Fixed and Finite Time Stability

These notes summarise the results from Kunal Garg’s papers on Fixed and Finite Time Stability. The notation is changed slightly, to a form that I prefer. Any mistakes in these notes are almost certainly mine. Please let me know if you spot anything weird.

The key references are:

  • [Ref 1] “Characterization of Domain of Fixed-time Stability under Control Input Constraints”, Kunal Garg and Dimitra Panagou
  • [Ref 2] Kunal Garg’s PhD Thesis

Definitions

These definitions are from [Ref 2].

Consider the autonous system

\[\dot x = f(x)\]

with state \(x \in R^n\), with continuous function \(f : R^n \rightarrow R^n\) with \(f(0) = 0\). We will assume that the solution exists, and is unique.

The equilibrium point \(x= 0\) is finite-time stable (FTS), if the equlibirum point is stable, and there exists and there exists an open neighborhood \(N\) of the origin, such that for all \(x(0) \in N \backslash \{0\}\) there exists a \(T(x(0)) \in (0, \infty)\) such that

\[\lim_{t\rightarrow T(x(0))} x(t) = 0\]

The equilibrium point is fixed time stable (FxTS) from a domain \(D\), if the equilibrium point is stable and if \(T \in (0, \infty)\) can be chosen independently of \(x(0)\) such that for all \(x(0) \in D\)

\[\lim_{t\rightarrow T} x(t) = 0\]

Theorem 1 [FTS]

Taken from [Ref 2].

Suppose there exists a positive definite, continuously differentiable function \(V : D \rightarrow R\), \(\alpha >0\), \(\mu > 1\) such that

\[\dot V(x) \leq -\alpha V(x)^{1-1/\mu}, \quad \forall x \in N \backslash \{0\}\]

where \(N \subset D\) is an open neighborhood of the origin.

Then the origin is finite time stable from the domain \(D\), with a settling time

\[T(x_0) \leq \frac{\mu}{\alpha} V_0^{1/\mu}\]

where \(V_0 = V(x(0))\).

Notice that in the limit that \(V_0 \rightarrow \infty\), the settling time also approaches infinity, i.e., \(T(x_0) \rightarrow \infty\). Therefore, no fixed-time stability conditions can be provided.

Rough Proof:

We can use a comparison system \(h(t)\), where \(h(0) = h_0 = V_0\). Then we impose

\[\dot h = -\alpha h^{1-1/\mu}\]

which implies that \(h(t) \geq V(x(t))\) for all \(t \geq 0\). Integrating both sides of the equation we get

\[\begin{align*} \int_{h_0}^{h_f} \frac{1}{ h^{1-1/\mu}} dh &= -\alpha \int_{0}^{t_f} dt\\ [\mu h^{1/\mu} ]|_{h_0}^{h_f} &= -\alpha t_f\\ h_f^{1/\mu} - h_0^{1/\mu} &= -\frac{\alpha}{\mu} t_f \end{align*}\]

therefore, if \(h_f \rightarrow 0\) and \(t_f \geq T\), we have

\[T \leq \frac{\mu}{\alpha} h_0^{1/\mu}\]

Theorem 2 [FxTS 1]

Taken from [Ref 2].

Suppose there exists a positive definite, continuously differentiable, radially unbounded function \(V: R^n \rightarrow R\) such that

\[\dot V(x) \leq -\alpha_1 V(x)^{\gamma_1} - \alpha_2 V(x)^{\gamma_2}, \quad \forall x \in R^n \backslash \{0\}\]

where \(\alpha_1 > 0, \alpha_2 > 0\), \(\gamma_1 >1\), \(0 < \gamma_2 < 1\).

Then the origin is globally fixed time stable, with settling time

\[T \leq \frac{1}{\alpha_1(\gamma_1 - 1)} + \frac{1}{\alpha_2(1-\gamma_2)}\]

By selecting \(\gamma_1 = 1 + 1/\mu\) and \(\gamma_2 = 1 - 1/\mu\), a (strictly) tigther upperbound on \(T\) is

\[T \leq \frac{\mu}{\sqrt{\alpha_1 \alpha_2}} \frac{\pi}{2}\]

Proof:

The proof will be a special case of the next theorem.

