The problem is to design $u : \mathbb{ R } ^n \mapsto \mathbb{ R } ^m$ such that the system

$$\dot x(t)=Ax(t)+B u (x(t))$$

is asymptotically stable $\Rightarrow$ $$x(t)\to 0 \quad \text{ as } \quad t\to+\infty$$

The problem is to design $u: \mathbb{ R }^n \mapsto \mathbb{ R }^m$ such that the system

$$\dot x(t)=Ax(t)+B u(x(t))$$ is finite-time stable $\Rightarrow$ $$\forall x(0)\in \mathbb{ R }^{n}, \quad \exists T=T(x(0))\geq 0 \quad : \quad x(t)=\mathbf{ 0}, \quad \forall t\geq T$$

The problem is to design $u: \mathbb{ R }^n \mapsto \mathbb{ R }^m$ such that the system $$\dot x(t)=Ax(t)+B u(x(t))$$ is fixed-time stable $\Rightarrow$ $$\exists T_{\max}>0 \quad : \quad x(t)=\mathbf{ 0 }, \quad \forall t\geq T_{\max}, \quad \forall x(0)\in \mathbb{ R }^{n}$$

The problem is to design a system (asymptotic ${\color{red}{observer}}$ )

$$\dot z(t)=f(z(t),p(t),y(t)), \quad \quad f:\mathbb{ R } ^n\times \mathbb{ R } ^n \times \mathbb{ R } ^k \to \mathbb{R}^n$$ such that $\|z(t)-x(t)\| \to 0 \quad \text{ as } t\to +\infty$

The problem is to design a system (finite-time observer)

$$\dot z(t)=f(z(t),p(t),y(t)), \quad \quad f:\mathbb{ R } ^n\times \mathbb{ R } ^n \times \mathbb{ R } ^k \to \mathbb{ R } ^n$$

such that $\forall x(0)\in \mathbb{ R } ^{n},\quad \exists T=T(x(0))\geq 0 \quad : \quad \|z(t)-x(t)\| = 0, \forall t\geq T$

The problem is to design a system (fixed-time observer)

$$\dot z(t)=f(z(t),p(t),y(t)), \quad \quad f:\mathbb{ R } ^n\times \mathbb{ R } ^n \times \mathbb{ R } ^k \to \mathbb{ R } ^n$$

such that $\quad \exists T_{\max}>0 \quad : \quad \|z(t)-x(t)\| = 0, \quad \forall t\geq T_{\max}, \quad \forall x(0)\in \mathbb{ R } ^{n}.$

$x\mapsto \lambda x \; ({\color{BlueViolet}{standard\; dilation}})$                             $f(\lambda x)=\lambda^{\nu} f(x), \;\; \;\; \forall s,x \;\; \;\; ({\color{BlueViolet}{symmetry}})$

$\lambda>0$ - scaling factor                                            $\nu\in \mathbb{ R }$- homogeneity degree

$x \mapsto\mathbf{d}(s)x$    $({\color{BlueViolet}{generalized \;\; \;\; dilation}})$                                $f(\mathbf{ d }(s)x)=e^{\nu s} f(x),$     $({\color{BlueViolet}{symmetry}})$

${\color{BlueViolet}{Limit \;\; property}}: \;\;\; \lim\limits_{s\to -\infty}\|\mathbf{ d }(s)x\|\!=\!0, \;\; \;\;$                              $\lim\limits_{s\to +\infty}\|\mathbf{ d } (s)x\|\!=\!+\infty,\;\; \forall x\!\neq\! \mathbf{ 0 }$

A continuous linear $\textbf{dilation}$ in $\mathbb{ R } ^n$ is a matrix-valued function given by

$\mathbf{ d } (s)=e^{sG_{\mathbf{ d } }}=\sum\limits_{i=0}^{+\infty}\tfrac{s^iG_{\mathbf{ d } }^i}{i!}, s\in \mathbb{ R }$

where $G_{\mathbf{ d } }\in \mathbb{ R } ^{n\times n}$ is an $\textit{anti-Hurwitz matrix}$ called a $\textbf{generator}$ of $\mathbf{d}$.

$\textit{Is there any potential advantage of a homogeneous system versus a linear one?}$

Let $\|\cdot\|$ be a norm in $\mathbb{ R } ^n$ and $\mathbf{d}$ be a $\textit{dilation}$ in $\mathbb{ R } ^n$.

• A dilation $\mathbf{d }$ is said to be monotone with respect to $\|\cdot\|$ if the function $s\mapsto \|\mathbf{d}(s)x\|$ is strictly increasing for any $x\neq \mathbf{0}$.
• A dilation $\mathbf{d }$ is said to be strictly monotone with respect to $\|\cdot\|$ if $\exists \beta>0:\|\mathbf{ d } (s)\|\leq e^{\beta s},\forall s\leq 0$.

