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# Robustness of systems homogeneous with respect to a part of variables

Denis Efimov, Ilya Kolmanovsky

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## To cite this version:

Denis Efimov, Ilya Kolmanovsky. Robustness of systems homogeneous with respect to a part of variables. Automatica, In press. hal-05342313

HAL Id: hal-05342313

https://inria.hal.science/hal-05342313v1

Submitted on 2 Nov 2025

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# Robustness of systems homogeneous with respect to a part of variables

Denis Efimov<sup>a</sup>, Ilya Kolmanovsky<sup>b</sup>

<sup>a</sup>Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France

<sup>b</sup>Department of Aerospace Engineering, the University of Michigan, US

## Abstract

The paper studies robust stability properties in presence of external perturbations for recently introduced partially homogeneous dynamical systems (for which the dilation is applied only to a part of the state variables). The results can be utilized to extend the approach, first introduced by [18] in the setting of exponential stabilization, to construct time-dependent control laws for nonholonomic systems which not only provide accelerated convergence (finite-time or nearly fixed-time) but also guarantee robustness to unmeasured disturbances. Examples of input-to-output stabilization of a nonholonomic integrator are reported.

## 1 Introduction

In the field of nonlinear control and observer design, the convergence rates of estimation or regulation errors to zero play a crucial role in meeting desired specifications. Several advanced nonlinear control and estimation approaches have been developed to ensure various convergence properties, including asymptotic, rated exponential, finite-time, or fixed-time convergence, see e.g., [3, 9, 14, 15, 29, 34]. Another important closed-loop property is the robustness to exogenous perturbations and measurement noise, which is frequently studied using the theory of input-to-state stability, see e.g., [6, 31].

Systems with homogeneous dynamics constitute an intriguing class of strongly nonlinear systems for which the global behavior mirrors the local one, and accelerated convergence rates and robustness can be achieved. The homogenization of closed-loop systems through a suitable dilation of control laws has been explored to transform local asymptotic stabilizing control laws into ones with rated convergence in [18], where it was demonstrated that exponential convergence decay can be achieved for a class of nonholonomic systems through the homogenization of closed-loop systems. Subsequent publications in [24], [21, 22, 40, 41] employed such a homogenization approach for finite/fixed-time stabilization of linear autonomous systems using the implicit Lyapunov function (LF) approach described in [16] and [1].

In our recent work [7], building on the insights from the extension of the theory of homogeneous systems to time-varying models in [25, 26], the time-varying feedback stabilization

approach introduced in [18] has been extended to finite-time and fixed-time stabilization. This extension involves the theory of homogeneous systems being expanded to partial symmetry, where only a part of the state is divided, and the variables representing generator dynamics, which inform time-dependence of the control law, are not scaled. The stability and convergence need only be considered with respect to a part of the state variables and the concept of (uniform) output stability, as defined in [13, 28, 30, 33], can be exploited in the analysis.

The conventional homogeneous systems are known for their advantageous robustness properties established using the (integral) input-to-state stability properties, see e.g., [2, 4, 39]. In the case of partial homogeneity, input-to-output stability properties studied in [13, 30] need to be considered instead, and exactly this question is investigated in this note. In particular, we show that this enables the robustification of the control laws obtained via the approach of [18]. For an illustration, we will consider the problem of robust stabilization of a nonholonomic integrator using time-varying feedback (unlike [35, 36] in our case it is not required that the time-varying gains escape to infinity in a prescribed time).

The paper is organized as follows. The background on stability and homogeneity is reviewed in Section 2. The control problem statement is introduced in Section 3. The robustness properties of partially homogeneous stable systems are investigated in Section 4. The control design is described in Section 5. The results of numerical experiments are reported in Section 6.
Notation

• $\mathbb{R}_+ = \{x \in \mathbb{R} : x \ge 0\}$, where $\mathbb{R}$ is the set of real numbers.

• $|\cdot|$ denotes the absolute value in $\mathbb{R}$, $\|\cdot\|$ is used for the Euclidean norm on $\mathbb{R}^n$.

• For a (Lebesgue) measurable function $d: \mathbb{R}_+ \to \mathbb{R}^s$ and $[t_0, t_1) \subset \mathbb{R}_+$, define the norm $\|d\|_{[t_0, t_1)} = \text{ess sup}_{t \in [t_0, t_1]} \|d(t)\|$, then $\|d\|_{\infty} = \|d\|_{[0, +\infty)}$ and the set of functions with the property that $\|d\|_{\infty} < +\infty$ is denoted by $\mathcal{L}_{\infty}^s$ (i.e., this is the set of essentially bounded measurable functions).

• For a function $d: \mathbb{R}_+ \to \mathbb{R}^s$ and a constant $a > 0$, the notation $d(a\cdot)$ corresponds to the function $d$ applied to the argument scaled by $a$.

• The class of continuous functions $X \to Y$ is denoted by $C(X, Y)$ for two metric spaces $X$ and $Y$.

• A finite series of integers 1, 2, ..., $n$ is denoted by $\overline{1, n}$.

In this work we will use the following comparison functions:

• A function $\sigma \in C(\mathbb{R}_+, \mathbb{R}_+)$ belongs to class $\mathcal{K}$ if it is strictly increasing and $\sigma(0) = 0$; it belongs to class $\mathcal{K}_{\infty}$ if it is also unbounded.

• A function $\beta \in C(\mathbb{R}_+ \times \mathbb{R}_+, \mathbb{R}_+)$ belongs to class $\mathcal{KL}$ if $\beta(\cdot, r) \in \mathcal{K}$ for any $r \in \mathbb{R}_+$ and $\beta(r, \cdot)$ is decreasing to zero for any fixed $r > 0$. We will say that such a function $\beta$ is from class $e\mathcal{KL}$ if $\beta(s, r) = cse^{-ar}$ for some $c > 0$ and $a > 0$.

• A function $\chi \in C(\mathbb{R}_+ \times \mathbb{R}_+, \mathbb{R}_+)$ belongs to class $\mathcal{FKL}$ if $\chi(\cdot, 0) \in \mathcal{K}$, $\chi(r, \cdot)$ is decreasing to zero for any fixed $r > 0$ and there is $T \in \mathcal{K}$ such that $\chi(r, t) = 0$ for all $t \ge T(r)$ and all $r \in \mathbb{R}_+$.

• A function $\chi \in \mathcal{FKL}$ with $\sup_{r \ge 0} T(r) < +\infty$ belongs to the class $\mathcal{F}x\mathcal{KL}$.

• Finally, $\chi \in C(\mathbb{R}_+ \times \mathbb{R}_+, \mathbb{R}_+)$ belongs to class $n\mathcal{F}x\mathcal{KL}$ if $\chi(\cdot, 0) \in \mathcal{K}$, $\chi(r, \cdot)$ is decreasing to zero for any fixed $r > 0$, and there is $T \in \mathcal{K}$ such that $\chi(r, t) \le \rho$ for any $\rho > 0$, for all $t \ge T(\rho^{-1})$ and all $r \in \mathbb{R}_+$.

## 2 Preliminaries

The standard stability notions and their definitions can be found in [15].

