On Finite- and Fixed-Time State Estimation for Uncertain Linear Systems

Abstract

In this chapter different approaches are provided to estimate the state of some classes of strongly observable linear systems with some parametric uncertainties and unknown inputs. A family of homogeneous observers and a fixed-time sliding-mode observer are introduced to solve this problem. The finite-time and fixed-time convergence properties and the synthesis of these observers are described along this chapter. Moreover, an unknown input identification approach is also introduced. Simulation results illustrate the performance of these state estimation approaches.

This chapter contains material reprinted from Automatica, Vol 87, Héctor Ríos and Andrew R. Teel, A hybrid fixed-time observer for state estimation of linear systems, Pages 103–112, Copyright (2018), with permission from Elsevier. This chapter also contains material reprinted from A. Gutiérrez, Héctor Ríos and Manuel Mera, Robust output-regulation for uncertain linear systems with input saturation, IET Control Theory & Applications. Copyright (2023) The Institution of Engineering and Technology. The Institution of Engineering and Technology is registered as a Charity in England & Wales (no 211014) and Scotland (no SC038698).

Appendix

This appendix collects some required preliminaries and the proofs of the given results.

1.1 Stability Notions

Consider the following nonlinear system

$$\begin{aligned} \dot{x} = f(x,w), \end{aligned}$$

(44)

where $x\in \mathbb {R}^n$ is the state and $w\in \mathbb {R}^p$ is the external disturbance such that $w\in \mathscr {W}=\{w\in \mathscr {L}_{\infty }:||w||_{\infty }\le \overline{w}\}$, with $\overline{w}\in \mathbb {R}_{>0}$. The function $f:\mathbb {R}^n \times \mathbb {R}^p \rightarrow \mathbb {R}^n$ is a locally Lipschitz continuous function such that $f(0,0)=0$. The solutions of system (44) are denoted as $x(t,x_0,w)$, with $x(0)=x_0$.

Definition 1

[20, 31]. The origin of system (44) is said to be: Stable if for any $\epsilon >0$ there is $\delta (\epsilon )$ such that for any $x_{0}\in \varOmega \subset \mathbb {R}^n$ the solutions are defined for all $t\ge 0$ and, if $||x_{0}||\le \delta (\epsilon )$, then $||x(t,x_{0}),w||\le \epsilon$, for all $t\ge 0$ and all $w\in \mathscr {W}$; Asymptotically Stable (AS) if it is Stable and for any $\epsilon >0$ there exists $T(\epsilon ,\kappa )\ge 0$ such that for any $x_{0}\in \varOmega$, if $||x_{0}||\le \kappa$, then $||x(t,x_{0},w)||\le \epsilon$, for all $t\ge T(\epsilon ,\kappa )$ and all $w\in \mathscr {W}$; Finite-Time Stable (FTS) if it is AS and for any $x_{0}\in \varOmega$ there exists $0\le T^{x_{0}}<+\infty$ such that $x(t,x_{0})=0$, for all $t\ge T^{x_{0}}$ and all $w\in \mathscr {W}$¹; and Fixed-Time Stable (FxTS) if it is FTS and the settling-time function $T(x_0)$ is bounded, i.e., $\exists T^+ >0:T(x_0)\le T^+$, for all $x_{0}\in \varOmega$.

If $\varOmega =\mathbb {R}^n$; then, the origin of system (44) is said to be globally Stable (GS), AS (GAS), FTS (GFTS), or FxT (GFxTS), respectively.

Before introducing a definition of finite and fixed-time input to state stability, let us introduce some useful functions. A continuous function $\alpha :{\mathbb {R}}_{\ge 0}\rightarrow {\mathbb {R}}_{\ge 0}$ belongs to class $\mathscr {K}$ if it is strictly increasing and $\alpha (0)=0$; it belongs to class $\mathscr {K}_{\infty }$ if it is also unbounded. A continuous function $\beta :\mathbb {R}_{\ge 0}\times \mathbb {R}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}$ belongs to class $\mathscr{K}\mathscr{L}_{T}$ if for each fixed s, $\beta (\cdot ,s)\in \mathscr {K}$, and for each fixed r there exists $0<T(r)<\infty$ such that $\beta (r,s)$ is decreasing to zero with respect to $s<T(r)$, and $\beta (r,s)=0$, for all $s\ge T(r)$.