A linear continuous dilation $\mathbf{ d }$ is strictly monotone with respect to the weighted Euclidean norm

$\|x\|=\sqrt{x^{\top}Px}, \quad x\in \mathbb{ R } ^n,$

where $0\prec P=P^{\top}\in \mathbb{ R } ^{n\times n}$, $\textbf{if and only if }$ the following linear matrix inequality holds $$PG_{\mathbf{ d } }+G_{\mathbf{ d } }^{\top} P\succ 0, \quad \quad \quad \quad \quad \quad (3)$$ where $G_{\mathbf{ d } }\in \mathbb{ R } ^{n}$ is the generator of the dilation $\mathbf{ d }$. Moreover, one has $$\!\begin{array}{c} \beta \|\mathbf{ d } (s)z\|^2\leq\frac{ \frac{d}{ds} \|\mathbf{ d } (s)z\|^2}2\leq \alpha \|\mathbf{ d } (s)z\|^2, \quad \forall s\in \mathbb{ R } , \forall z\in \mathbb{ R } ^n, \\ e^{\alpha s}\!\leq\!\lfloor \mathbf{ d } (s)\rfloor\!\leq\!\|\mathbf{ d } s)\|\!\leq e^{\beta s},\quad \forall s\!\leq\! 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad (4) \\ e^{\beta s}\!\leq\!\lfloor \mathbf{ d } (s)\rfloor\!\leq\!\|\mathbf{ d } (s)\|\!\leq e^{\alpha s}, \quad \forall s\!\geq\! 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \end{array}$$ where $$\alpha=\frac{1}{2}\lambda_{\max}\left(P^{\frac{1}{2}} G_{\mathbf{ d } }P^{-\frac{1}{2}}+P^{-\frac{1}{2}}G^{\top}_{\mathbf{ d } } P^{\frac{1}{2}}\right)>0,$$ $$\beta=\frac{1}{2}\lambda_{\min}\left(P^{\frac{1}{2}} G_{\mathbf{ d } }P^{-\frac{1}{2}}+P^{-\frac{1}{2}}G^{\top}_{\mathbf{ d } } P^{\frac{1}{2}}\right)>0.$$

A vector space over the filed $\mathbb{ R }$ of reals is a set $\mathbb{V}$ together with two operations:

• ${\color{blue}{vector \;\; addition\;\; }} \;\; \mathbb{V} \times \mathbb{V} \to \mathbb{V}$ denoted by ${\color{blue}{v+w}}$ for $v,w\in \mathbb{V}$.

• ${\color{red}{multiplication\;\; by\;\; a \;\; scalar\;\; } } \;\; \mathbb{ R } \!\times\!\mathbb{V} \to \mathbb{V}$ denoted by ${\color{red}{\alpha \cdot v}}$ for $\alpha\!\in\!\mathbb{ R }$, $v\!\in\!\mathbb{V}$.

satisfying certain axioms

• $\textit{Associativity}: u+(v +w)=(u+v)+w$

• $\textit{Distributivity}: (\alpha+\beta)u=\alpha\cdot u+\beta\cdot v$

• ...

$\textit{Is it possible to construct a vector space using a }$ $\underline{generalized \;\; dilation}$ $\textit{ as a multiplication by a scalar?}$

Let a linear continuous dilation $\mathbf{d}$ in $\mathbb{R}^n$ be monotone with respect to a norm $\|\cdot\|$. A function $\|\cdot\|_{\mathbf{d}}:\mathbb{R}^n\to \mathbb{R}_+$ defined as $\|\mathbf{0}\|_{\mathbf{d}}=0$ and
$$\|x\|_{\mathbf{ d } }\!=\!e^{s_x} \quad \text{ where } \quad s_x\in \mathbb{ R }: \| \mathbf{ d } (-s_x)x\|\!=\!1 \quad \text{ for } \quad x\neq\mathbf{ 0 }$$ is called the canonical $\mathbf{d}$-homogeneous norm.

If a linear continuous dilation $\mathbf{ d }$ in $\mathbb{ R } ^n$ is strictly monotone with respect to the norm $\|x\|=\sqrt{x^{\top} Px}, P\succ 0$ then

1. $\|\cdot\|_{\mathbf{ d } } : \mathbb{ R } ^n\to \mathbb{ R } _+$ is single-valued and $\|\cdot\|_{\mathbf{ d } }\in C^{\infty}(\mathbb{ R } ^n)$;

2. $x\neq \mathbf{0} \Leftrightarrow \|x\|_{\mathbf{ d } }\neq \mathbf{ 0}$;

3. $\|\pm \mathbf{ d } (s)x\|_{\mathbf{ d } }=e^{s}\|x\|_{\mathbf{ d } }$, $\forall s\in \mathbb{ R } , \forall x\in \mathbb{ R } ^n$;