### 2.1 Robust output stability

Consider a dynamic system represented by a model,

$$ 
\begin{aligned}
\dot{x}(t) &= f(x(t), d(t)), \quad t \ge 0, \\
y(t) &= h(x(t)),
\end{aligned} 
\quad (1) 
$$

where $x(t) \in \mathbb{R}^n$ is the state vector, $d(t) \in \mathbb{R}^s$ is the vector of external inputs, $d \in \mathcal{L}_{\infty}^s$, and $y(t) \in \mathbb{R}^p$ is the output; $f \in C(\mathbb{R}^n \times \mathbb{R}^s, \mathbb{R}^n)$ is such that the existence and uniqueness of solutions of (1) in forward time is guaranteed; $h: \mathbb{R}^n \to \mathbb{R}^p$ is a continuously differentiable function; $f(0, 0) = 0$ and $h(0) = 0$. The unique maximal solution of the system (1)

for an initial condition $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_{\infty}^s$ is denoted by $X(t, x_0, d)$ for those $t \ge 0$ for which the solution exists; then $Y(t, x_0, d) = h(X(t, x_0, d))$.

The system (1) is called forward complete if for any $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_{\infty}^s$ the solution $X(t, x_0, d)$ exists for all $t \ge 0$.

**Definition 1** ([30]) A forward complete system (1) is input-to-output stable (IOS) if there exist $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$ such that the inequality,

$$ \|Y(t, x_0, d)\| \le \beta(\|x_0\|, t) + \gamma(\|d\|_{[0, t)}), \quad \forall t \ge 0 $$

holds for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_{\infty}^s$. This system is state-independent input-to-output stable (SIIOS) if this estimate can be strengthened to the inequality

$$ \|Y(t, x_0, d)\| \le \beta(\|h(x_0)\|, t) + \gamma(\|d\|_{[0, t)}), \quad \forall t \ge 0 $$

being satisfied for all $x_0 \in \mathbb{R}^n$ and $d \in \mathcal{L}_{\infty}^s$.

**Definition 2** An IOS (SIOS) system (1) is called FT/FxT/nFxT-IOS(SIOS) if $\beta \in \mathcal{FKL}/\mathcal{F}x\mathcal{KL}/n\mathcal{F}x\mathcal{KL}$, respectively.

Restricting the admissible values for the input to the set $D = \{d \in \mathcal{L}_{\infty}^s : \|d\|_{\infty} \le d\}$, where $0 < d < +\infty$, the respective uniform output stability notions can be defined.

**Definition 3** ([30]) A forward complete system (1) is uniformly output stable (uOS) with respect to the inputs in $D$ if there exists $\beta \in \mathcal{KL}$ such that the property

$$ \|Y(t, x_0, d)\| \le \beta(\|x_0\|, t), \quad \forall t \ge 0 $$

holds for all $x_0 \in \mathbb{R}^n$ and $d \in D$. It is state-independent uniformly output stable (SIuOS) with respect to the inputs in $D$ if this estimate can be strengthened to

$$ \|Y(t, x_0, d)\| \le \beta(\|h(x_0)\|, t), \quad \forall t \ge 0 $$

which must be satisfied for all $x_0 \in \mathbb{R}^n$ and $d \in D$.

The uniformity in the definition above is understood as independence of the stability properties from the presence of inputs taking values in a given bounded set.

**Definition 4** ([7]) An uOS system (1) is called uFTOS/ uFxTOS/unFxTOS if $\beta \in \mathcal{FKL}/\mathcal{F}x\mathcal{KL}/n\mathcal{F}x\mathcal{KL}$.

In the original input-to-output stability framework of [30], the function $f$ is required to be locally Lipschitz continuous. In the papers by [37, 38], considering the case $d \equiv 0$, the output stability characterizations have been introduced for the case when $f \in C(\mathbb{R}^n \times \mathbb{R}^s, \mathbb{R}^n)$ and is locally Lipschitz continuous on $\mathbb{R}^n \setminus \{x \in \mathbb{R}^n : h(x) = 0\} \times \mathbb{R}^s$. These relaxations allowed the accelerated output convergence rates to be defined in [37] (a finite-time convergence needs the system to be non-Lipschitz continuous at least at zero).

If $h(x) = x$, then FT/FxT/nFxT-IOS/SIIOS properties reduce to the conventional finite/fixed-time input-to-state stability, see e.g., [9, 10, 17].
**Definition 5** ([30]) A system (1) is uniformly bounded input bounded state stable (UBIBS) if it is forward complete and there exists $\sigma \in K$ such that

$$ \|X(t, x_0, d)\| \leq \max\{\sigma(\|x_0\|), \sigma(\|d\|_{[0,t)})\}, \forall t \geq 0 \\ \text{for all } x_0 \in \mathbb{R}^n \text{ and } d \in \mathcal{L}_{\infty}^s. $$

The introduced above output stability properties (as IOS/SIIOS and their accelerated versions) can be effectively established using the Lyapunov function method.

**Definition 6** ([30]) A smooth function $V: \mathbb{R}^n \to \mathbb{R}_+$ is called IOS-LF if for some $\alpha_1, \alpha_2 \in K_\infty$, $\chi \in K$ and $\alpha_3 \in K\mathcal{L}$:

$$ \alpha_1(\|h(x)\|) \leq V(x) \leq \alpha_2(\|x\|), \forall x \in \mathbb{R}^n; \\ V(x) \geq \chi(\|d\|) \Rightarrow $$

$$ \frac{\partial V(x)}{\partial x} f(x, d) \leq -\alpha_3(V(x), \|x\|), \forall x \in \mathbb{R}^n, \forall d \in \mathbb{R}^s. $$

Moreover, $V$ is called SIIOS-LF if for some $\alpha_1, \alpha_2 \in K_\infty$, $\chi \in K$ and $\alpha_4 \in K$:

$$ \alpha_1(\|h(x)\|) \leq V(x) \leq \alpha_2(\|h(x)\|), \forall x \in \mathbb{R}^n; \\ V(x) \geq \chi(\|d\|) \Rightarrow $$

$$ \frac{\partial V(x)}{\partial x} f(x, d) \leq -\alpha_4(V(x)), \forall x \in \mathbb{R}^n, \forall d \in \mathbb{R}^s. $$

**Definition 7** A SIIOS-LF is called FT-SIIOS-LF, nFxt-SIIOS-LF or Fxt-SIIOS-LF if $\alpha_4(s) \geq cs^a$, $\alpha_4(s) \geq cs^b$ or $\alpha_4(s) \geq c(s^a + s^b)$, respectively, for $c > 0$, $a \in [0, 1)$ and $b > 1$. Finally, an IOS-LF is FT/nFxt/Fxt-IOS-LF if for each $r > 0$ there exist $c_r > 0$, $c_0 > 0$, $a \in [0, 1)$ and $b > 1$ such that $\alpha_3(s, r) \geq c_r s^a$, $\alpha_3(s, r) \geq c_0 s^b$ or $\alpha_3(s, r) \geq c_r s^a + c_0 s^b$, correspondingly.

**Theorem 1** ([30]) A UBIBS system (1) with locally Lipschitz continuous $f$ is IOS/SIIOS if and only if it admits an IOS/SIIOS-LF.

The above Lyapunov conditions have been extended in [13] to the case of non-UBIBS systems. For SIIOS case, UBIBS property can be replaced by forward completeness. For the sufficiency of existence of these LFs for IOS/SIIOS properties, the local Lipschitzness of $f$ can be omitted.