Definition 2

[7, 20]. The system (44) is said to be Input-to-State Stable (ISS), with respect to w, if there exist some functions $\beta \in \mathscr{K}\mathscr{L}$ and $\gamma \in \mathscr {K}$, such that any solution $x(t,x_0,w)$, for any $x_0\in \mathbb {R}^n$ and any $w\in \mathscr {W}$, satisfies

$$\begin{aligned} ||x(t,x_0,w)|| \le \beta (||x_0||,t) + \gamma (||w||_\infty ), \ \forall t\ge 0. \end{aligned}$$

The system (44) is said to be Finite-Time ISS (FT-ISS) if $\beta \in \mathscr{K}\mathscr{L}_T$, and Fixed-Time ISS (FxT-ISS) if it is FT-ISS and $T(x_0)\le T^+$.

1.2 Homogeneity

Some notions related to homogeneity are introduced. For any $r_{j}>0$, $j=\overline{1,n}$ and $\lambda >0$, define the dilation matrix $\varLambda _{r}(\lambda ):=\text {diag}(\lambda ^{r_{1}},\ldots ,\lambda ^{r_{n}})$ and the vector of weights $r:=(r_{1},\ldots ,r_{n})^{T}$. Let us introduce the following homogeneity definition.

Definition 3

[3]. The function $g:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ is called r-homogeneous, if there exists $d\in \mathbb {R}$ such that $g(\varLambda _{r}(\lambda )x)=\lambda ^{d}g(x)$, for all $(x,\lambda )\in {\mathbb {R}}^{n}\times {\mathbb {R}}_{>0}$. The vector field $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}$ is called r-homogeneous, if there exists $d\ge -\min _{1\le j\le n}r_{i}$ such that $f(\varLambda _{r}(\lambda )x)=\lambda ^{d}\varLambda _{r}(\lambda )f(x)$, for all $(x,\lambda )\in {\mathbb {R}}^{n}\times {\mathbb {R}}_{>0}$. The constant d is called the degree of homogeneity, i.e., $\textrm{deg}(g)=d$ or $\textrm{deg}(f)=d$.

Note that a differential equation $\dot{x}=f(x)$ with homogeneity degree d is invariant with respect to the time-coordinate transformation $(t,x)\mapsto (\lambda ^{-d}t,\varLambda _{r}(\lambda )x)$. Then, defining $\textrm{deg}(t)=-d$, it is possible to call the differential equation itself homogeneous with degree d. It is worth mentioning that Definition 3 also applies for set-valued maps and differential inclusions [3].

Now, the following result, given by [8], represents the main application of homogeneity to finite-time stability and finite-time stabilization.

Theorem 3

[8]. Let $f:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}$ be a continuous r-homogeneous vector field with a negative degree. If the origin of the system (44), with $w\equiv 0$, is locally AS then it is GFTS.

Define an extended auxiliary vector field F for system (44) as follows

$$ F(x,w) = \left( \begin{array}{cc} f^T(x,w) & 0_p^T \end{array}\right) ^T \in \mathbb {R}^{n+p}, $$

where $0_p \in \mathbb {R}^{p}$ is a zero vector of dimension p. The following Theorem describes the ISS properties of the system (44) in terms of its homogeneity.

Theorem 4

[7]. Let the extended vector field F be r-homogeneous with degree $d \ge - \min r_i$, with $i=\overline{1,n}$, for vectors of weights $r=(r_1,\ldots ,r_n)>0$ and $\tilde{r}=(\tilde{r}_1,\ldots ,\tilde{r}_p)\ge 0$, i.e., $f(\varLambda _r(\lambda )x,\varLambda _{\tilde{r}}(\lambda )w) = \lambda ^d \varLambda _r f(x,w)$ holds. Let the system (44) be GAS for $w\equiv 0$. Then, the system (44) is ISS if $\min \tilde{r}_j > 0$, with $j=\overline{1,p}$.

Define the set $\mathbb {S}_r = \lbrace x\in \mathbb {R}^n: ||x||_r = 1 \rbrace$. Then, an extension of the previous Theorem for the case when $f:\mathbb {R}^n \times \mathbb {R}^p \rightrightarrows \mathbb {R}^n$ is a set-valued map is given by the following Theorem.

