Asymptotic Stability for Humans
Wednesday, 27 April 2022 (#f4a288b)
Wednesday, 27 April 2022 (#f4a288b)
🚧 TODO: 🚧 Use historical perspective to show that the first concept that emerges was attractivity (named as “stability” then). Try to put the introduction of pathological “attractive-but-not-stable” in this historical picture (Vinograd (1957): https://zbmath.org/?q=an%3A0078.08001.). Use the historical perspective as an indicator of the level of difficulty (later is harder). Also use “common references” such as Wikipedia: https://en.wikipedia.org/wiki/Lyapunov_stability
🚧 TODO: 🚧 Emphasize that our definition makes clear the attractivity concept is 90% of the way and need only a small strenghtening, of the same flavor. We can’t say the same for the classic definition (at the very least, this is much less obvious!)
Asymptotic stability is a cornerstone of control theory and engineering. There is little doubt in my opinion that if only one notion among stability, attractivity and asymptotic stability, should be taught to control engineers, it should be asymptotic stability. Therefore, asymptotic stability, not attractivity nor stability, should be the only focus of time and complexity-bound control engineering lectures. As heartbreaking as this conclusion can be, most experienced lecturers know that chosing means eliminating (“choisir, c’est renoncer”1).
However, the classic approach (see e.g. Khalil (2002) p. 112, Sontag (1998) p. 211) introduces asymptotic stability as a strengthening of the concept of stability, with a composite definition that required stability and attractivity. It has several didactic drawbacks:
It requires to understand the concept of stability, which is arguably hard (at the very least much harder than attractivity). Using it as a foundation to define asymptotic stability makes automatically asymptotic stability at least equally as hard.
Using a composite definition for asymptotic stability makes it a derived concept, which appears as a second-class citizen of control theory instead of the first-class citizen that it should be.
This definition fails to address in simple terms what are in my opinion the shortcoming of the concept of attractivity and why it needs to be strengthened: a catastrophic failure to ensure a common speed of convergence, even for arbitrary close initial states.
In this document we follow a different path: we introduce attractivity, expose it shortcomings, then introduce a definition of asymptotic stability as an independent concept which is
an obvious strenghtening of attractivity (and clearly solves its issues),
of reasonable complexity (harder than attractivity, but simpler than stability),
equivalent to the classic definition.
To the best of my knowledge, my definition is original, but I have not performed an extensive research on this subject. Suffice to say that I have never been exposed to it in my earlier studes; if the concept is not new (I seriously doubt it is) at the very least it is not in my opinion popular enough given its didactic potential.
🚧 TODO: 🚧 explain document meant for lecturers, not students. Relate this to its organisation (relatively standalone, level of details, videos, extra work to adapt, not “off-the-shelf” use, etc.). And thus explain the plan of the document (with links).
An (time-invariant) vector field is a \(\mathbb{R}^n\)-valued function \(f\) whose domain \(\mathrm{dom} \, f\) is an open subset \(U\) of \(\mathbb{R}^n\) (for some natural number \(n\)): \[ f: U \subset \mathbb{R}^n \to \mathbb{R}^n, \qquad \partial U \cap U= \varnothing. \]
A vector field defines a unique (time-invariant) dynamical system denoted \(\dot{x} = f(x)\). Any element of the domain of definition of \(f\) is a (valid) state of the dynamical system.
An vector field \(f\) and a valid state \(x_0\) define a unique initial-value problem (IVP) denoted \(\dot{x} = f(x)\), \(x(0) = x_0\).
A (forward) solution of the IVP \(\dot{x} = f(x)\), \(x(0) = x_0\) is an absolutely continuous and \(\mathbb{R}^n\)-valued function \(x\) defined on \(\left[0, \tau\right[\) for some \(\tau \in \left]0, +\infty\right]\) such that \[ x(t) = x_0 + \int_0^t f(x(s)) \, ds, \qquad 0 \leq t < \tau. \] When we wish to emphasize the role of the initial state \(x_0\), we denote the solution \(x(t, x_0)\) and then call the application \(x\) the flow of the dynamical system. We also consider solutions associated with a set \(X_0 \subset \mathrm{dom} \, f\) of initial states: we denote \(x(t, X_0)\) the image of \(X_0\) by the flow at time \(t\): \[ x(t, X_0) := \{x(t, x_0) \; | \; x_0 \in X_0\}. \]
A solution of the IVP \(\dot{x} = f(x)\), \(x(0) = x_0\) is maximal if no other solution is a strict extension of it.