**Theorem 2** A forward complete system (1) is FT/nFxt/Fxt-SIIOS if it admits a FT/nFxt/Fxt-SIIOS-LF, and a UBIBS system (1) is FT/nFxt/Fxt-IOS if it has a FT/nFxt/Fxt-IOS-LF.

**PROOF.** This follows as a consequence of Theorem 1 and the standard comparison lemmas, along the same lines as in [9, 10, 17].

The respective Lyapunov characterizations of the uniform output stability properties with accelerated convergence (i.e., uFTOS/uFxtOS/unFxtOS) can be found in [12, 37, 38].

## 2.2 Homogeneity

For any $r_i > 0$, $i = 1, \dots, n$ and $\lambda > 0$, define the vector of weights $\mathbf{r} = [r_1, \dots, r_n]$ and the dilation matrix $\Lambda_r(\lambda) = \text{diag}\{\lambda^{r_i}\}_{i=1}^n$; $r_{\min} = \min_{i=1, n} r_i$ and $r_{\max} = \max_{i=1, n} r_i$.

**Definition 8** ([9, 42]) A function $g: \mathbb{R}^n \to \mathbb{R}$ is called r-homogeneous, if for some $\nu \in \mathbb{R}$ the relation

$$ g(\Lambda_r(\lambda)x) = \lambda^{\nu} g(x) $$

holds for any $x \in \mathbb{R}^n$ and all $\lambda > 0$.

A vector field $f: \mathbb{R}^n \to \mathbb{R}^n$ is called r-homogeneous, if for some $\nu \geq -r_{\min}$ the relation

$$ f(\Lambda_r(\lambda)x) = \lambda^{\nu} \Lambda_r(\lambda) f(x) $$

holds for any $x \in \mathbb{R}^n$ and all $\lambda > 0$.

In both cases, the constant $\nu$ is called the degree of homogeneity.

A dynamical system

$$ \dot{x}(t) = f(x(t)), \quad t \geq 0, \quad x(t) \in \mathbb{R}^n $$

is called r-homogeneous of degree $\nu$ if this property is satisfied for $f$ in the sense of Definition 8.

For any $x \in \mathbb{R}^n$ and $\varpi \geq r_{\max}$, a homogeneous norm can be defined as follows

$$ \|x\|_r = \left( \sum_{i=1}^{n} |x_i|^{\varpi/r_i} \right)^{1/\varpi}. $$

For all $x \in \mathbb{R}^n$, its Euclidean norm $\|x\|$ is related with the homogeneous one:

$$ \underline{\sigma}_r(\|x\|_r) \leq \|x\| \leq \bar{\sigma}_r(\|x\|_r) $$

for some $\underline{\sigma}_r, \bar{\sigma}_r \in K_\infty$, as shown in [11]. In the following, due to this “equivalence”, stability analysis with respect to the norm $\|\cdot\|$ can be substituted with analysis for the norm $\|\cdot\|_r$. The homogeneous norm has an important property: It is r-homogeneous of degree 1, that is $\|\Lambda_r(\lambda)x\|_r = \lambda\|x\|_r$ for all $x \in \mathbb{R}^n$ and $\lambda > 0$. In addition, for any continuous r-homogeneous function $g: \mathbb{R}^n \to \mathbb{R}$ of degree $\nu$, the following relations are satisfied:

$$ \underline{c}\|x\|_r^{\nu} \leq g(x) \leq \bar{c}\|x\|_r^{\nu} $$

for any $x \in \mathbb{R}^n$ and $c = \min_{\|\xi\|_r = 1} g(\xi)$, $\bar{c} = \max_{\|\xi\|_r = 1} g(\xi)$.

## 2.3 Partial homogeneity

Following [7], consider a nonlinear system admitting a decomposition of the state vector into output dynamics and the remaining variables:

$$ \dot{y}(t) = F(y(t), z(t)), \quad (2) $$

$$ \dot{z}(t) = G(y(t), z(t)), $$

where $y(t) \in \mathbb{R}^p$ and $z(t) \in \mathbb{R}^s$ are the components of the state vector $x(t) = [y(t)^\top \ z(t)^\top]^\top \in \mathbb{R}^n$ with $n = p + s$,
$F \in C(\mathbb{R}^p \times \mathbb{R}^s, \mathbb{R}^p)$ and $G \in C(\mathbb{R}^p \times \mathbb{R}^s, \mathbb{R}^s)$ are locally Lipschitz continuous functions on $\mathbb{R}^p \setminus \{0\} \times \mathbb{R}^s$, $F(0, z) = 0$ for all $z \in \mathbb{R}^s$. For specified initial conditions $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, we denote the respective solutions of (2) by $Y(t, y_0, z_0)$ and $Z(t, y_0, z_0)$ for those $t \ge 0$ for which these solutions exist. For brevity of exposition, we assume in the sequel that these solutions are defined for all $t \ge 0$, i.e., (2) is forward complete.

**Definition 9** The system (2) is called partially **r-homogeneous** if there exists a vector of weights $\mathbf{r} = [r_1, ..., r_p]$ with $r_i > 0$, $i = 1,p$ such that the following properties,

$F(\Lambda_r(\lambda)y, z) = \lambda^\nu \Lambda_r(\lambda)F(y, z)$, $G(\Lambda_r(\lambda)y, z) = \lambda^\nu G(y, z)$, hold for all $\lambda > 0$, $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$ and some $\nu \ge -r_{\min}$, which is called the degree of partial homogeneity.

Hence, according to Definition 8, $F$ is **r-homogeneous** vector field of degree $\nu$ and $G$ is **r-homogeneous** vector function of degree $\nu$ considering $z$ as a constant.

Partially homogeneous systems demonstrate the most important and fundamental feature of homogeneous dynamics, which is scaling of solutions under time reparametrization:

**Proposition 1 ([7])** For a partially **r-homogeneous** system (2) of degree $\nu$, if $Y(t, y_0, z_0)$ and $Z(t, y_0, z_0)$ are solutions of (2) for some $y_0 \in \mathbb{R}^p \setminus \{0\}$ and $z_0 \in \mathbb{R}^s$, then for any $\lambda > 0$:

$$ 
\begin{aligned}
Y(t, \Lambda_r(\lambda)y_0, z_0) &= \Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0), \\
Z(t, \Lambda_r(\lambda)y_0, z_0) &= Z(\lambda^\nu t, y_0, z_0).
\end{aligned}
$$

This result implies that the global stability properties can be concluded from the local ones, see [7]. Moreover, as in the case of the standard homogeneity, the degree $\nu$ can be used to determine the convergence rates in the system:

**Theorem 3 ([7])** For a partially **r-homogeneous** system (2) of degree $\nu$, if $\|Z(t, y_0, z_0)\| \le \sigma(\|z_0\|) + \sigma_0$ for all $t \ge 0$, all $y_0 \in \mathbb{R}^p$ and $z_0 \in \mathbb{R}^s$, for some $\sigma \in K$ and $\sigma_0 \ge 0$, then a uOS system (2) is uFTOS or unFxTOS provided that the homogeneity degree $\nu$ is negative or positive, respectively, while for $\nu = 0$ the convergence rate of $Y(t, y_0, z_0)$ to the origin is exponential.