4. $\|x\|\!=\!1 \!\Leftrightarrow\! \|x\|_{\mathbf{ d } }\!=\!1$, $\|x\|\!<\!1\!\Leftrightarrow\! \|x\|_{\mathbf{ d } }\!<\!1$ and $\|x\|\!>\!1\! \Leftrightarrow\! \|x\|_{\mathbf{ d } }\!>\!1$;

5. for $0<2\beta = \lambda_{\min}\left( PG_{\mathbf{ d } }+G_{\mathbf{ d } }^{\top}P \right)\leq \lambda_{\max}\left( PG_{\mathbf{ d } }+G_{\mathbf{ d } }^{\top}P \right)= 2\alpha$ one has $$\left| \|x_1\|_{\mathbf{ d } }^{\alpha}-\|x_2\|_{\mathbf{ d } }^{\alpha}\right|\leq \|x_1-x_2\|, \quad \forall x_1,x_2\in B,\quad \quad \quad \quad \quad (5)$$ $$\left| \|x_1\|_{\mathbf{ d } }^{\beta}-\|x_2\|_{\mathbf{ d } }^\beta\right|\leq \|x_1-x_2\|, \quad \forall x_1,x_2\in \mathbb{ R } ^n\backslash B,\quad \quad \quad (6)$$ where $B=\{x:\in\mathbb{ R } ^n: \|x\|\leq 1\}$ is the unit ball in $\mathbb{ R } ^n$.

6. the derivative of $\|\cdot\|_{\mathbf{ d } }$ is given by $$\frac{\partial \|x\|_{\mathbf{ d } }}{\partial x}=\|x\|_{\mathbf{ d } }\tfrac{x^{\top } \mathbf{ d } ^{\top}(-\ln\|x\|_{\mathbf{ d } }) P \mathbf{ d } (-\ln\|x\|_{\mathbf{ d } })}{x^{\top } \mathbf{ d } ^{\top}(-\ln\|x\|_{\mathbf{ d } })PG_{\mathbf{ d } }\mathbf{ d } (-\ln\|x\|_{\mathbf{ d } })x}, \quad x\neq \mathbf{ 0} . \quad \quad \quad (7)$$

Let a linear continuous dilation $\mathbf{ d }$ in $\mathbb{R}^n$ be strictly monotone with respect to the norm $\|x\|=\sqrt{x^{\top} Px}$. The mapping $\Phi : \mathbb{ R } ^n \to \mathbb{ R } ^n$ given by

$$\Phi(x)=\|x\|_{\mathbf{ d } } \mathbf{ d } (-\ln \|x\|_{\mathbf{ d } }) x, \quad x\in \mathbb{ R } ^n\backslash\{\mathbf{0}\} \quad \quad \quad (8)$$

with $\Phi(\mathbf{ 0 } )=\mathbf{ 0 }$ is a homeomorphism in $\mathbb{ R } ^n$, its inverse has the form

$$\Phi^{-1}(z)=\|z\|^{-1}\mathbf{ d } (\ln \|z\|)z, \quad z\in \mathbb{ R } ^n\backslash\{\mathbf{0}\}$$

with $\Phi^{-1}(\mathbf{ 0 } )=\mathbf{ 0 }$ by continuity.

Let a linear dilation $\mathbf{ d }$ be strictly monotone with respect to the norm $\|x\|=\sqrt{x^{\top} Px}$. Let an addition of vectors ${\color{blue}{ \tilde{\mathbf{+}}}}    : \mathbb{ R } ^n\times \mathbb{ R } ^n \to \mathbb{ R } ^n$ and a multiplication by a scalar ${\color{red}{\tilde{\cdot}}}    : \mathbb{ R } \times \mathbb{ R } ^n\to \mathbb{ R } ^n$ be defined as follows

• $x{\color{blue}{\tilde{\mathbf{ + }}}}     y:=\Phi^{-1}(\Phi(x)+\Phi(y))$, where $x,y\in \mathbb{ R } ^n$,

• $\lambda{\color{red}{\tilde{\cdot}}}     x:= \texttt{sign}(\lambda)\mathbf{ d } (\ln |\lambda|)x$, where $\lambda\in \mathbb{ R }$, $x\in \mathbb{ R } ^n$.

Then the set $\mathbb{ R } ^n$ together with the operations ${\color{blue}{ \mathbf{ + }}}$ and ${\color{red}{\tilde{\cdot}}}$ is a linear vector space $\mathbb{ R } ^n_{\mathbf{ d } }$ with the norm $\|\cdot\|_{\mathbf{ d } }$.