Theorem 3 [FxTS 2]

Taken from [Ref 2].

Let \(V: R^n \rightarrow R\) be a continuously differentiable, positive definite, radially unbounded function satisifying

\[\dot V(x) \leq -\alpha_1 V(x)^{1+1/\mu} - \alpha_2 V(x)^{1-1/\mu} + \delta_1 V(x), \quad \forall x \in R^n \backslash \{0\}\]

where \(\alpha_1 > 0, \alpha_2 >0 , \mu > 1, \delta_1 \in R\).

Then there exists a neighborhood \(D \subset R^n\) of the origin such that

  • If \(\frac{\delta_1}{2 \sqrt{\alpha_1 \alpha_2}} \leq 0\), the origin is globally fixed time stable, (i.e., \(D = R^n\)) with settling time
\[T \leq \frac{\mu}{\sqrt{\alpha_1 \alpha_2}} \frac{\pi}{2}\]
  • If \(0 \leq \frac{\delta_1}{2 \sqrt{\alpha_1 \alpha_2}} < 1\), the origin is globally fixed time stable, (i.e., \(D = R^n\)) with settling time
\[T \leq \frac{\mu}{\alpha_1 k_1} \left( \frac{\pi}{2} - \tan^{-1} k_2 \right)\]
  • If \(\frac{\delta_1}{2 \sqrt{\alpha_1 \alpha_2}} \geq 1\), the origin is fixed time stable from a domain
\[D = \left\{ x : V(x)^{1/\mu} \leq k \left(\frac{\delta_1 - \sqrt{\delta_1^2 - 4 \alpha_1 \alpha_2}}{2 \alpha_1}\right)\right\}\]

with settling time

\[T = \frac{\mu}{\alpha_1 (b-a)} \left( \log \left(\frac{b-ka}{a-ka}\right)- \log\left(\frac{b}{a}\right)\right)\]

where \(k \in (0, 1)\) can be chosen arbitrarily, and \(a < b\) are the solutions of \(\alpha_1 z^2 - \delta_1 z + \alpha_2 = 0\) and \(k_1 = \sqrt{\frac{4 \alpha_1 \alpha_2 - \delta_1^2}{4 \alpha_1^2}}\) and \(k_2 = \frac{-\delta_1}{\sqrt{4 \alpha_1 \alpha_2 - \delta_1^2}}\)

I will try to simplify this theorem, purely algebraically.

Theorem 4 [FxTS 3]

Let \(V: R^n \rightarrow R\) be a continuously differentiable, positive definite, radially unbounded function satisifying

\[\dot V(x) \leq -\alpha V(x)^{1+1/\mu} - \alpha V(x)^{1-1/\mu} + 2 \alpha r V(x), \quad \forall x \in R^n \backslash \{0\}\]

where \(\alpha > 0, \mu > 1, r \in R\).

Then there exists a neighborhood \(D \subset R^n\) of the origin such that

  • If \(r \leq 0\), the origin is globally fixed time stable, (i.e., \(D = R^n\)) with settling time
\[T \leq \frac{\mu \pi}{2 \alpha}\]
  • If \(0 \leq r < 1\), the origin is globally fixed time stable, (i.e., \(D = R^n\)) with settling time
\[T \leq \frac{\mu}{\alpha \sqrt{1 - r^2}} \left( \frac{\pi}{2} + \tan^{-1} \left(\frac{r}{\sqrt{1 - r^2}}\right) \right)\]
  • If \(r \geq 1\), the origin is fixed time stable from a domain
\[D = \left\{ x : V(x)^{1/\mu} \leq k \left(r - \sqrt{ r^2 - 1}\right)\right\}\]

with settling time

\[T = \frac{\mu}{2 \alpha \sqrt{r^2-1}} \log \left(\frac{2 k}{1-k} r \left(\sqrt{r^2-1}-r\right) + \frac{1+k}{1-k}\right)\]

where \(k \in (0, 1)\) can be chosen arbitrarily.

Notice that larger \(k\) will have larger domains of attraction but larger settling times. As \(k\rightarrow 1\), \(T\rightarrow \infty\). As \(k\rightarrow 0\), \(T \rightarrow 0\).