In such a case, the uFxTOS can be obtained recalling the ideas of bi-limit homogeneity introduced in [2]: If a uOS system admits approximation by a partially homogeneous model of negative degree close to the origin, and by another partially homogeneous model of positive degree for the output norm large enough, then stability of these approximate dynamics provide a local uFTOS and practical unFxTOS properties for the original system, and their combination leads to the desired characteristics.

It is also possible to extend the results of [27, 42] on existence of homogeneous LFs to the case of partial homogeneity:

**Theorem 4 ([7, 8])** For a partially **r-homogeneous** system (2) of degree $\nu$, it is SIuOS if and only if there is a smooth on $\mathbb{R}^p \setminus \{0\} \times \mathbb{R}^s$ LFW : $\mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$, $C^k$ on $\mathbb{R}^p \times \mathbb{R}^s$ for any integer $k \ge 1$, such that for some $c_1, c_2, c_3 > 0$ and all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$:

$$ 
\begin{aligned}
c_1 \|y\|_r^\mu &\le W(y, z) \le c_2 \|y\|_r^\mu, \\
\frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) &\le -c_3 \|y\|_r^{\nu+\mu}, \\
W(\Lambda_r(\lambda)y, z) &= \lambda^\mu W(y, z), \quad \forall \lambda > 0, \\
\text{where } \mu > kr_{\max}.
\end{aligned}
$$

A similar, but a less constructive result can be formulated for uOS case:

**Theorem 5 ([7, 8])** For a **r-partially homogeneous** system (2) of degree $\nu$, if $F, G$ are locally Lipschitz continuous vector functions, the UBIBS system (2) is uOS if and only if there is a LFW : $\mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$, $C^k$ on $\mathbb{R}^p \times \mathbb{R}^s$ for any integer $k \ge 1$, such that for some $\tilde{\alpha}_1, \tilde{\alpha}_2 \in K_\infty$ and all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$:

$$ 
\begin{aligned}
\tilde{\alpha}_1 (\|y\|) &\le W(y, z) \le \tilde{\alpha}_2 (\|x\|), \\
W(y, z) \ne 0 &\Rightarrow \frac{\partial W(y, z)}{\partial y} F(y, z) + \frac{\partial W(y, z)}{\partial z} G(y, z) < 0, \\
W(\Lambda_r(\lambda)y, z) &= \lambda^\mu W(z, y), \quad \forall \lambda > 0, \\
\text{where } x &= [y^\top z^\top]^\top \text{ and } \mu > kr_{\max}.
\end{aligned}
$$

### 3 Problem statement

Consider the problem of robust and accelerated stabilization (in the sense of convergence rates given in Definitions 1 and 2) of a controllable system of the form:

$$ \dot{y}(t) = X_0(y(t)) + \sum_{i=1}^{m} X_i(y(t))u_i(t) + d(t), \quad t \ge 0, \quad (3) $$

where $y(t) \in \mathbb{R}^p$ is the state of (3), $u(t) \in \mathbb{R}^m$ is the control to be designed, $d(t) \in \mathbb{R}^p$ is the exogenous disturbance, $d \in \mathcal{L}_\infty$; $X_j \in C(\mathbb{R}^p, \mathbb{R}^p)$ for $j = 0, m$. If $X_0(\cdot) = 0$, then (3) becomes driftless; such models frequently appear in the study of mechanical systems with nonholonomic constraints as described in [18]. Since asymptotic stabilization of systems of this kind by a continuous feedback $u(t) = \kappa(y(t))$ (with a continuous function $\kappa : \mathbb{R}^p \to \mathbb{R}^m$) is impossible [5], time-varying controls $u(t) = \kappa(t, y(t))$ (with continuous $\kappa : \mathbb{R} \times \mathbb{R}^p \to \mathbb{R}^m$) are frequently used, see e.g., [18, 23, 32] (for the case $d \equiv 0$). In such a setting, the time dependence can be modeled by introducing auxiliary state variables $z(t) \in \mathbb{R}^s$ in the system dynamics, for which there is no requirement for convergence. Hence, the problem of output $y(t)$ stabilization can be analyzed for an autonomous extended model with the combined state $x(t) = [y^\top(t) z^\top(t)]^\top$. Other situations, when
additional dynamics can appear, correspond to the trajectory tracking, adaptive control and estimation, or disturbance compensation scenarios, then $z(t)$ (or a function of it) becomes the reference, an estimate or the internal model state. Therefore, in such a case the studied problem consists in upgrade of a local time-varying asymptotically stabilizing control law for (3) with $d \equiv 0$ to a global accelerated (exponential, finite- or fixed-time) and robust in $d \in \mathcal{L}_{\infty}^p$ regulator.

In [18] the theory of homogeneous systems was applied to design a feedback law guaranteeing exponential stability of the closed-loop driftless dynamics at the origin. The method is based on scaling existing feedback functions, which provide only asymptotic (non-exponential, and possibly only local) stabilization. A homogeneous norm was used for scaling, similarly as it has been suggested for linear systems in [24], or using implicit LFs in [16, 21, 22, 40, 41]. In [7], utilizing similar ideas and extending the theory of homogeneous systems to the case of partial symmetry and stabilization, finite/fixed-time stabilizing feedbacks are synthesized for $d \equiv 0$. In the present contribution we will extend this result to the case when $d \neq 0$.

## 4 Robustness of partially homogeneous systems

Consider an extension of the system (2) with the inputs:

$$ 
\begin{aligned}
\dot{y}(t) &= F(y(t), z(t), d(t)), \\
\dot{z}(t) &= G(y(t), z(t), d(t)),
\end{aligned}
\quad (4) $$

where as before $y(t) \in \mathbb{R}^p$ and $z(t) \in \mathbb{R}^s$ are the components of the state vector, $d(t) \in \mathbb{R}^q$ is the exogenous input, $d \in \mathcal{L}_{\infty}^q$; $F \in C(\mathbb{R}^p \times \mathbb{R}^s \times \mathbb{R}^q, \mathbb{R}^p)$ and $G \in C(\mathbb{R}^p \times \mathbb{R}^s \times \mathbb{R}^q, \mathbb{R}^s)$ are continuous functions ensuring existence of solutions for (4), and $F(0, z, 0) = 0$ for all $z \in \mathbb{R}^s$.

We will need the following restriction for (4) that covers the partial homogeneity of the system for $d = 0$:

**Assumption 1** There exist vectors of weights $\mathbf{r} = [r_1, ..., r_p]$ and $\tilde{\mathbf{r}} = [\tilde{r}_1, ..., \tilde{r}_q]$ with $r_i > 0$, $i = 1, p$ and $\tilde{r}_j > 0$, $j = 1, q$ such that the following properties,

$$ 
\begin{aligned}
F(\Lambda_r(\lambda)y, z, \Lambda_{\tilde{r}}(\lambda)d) &= \lambda^\nu \Lambda_r(\lambda) F(y, z, d), \\
G(\Lambda_r(\lambda)y, z, \Lambda_{\tilde{r}}(\lambda)d) &= \lambda^\nu G(y, z, d),
\end{aligned}
$$

hold for all $\lambda > 0$, $y \in \mathbb{R}^p$, $z \in \mathbb{R}^s$, $d \in \mathbb{R}^q$ and some $\nu \ge -r_{\min}$.