Let a linear continuous dilation $\mathbf{d}$ be strictly monotone with respect to the weighted Euclidean norm $\|x\|=\sqrt{x^{\top}Px}$. Let $\|\cdot\|_{\mathbf{d}}$ be the canonical homogeneous norm induced by $\|\cdot\|$. The mapping $\mathbb{R}^n_{\mathbf{d}} \times \mathbb{R}^n_{\mathbf{d}} \mapsto \mathbb{R}$ given by $$\langle x,y\rangle_{\mathbf{d}}= \Phi^{\top}(x) P \Phi(y)$$ is an inner product on $\mathbb{R}^n_{\mathbf{d}}$, where $\Phi$ is given by (8).

A    $\underline{function }$    $h: \mathbb{ R }^n \to \mathbb{ R }$ is said to be $\mathbf{ d }$-homogeneous of a degree $\nu\in \mathbb{ R }$ if

$$h(\mathbf{ d } (s)x)=e^{\nu s}h(x) \quad \text{ for } \quad s\in\mathbb{ R }, \quad x\in \mathbb{ R }^n, \quad \quad \quad \quad \quad \quad (9)$$

where $\mathbf{ d }$ is a dilation in $\mathbb{ R }^n$.

Let $h \in \mathcal{H}_{\mathbf{ d }}(\mathbb{ R } ^n)$ be such that $\sup_{x\in S} |h(x)|<+\infty$.

• If $\deg_{\mathbf{ d }}(h)>0$ then $h$ is bounded on any $\mathbf{ d }$-homogeneous ball $B_{\mathbf{ d }}(r)$ and
  $h(x)\to 0$ as $x\to 0$

• If $\deg_{}(h)\!<\!0$ then $h$ is bounded on any set $\mathbb{ R } ^n\backslash B_{\mathbf{ d }}(r)$ with $r\!>\!0$ and
  $h(x)\to 0$ as $x\to \infty$

• If $\deg_{\mathbf{ d }}(h) =0$ then $h$ is uniformly bounded on $\mathbb{ R } ^n$ and, moreover,
  $h\equiv \textrm{const}$ provided that $h$ is continuous at zero.

If $h \in \mathcal{H}_{\mathbf{ d }}(\mathbb{ R } ^n)$ is differentiable on $\mathbb{ R } ^n \backslash \{ \mathbf{ 0 } \}$ then

$$\frac{\partial h(x)}{\partial x} G_{\mathbf{ d }}x=\deg_{\mathbf{ d }}(h) \cdot h(x), \quad \forall x\neq \mathbf{ 0 },n \quad \quad \quad (10)$$

where $G_{\mathbf{ d }}\in \mathbb{ R } ^{n\times n}$ is a generator of the linear dilation $\mathbf{ d }$.

A $\underline{vector \; field}$     $f:\mathbb{ R } ^n \!\to\! \mathbb{ R } ^n$ is said to be $\mathbf{ d }$-homogeneous of degree $\mu\!\in\! \mathbb{ R }$ if

$$f(\mathbf{ d }(s)x)=e^{\mu s}\mathbf{ d }(s)f(x) \quad \text{ for all } \quad s\in\mathbb{ R } , \quad u\in \mathbb{ R } ^n, \quad \quad \quad (11)$$

where $\mathbf{ d }$ is a dilation in $\mathbb{ R } ^n$.

Let $f\in \mathcal{F}_{\mathbf{ d } }(\mathbb{ R } ^n)$ be such that $M:=\sup_{x\in S}\|f(x)\|<+\infty$

• If $\deg_{\mathbf{ d } }(f)+\beta>0$ then $f$ is bounded on $B_{\mathbf{ d } }(r)$ and
  $\|f(x)\|\to 0$ as $\|x\|\to 0$

• f $\deg_{\mathbf{ d } }(f)+\alpha<0$ then $f$ is bounded on $\mathbb{ R } ^{n}\backslash B_{\mathbf{ d } }(r)$ and
  $\|f(x)\|\to 0$ as $\|x\|\to +\infty$

• If $\deg_{\mathbf{ d } }(f)+\beta=\deg_{\mathbf{ d } }(f)+\alpha=0$ then $f$ is uniformly bounded on $\mathbb{ R } ^{n}$,

where $\alpha=\lambda_{\max}\left(P^{1/2}G_{\mathbf{ d } }P^{-1/2}+P^{-1/2}G_{\mathbf{ d } }P^{1/2}\right)$ and $\beta=\lambda_{\min}\left(P^{1/2}G_{\mathbf{ d } }P^{-1/2}+P^{-1/2}G_{\mathbf{ d } }P^{1/2}\right)$.

