Lately, linear discrete-time control theory has start to appear in areas far beyond the ordinary. As a byproduct, papers start to surface which claim that stability of linear discrete-time systems is characterized by .
The confunsion is complete by calling this object the To continue, define two sets; and .
Since we have . Now, the main question is, Then , but , hence fails to hold. A setting where the aforementioned property of might help is Robust Control, say we want to assess if some algorithm rendered a compact convex set , filled with 's, stable. As highlighted before, we could just check if all extreme points of are members of , which might be a small and finite set. Thus, computationally, it appears to be attractive to consider over the generic . As a form of critique, one might say that is a lot We will stress that by approximating with , regardless of their volumetric difference, we are ignoring a full Then , and both and have .
We observe that indeed is contractive, for any initial condition on , we move strictly inside the sphere, whereas for , when starting from the same initial condition, we take a detour outside of the sphere before converging to . So although and have the same spectrum, they parametrize In our deterministic setting this would mean that we would confine our statespace to a (solid) sphere with radius , instead of .
Moreover, in linear optimal control, the resulting closed-loop system is usually not contractive.
Think of the infamous pendulum on a cart. Being energy efficient has usually nothing to do with taking the shortest, (update June 06): As suggested by Pedro Zattoni, we would like to add some nuance here. As highlighted in the first 12 minutes of this lecture by Pablo Parrilo, if is stable, then, one can always find a (linear) change of coordinates such that the transformed system matrix is contractive. Although a very neat observation, you have to be careful, since your measurements might not come from this transformed system. So either you assume merely that is stable, or you assume that is contractive, but then you might need to find an additional map, mapping your states to your measurements. : Recall, . Then, let be some eigenvector of . Now we have . Since this eigenvalue is arbitrary it follows that . Then, , yet, . For the full set of conditions on such that see this paper by Goldberg and Zwas. |