Under these restrictions, the solutions of (4) stay interrelated after dilation of initial conditions for $y$ and time rescaling in the input:

**Proposition 2** Under Assumption 1, if $Y(t, y_0, z_0, d(\cdot))$ and $Z(t, y_0, z_0, d(\cdot))$ are unique solutions of (4) for some $y_0 \in \mathbb{R}^p \setminus \{0\}$, $z_0 \in \mathbb{R}^s$ and $d \in \mathcal{L}_{\infty}^q$, with $d(t) = 0$ only at isolated time instants $t \in \mathbb{R}_+$, then for any $\lambda > 0$:

$$ 
\begin{aligned}
Y(t, \Lambda_r(\lambda)y_0, z_0, \Lambda_{\tilde{r}}(\lambda)d(\lambda^\nu)) &= \Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu)), \\
Z(t, \Lambda_r(\lambda)y_0, z_0, \Lambda_{\tilde{r}}(\lambda)d(\lambda^\nu)) &= Z(\lambda^\nu t, y_0, z_0, d(\lambda^\nu)).
\end{aligned}
$$

**PROOF.** Let $y_0 \in \mathbb{R}^p$, $z_0 \in \mathbb{R}^s$ and $d \in \mathcal{L}_{\infty}^q$ be given, while $Y(t, y_0, z_0, d(\cdot))$ and $Z(t, y_0, z_0, d(\cdot))$ be the corresponding solutions to (4). We need to show that for any $\lambda > 0$, if we scale the output initial conditions as $\Lambda_r(\lambda)y_0$ and apply the input with dilated amplitude and scaled time $\Lambda_{\tilde{r}}(\lambda)d(\lambda^\nu t)$, then the new solutions will be $\Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot))$ and $Z(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot))$, respectively. To this end, we need to check that the latter expressions are indeed the solutions of (4):

$$ 
\begin{aligned}
\frac{d\Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot))}{dt} &= \lambda^\nu \Lambda_r(\lambda) \frac{dY(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot))}{d\lambda^\nu t} \\
&= \lambda^\nu \Lambda_r(\lambda) F\left(Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), \right. \\
&\qquad \left. Z(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), d(\lambda^\nu t)\right) \\
&= F\left(\Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), \right. \\
&\qquad \left. Z(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), \Lambda_{\tilde{r}}(\lambda)d(\lambda^\nu t)\right);
\end{aligned}
$$

$$ 
\begin{aligned}
\frac{dZ(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot))}{dt} &= \lambda^\nu \frac{dZ(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot))}{d\lambda^\nu t} \\
&= \lambda^\nu G\left(Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), \right. \\
&\qquad \left. Z(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), d(\lambda^\nu t)\right) \\
&= G\left(\Lambda_r(\lambda)Y(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), \right. \\
&\qquad \left. Z(\lambda^\nu t, y_0, z_0, d(\lambda^\nu \cdot)), \Lambda_{\tilde{r}}(\lambda)d(\lambda^\nu t)\right).
\end{aligned}
$$

The solution uniqueness now implies the desired result (since the input $d$ vanishes at isolated time instants only, for the case of $\nu < 0$ the solutions cannot stay during a time interval at the submanifold $y = 0$, where the non-uniqueness issue may arise).

Using such a symmetry with the input we can establish the robust stability conditions for (4).

**Theorem 6** Let Assumption 1 be satisfied, and suppose the system (4) admits a forward invariant compact set $\mathcal{Z} \subset \mathbb{R}^s$ while being SIuOS for $d \equiv 0$, then there exist $\beta \in \mathcal{KL}$ and $\gamma \in \mathcal{K}$ such that

$$ \|y(t)\| \le \beta(\|y(0)\|, t) + \gamma(\|d\|_{[0,t)}), \quad \forall t \ge 0 $$

holds for all $y(0) \in \mathbb{R}^p$, $z(0) \in \mathcal{Z}$ and $d \in \mathcal{L}_{\infty}^q$. Moreover, $\beta \in \mathcal{FKL}$ if $\nu < 0$, $\beta \in \mathcal{eKL}$ if $\nu = 0$ and $\beta \in \mathcal{nFKL}$ if $\nu > 0$.

**PROOF.** Under Assumption 1, the system (4) is partially $\mathbf{r}$-homogeneous and SIuOS for $d = 0$. Then, according to Theorem 4, there is a LF $W: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$, smooth on $\mathbb{R}^p \setminus \{0\} \times \mathbb{R}^s$, $C^k$ on $\mathbb{R}^p \times \mathbb{R}^s$ for an integer $k \ge 2$, such
that for some $c_1, c_2, c_3 > 0$ and all $y \in \mathbb{R}^p$ and $z \in \mathbb{R}^s$:

$$c_1 \|y\|_r^\mu \le W(y, z) \le c_2 \|y\|_r^\mu,$$

$$\frac{\partial W(y, z)}{\partial y} F(y, z, 0) + \frac{\partial W(y, z)}{\partial z} G(y, z, 0) \le -c_3 \|y\|_r^{\nu+\mu},$$

$$W(\Lambda_r(\lambda)y, z) = \lambda^\mu W(y, z), \forall \lambda > 0,$$

where $\mu > kr_{\max}$. Let us show that $W$ is actually an SIIOS-LF considering $z \in \mathcal{Z}$. Clearly, $W$ admits the required lower and upper bounds, and we need to establish the desired estimate for its time derivative calculated along the solutions of (4):

$$\dot{W} := \frac{\partial W(y, z)}{\partial y} F(y, z, d) + \frac{\partial W(y, z)}{\partial z} G(y, z, d)$$

$$\le -c_3 \|y\|_r^{\nu+\mu} + \frac{\partial W(y, z)}{\partial y} (F(y, z, d) - F(y, z, 0))$$

$$+ \frac{\partial W(y, z)}{\partial z} (G(y, z, d) - G(y, z, 0))$$

$$= -c_3 \|y\|_r^{\nu+\mu}$$

$$+ \|y\|_r^{\nu+\mu} \frac{\partial W(\tilde{y}, z)}{\partial \tilde{y}} (F(\tilde{y}, z, \Lambda_{\tilde{r}}^{-1}(\|y\|_r)d) - F(\tilde{y}, z, 0))$$

$$+ \|y\|_r^{\nu+\mu} \frac{\partial W(\tilde{y}, z)}{\partial z} (G(\tilde{y}, z, \Lambda_{\tilde{r}}^{-1}(\|y\|_r)d) - G(\tilde{y}, z, 0))$$

for all $y \in \mathbb{R}^p \setminus \{0\}$, $z \in \mathbb{R}^s$ and $d \in \mathbb{R}^q$, where $\tilde{y} = \Lambda_r(\|y\|_r^{-1})y \in S_r^p = \{\tilde{y} \in \mathbb{R}^p : \|\tilde{y}\|_r = 1\}$. Considering only initial conditions for $z$ in the forward invariant and compact set $\mathcal{Z}$, we have that $z(t) \in \mathcal{Z}$ for all $t \ge 0$. Note that there exist $a_y > 0$ and $a_z > 0$ such that

$$\left\| \frac{\partial W(\tilde{y}, z)}{\partial \tilde{y}} \right\| \le a_y, \quad \left\| \frac{\partial W(\tilde{y}, z)}{\partial z} \right\| \le a_z$$

for all $\tilde{y} \in S_r^p$ and $z \in \mathcal{Z}$. Moreover, by continuity of the functions $F$ and $G$, there exists $\chi > 0$ such that