Need to say explicitly why we are talking about this somewhere. We are actually using well-posedness in our equivalence proofs.
A dynamical system \(\dot{x} = f(x)\) is well-posed if for any initial state the corresponding IVP has a unique maximal solution \(x\) which depends continuously on the initial state: if \(x(t, x_0)\) is defined on \([0, \tau]\), then for \(x_0'\) close enough to \(x_0\), \(x(t, x'_0)\) is also defined on \([0, \tau]\) and \[ \lim_{x_0' \to x_0} \left[\max_{t \in [0, \tau]} \|x(t, x'_0) - x(t, x_0)\|\right] = 0. \]
If the vector field \(f\) is continuously differentiable, \[ \frac {\partial f_i}{\partial x_j} \mbox{ exists and is continuous}, \qquad i, j=1,\dots, n \] then the dynamical system \(\dot{x} = f(x)\) is well-posed.
An equilibrium \(x_*\) of a dynamical system \(\dot{x} = f(x)\) is a state such that \(x: t \in \left[0, +\infty\right[ \to x_*\) is a solution of \(\dot{x} = f(x)\), \(x(0) = x_*\).
Equivalently, a state \(x_*\) is an equilibrium if and only if \(f(x_*) = 0\).
An equilibrium \(x_*\) of a well-posed system \(\dot{x} = f(x)\) is (globally) attractive if for any \(x_0\) in the domain of definition of \(f\), the solution \(x(t, x_0)\) (exists for any \(t\geq 0\) and) tends to \(x_*\) as \(t\) tends to \(+\infty\). \[ \forall \, x_0 \in \mathrm{dom} \, f, \; \lim_{t \to +\infty} x(t, x_0) = x_*. \]
The time-invariant system \[\begin{equation} \dot{x}^1 = -2x^1+x^2, \; \dot{x}^2 = -2x^2+x^1, \qquad x = (x^1, x^2) \label{LTI} \end{equation}\] has a single equilibrium \(x_* = (0, 0)\). The auxiliary variables \(y^1:= x^1+x^2\) and \(y^2:=x^1 - x^2\) satisfy the decoupled differential equations \(\dot{y}^1 = -y^1\) and \(\dot{y}^2 = -3 y^2\), the solution associated to an initial state \(x_0 = (x_0^1, x_0^2)\) is \[ y^1(t) = (x_0^1 + x_0^2)e^{-t}, \; y^2(t) = (x_0^1 - x_0^2)e^{-3t} \] and thus \[\begin{equation} \left[ \begin{array}{c} x^1 \\ x^2 \end{array} \right](t) = 1/2 \left[ \begin{array}{c} x_0^1 + x_0^2 \\ x_0^1 + x_0^2 \end{array} \right]e^{-t} + 1/2 \left[ \begin{array}{c} x_0^1 - x_0^2 \\ x_0^2 - x_0^1 \end{array} \right]e^{-3t}. \label{LTI-solutions} \end{equation}\] It is therefore clear that for every initial state \(x_0\), we have \[ \lim_{t \to +\infty} x(t, x_0) = x_*. \] Hence, the equilibrium is attractive.
The equations \[\begin{equation} \dot{x}_1 = \frac{x_1^2(x_2 - x_1)+x_2^5}{(x_1^2+x_2^2)[1+ (x_1^2 + x_2^2)^2]} \label{Vinograd-1} \end{equation}\]
\[\begin{equation} \dot{x}_2 = \frac{x_2^2 (x_2 - 2x_1)}{(x_1^2+x_2^2)[1+ (x_1^2 + x_2^2)^2]} \label{Vinograd-2} \end{equation}\]
describe a time-invariant system introduced by Vinograd (1957). The origin \((0,0)\) is the single equilibrium of the system and it is attractive.
