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Often when I read papers on control systems, I see a reference to a state space being defined as something in R(n) manifold.

My question is why is state space referred with respect to manifolds? Defining a state space as something in R(n) Euclidian space seems to give me a good picture of what a state space is. What more information is conveyed by referring a SS to a manifold over referring it to an Euclidian space? My understanding of a manifold is as in the wiki page.

Also, another term I find is when talking about a linear transformation between one R(n) space to another R(n) space is the term "Diffeomorphism". I googled and I understand Diffeomorphism is some form sort of transformation between spaces with one-to-one relation. What extra information does terms like "Diffeomorphism" mean when a "linear invertible transformation" could be used in its place?

This is my first question in Math stackexchange and I am not sure if this is a right forum. Let me know if this question should go here.

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  • $\begingroup$ It depends a lot on the paper. In some cases, these terms actually are meaningful (e.g. the system may have constraints, and thus live on a submanifold of the full space that is modeled). Sometimes, it just makes everything sound fancier... any paper in particular puzzling you? $\endgroup$
    – Steve Heim
    Sep 26, 2019 at 20:28
  • $\begingroup$ No paper in particular but there had been many. I am working in linearization of a non-linear system using feedback. In these papers I typically find terms like manifolds and diffeomorphism used typically in introduction when the non-linear systems are introduced $\endgroup$
    – jrvinayak
    Sep 27, 2019 at 8:05
  • $\begingroup$ Welcome. +1 Most questions here are more focused than yours, but yours is excellent. It was clear enough to get an excellent answer that satisfied you. $\endgroup$ Sep 27, 2019 at 16:38

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Since an $n$-dimensional Riemannian manifold can be identified with a subset of $\Bbb R^m$ for some $m \ge n$, you could indeed always consider state spaces to be such a subset. But there are some issues with doing so.

First of all $m$ usually has to be larger than $n$, which means you end up with extra variables needed to represent your state, and are forced to deal with the formulas needed to express the values of those extra variables in terms of a subset of size $n$ that are allowed to wander freely. For example, if your state space happens to be a sphere, then there are two degrees of freedom for your state to change. But you can't have a sphere in $\Bbb R^2$. At a minimum, it has to be in $\Bbb R^3$, and you have three variables, $x, y, z$, and an extra relationship between them: $x^2 + y^2 + z^2 = 1$.

Of course in this specific example, you could use spherical coordinates $\rho, \theta, \phi$, where the extra relationship is trivial, $\rho = 1$, so you just ignore $\rho$ and consider only the variables $\theta, \phi$. But if you need to differentiate, there is a problem - most places are fine, but not at $\phi = 0$ or $\phi = \pi$. The coordinate system is singular at these points, even though the sphere itself is not. You have to switch to a different coordinate system to differentiate here. --- And in doing so, you are no longer working with a subset of $\Bbb R^n$, but rather, with a Riemannian manifold.

Switching coordinate systems brings in a lot of mechanics - transformations, and how those transformations affect tangent vectors and linear transformations of those vectors and differentiation and integration. When you follow it this way, there is a lot to keep track of. This is the reason manifolds were invented - to provide a general framework for expressing these sorts of constructions, and proving general results about them instead of having to prove them again and again for each specific case. To study what is possible, and why, and was is not possible, and why. To build up experience and intuition in these situations.

State spaces tend to come with properties that naturally make them manifolds. You can move between nearby states by adjusting the values of a finite number $n$ of parameters. Very commonly, there exists a set of parameters that is capable of smoothly representing every possible state. When this happens, then the state space is just a nice open subset of $\Bbb R^n$, and that is all you need. But there are occasional state spaces where no set of parameters is capable of representing every possible state. If you following a path through state space, at some point you need to switch from one set of parameters to another. Such state spaces are naturally manifolds. The different sets of parameters are different coordinate systems (aka, "charts"), and when you patch them together, that is exactly what a manifold is. There is often also a way of expressing distance in this state space, which is what makes the manifold "Riemannian" (named after the great mathematician Bernhard Riemann).

If you insist on putting this manifold inside $\Bbb R^m$, then you find that you are no longer studying just your state space (intrinsic properties), but also how that state space was stuffed ("embedded" is the term-of-art) into $\Bbb R^m$ (extrinsic properties). It is actually quite difficult to separate out the properties of the state space from those of the embedding. For example, I have had more than one conversation about why the circle has no curvature. (The intrinsic concept of curvature requires 2 dimensions to support, but the circle is 1 dimensional. The curvature you see in the circle is not an intrinsic property, but rather an extrinsic one foisted on it by its embedding in the plane.)

So if you want to study state spaces in general, without picking up properties that are actually foreign to them, you need to think of them as manifolds: surfaces that may undulate or turn back on themselves, or even twist as in a Moebius strip; and their higher dimensional counterparts.

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  • $\begingroup$ This is truely one of the most satisfying answers I've found in MSE! I feel like I read the introduction of a well written book. Thank you. $\endgroup$
    – Hermis14
    Dec 25, 2021 at 11:56
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It is very simples.

Every linear differential equation have a state-space model representation:

dx/dt = Ax+Bu

where x is the state-variable, u is the system's input, A and B are matrices.

For example, in a RL circuit, one would like to calculate voltage over L and current over R. Thus, x = [v ; i] (column vector) and A,B are regarding RL parameters. The state-space variable don't show the time explicitly, so we should plot it in a chart/graph with two axes ("v" and "i") wich is a parametrized curve f : R -> R² (time -> (v,i)).

You can thought about this like the following: the set of curves f are contained in a surface in Rn. automatically, this is a manifold. The matrix A is someking like a metric tensor that defines de curvature of the f curve. So, a central problem in electrical engineering is to calculate A so that the set of curves f are all inside some ball regarding some equivalence class set, wich implicates the behavior of the circuit under other equivalence set of inputs "u". This improves the controlling.

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