$$\|\Lambda_{\tilde{r}}^{-1}(\|y\|_r)d\|_{\tilde{r}} \le \chi^{-1} \Rightarrow$$

$$\|F(\tilde{y}, z, \Lambda_{\tilde{r}}^{-1}(\|y\|_r)d) - F(\tilde{y}, z, 0)\| \le \frac{c_3}{3a_y},$$

$$\|G(\tilde{y}, z, \Lambda_{\tilde{r}}^{-1}(\|y\|_r)d) - G(\tilde{y}, z, 0)\| \le \frac{c_3}{3a_z}$$

for all $\tilde{y} \in S_r^p$, $z \in \mathcal{Z}$ and $d \in \mathbb{R}^q$. Note that $\|\Lambda_{\tilde{r}}^{-1}(\|y\|_r)d\|_{\tilde{r}} = \|y\|_r^{-1}\|d\|_{\tilde{r}}$, hence,

$$\|y\|_r \ge \chi \|d\|_{\tilde{r}} \Rightarrow \dot{W} \le -\frac{c_3}{3} \|y\|_r^{\nu+\mu}$$

or

$$W^{\mu-1} \ge c_2^{\mu-1} \chi \|d\|_{\tilde{r}} \Rightarrow \dot{W} \le -\frac{c_3}{3c_2^{\mu}} W^{\frac{\nu+\mu}{\mu}}$$

for all $y \in \mathbb{R}^p$, $z \in \mathcal{Z}$ and $d \in \mathbb{R}^q$, which implies that an SIIOS-like estimate is satisfied, as desired. Indeed, define the time interval of existence of a solution $y(t)$ by $\mathcal{T} \subseteq \mathbb{R}_+$ ($z(t)$ is well defined and bounded for all $t \ge 0$ belonging to $\mathcal{Z}$), then $\mathcal{T} = \mathcal{T}_1 \cup \mathcal{T}_2$, with $\mathcal{T}_1 = \{t \in \mathcal{T} : W^{\mu-1}(t) > c_2^{\mu-1} \chi \sup_{s \in [0,t]} \|d(s)\|_{\tilde{r}}\}$ and $\mathcal{T}_2 = \{t \in \mathcal{T} : W^{\mu-1}(t) \le c_2^{\mu-1} \chi \sup_{s \in [0,t]} \|d(s)\|_{\tilde{r}}\}$, where we denoted $W(t) = W(y(t), z(t))$. Hence, for any interval $(t_s, t_e) \subset \mathcal{T}_1$ there is $\beta \in \mathcal{KL}$ ($\beta \in \mathcal{FKL}$ or $nF \times \mathcal{KL}$ for $\nu < 0$ or $\nu > 0$,

correspondingly, since $\frac{\nu+\mu}{\mu} < 1$ or $\frac{\nu+\mu}{\mu} > 1$ in these cases, see the standard finite/fixed-time comparison lemma results in [9]; for $\nu = 0$, we have $\frac{\nu+\mu}{\mu} = 1$, and the exponential convergence is recovered) such that $W(t) \le \beta(W(t_s), t-t_s)$ for all $t \in (t_s, t_e)$, and $\beta(s, 0) = s$ due to strict decay of $W(t)$ (by definition and continuity, we have that either $t_s = 0$ or $W(t_s) \le c_2 \chi^\mu \sup_{s \in [0,t_s]} \|d(s)\|_{\tilde{r}}^\mu$ for $t_s > 0$. Inversely, for any isolated time instant, or an interval $[t'_s, t'_e] \in \mathcal{T}_2$, we have that $W(t) \le c_2 \chi^\mu \sup_{s \in [0,t_s]} \|d(s)\|_{\tilde{r}}^\mu$ for all $t \in [t'_s, t'_e]$. Combining these estimates we get $W(t) \le \beta(W(0, t) + c_2 \chi^\mu \sup_{s \in [0,t_s]} \|d(s)\|_{\tilde{r}}^\mu$ for all $t \in \mathcal{T} = \mathbb{R}_+$.

As we can observe from the proof of this theorem, the asymptotic gain from the input $d$ to the output $y$ is linear in the corresponding homogeneous norms.

## 5 Control design

We now exploit the theoretical findings of Section 4 to robustify and accelerate time-varying stabilizing feedback laws for a class of systems (3). The proposed approach follows the ideas of [18] as in [7]. We need the following assumptions:

**Assumption 2** There exists a vector of weights $\mathbf{r} = [r_1, ..., r_p]$ with $r_i > 0$, $i = \bar{1}, p$ such that the property for

$$X_j(\Lambda_r(\lambda)y) = \lambda^{\nu_j} \Lambda_r(\lambda)X_j(y), \quad \nu_j \ge -r_{\min}, \quad j = \bar{0}, \bar{m}$$
holds for all $\lambda > 0$ and $y \in \mathbb{R}^p$.

Formally, Assumption 2 does not imply any kind of homogeneity of (3).

**Assumption 3** There is a locally Lipschitz continuous vector field $g: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}^s$ such that the system

$$\dot{z}(t) = g(y(t), z(t)), \quad t \ge 0, \qquad (5)$$

with $z(t) \in \mathbb{R}^s$ and $y \in \mathcal{L}_\infty$, admits a compact invariant set $\mathcal{Z} \subset \mathbb{R}^s$.

A signal generator system (5) defined in Assumption 3 will be subsequently used to induce the time dependence in the control laws:

**Assumption 4** There exist locally Lipschitz continuous functions $U_i: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}$, $i = \bar{1}, m$ and a smooth function $V: \mathbb{R}^p \times \mathbb{R}^s \to \mathbb{R}_+$ such that for some $\alpha_1, \alpha_2 \in \mathcal{K}_\infty$ and $\alpha_3 \in \mathcal{K}$,

$$\alpha_1(\|y\|) \le V(y, z) \le \alpha_2(\|y\|),$$

$$\frac{\partial V(y, z)}{\partial y} f(y, z, 0) + \frac{\partial V(y, z)}{\partial z} g(y, z) \le -\alpha_3(\|y\|)$$

hold for all $y \in \mathcal{X} \subset \mathbb{R}^p$ (with $0 \in \text{int}(\mathcal{X})$), $z \in \mathcal{Z}$, where

$$f(y, z, d) = X_0(y) + \sum_{i=1}^{m} X_i(y) U_i(y, z) + d.$$
This assumption states that for the system (3), (5) there exist non-autonomous (dependent on $z$) control laws $U_i$, $i = \overline{1,m}$ that render the closed-loop dynamics SIuOS for $d=0$ with respect to the output $y$ at least locally in $\mathcal{X} \times \mathcal{Z}$ with a known Lyapunov function $V$. This property implies a local SIHOS for the system (3), (5) closed by the feedbacks $U_i$, $i = \overline{1,m}$ for the initial conditions in $\mathcal{X} \times \mathcal{Z}$ and for sufficiently small disturbances $d$.