🚧 TODO 🚧
Let \(a, b\) be two points of \(\mathbb{R}^n\) and \(A, B\) be two sets of \(\mathbb{R}^n\). The distance between \(a\) and \(b\) is defined by \[ d(a, b) := \|b - a\|, \] the distance between \(A\) and \(b\) and \(a\) and \(B\) respectively by \[ d(A, b) := \sup_{a \in A} d(a, b)\, \qquad d(a, B) := \sup_{b \in B} d(a, b) \] and the distance between \(A\) and \(B\) by \[ d(A, B) := \inf_{a \in A} d(a, B)= \inf_{a \in A} \inf_{b \in B} d(a, b). \]
However, this distance between two sets does not measure in an adequate way how far the two sets differ from each other. For example if \(A\) and \(B\) share points then \(d(A, B)=0\) even if \(A\) is very small and \(B\) very large.
For this task, the (Pompeiu-)Hausdorff distance is more appropriate \[ d_H(A, B) := \max \left\{ \sup_{a \in A} d(a, B), \sup_{b \in B} d(A, b) \right\} \] (see e.g. Kuratowski (1966), p. 214). It measures how far the set \(A\) deviates from the set \(B\) and how far the set \(B\) deviates from the set \(A\) and returns the largest of these numbers. If we consider only non-empty bounded and closed (compact) subsets of \(\mathbb{R}^n\), this Hausdorff “distance” is actually a proper distance.
We can define limits of sets using the Hausdorff distance: we say that a time-dependent set \(A(t)\) tends to a set \(B\) when \(t\) tends to \(+\infty\), noted \[ \lim_{t \to +\infty} A(t) = B, \] whenever \(\lim_{t \to +\infty} d_H(A(t), B) = 0.\)
To study asymptotic stability, we only need to consider the Hausdorff distance \(d_H(A, B)\) when \(B\) has a single element \(b\). In this case, the Hausdorff distance has a simple geometric interpretation.
The Hausdorff distance between \(A\) and \(\{b\}\) is the radius of the smallest circle centered at \(b\) which encloses \(A\) \[ d_H(A, {b}) = \sup_{a \in A} \|a - b\| = \min \, \{r \geq 0 \; | \; \mbox{for any }\, a \in A, \, \|a - b\| \leq r\} \] (or \(+\infty\) if \(A\) is unbounded i.e. no circle of finite radius encloses it).
If \(B =\{b\}\) , we have \[ \sup_{a \in A} d(a, B) = \sup_{a \in A} d(a, b) \geq \inf_{a \in A} d(a, b) = d(A, b) = \sup_{b' \in B} d(A, b'), \] thus \[ \max \left\{ \sup_{a \in A} d(a, B), \sup_{b \in B} d(A, b) \right\} = \sup_{a \in A} d(a, b) \] and consequently \[ d_H(A, \{b\}) = \sup_{a \in A} d(a, b) = \sup_{a \in A} \|a - b\|. \]
An equilibrium \(x_*\) of a well-posed system \(\dot{x} = f(x)\) is (globally) asymptotically stable if for any state \(x_0\) there is a (small enough) closed ball of states of positive radius \(r\) centered at \(x_0\) \[ \bar{B}(x_0, r) := \{x \in \mathbb{R}^n \; | \; \|x - x_0\| \leq r\} \] whose image by the flow at time \(t\) (exists for every \(t\geq 0\) and) tends to \(\{x_*\}\) as \(t\) tends to \(+\infty\). \[ \forall \, x_0 \in \mathrm{dom} \, f, \; \exists \, r > 0, \; \lim_{t \to +\infty} x(t, \bar{B}(x_0, r)) = \{x_*\}. \]
A set \(A\) of \(\mathbb{R}^n\) is compactly included in a set \(B\) of \(\mathbb{R}^n\), noted \(A \Subset B\), if \(A\) is bounded and its closure \(\bar{A} := \{x \in \mathbb{R}^n \; | \; d(x, A)=0\}\) is included in \(B\).
An equilibrium \(x_*\) of a well-posed system \(\dot{x} = f(x)\) is (globally) asymptotically stable if and only if the image of any set of states compactly included in \(\mathrm{dom} \, f\) by the flow at time \(t\) (exists for every \(t\geq 0\) and) tends to \(\{x_*\}\) as \(t\) tends to \(+\infty\). \[ \forall \, X_0 \Subset \mathrm{dom} \, f, \; \lim_{t \to +\infty} x(t, X_0) = \{x_*\}. \]
Since a closed ball of states is compactly included in \(\mathrm{dom} \, f\), the above criteria clearly implies asymptotic stability.