Theorem 5

[6]. Let the discontinuous extended vector field F be r-homogeneous with degree $d \ge - \min r_i$, with $i=\overline{1,n}$, for vectors of weights $r=(r_1,\ldots ,r_n)>0$ and $\tilde{r}=(\tilde{r}_1, \ldots ,\tilde{r}_p)\ge 0$, i.e., $f(\varLambda _r(\lambda )x,\varLambda _{\tilde{r}}(\lambda )w) = \lambda ^d \varLambda _r f(x,\xi )$ holds. Let the system (44) be GAS for $w\equiv 0$ and also

$$ \sup _{\forall z \in \mathbb {S}_r}||f(z,w) - f(z,0)|| \le \sigma (||w||), $$

for all $w\in \mathscr {W}$ and some $\sigma \in \mathscr {K}_\infty$. Then, the system (44) is ISS if $\min \tilde{r}_j > 0$, with $j=\overline{1,p}$. Moreover, if $d<0$, then system (44) is FT-ISS.

1.3 Implicit Lyapunov Function

The following theorems provide the background for asymptotic and finite-time stability analysis, respectively, of (44) using the Implicit Lyapunov Function (ILF) Approach, with $w\equiv 0$, [30].

Theorem 6

[30]. If there exists a continuous function $G:\mathbb {R}_{\ge 0}{\times }\mathbb {R}^{n}\rightarrow \mathbb {R},$ $(V,x)\mapsto G(V,x),$ satisfying the following conditions:

1) G is continuously differentiable outside the origin for all positive $V\in \mathbb {R}_{\ge 0}$ and for all $x\in \mathbb {R}^{n}\backslash \{0\}$;

2) for any $x\in \mathbb {R}^{n}\backslash \{0\}$ there exists $V\in \mathbb {R}_{+}$ such that $G(V,x)=0$;

3) let $\varPhi =\{(V,x)\in \mathbb {R}_{+}\times \mathbb {R}^{n}\backslash \{0\}:G(V,x)=0\},$ then, $\lim _{||x||\rightarrow 0}V=0^{+},\;\lim _{V\rightarrow 0}||x||=0,\;\lim _{||x||\rightarrow \infty }V=+\infty$, for all $(V,x)\in \varPhi$;

4) the inequality $\frac{\partial G(V,x)}{\partial V}<0$ holds for all $V\in \mathbb {R}_{+}$ and for all $x\in \mathbb {R}^{n}\backslash \{0\}$;

5) $\frac{\partial G(V,x)}{\partial x}f(x)<0$ holds for all $(V,x)\in \varPhi$; then the origin of (44) is GAS.

Theorem 7

[30]. If there exists a continuous function $G:\mathbb {R}_{\ge 0}\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ that satisfies the conditions 1–4 of Theorem 6, and there exist $c>0$ and $0<\mu <1$, such that

$$ \frac{\partial G(V,x)}{\partial x}f(x)\le cV^{1-\mu }\frac{\partial G(V,x)}{\partial V}, $$

holds for all $(V,x)\in \varPhi$, then the origin of (44) is GFTS and $T(x_{0})\le \frac{V_{0}^{\mu }}{c\mu }$ is the settling time function, where $G(V_{0},x_{0})=0$.

1.4 Observability and Strong Observability

Some definitions for strong observability, invariant zeros and relative degree are introduced in this section for the system (1)–(2), considering that $\varDelta = 0$, (see, e.g., [16, 42]).

Definition 4

[16, 42]. The system (1)–(2) is called Strongly Observable if for any initial condition x(0) and every $\varphi$, the identity $y\equiv 0$, implies that also $x\equiv 0$.

Definition 5

[16, 42]. The complex number $s_{0}\in \mathscr {C}$ is called an Invariant Zero of the triple (A, D, C) if $\textrm{rank}(R(s_{0}))<n+\textrm{rank}(D)$, where R(s) is the Rosenbrock matrix of system (1)–(2).

In the case when $D=0$, the notion of strong observability coincides with observability. Finally, the definition of relative degree is introduced.

Definition 6

[16]. The output y is said to have a relative degree $\sigma$ with respect to the input w if

$$\begin{aligned} CA^{k}D =0,\ \forall k<\sigma -1, \ CA^{\sigma -1}D \ne 0. \end{aligned}$$

Note that, according to [16], the following statements are equivalent:

• 1. The system (1)–(2) is Strongly Observable.

• 2. The triple (A, C, D) has no invariant zeros.

• 3. The output of the system (1)–(2) has relative degree n with respect to w.