Following [7], assume that the set $\mathcal{V} = \{y \in \mathbb{R}^p, z \in \mathbb{R}^s : V(y, z) \le 1\}$ belongs to the interior of $\mathcal{X} \times \mathcal{Z}$ (this can be always guaranteed by scaling $V$). Define a map $\psi : \Omega \to (0, +\infty)$, where $\Omega = \mathbb{R}^p \setminus \{0\} \times \mathcal{Z}$, as a solution of the equation:

$$V(\Lambda_r(\psi(y, z))y, z) = 1. \quad (6)$$

Such a $\psi(y, z)$ always exists since $V$ is assumed to be positive definite and radially unbounded in $y$, while the transformation $\Lambda_r(\psi(y, z))y$ linearly scales the homogeneous norm of $y$ (i.e., $\|\Lambda_r(\psi(y, z))y\|_r = \psi(y, z)\|y\|_r$) and the homogeneous norm is equivalent to the standard one $\| \cdot \|$. According to the Implicit Function Theorem, $\psi$ is locally uniquely defined and continuously differentiable in $\Omega$ provided that

$$\frac{\partial V(y, z)}{\partial y} \mathcal{H}y > 0, \forall (y, z) \in \partial \mathcal{V} \quad (7)$$

with $\mathcal{H} = \text{diag}\{r_k\}_{k=1}^p$, and obviously $\psi(y, z) = 1$ for $(y, z) \in \partial \mathcal{V}$. Due to these regularity properties and positive definiteness of $V$, $\psi(\Lambda_r(\lambda)y, z) = \lambda^{-1}\psi(y, z)$ for all $\lambda > 0$, all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$, i.e., $\psi^{-1}$ is $\mathbf{r}$-homogeneous of degree 1 considering $z$ as a constant, and it can be viewed as a partially homogeneous norm with respect to $y$.

**Remark 1** To calculate the value of $\psi(y, z)$, a bisection algorithm from [19] can be used. Typically, the required number of iterations to achieve a reasonable accuracy in $\psi$ and the associated computational load are small.

As in [7], recalling assumptions 2, 3 and 4, let us define the control laws for the system (3) as

$$u_i(y, z) = \begin{cases} K_i(y, z) & \text{if } y \ne 0, \\ 0 & \text{if } y = 0, \end{cases} \quad i = \overline{1,m}, \quad (8)$$

$$K_i(y, z) = \psi^{-\nu_0+\nu_i}(y, z)U_i(\Lambda_r(\psi(y, z))y, z),$$

and the accelerated generator dynamics as

$$\dot{z}(t) = G(y(t), z(t)) \quad (9)$$

$$= \begin{cases} \psi^{-\nu_0}(y(t), z(t))g(\Lambda_r(\psi(y, z))y, z(t)) & \text{if } y \ne 0, \\ 0 & \text{if } y = 0. \end{cases}$$

Note that the control laws (8) are locally Lipschitz continuous on $\Omega$ provided that

$$\nu_0 \ge \max_{i=1,m} \nu_i \quad (10)$$

and partially homogeneous:

$$K_i(\Lambda_r(\lambda)y, z) = \lambda^{\nu_0-\nu_i} K_i(y, z)$$

for all $\lambda > 0$ and all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$. Similarly, the accelerator dynamics (9) are also locally Lipschitz continuous on $\Omega$ for $\nu_0 \ge 0$ and partially homogeneous:

$$G(\Lambda_r(\lambda)y, z) = \lambda^{\nu_0} G(y, z)$$

for all $\lambda > 0$, $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$. Denote the closed-loop dynamics (3), (8) as

$$F(y, z, d) = X_0(y) + \sum_{i=1}^{m} X_i(y) K_i(y, z) + d,$$

then due to the established above symmetry relations for $K_i$ and Assumption 2,

$$F(\Lambda_r(\lambda)y, z, \Lambda_{\tilde{r}}(\lambda)d) = \lambda^{\nu_0} \Lambda_r(\lambda) F(y, z, d)$$

for all $\lambda > 0$, all $(y, z) \in \mathbb{R}^p \times \mathcal{Z}$ and $d \in \mathbb{R}^p$, where $\tilde{r}_i = r_i + \nu_0$ for $i = \overline{1,p}$ (Assumption 1 is verified). Finally, define

$$W(y, z) = \begin{cases} \psi^{-1}(y, z) & \text{if } y \ne 0, \\ 0 & \text{if } y = 0 \end{cases} \quad (11)$$

as an implicit Lyapunov function candidate. Its time derivative along the trajectories of (3), (8), (9) with $d=0$ takes the following form for $\xi = \Lambda_r(\psi)y \in \partial \mathcal{V}$ (hence, $\psi(\xi, z) = 1$ and $F(\xi, z, 0) = X_0(\xi) + \sum_{i=1}^m X_i(\xi)\psi^{-\nu_0+\nu_i}(\xi, z) U_i(\Lambda_r(\psi(\xi, z))\xi, z) = f(\xi, z, 0)$ and $G(\xi, z) = g(\xi, z)$):

$$\begin{aligned} & \frac{\partial W(y, z)}{\partial y} F(y, z, 0) + \frac{\partial W(y, z)}{\partial z} G(y, z) \\ & = \psi^{-1-\nu_0}(y, z) \frac{\frac{\partial V(\xi, z)}{\partial y} F(\xi, z, 0) + \frac{\partial V(\xi, z)}{\partial z} G(\xi, z)}{\frac{\partial V(\xi, z)}{\partial x} \mathcal{H}\xi} \\ & \le -\psi^{-1-\nu_0}(y, z) \frac{\alpha_3 (\|\xi\|)}{\frac{\partial V(\xi, z)}{\partial x} \mathcal{H}\xi} \end{aligned}$$

for all $(y, z) \in \Omega$. Since $\frac{\partial V(\xi, z)}{\partial \tilde{r}_i} \mathcal{H}\xi \le \eta(\|\xi\|)$ due to the properties of $V$ for some $\eta \in \mathcal{K}$, the latter estimate implies SIuOS property for (3), (8), (9) with $d=0$. Therefore, the system (3), (8), (9) verifies all conditions of Theorem 6:

**Theorem 7** Let (7) and assumptions 2 (with (10)) and 3, 4 be satisfied. Then for $\nu_0 \ge 0$ there exist $\beta \in \mathcal{KL}$ and $\gamma > 0$ such that

$$\|y(t)\|_r \le \beta(\|y(0)\|_r, t) + \gamma \sup_{s \in [0, t)} \|d(s)\|_{\tilde{r}}, \forall t \ge 0$$

holds for all $y(0) \in \mathbb{R}^p$, $z(0) \in \mathcal{Z}$ and $d \in \mathcal{L}_\infty$. Moreover, $\beta \in \mathcal{eKL}$ if $\nu_0 = 0$ and $\beta \in \mathcal{nFxKL}$ if $\nu > 0$.

In [18] such a theorem was obtained for $\nu_0 = 0$ and $d=0$ only. In [7], $\nu_0 \ne 0$ was considered in the disturbance-free case.

**Remark 2** If $\nu_0 < 0$ then the modified generator dynamics (9) become discontinuous for $y=0$ (we have $\lim_{\|y\| \to 0} \psi^{-1}(y, \cdot) = 0$). In the absence of disturbances, and in the case $\nu_0 < 0$, we can consider the system behavior just while $y(t) \ne 0$ as $y=0$ is the invariant subspace, see [7]. However, for $d \ne 0$, a perturbation may force the
system to reach the set where $y = 0$ at a finite instant of time, and next to leave such a set posing thereby a problem with the definition of the solution $z(t)$ in (9). In fact, the uniqueness of solutions for (9) is lost in such a case, and we can select any value in $\mathcal{Z}$ for the variable $z$ while leaving $y = 0$ (it is the generator state, and its value is in our hands, while the control equals zero on $y = 0$). Under such a convention, the result of Theorem 7 also “holds” in the case $\nu_0 < 0$ with $\beta \in \mathcal{F}\mathcal{K}\mathcal{L}$.

