Consistent Discretization of a Class of Predefined-Time Stable Systems

The following definition was inspired on the comparison class $\mathcal{K}$ functions (Kellett, 2014, Definition 1)

Consider (the scalar form of) the class of systems introduced by Aldana-López et al. (2019) and Jiménez-Rodríguez et al. (2019)

where $x:\mathbb{R}_{\geq 0}\to\mathbb{R}$ is the state of the system, $\rho_{1}>0$ and $0\leq\rho_{2}<1$ are tunable parameters, $\kappa\in\mathcal{K}^{1}$ is continuously differentiable, and $\text{sign}\left(\cdot\right)$ stands for the signum function given by

From the theory of ordinary differential equations, (1) is a separable first-order equation, whose solution $x(t)$ satisfies

Applying the change of variable $\xi\to\kappa(\left\lvert\xi\right\rvert)^{1-\rho_{2}}\text{sign}\left(\xi\right)$ in the left side integral and integrating both sides of the above equation, it yields

for $\kappa(\left\lvert x^{0}\right\rvert)^{1-\rho_{2}}-\frac{t}{\rho_{1}}\geq 0$. From the above, the solution of (1) is

where $\kappa^{-1}$ stands for the inverse of the function $\kappa\in\mathcal{K}$.

Hence, one can conclude that the origin $x=0$ of (1) is finite-time stable and the settling-time function is $T(x^{0})=\rho_{1}\kappa\left(\left\lvert x^{0}\right\rvert\right)^{1-\rho_{2}}$. But the settling-time function $T(x^{0})=\rho_{1}\kappa\left(\left\lvert x^{0}\right\rvert\right)^{1-\rho_{2}}<\rho_{1}$ is bounded, then the origin $x=0$ of (1) is, in fact, fixed-time stable.

Moreover, for any desired upper bound of the settling-time function $T_{c}$, there exists an appropriate selection of the parameter $\rho_{1}=T_{c}$ such that the convergence time will always be less than $T_{c}$ for any initial condition $x^{0}$. This property is known as predefined-time stability (see Appendix A).

Consider the continuous-time system (1) and let $x_{k}=x(kh)$, where $h>0$ is the sampling step size, and $k\in\mathbb{N}$.

To obtain the exact discretization of system (1), one can apply the same procedure as in Subsection 2.2, integrating the left side between the samples $x_{k}$ and $x_{k+1}$, and the right side between $kh$ and $(k+1)h$, respectively, obtaining

Recall the transformation

used in Subsection 2.2 to find the solutions of (1). This transformation is invertible, and its inverse $y_{\rho_{2}}^{-1}:(-1,1)\to\mathbb{R}$ is given by

Using (4) in the left side integral of (3), one obtains

Hence, integrating both sides of the above expression and having in mind that $\text{sign}\left(x_{k+1}\right)=\text{sign}\left(x_{k}\right)$ for the first-order system (1), the above equation turns into

Finally, let $\alpha\in\mathbb{R}_{+}$ and consider the discontinuous function $\theta_{\rho_{2},\alpha}:\mathbb{R}\to(-\alpha-1,\alpha+1)$ defined by $\theta_{\rho_{2},\alpha}(x)=\left(\kappa(\left\lvert x\right\rvert)^{1-\rho_{2}}+\alpha\right)\text{sign}\left(x\right)$. The inverse of $\theta_{\rho_{2},\alpha}$ is the continuous function $\theta_{\rho_{2},\alpha}^{-1}:(-\alpha-1,\alpha+1)\to\mathbb{R}$ given by

Thus, replacing (6) in (5), it yields

with $x_{0}=x^{0}$.

This following theorem summarizes the previous constructive result:

Theorem 6 has the following attractive corollary:

The solution of the continuous-time system (1) satisfies

Moreover, from Theorem 6, we can establish for the solution of the discrete-time system (7) that

which also holds for $kh\geq\rho_{1}$, given that $\kappa\in\mathcal{K}$.