Conversly, let \(X_0\) be a set compactly included in \(\mathrm{dom} \, f\). For every \(x_0 \in \bar{X}_0,\) there is a positive radius \(r(x_0)\) such that \[ \lim_{t \to +\infty} x(t, \bar{B}(x_0, r(x_0))) = \{x_*\}. \] The collection of all open balls \(B(x_0, r(x_0))\), when \(x_0 \in \bar{X}_0\) is an open cover of the compact set \(\bar{X}_0\), thus there is a finite sub-collection \(B_1\), \(B_2\), \(\dots\), \(B_p\) such that \[ \bar{X}_0 \subset B_1 \cup B_2 \cup \dots \cup B_p. \] Since for any \(k=1,\dots, p\), we have \(\lim_{t \to +\infty} x(t, B_k) = \{x_*\}\), we have \[ \lim_{t \to +\infty} d_H(B_k, \{x_*\}) \to 0. \] Since \(X_0 \subset \bar{X}_0 \subset B_1 \cup B_2 \cup \dots \cup B_p\), \[ d_H(X_0, \{x_*\}) \leq d_H(\bar{X}_0, \{x_*\}) \leq \max_{k=1...p} d_H(B_k, \{x_*\}) \] and thus \[ \lim_{t \to +\infty} d_H(X_0, \{x_*\}) \to 0. \]
The origin is an asymptotically stable equilibrium of the system \[ \dot{x}^1 = -2x^1+x^2, \; \dot{x}^2 = -2x^2+x^1, \qquad x = (x^1, x^2) \] Indeed, the closed-form solution \(\eqref{LTI-solutions}\) yields \[ \|x(t, x_0)\| \leq 2 \|x_0\| e^{-t}. \] A set \(X_0\) is compactly included in \(\mathbb{R}^n\) if and only if it is a bounded subset of \(\mathbb{R}^n\). For any such set, \[ d_H(x(t, X_0), 0\}) \leq 2 \sup_{x_0 \in X_0} \|x_0\| e^{-t}. \] Hence \(x(t, X_0) \to \{0\}\) when \(t \to +\infty\): the origin is asymptotically stable.
🚧 TODO 🚧
Attractivity + Stability \(\Leftrightarrow\) Asymptotic Stability 🚧 TODO 🚧
🚧 TODO 🚧
Obvious (by design).
Let’s assume that the system is asymptotically stable. Let \(r_1 > 0\) such that the closed ball \(B_1\) of radius \(r_1\) centered at \(x_e\) is included in \(\mathrm{dom} \, f\). \[ B_1 := \{x \in \mathbb{R}^n \; | \; \|x - x_e\| \leq r_1 \} \subset \mathrm{dom} \, f. \]
It is bounded, its closure is included in \(\mathrm{dom} \, f\) and it is also a neighbourhood of \(x_e\). Since the the system is asymptotically stable, \(x(t, B_1)\) is defined for any \(t\geq 0\) and there is a \(\tau \geq 0\) such that for any \(t \geq \tau\), the image of \(B_1\) by \(x(t, \cdot)\) is included in itself \[ t\geq \tau \, \Rightarrow \, x(t, B_1) \subset B_1. \]
Additionally, the system is well-posed, hence there is a \(r_2 > 0\) such that for any \(x_0\) in the closed ball \(B_2\) of radius \(r_2\) centered at \(x_e\) is included in \(\mathrm{dom} \, f\) and any \(t \in [0, \tau]\), we have \(\|x(x_0, t) - x(x_e, t) \| \leq r_1.\) Since \(x_e\) is an equilibrium, \(x(t, x_e) = x_e\), thus \(\|x(x_0, t) - x_e \| \leq r_1.\) Equivalently, \[ 0\leq t \leq \tau \, \Rightarrow \, x(t, B_2) \subset B_1. \]
Note that since \(x(0, B_2) = B_2\), this inclusion yields \(B_2 \subset B_1\). Thus, for any \(t \geq 0\), either \(t\in [0, \tau]\) and \(x(t, B_2) \subset B_1\), or \(t\geq \tau\) and since \(B_2 \subset B_1\), we have \(x(t, B_2) \subset x(t, B_1) \subset B_1\).