If $f\in \mathcal{F}_{\mathbf{ d } }(\mathbb{ R } ^n)$ is differentiable on $\mathbb{ R } ^n\backslash\{\mathbf{0} \}$ then

$$\label{eq:hom_op_deriv_relation_Rn} \frac{\partial f(x)}{\partial x} G_{\mathbf{ d } }x=(\deg_{\mathbf{ d } }(f) I_n+G_{\mathbf{ d } })f(x) \text{ for all } x\in \mathbb{ R } ^n\backslash\{\ \mathbf{ 0 } \}. \quad \quad \quad (12)$$

The following three claims are equivalent ( [ 15 ] , [ 20 ] ) :

1. a vector field $x\mapsto Ax$ with $x\in \mathbb{ R } ^n$ and $A\in \mathbb{ R } ^{n\times n}$ is $\mathbf{ d }$-homogeneous of degree $\mu\neq 0$ with respect to a linear continuous dilation $\mathbf{ d }$ in $\mathbb{ R } ^n$;

2. there exists an anti-Hurwitz matrix $G_{\mathbf{ d } }\in \mathbb{ R } ^{n\times n}$ such that $\begin{equation}\label AG_{\mathbf{ d } }=(\mu I+G_{\mathbf{ d } })A; \quad \quad \quad (13) \end{equation}$

3. the matrix $A$ is nipotent.

How to find $G_{\mathbf{ d } }$ for a given $\mu\neq 0$?

Let $\mathbf{ d }$ be a linear dilation and let a linear vector field $f:\mathbb{ R }^n\to \mathbb{ R }^n$ given by

$$f(x)=Ax, \quad x\in \mathbb{ R }^n, \quad A\in \mathbb{ R }^{n\times n}$$

be $\mathbf{ d }$-homogeneous of degree $\mu\in\mathbb{ R }^n$ (i.e., $A\mathbf{ d } (s)=e^{\mu s}\mathbf{ d } (s)A$) then

$$e^{tA}\mathbf{ d } (s)=\mathbf{ d } (s)e^{tAe^{\mu s}}, \quad \forall s\in \mathbb{ R }, \quad \forall t\in \mathbb{ R }.$$

Let $\mathbf{ d }$ be a dilation in $\mathbb{ R }^n$ and the vector field $f$ be $\mathbf{ d }$-homogeneous of degree $\mu\in \mathbb{ R }$. If $x(t,x_0)$ with $t\in [0, T]$ is a solution of the system (14) with the initial condition $x(0)=x_0$ then, for any $s\in \mathbb{ R }$,

$$x(t,\mathbf{ d } (s)x_0)=\mathbf{ d } (s)x(e^{\mu s}t,x_0), \quad t\in [0, e^{-\mu s}T]$$

is a solution of (14) with the $\underline{scaled}$ initial condition $x(0)=\mathbf{ d } (s)x_0$.

Let $f$ be a continuous $\mathbf{ d }$-homogeneous vector field of degree $\mu\leq 0$ then the system $\dot x=f(x)$ is $complete .$

The system (14) is said to be locally (globally) uniformly $\textit{Lyapunov stable}$ if ,

$$\exists \varepsilon\in \mathcal{K} \quad : \quad |x(t,x_0)|\leq \varepsilon(|x_0|), \quad \forall t\geq 0, \quad \forall x_0\in \Omega,$$

where $\Omega$ is a neighborhood of the origin (resp., $\Omega=\mathbb{R}^n$)

Let $f:\mathbb{ R} ^n\to\mathbb{ R }^n$ be a continuous $\mathbf{ d }$-homogeneous vector field. The system (14) is Lyapunov stable if and only if it has a positively invariant, bounded neighborhood of the origin.

The system (14) is said to be locally (globally) uniformly $\textit{asymptotically stable}$ if $\exists \beta\in \mathcal{KL}$

$$\|x(t,x_0)\|\leq \beta(|x_0|,t), \quad \forall t\geq 0, \quad \forall x_0\in \Omega,$$

where $\Omega$ is a neighborhood of the origin (resp., $\Omega=\mathbb{ R} ^n$)

Let a continuous vector field $f:\mathbb{ R } ^n\to \mathbb{ R } ^n$ be $\mathbf{ d }$-homogeneous of degree $\mu$. Let $m>0$ be an arbitrary positive number.

The system (14) is asymptotically stable if and only if there exists a positive definite homogeneous function $V\in C(\mathbb{ R } ^n)\cap C^{\infty}(\mathbb{ R } ^n\backslash\mathbf{ 0 } )$ of degree $m$:

$$\dot V(x)\leq -\rho V^{1+\frac{\mu}{m}}(x), \quad \forall x\in \mathbb{ R } ^n\backslash\{\mathbf{ 0 } \}, \quad \quad \quad (15)$$

where $\rho>0$ is some number.

Let $f$ be $\mathbf{ d }$-homogeneous of degree $\mu \!\in\! \mathbb{ R }$. If (14 ) is locally asymptotically stable then it is

• globally uniformly $\textbf{finite-time stable}$ for $\mu<0$ with a settling-time function $T$ being continuous at $x=0$;

• globally uniformly $\textbf{nearly fixed-time stable}$ for $\mu>0$.

