Stable response for an unstable system

I have a stable system transfer function wich I'm trying to control with a PID controller. The funcion is:

2.494e-008 s^4 + 8.735e-010 s^3 - 5.82e-016 s^2

-------------------------------------------------

s^6 + 0.05155 s^5 + 0.001775 s^4 + 4.491e-005 s^3

and I'm trying to control it with this PID:

9e006 s^2 + 8e006 s + 8e5

-------------------------------------

               s

When I do that, I obtain an unstablesystem transer function, with one pole with positive real part, but when I simulate it for a step, I get a stable response. Anyone know what's going on?

    G(1)=tf([2.494e-008 8.735e-010 -5.82e-016 0 0],[1 0.05155 0.001775 4.491e-005 0 0 0]);
    Kp=8e6;
    Ki=8e5;
    Kd=9e6;
    l=tf([Kd Kp Ki],[1 0]);
    n=feedback(l*G(1),1);
    step(n,t)

I get this plot as a response: http://tinypic.com/r/1fv6sp/8

Which is great, but does not match the transfer function.

Answer 1

[Yu Jiang]

I have computed the poles of the closed-loop system by executing the following command in the MATLAB command line.

>> [z,p,k] = zpkdata(n);
>> p{:}

Note that I use the function zpkdata (See Documentation) to read the poles of the system. The results are shown below

ans =

     0.0000 + 0.0000i
     0.0000 + 0.0000i
    -0.0670 + 0.4265i
    -0.0670 - 0.4265i
    -0.1071 + 0.0000i
    -0.0350 + 0.0000i
     0.0000 + 0.0000i

It seems that there are no eigenvalues with positive real parts.

Even if the system is unstable, it does not necessarily mean its step response will diverge. Here is an example

>> A = [1 1; 0 -1]; B = [0;1]; C= [0, 1]; D=0;
>> sys = ss(A,B,C,D);
>> step(sys)

The step response is convergent. However, the system is unstable since A contains 1 as an eigenvalue.

[Daniela]

To me it shows the system has the pole 6.6627e-7.

But the example with the unstable system really helped. Thanks!

[Daan van Vrede]

@Yu Yiang, regarding your statement: "Even if the system is unstable, it does not necessarily mean its step response will diverge. Here is an example".

In this case there is a pole/zero cancellation. The positive (i.e. unstable) pole cancels out. Taking that into account, the system is actually stable (since only the -ve pole is left over) and hence there is no surprise that the step response does not diverge. The matlab function minreal() can be used to evaluate pole/zero cancellation: https://nl.mathworks.com/help/control/ref/minreal.html

[Rohan Madnani]

Is the converse also true? "Even if a system is stable, it does not necessarily mean that its step response will converge", because I am facing this issue. My tranfer function is stable as per Bode plot, but the step response is diverging. Kindly help