To summarize our findings: we have established that for any \(r_1 > 0\) such that \(B_1 \subset \mathrm{dom} \, f\) there is a \(r_2 > 0\) such that \[ t\geq 0 \, \Rightarrow x(t, B_2) \subset B_1. \] Therefore that the system is stable.
ℹ️ We prove directly the stronger version of A.S.
Let \(X_0\) be a bounded set whose closure is included in \(\mathrm{dom} \, f\). We assume that additionally \(X_0\) is closed (otherwise we substitute the closure of \(X_0\) to \(X_0\)).
Let \(r_1 > 0\) and \(B_1\) be the closed ball of radius \(r_1\) centered on \(x_e\). Since the system is stable, there is a radius \(r_2 > 0\) such that for any \(t \geq 0\), \(x(t, B_2) \subset B_1\). Let \(r_3 := r_2/2\) and \(B_3\) be the ball of radius \(r_3\) centered on \(x_e\).
Let \(x_0 \in X_0\) ; since \(x_e\) is attractive, there is a \(\tau \geq 0\) such that for any \(t\geq \tau,\) \(x(t, x_0) \in B_3\). Since the system is well-posed, there is a radius \(r_4 > 0\) such that for any \(t \in [0, \tau],\) and any \(x_1\) in the ball of radius \(r_4\) centered on \(x_0\), \(\|x(t, x_1) - x(t, x_0)\| \leq r_3\). Consequently, \(\|x(\tau, x_1) - x(\tau, x_0)\| \leq r_3\) and thus \[ \begin{split} \|x(\tau, x_1) - x(\tau, x_e)\| &\leq \|x(\tau, x_1) - x(\tau, x_0)\| + \|x(\tau, x_0) - x(\tau, x_e)\| \\ &\leq r_3 + r_3 \\ &= r_2 \end{split} \] and thus \(x(\tau, x_1) \in B_2\). Since \(x(t, B_2) \subset B_1\) for any \(t \geq \tau\), for any such \(t\) we have \[ x(t, x_1) = x( t-\tau, x(\tau, x_1)) \in B_1. \]
At this stage, we have proven that for any \(r_1 > 0\) and any \(x_0 \in X_0,\) there is a \(\tau(x_0) > 0\) and a \(r_4(x_0)>0\) such that \[ \|x_1 - x_0\| \leq r_4(x_0) \; \wedge \; t\geq \tau(x_0) \; \Rightarrow \; \|x(t, x_1) - x_e\| \leq r_1. \] The collection of open balls centered on \(x_0\) and of radius \(r_0(x_0)\), indexed by \(x_0 \in X_0\) is an open cover of the closed set \(X_0\), thus there is a finite collection \(x_0^1, \, x_0^2, \dots, \, x_0^m\) of points of \(X_0\) such that the open balls centered on \(x_0^k\) with radius \(r_4(x_0^k)\) cover \(X_0\). Consequently, for any \(x_0 \in X_0\), there is a \(k \in \{1, \dots, m\}\) such that \(\|x_0 - x_0^k\| \leq r_4(x_0^k)\) and thus \(\|x(t, x_0) - x_e\| \leq r_1\) when \(t \geq \tau(x_0^k)\). Consequently, for any \(x_0 \in X_0\), if \[ t\geq \tau := \max_{i=1,\dots,m} \tau(x_0^k), \] then \(\|x(t, x_0) - x_e\| \leq r_1\). Thus, the equilbrium is asymptotically stable.
↩︎L’erreur de ma vie fut dès lors de ne continuer longtemps aucune étude, pour n’avoir su prendre mon parti de renoncer à beaucoup d’autres. — N’importe quoi s’achetait trop cher à ce prix-là, et les raisonnements ne pouvaient venir à bout de ma détresse. Entrer dans un marché de délices, en ne disposant (grâce à Qui ?) que d’une somme trop minime ; en disposer ! Choisir, c’était renoncer pour toujours, pour jamais, à tout le reste — et la quantité nombreuse de ce reste demeurait préférable à n’importe quelle unité.
André Gide, Les nourritures terrestres