A system (16) is said to be Input-to-State Stable (ISS) with respect to $q\in L^{\infty}(\mathbb{ R } ,\mathbb{ R } ^{k})$ if there exist $\beta\in \mathcal{KL}$ and $\gamma \in \mathcal{K}$ such that

$$\|x(t)\|\leq \beta(\|x_0\|,t)+\gamma(\|q\|_{L^{\infty}((t_0,t),\mathbb{ R } ^k)}). \quad \quad \quad (17)$$

Let $\mathbf{ d } _q$ be a dilation in $\mathbb{ R } ^n$ and $\mathbf{ d } _q$ be a dilation in $\mathbb{ R } ^m$ such that $\exists \mu\in \mathbb{ R } :$

$$f(\mathbf{ d } _x(s)x,\mathbf{ d } _q(s)q)=e^{\mu s}\mathbf{ d } _x(s)f(x,q) \quad \forall x\in \mathbb{ R } ^m, \forall q\in \mathbb{ R } ^m, \forall s\in \mathbb{ R } .$$

If the system $\dot x=f(x,\mathbf{0 } )$ is asymptotically stable then the system (16) is ISS.

Given $\mu\in \mathbb{ R }$ the control aim is to design a dilation $\mathbf{ d }$ in $\mathbb{ R } ^n$ and a feedback control $u:\mathbb{ R } ^n\mapsto \mathbb{ R } ^m$ such that the closed-loop system is

1. $\mathbf{ d }$-homogeneous of degree $\mu$, i.e., for $f(x)=Ax+Bu(x)$ one holds
   $$f(\mathbf{ d } (s)x)=e^{\mu s}\mathbf{ d } (s)f(x), \quad \forall x\in \mathbb{ R } ^n, \quad \forall s\in \mathbb{ R } ;$$

2. asymptotically stable ($\Rightarrow$ finite-time stability for $\mu<0$).

The system (18) is homogeneously stabilizable with $\mu\neq 0$ $\textbf{if and only if }$ the pair $\{A,B\}$ is controllable . For any controllable pair $\{A, B\}$ one holds

1. the linear algebraic equation $$AG_0-G_0A+BY_0=A, \quad G_0B=\mathbf{ 0} \quad \quad \quad (19)$$
   has a solution $Y_0\in\mathbb{ R } ^{m\times n}$, $G_0\in \mathbb{R}^{n\times n}$ and for any solution one hold

   • the matrix $G_0-I_n$ is invertible is invertible;

   • $G_{\mathbf{d}}\!=\!I_n\!+\!\mu G_0$ is anti-Hurwitz for $\mu\!\leq 1/ n^*$, where $n^*\!\in\! \mathbb{N}$ is a minimal number such that $\texttt{rank}[B,AB,...,A^{n^*-1}B]=n$;

   • the matrix $A_0=A+BK_0$ is nilpotent, $K_{0}\!=\!Y_0(G_0-I_n)^{-1}\!$ and $A_0G_{\mathbf{d}}\!=\!(G_{\mathbf{d}}\!+\!\mu I_n)A_0,G_{\mathbf{d}}B\!=\!B;$ (20)

2. the linear algebraic system
   $$\begin{array}{c} A_0X\!+\!XA_0^{\top}\!\!+\!BY\!+\!Y^{\top}\!B^{\top}\!\!+\!\rho(G_{\mathbf{ d } }X\!+\!XG_{\mathbf{ d } }^{\top})\!=\!\mathbf{ 0 } , \\ G_{\mathbf{ d } }X\!+\!XG_{\mathbf{ d } }^{\top}\!\succ\! 0, \quad X\!=\!X^{\top}\!\succ\! 0 \end{array} \quad \quad \quad (21)$$
   has a solution $X\!\in\! \mathbb{R}^{n\times n}$, $Y\!\in\! \mathbb{R}^{m\times n}$ for any $\rho\!\in\! \mathbb{ R } _+$;

3. the homogeneous norm $\|\cdot\|_{\mathbf{d}}$ induced by $\|x\|=\sqrt{x^{\top} X^{-1}x}$ is a Lyapunov function of the system (18) with the feedback law

   $$u(x)\!=\!K_0x+\|x\|_{\mathbf{d}}^{1+\mu}K\mathbf{d}(-\ln \|x\|_{\mathbf{d}})x, \quad K\!=\!YX^{-1}, \quad \quad \quad (22)$$

   where $\mathbf{d}$ is a dilation generated by $G_{\mathbf{d}}$; moreover,

   $$\tfrac{d}{dt} \|x\|_{\mathbf{d}} = -\rho \|x\|^{1+\mu}_{\mathbf{d}}, \quad x\neq \mathbf{0}; \quad \quad \quad (23)$$

4. $u \in C^{\infty}(\mathbb{ R } ^{n}\backslash\{\mathbf{ 0 } \})$ and $|u(x)| \leq K_0|x| +\lambda_{\max}(X)\|x\|_{\mathbf{ d } }^{1+\mu}, \forall x\in \mathbb{ R } ^n$, $\forall \mu\geq -1$;

5. the system (18), (22) is $\mathbf{ d }$-homogeneous of degree $\mu$.

