS')

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And this fails.

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clc;clear;

J = 3.2284E-6;

b = 3.5077E-6;

K = 0.0274;

R = 4;

L = 2.75E-6;

A = [-b/J K/J

-K/L -R/L];

B = [0

1/L];

C = [1 0];

D = 0;

sys=ss(A,B,C,D);

step(sys)

title('Continuous SS')

x=[0 0]';

u=heaviside(1)

Ts=0.1;

t=Ts:Ts:140;

for i=1:1400

x=(A*Ts+eye(2))*x+B*Ts*u;

y(i)=x(1);

end

plot(t,y)

title('Discrete SS')

And this problem disturbs me to make state observer, ESO, disturbance compensate at my project by simulink.

Answer 1

[Paul]

Hi SUENGTAE,

In the first case we have:

J = 0.01;
b = 0.1;
K = 0.01;
R = 1;
L = 0.5;
A = [-b/J   K/J
    -K/L   -R/L];

The eigenvalues of the A matrix are

eig(A)

ans = 2×1

-9.9975 -2.0025

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The eigenvalues of the discretized system are

Ts = 0.1;
eig(eye(2)+A*Ts)

ans = 2×1

0.0003 0.7997

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The sampling period, Ts, is small enough so that the discretized system is stable (all poles are inside the unit circle), though Ts = 0.1 still feels too large to accurately represent the dynamics when the fastest pole has time constant of 1/9.9975, which is also 0.1.

In the other case

J = 3.2284E-6;
b = 3.5077E-6;
K = 0.0274;
R = 4;
L = 2.75E-6;
A = [-b/J   K/J
    -K/L   -R/L];
eig(A)

ans = 2×1

1.0e+06 * -0.0001 -1.4545

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●

The dynamics are much faster and the sampling period isn't nearly small enough to maintain stability after discretization

eig(eye(2)+A*Ts)

ans = 2×1

1.0e+05 * -0.0000 -1.4545

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[SUENGTAE]

Paul, Thank you!

I verified my code according to your great advice.

Unfortunately, I have more extreme system coefficient than case 2.

I tried to change simulink solver's sample time smaller and smaller.

Improved results in the graph but still it can't get stability even at Ts=1e-9.

If it requires smaller sampling time than 1e-9 or something, simulink and my processor(ESP-32) seems not to be able to endure that condition.

Anyway, Thank you for your help.