I'm having some problems figuring out a system that it's stable for some values of its physical properties but unstable for others.
Even a basic example would help a lot, a dimension 2 is totally fine.
I do not have problems figuring out what mathematically robust stability means, but I can't find any physical example.
actually what I'm looking for is more likely a system that has a finite robust stability margin. outside references are good too, thanks in advance.
It depends what is meant by "stable". A pot of water at room temperature is "stable" in that all the water is in liquid form. If you boil the water, it becomes "unstable" because the water is in flux between liquid and gaseous form. The physical property in flux is temperature. You could also keep the temperature constant and vary the air pressure if the system. Drop air pressure low enough and the water begins to boil.
A stable property of the same system could be salinity. It doesn't mater how much salt you add, it's not going to change the water to gaseous form. Again, it depends on what you mean by "stable".
You can actually cook up your own system for testing. Notice that most physical systems that would be intuitive for you are inherently stable. The unstable ones are deliberately destabilized for example air-bearing-based platforms etc. that needs active control from the moment of initialization. Hence physical connection to an unstable system is often not so intuitive (unless you have the model of JSF).
Instead you can test yourself with the basics of feedback control. The first message that every control engineer is taught from the start is that feedback is useful for a lot of stuff that I'll skip but one extremely important disadvantage is that it can destabilize a perfectly stable system.
Here is a very simple matlab script that shows how high gain feedback can lead to unstable system. The model is the typical mass-spring-damper system (or an RLC circuit if you wish)
m = 30;b = 2;k = 500;
G = tf(1,[m,b,k]);
Act = tf(1,[1 10]);
K = 150;
P = feedback(Act*G,K);
if isstable(P)
disp('Nominally stable')
end
b_mesh = 0.005:0.1:2;
k_mesh = 10:50:1000;
stab_flag=zeros(length(b_mesh),length(k_mesh));
for i=1:length(b_mesh)
for j = 1:length(k_mesh)
G = tf(1,[m,b_mesh(i),k_mesh(j)]);
P = feedback(Act*G,K);
stab_flag(j,i) = isstable(P);
end
end
[X,Y] = meshgrid(b_mesh,k_mesh);
surf(X,Y,stab_flag)
xlabel('damping'),ylabel('stiffness')
As we see here with z axis being the boolean stable or unstable, for small enough damping values our high gain feedback steers our system to instability. Notice that it is the actuator dynamics that I've introduced that makes this example work (which is another message for the undergrad readers if they stumble upon this). If you remove Act or make it equal to 1, it will not be unstable (theoretically).
Hence the message is that this particular controller does not robustly stabilize the system for certain damping values close to zero.
Or the closed-loop system P is not robustly stable.
Most natural systems are robustly stable (as you'd expect, because the less robust ones probably died off! haha). However, while it may feel contrived, it is easy to come up with a physical albeit man-made example. Consider a bar pivoted at its center and with a mass that can be placed anywhere along its length.
http://postimg.org/image/d7xhi84qp/
At the instance shown, it is clear that for the system parameter r, if
r > L/2 stable
r < L/2 unstable
Of course, saying "stable" means nothing without reference to equilibrium points. Imagine the pivoted bar with the mass hanging downwards at its nice stable equilibrium point. As you vary r, the stability of this equilibrium point changes. It can go from being asymptotically stable, to a center, to unstable just by varying r.
As for a practical example. Well, I'd say most "good ideas" are stable in the parameters you'd expect to adjust, and so any example of a "practical" robustly unstable system will just sound like a bad idea. However, perhaps one should think not of the parameters you'd expect to adjust, but rather the ones you just don't know with high certainty.
How about the inertia matrix of a rotating space craft? If you are really wrong about what its values are, you could perform a maneuver that goes unstable because it forced rotation about the intermediate rotation axis.
How about the location of the center of lift and center of mass of an aircraft? If the center of lift isn't behind the center of mass, pitch control will become unstable (fighter jets love doing this for high speed maneuvers).
Also, your question is not exactly software related... Find a control-theory forum instead!