Given $\mu\in \mathbb{ R }$ we need to design a dilation $\mathbf{ d }$ in $\mathbb{ R } ^n$ and an observer:

$$\dot z=Az+p+g(Cz-y), \quad z(0)=\mathbf{ 0 } \quad g: \mathbb{ R } ^{k}\mapsto \mathbb{ R } ^n$$

such that the error equation

$$\dot \epsilon=A\epsilon+g(C\epsilon), \quad \epsilon=z-x$$

is

$\mathbf{ d }$-homogeneous of degree $\mu$, i.e., for $f(\epsilon)=A\epsilon+g(C\epsilon)$ one holds $$f(\mathbf{ d } (s)\epsilon)=e^{\mu s}\mathbf{ d } (s)f(\epsilon), \quad \forall \epsilon\in \mathbb{ R } ^n, \quad \forall s\in \mathbb{ R } ;$$

asymptotically stable.

The system (24) is homogeneously observable with degree $\mu \neq 0$ $\textbf{if and only if}$ the pair $\{A,C\}$ is observable. For any observable pair $\{A,C\}$ one holds

1. the linear algebraic equation
   $$AG_0-G_0A+Y_0C=A, \quad CG_0=\mathbf{0} \quad \quad \quad (25)$$
   has a solution $Y_0\in \mathbb{R}^{n\times k}$, $G_0\in \mathbb{R}^{n\times n}$ and for any solution one holds
   • the matrix $G_0+I_n$ is invertible

   • the matrix $G_{\mathbf{d}}=I_n+\nu G_0$ is anti-Hurwitz for $\nu\geq -1/\tilde n$, where $\tilde n$ is a minimal natural number such that
     $\texttt{rank}\left[\begin{smallmatrix}C\\CA\\...\\CA^{n-1}\end{smallmatrix}\right]=n.$

   • the matrix $A_0=A+L_0C$ is nilpotent, $L_0=-(G_{0}+I_n)^{-1}Y_0$ and
     $A_0G_{\mathbf{d}}=(\nu I_n +G_{\mathbf{d}})A_0, \quad CG_{\mathbf{d}}=C;$

2. the algebraic system $$\! \!\!\!\!\!\! \begin{array}{c} PA_0\!+\!A_0^{\top}\!P\!+\!YC\!+\!C^{\top}\!Y^{\top}\!\!+\!\rho(PG_{\mathbf{d}}\!+\!G_{\mathbf{d}}^{\top}P)\!=\!\mathbf{ 0 } , \\ PG_{\mathbf{d}}+G_{\mathbf{d}}^{\top}P\succ 0, \quad P=P^{\top}\succ 0 \end{array} \quad \quad \quad (26) \!\!\!\!\!\!$$ has a solution $P\in \mathbb{ R } ^{n\times n}$, $Y\in \mathbb{ R } ^{k\times n}$ for any $\rho\in \mathbb{ R } _+$.

3. the canonical homogeneous norm $\|\cdot\|_{\mathbf{ d } }$ induced by the weighted Euclidean norm $\|\epsilon\|=\sqrt{\epsilon^{\top}P\epsilon}$ is a Lyapunov function (at least if $\nu$ is close to $0$) for the error equation (28) of the observer $$\dot z\!=\!Az\!+\!p\!+\!\left( L_0\!+ {\color{blue}{\!{ {|Cz\!-\!y|^{\nu-1}\mathbf{ d } (\ln |Cz-y|)}} } } L\right)(Cz\!-\!y), \quad z(0)\!=\!\mathbf{ 0 } , \quad \quad \quad (27)$$ where $L=P^{-1}Y$;
   • the error equation $$\dot \epsilon=(A_0+|C\epsilon|^{\nu-1}\mathbf{ d} (\ln |C\epsilon|)LC)\epsilon, \quad \epsilon=z-x \quad \quad \quad (28)$$ is continuous for $\nu>-1/n^*$ and discontinuous for $\nu=-1/n^*$;