The condition (7) can be relaxed by using the concept of generalized linear homogeneity (see [9, 20]) instead of the weighted one (all proofs can be repeated with minor changes).

Finally, the proposed controls are also robust with respect to the measurement noises:

**Corollary 1** Let all conditions of Theorem 7 be satisfied, while the variable $y(t)$ is available with a noise $v \in \mathcal{L}_{\infty}^p$, and the control algorithm takes the form:

$$u_i(t) = u_i(y(t) + v(t), z(t)), \quad i = \overline{1, m}, \\ \dot{z}(t) = G(y(t) + v(t), z(t)),$$

where the functions $u_i$ and $G$ are defined in (8), (9). Then for $\nu_0 \ge 0$ there exist $\beta \in \mathcal{K}\mathcal{L}$ and $\gamma, \ell > 0$ such that

$$\|y(t)\|_r \le \beta(\|y(0)\|_r, t) + \gamma \sup_{s \in [0,t)} \|d(s)\|_{\tilde{r}} \\ + \ell \sup_{s \in [0,t)} \|v(s)\|_r, \quad \forall t \ge 0$$

holds for all $y(0) \in \mathbb{R}^p$, $z(0) \in \mathcal{Z}$, $d \in \mathcal{L}_{\infty}^p$ and $v \in \mathcal{L}_{\infty}^p$. Moreover, $\beta \in \mathcal{E}\mathcal{K}\mathcal{L}$ if $\nu_0 = 0$ and $\beta \in \mathcal{n}\mathcal{F}\mathcal{x}\mathcal{K}\mathcal{L}$ if $\nu > 0$.

**PROOF.** It is clear that for the extended input, including $d$ and $v$, the closed-loop system is SIuOS in the disturbance-free case, while due to placement of the measurement noise, Assumption 1 is verified for $v$ considering dilation with the weights $r$. Next, all steps of the proof coincide with the ones of Theorem 7.

The effect of implementation of the control law using $\psi$ derived with some numerical errors (see Remark 1) can be treated as an additive perturbation, then applying a similar analysis (the function $\psi^{-1}$ is partially homogeneous of degree 1), the robustness of the closed-loop system to these computational inaccuracies can be established.

# 6 Robust control of a nonholonomic integrator

For an illustration, consider an example of (3) with $p = 3$, $m = 2$ and

$$X_0(y) = 0, X_1(y) = [1 \ 0 \ y_2]^\top, X_2(y) = [0 \ 1 \ 0]^\top,$$

which verifies Assumption 2 for $r = [1, 1, 2]$, $\nu_1 - \nu_2 = -1$ and $\nu_0 \ge -1$. This is the classical third order nonholonomic system in power form. Take

$$g(z) = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} z$$

with $z \in \mathbb{R}^2$ satisfying Assumption 3 for $\mathcal{Z} = \{z \in \mathbb{R}^2 : \|z\| = 1\}$. In [32] (see also [18]) it has been shown that the control law

$$U_1(y, z) = -y_1 + y_3z_1, \quad U_2(y, z) = -y_2 + y_3^2 z_2$$

stabilizes asymptotically this system at the origin with the Lyapunov function given by

$$V(y, z) = \frac{1}{2} \left[ \left(y_1 - \frac{y_3}{2}(z_1 - z_2)\right)^2 + \left(y_2 - \frac{y_3^2}{2}(z_1 + z_2)\right)^2 + y_3^2 \right], \\ \dot{V} = -(y_1 - \frac{y_3}{2}(z_1 - z_2))^2 - (y_2 - \frac{y_3^2}{2}(z_1 + z_2))^2 + \epsilon(y, z),$$

where $\epsilon(y, z)$ contains high-order terms. Hence, Assumption 4 is also true for $\mathcal{X} = \{y \in \mathbb{R}^3 : \|y\| \le \epsilon\}$ with $\epsilon > 0$ being sufficiently small. In [18] it has been claimed that $\epsilon > 1$ and (7) holds. Therefore, the conditions of Theorem 7 are verified for $\nu_0 = -0.5$ providing FT-SIOS or for $\nu_0 = 0.5$ ensuring nFtT-SIOS for the control laws (8), (9):

$$K_1(y, z) = \psi^{-\nu_0}(y, z)(-y_1 + \psi(y, z)y_3\dot{z}_1), \\ K_2(y, z) = \psi^{-\nu_0}(y, z)(-y_2 + \psi^3(y, z)y_3^2z_2), \\ G(y, z) = \begin{cases} \psi^{-\nu_0}(y, z) \begin{bmatrix} z_2 \\ -z_1 \end{bmatrix} & \text{if } y \neq 0, \\ 0 & \text{if } y = 0, \end{cases}$$

where $\psi = \psi(y, z)$ is the solution of the equation:

$$\psi^2[(y_1 - \psi \frac{y_3}{2}(z_1 - z_2))^2 + (y_2 - \psi^3 \frac{y_3^2}{2}(z_1 + z_2))^2 + \psi^2 y_3^2] = 2.$$

The results of MATLAB simulations of the closed-loop system for

$$d(t) = 0.1 \begin{bmatrix} \sin(5t) \\ \cos(\pi t) \\ \sin(10t) \end{bmatrix},$$

and using ode45 are presented in Figures 1 and 2 for $\nu_0 = -0.5$ (the value of $\psi$ was saturated while $y$ approached zero) and $\nu_0 = 0.5$, respectively. The algorithm from [19] was used to calculate $\psi$ with 5 bisection iterations on each time step. On the plots, the state norm $\|y\|_r$ (with $\varpi = 1$) in logarithmic scale, the generator output $z_1$ and the controls $u$ are shown versus the time $t$. As it can be observed, the state norm converges to a vicinity of zero in the presence of disturbances (the results of disturbance-free simulations demonstrating the accelerated decay can be found in [7]). Note that the frequency of oscillations of the generator
<figure><img src="image_4.png" /><figcaption>Fig. 1. The results of robust finite-time stabilization: The time histories of $||y||_r$, $z_1$ and $u$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. The results of robust nearly fixed-time stabilization: The time histories of $||y||_r$, $z_1$ and $u$.</figcaption></figure>

variables varies over time while the controls are converging to zero. As confirmed by additional simulations, the size of the neighborhood to which the convergence takes place follows the decrease in the disturbance magnitude, consistently with the theoretical results. Note also that the original algorithm from [32] is local and without robustness guarantees, demonstrating a slower convergence.

# 7 Conclusion

The paper further develops a recently proposed notion of homogeneity with respect to a part of the state variables by proposing the results, which characterize the related robust stability properties in the presence of external perturbations. These results can be applied to accelerate and to ensure robustness of the time-varying control laws previously proposed in [18] for nonholonomic systems. Examples of finite-time and nearly fixed-time robust stabilization of a nonholonomic integrator were reported.

# Acknowledgement

The second author would like to acknowledge the support of the National Science Foundation under award number ECCS-1931738.

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