MATLAB: I have a state space system like this and I want to Augment the discrete-time plant model with an integrator, and check controllability and observability of the augmented model.

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I have a state space system like this and I want to Augment the discrete-time plant model with an integrator, and check controllability and observability of the augmented model.

Best Answer

Integral in discrete time using Forward Euler method;

x(k+1) = x(k) + dt*u;
y(k)   = x(k);

Other methods can be found here.

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MATLAB: Controllable and observable canonical form

The tf2ss function wants a transfer function as input, not a system object.

Try this:

[A B C D]=tf2ss(Den,Num)
A =
    -5    -6
     1     0
B =
     1
     0
C =
     1     1
D =
     0

EDIT —

To get the state space representation from a system object, just use the ss funciton:

[A B C D] = ss(s);

MATLAB: How accurate is the numerical integrator, cumtrapz

There is no bound to the error of cumtrapz()

Consider f = x^3-sin(x) over 0 to 5, with step size 1/1000, then the numeric error from the trapazoid rule is about 6E-6.

Now consider f = 1000*cot(Pi*x-2) over 0 to 1/2, with step size 1/1000, then the numeric error from the trapazoid rule is about 1E-4 . This is 1/10th of the range that produced the 6E-6 error for the first function, so if it were range alone that made the difference than you would expect around 6E-7 error. If you were to then examine and say "Oh but the multiplier is 1000 times more" then that would at get you to about 6E-4 error, more than the 1E-4 observed. We can thus determined that there must be more factors than just constant multipliers or length of the range: the numeric error must depend upon the shape of the function itself.

Now consider f = cot(Pi*x-2) over 0 to 5, with step size 1/1000, then the numeric error from the trapazoid rule is undefined, as the calculation would involve adding values that range from -inf to +inf and adding -inf to +inf is undefined. So it definitely depends upon the function being evaluated.

You can calculate the error symbolically. The trapazoid rule estimate of the integral of F from A to B step size H is

(symsum(F(k*H), k, 1, B/H-1)+(1/2)*F(A)+(1/2)*F(B))*H

from which you can subtract int(F,A,B) to get the error.

Best Answer

Related Solutions

MATLAB: Controllable and observable canonical form

MATLAB: How accurate is the numerical integrator, cumtrapz

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