   • the error system (27) is $\mathbf{ d }$-homogeneous of degree $\nu$,

$$u_j=K_0x_j+\|x_j\|^{1+\mu}_{\mathbf{ d } } K\mathbf{ d } (-\ln \|x_j\|_\mathbf{ d } )x_j$$

$$u_j=K_j(I_n-(t_{j+1}-t_j)(A+BK_j ))^{-1}x_j, \quad K_j=K_0+\|x_j\|^{1+\mu}_{\mathbf{ d } } K\mathbf{ d } (-\ln \|x_j\|_\mathbf{ d } )$$

$$z_{j+1}=z_j+(t_{j+1-t_j})\left(Az_j\!+\!p_j\!+\!(L_0\!+\!|Cz_j\!-\!y_j|^{\nu-1} \mathbf{d}(\ln |Cz_j\!-\!y_j|)L)(Cz_j\!-\!y_j)\right)$$

$$z_{j+1}=Q_j^{-1}\left(z_j+(t_{j+1}-t_j)\left(p_j-L_j y_j\right)\right)$$

where $Q_j=I_n-h(A+L_jC)$ and

$$L_j=L_0+|Cz_j-y_j|^{\nu-1} \mathbf{d}(\ln |Cz_j-y_j|)L.$$

Let a linear continuous dilation $\mathbf{d}$ have a diagonalizable generator, i.e., $\exists J \in \mathbb{R}^{n\times n}, \det(J)\neq 0$ such that $$G_{\mathbf{d}}=J^{-1}\Lambda J, \quad \Lambda=\mathrm{diag}\{r_1,r_2,...,r_n\}\succ 0.$$ Then the function $\|\cdot\|_{\bar r} : \mathbb{R}^n\to \mathbb{R}$ defined as follows $$\|x\|_{\bar r}\!=\!\left(\Psi^{\top}\!(x) Q \Psi(x)\right)^{\frac{1}{2 \bar r}}\!, \quad \Psi(x)\!=\! \left(\! \begin{smallmatrix} \lfloor e_1^{\top}\!Jx\rceil^{\frac{\bar r}{r_1}}\\ \lfloor e_2^{\top}\!Jx\rceil^{\frac{\bar r}{r_2}}\\ ...\\ \lfloor e_n^{\top}\!Jx\rceil^{\frac{\bar r}{r_n}}\\ \end{smallmatrix}\! \right)\!, \quad \bar r>0$$ is a $\mathbf{d}$-homogeneous norm in $\mathbb{R}^n$ and $\Psi$ is a homeomorphism on $\mathbb{R}^n$, where $h_i=(0,...,1,...,0)^{\top}\in \mathbb{R}^n$ is the unit vector, $\lfloor \cdot\rceil^{\alpha}=|\cdot|^{\alpha} \mathrm{sign}(\cdot)$ is the power operation presering the sign and $0\prec Q=Q^{\top}\in \mathbb{R}^{n\times n}$. Moreover,

• $\|\cdot\|_{\bar r}$ is continuously differentiable on $\mathbb{R}^{n}\backslash\{\mathbf{0}\}$ provided that $\bar r\geq \max\{r_1,...,r_n\}$;
• $\|\cdot\|_{\bar r}$ is analytic on $\mathbb{R}^{n}\backslash\{\mathbf{0}\}$ provided that $\bar r= p_1 p_2 ... p_n$ and $r_i=\tfrac{p_i}{q}$, where $p_i,q\in \mathbb{N}$ are odd natural numbers, $i=1,2,...,n$.

Let a linear continuous dilation $\mathbf{d}$ in $\mathbb{R}^n$ be strictly monotone with respect to the weighted Euclidean norm $\|x\|=\sqrt{x^{\top}Px}$ with $P\succ 0$ satisfying $PG_{\mathbf{d}}+G_{\mathbf{d}}^{\top}P\succ 0$. Let $\|\cdot\|_{\mathbf{d}}$ be a canonical $\mathbf{d}$-homogeneous norm induced by $\|\cdot\|$ and $\|\cdot\|_{\bar r}$ be an arbitrary $\mathbf{d}$-homogeneous norm. Then for any $\epsilon>0$ there exist $N\in \mathbb{N}$, $\beta_{i}\in \mathbb{R}^{n}$, $\alpha_i,\gamma_i\in \mathbb{R},i=1,2,...,N$ and $\xi\in C(\mathbb{R}^n\backslash\{\mathbf{0}\};\mathbb{R})$ such that $|\xi(x)|\leq \epsilon,\forall x\in \mathbf{R}^n\backslash\{\mathbf{0}\}$ and $$\begin{equation} \|x\|_{\mathbf{d}}\!=\!\|x\|_{\bar r}\left(\sum_{i=1}^{N}\tfrac{\gamma_i}{1+e^{\alpha_i +\beta_i^{\top}\mathbf{d}(-\ln \|x\|_{\bar r})x}}+\xi\left(x\right)\right), \quad \forall x\!\in\! \mathbb{R}^n\backslash\{\mathbf{0}\}. \end{equation}$$