1.5 Hybrid Systems

Consider the following model of a hybrid system [17]

$$\begin{aligned} \dot{x} & =f(x,w),\ x\in \mathscr {C},\end{aligned}$$

(45)

$$\begin{aligned} x^{+} & =g(x,w),\ x\in \mathscr {D}, \end{aligned}$$

(46)

where $x\in \mathbb {R}^{n}$ is the state of the system changing according to the differential equation (45) while x is in the flow closed set $\mathscr {C}$, and it can change according to the difference equation (46) while x is in the jump closed set $\mathscr {D}$, $x^{+}\in \mathbb {R}^{n}$ represents the value of the state after an instantaneous change, and $w\in \mathscr {W}$. Let a hybrid arc $\phi (t,j)$ be a solution to the hybrid system (45)–(46), $\phi _{0}=\phi (0,0)$ be the initial condition, and $\textrm{dom}\phi$ denotes the hybrid time domain of $\phi (t,j)$, where solutions are parameterized by both $t\in \mathbb {R}_{\ge 0}$, the amount of time passed, and $j\in \mathbb {N}$, the number of jumps that have occurred. The subsets of $(t,j)\in \mathbb {R}_{\ge 0}\times \mathbb {N}$ that correspond to evolutions of the hybrid system (45)–(46) are called hybrid time domains (for more details see [17]).

Let us introduce, inspired by [28], a definition of finite and fixed-time attractiveness for a closed set $\mathscr {M}\subset \mathbb {R}^{n}$.

Definition 7

[34]. The closed set $\mathscr {M}$ is said to be Finite-Time Attractive (FTA) for (45)–(46) if for each initial condition $\phi _0$ there exists $T(\phi _{0})>0$, such that for any solution $\phi$ to (45)–(46) with $\phi _{0}\in \mathbb {R}^{n}$, $(t,j)\in \textrm{dom}\phi$ and $t+j\ge T(\phi _{0})$ imply $||\phi (t,j)||_{\mathscr {M}}=0$, where $T:\mathbb {R}^{n}\rightarrow \mathbb {R}_{\ge 0}$ is the settling-time function; Fixed-Time Attractive (FxTA) for (45)–(46) if it is FTA and $T(\phi _{0})$ is bounded by some number $T^+>0$.

A definition of finite and fixed-time input-to state stability is also introduced.

Definition 8

[34]. The system (45)–(46) is said to be FT-ISS, with respect to $\mathscr {M}$, if for each initial condition $\phi _0$ and every input $w\in \mathscr {W}$ there exist $T(\phi _{0})>0$ and some functions $\beta \in \mathscr{K}\mathscr{L}_{T}$ and $\gamma \in \mathscr {K}$, such that any solution $\phi$ to (45)–(46) with $(t,j)\in \textrm{dom}\phi$ satisfies

$$\begin{aligned} & ||\phi (t,j)||_{\mathscr {M}}\le \beta (||\phi _{0}||,t+j)+\gamma (||w||_{\infty }),\ \forall 0\le t+j<T(\phi _{0}),\\ & \qquad \quad ||\phi (t,j)||_{\mathscr {M}}\le \gamma (||w||_{\infty }),\ \forall t+j\ge T(\phi _{0}). \end{aligned}$$

The system (45)–(46) is said to be FxT-ISS if it is FT-ISS and $T(\phi _{0})$ is bounded by some number $T^+>0$.

Now, we provide some proofs of the given results.

1.6 Proof of Theorem 1 with $\mu =0$

The error dynamics (9), when $\mu =0$ and $\bar{\varDelta } x + \bar{D}w \equiv 0$, is given as

$$\begin{aligned} \dot{\varepsilon }=(A_{0}-K\bar{C})\varepsilon , \end{aligned}$$

(47)

where $K=(k_1,\ldots ,k_n)^T\in \mathbb {R}^n$ and $\bar{C}=CT^{-1}$. It is clear that such a system is linear and, since the pair $(A_0,\bar{C})$ is observable, there always exists $K\in \mathbb {R}^n$ such that the matrix $(A_{0}-K\bar{C})$ is Hurwitz. Therefore, system (47) is Globally Exponentially Stable (GES).

Define an extended auxiliary vector field F for system (9), when $\mu =0$, as follows:

$$ F(\varepsilon ,\xi ) = \left( \begin{array}{cc} f^T(\varepsilon ,\xi ) & 0_p^T \end{array}\right) ^T = \left( \begin{array}{cc} \varepsilon ^T(A_{0}-K\bar{C})^T + \xi ^T \varXi ^T & 0_{n+1}^T \end{array}\right) ^T \in \mathbb {R}^{n+(n+1)}, $$

where $\xi =(x^T,w^T)^T \in \mathbb {R} ^{n+1}$ and $\varXi = (\bar{\varDelta },\bar{D})\in \mathbb {R}^{n \times (n+1)}$. It is given that the extended vector field F is r-homogeneous with degree $d = 0$ for vectors of weights $r=(1,\ldots ,1)>0$ and $\tilde{r}=(1,\ldots ,1) > 0$, i.e., $f(\varLambda _r(\lambda )\varepsilon ,\varLambda _{\tilde{r}}(\lambda )\xi ) = \lambda ^d \varLambda _r f(\varepsilon ,\xi )$ holds.

