## Topological Considersations in Stability Analysis |17 November 2020|
In this post we discuss very classical, yet, highly underappreciated topological results. We will look at dynamical control systems of the form where is some finite-dimensional embedded manifold. As it turns out, topological properties of encode a great deal of fundamental control possibilities, most notably, if a To start the discussion, we need one important topological notion.
To say that a space is contractible when it has no holes (genus ) is wrong, take for example the sphere. The converse is true, any finite-dimensional space with a hole cannot be contractible. See Figure 1 below.
## Topological Obstructions to Global Asymptotic StabilityIn this section we highlight a few (not all) of the most elegant topological results in stability theory. We focus on results without appealing to the given or partially known vector field . This line of work finds its roots in work by Bhatia, SzegĂ¶, Wilson, Sontag and many others.
This theorem by Sontag states that the underlying topology of a dynamical system, better yet, the space on which the dynamical system evolves, dictates what is possible. Recall that linear manifolds are simply hyperplanes, which are contractible and hence, as we all know, global asymptotic stability is possible for linear systems. However, if is not contractible, there does not exist a globally asymptotically stabilizable continuous-time system on , take for example any sphere. The next example shows that one should not underestimate ‘‘
Indeed, the solution to (1) is given by . Since this group consists out of two disjoint components there does not exist a matrix , or continuous nonlinear map for that matter, which can make a vector field akin to (1) globally asymptotically stable. This should be contrasted with the closely related ODE , . Even for the path connected component , The theorem by Sontag obstructs the existence of a continuous global asymptotically stable vector field. This because the group is not simply connected for (This follows most easily from establishing the homotopy equivalence between and via (matrix) Polar Decomposition), hence the fundamental group is non-trivial and contractibility is out of the picture. See that if one would pick to be stable (Hurwitz), then for we have , however . Following up on Sontag, Bhat and Bernstein revolutionized the field by figuring out some very important ramifications (which are easy to apply). The key observation is the next lemma (that follows from intersection theory arguments, even for non-orientable manifolds).
Clearly, we have to ignore -dimensional manifolds here. Important examples of this lemma are the -sphere and the rotation group as they make a frequent appearance in mechanical control systems, like a robotic arm. Note that the boundaryless assumption is especially critical here. Now the main generalization by Bhat and Bernstein with respect to the work of Sontag is to use a
Intuitively, vector bundles can be thought of as manifolds with vector spaces attached at each point, think of a cylinder living in , where the base manifold is and each fiber is a line through extending in the third direction, as exactly what the figure shows. Note, however, this triviality only needs to hold locally. A concrete example is a rigid body with , for which a large amount of papers claim to Now we can state their main result
Indeed, compactness of can also be relaxed to not being contractible as we saw in the example above, however, compactness is in most settings more tangible to work with.
## Global IsomorphismsOne might wonder, given a vector field over a manifold , since we need contractibility of for to have a global asymptotically stable equilibrium points, how interesting are these manifolds? As it turns out, for , which some might call the high-dimensional topological regime, contractible manifolds have a particular structure. A topological space being
This definition is rather complicated to parse, however, we can give a more tangible description. Let be a non-compact topological space, then, is said to have
Now, most results are usually stated in the context of a piecewise linear (PL) topology. By appealing to celebrated results on triangulization (by Whitehead) we can state the following.
You might say that this is all obvious, as in Figure 3, we can always do it, also for lower-dimensions. However, there is famous counterexample by Whitehead, which provided lots of motivation for the community: ‘‘ In fact, this example was part of the now notorious program to solve the PoincarĂ© conjecture. All in all, this shows that contractible manifolds are not that interesting from a nonlinear dynamical systems point of view, compact manifolds is a class of manifolds providing for more of a challenge. ## Remark on the Stabilization of SetsSo far, the discussion has been on the stabilization of (equilibrium)
In rather loose terms, being a weak deformation retract of means that one can continuously ‘‘retract’’ points from and morph them into the set . So, this works for and , due to being punctured. See p.200 for more information on retracts. In Figure 4 below we show that since this lemma is mostly concerned with the local topology, the global topology is less important. For , the set is not a deformation retract of the set such that can never be a region of attraction, this should be contrasted with the setting on the Torus .
The beauty of all of this is that even if you are not so sure about the exact dynamical system at hand, the underlying topology can already provide for some answers. In particular, to determine, what can, or cannot be done. This post came about during a highly recommended course at EPFL. |