Hence, the result follows. $\blacksquare$

Consider the nonlinear transformation given by (4), and define $w(t)=y_{\rho_{2}}(x(t))$. Its time-derivative along the trajectories of system (1) is

since $\text{sign}\left(w\right)=\text{sign}\left(x\right)$.

Using the implicit or backward Euler method (as in (Drakunov and Utkin, 1989; Utkin, 1994)) with time step $h>0$ for the discretization of the transformed system (8), it yields

where $w_{k}=y_{\rho_{2}}(x_{k})$ and $x_{k}=x(kh)$. Rewriting the above in terms of $x_{k}$, one obtains (3), as before.

Consider the perturbed control system

where $x:\mathbb{R}_{\geq 0}\to\mathbb{R}$ is the state of the system, $u\in\mathbb{R}$ is the control input signal, and $\Delta:\mathbb{R}_{\geq 0}\times\mathbb{R}\to\mathbb{R}$ is an unknown perturbation term which is assumed to be bounded of the form $\sup_{(t,x)\in\mathbb{R}_{\geq 0}\times\mathbb{R}}\left\lvert\Delta(t,x)\right\rvert\leq\delta$, with $\delta\in\mathbb{R}_{+}$ known.

It is well known that, in continuous time, the perturbation term $\Delta(t,x)$ can only be entirely rejected by a discontinuous control term, since no conditions of smoothness, Lipschitz continuity neither continuity are assumed (Jiménez-Rodríguez et al., 2019).

According to the above, a suitable feedback predefined-time controller design is

where $\rho_{1}>0$, $\rho_{3}\geq\delta$, and $\kappa\in\mathcal{K}^{1}$ is continuously differentiable and such that $\kappa^{\prime}(0):=\kappa^{\prime}(0^{+})\geq\kappa^{\prime}(r)$ for all $r\in\mathbb{R}_{\geq 0}$. Note that (10) is an augmented gain version of the right side of (1) with $\rho_{2}=0$.

To verify the predefined-time convergence of the closed-loop system (9)-(10), note that

for $x\neq 0$. Reasoning as in Subsection 2.2, and using the comparison lemma in the above differential inequality, one can easily conclude that $\left\lvert x(t)\right\rvert=x(t)=0$ for $t\geq\rho_{1}$.

Consider transformation (4), with $\rho_{2}=0$, given by $z(t)=y_{0}(x(t))=\kappa(\left\lvert x(t)\right\rvert)\text{sign}\left(x(t)\right)$. Its derivative along the trajectories of the closed-loop system (9)-(10) is

where $\beta:=\frac{1}{\rho_{1}}+\rho_{3}\kappa^{\prime}(0)$, and $f(t,x)=\kappa^{\prime}(\left\lvert x\right\rvert)\Delta(t,x)$ is the perturbation term in the transformed coordinate $z$, which complies to $\sup_{(t,x)\in\mathbb{R}_{\geq 0}\times\mathbb{R}}\left\lvert f(t,x)\right\rvert\leq\kappa^{\prime}(0)\delta$.

Again, using the implicit Euler method with time step $h>0$ for the discretization of the system (11), it yields

where $f_{k}=f(kh,x_{k})$ is the perturbation term at the sample instants.

Hence, replacing the function $\theta_{0,h\beta}^{-1}$ (6) into (12), one obtains

with $x_{0}=x^{0}$.

The discrete-time system (13) provides a way of simulating the closed-loop continuous-time system (9)-(10). Moreover, it preserves the predefined-time stability behavior, as stated in the following proposition:

Taking absolute value in both sides of (13) and using the triangle inequality one gets,

On the other hand, from $\beta:=\frac{1}{\rho_{1}}+\rho_{3}\kappa^{\prime}(0)$, $\rho_{3}\geq\delta$, $\left\lvert f_{k}\right\rvert\leq\kappa^{\prime}(0)\delta$ and the above, the inequality

is obtained.

Finally, noticing that the right side of the above inequality is equal to the absolute value of the right side of (7) with $\rho_{2}=0$, and that it is a non-decreasing function of $x_{k}$, we get

applying Lemma 13. $\blacksquare$