Therefore, according to Theorem 4, since $\min \tilde{r}_j = 1 > 0$, with $j=\overline{1,n+1}$, and system (47), with $\mu =0$ and $\bar{\varDelta } x + \bar{D}w \equiv 0$, is GES; then the system (9), when $\mu =0$, is ISS with respect to $\xi$, and hence, it is ISS with respect to x and w.

1.7 Proof of Theorem 1 with $\mu =1$

The error dynamics (9), when $\mu =1$, $p_{11}=1$ and $\bar{\varDelta }_l =0$, $l=\overline{1,n-1}$, can be written as

$$\begin{aligned} \dot{\varepsilon }_{l} &=\varepsilon _{l+1} - k_{l} \lceil \varepsilon _{1}\rfloor ^{\frac{n-l}{n}}, \ \forall l= \overline{1,n-1}, \end{aligned}$$

(48)

$$\begin{aligned} \dot{\varepsilon }_{n} &=\bar{\varDelta }_n x + CA^{n-1}Dw - k_{n} \lceil \varepsilon _{1}\rfloor ^{0}. \end{aligned}$$

(49)

The previous dynamics, with $\bar{\varDelta }_n =0$ and $w \equiv 0$, is r-homogeneous with degree $d = -1$ for a vector of weights $r=(n,n-1,\ldots ,1)$. Note that this dynamics has the same structure as the HOSM differentiator [21]. Its negative homogeneity degree and the discontinuous term of the algorithm, i.e., $k_{n} \lceil \varepsilon _{1}\rfloor ^{0}$, ensure the robust and FT stability of $\varepsilon =0$ against any unknown input $w\in \mathscr {W}$ and $x\in \mathscr {X}$ whenever the gains $k_i$, with $i=\overline{1,n}$, are properly chosen.

Then, based on homogeneity and Lyapunov theory one can show that the dynamics given by (48) and (49) is GFTS. The following result is recalled (for details see [9]).

Theorem 8

[9]. System (48)–(49) admits the following strong, proper, smooth and r-homogeneous of degree m Lyapunov function

$$ V(z)=\sum _{l=1}^{n-1}\gamma _{l}Z_{l}(z_{l},z_{l+1})+\frac{\gamma _{n}}{m}|z_{n}|^{m}, $$

with $m\ge 2n-1$, some positive constants $\gamma _{i}>0$, $i=\overline{1,n},$ $z_{1}=\frac{\varepsilon _{1}}{1},\ z_{2}=\frac{\varepsilon _{2}}{k_{1}},\ldots ,$ $\ z_{n}=\frac{\varepsilon _{n}}{k_{n-1}},$ and

$$\begin{aligned} Z_{i}(z_{i},z_{i+1})=\frac{r_{i}}{m}|z_{i}|^{\frac{m}{r_{i}}}-z_{i}\left\lceil z_{i}\right\rfloor ^{\frac{m- r_{i}}{r_{i}+1}} +\left( \frac{m-r_{i}}{m}\right) |z_{i+1}|^{\frac{m}{r_{i+1}}},\ i=\overline{1,n}. \end{aligned}$$

Moreover, if $x\in \mathscr {X}$ and $w\in \mathscr {W}$, then there exist some positive constants $k_{i}$ and $\gamma _{i}$, $i=\overline{1,n},$ such that system (48)–(49) is GFTS.

Now, define an extended auxiliary vector field F for system (9), when $\mu =1$, as follows:

$$\begin{aligned} F(\varepsilon ,\xi ) &= \left( \begin{array}{cc} f^T(\varepsilon ,\xi ) & 0_p^T \end{array}\right) ^T \\ &= \left( \begin{array}{cc} \varepsilon ^TA_{0}^T + h^T\varphi - \chi \vert ^T_{\mu =1}(\varepsilon _1) + \xi ^T\varXi ^T & 0_{n+1}^T \end{array}\right) ^T \in \mathbb {R}^{n+(n+1)}, \end{aligned}$$

where $\varphi =\bar{\varDelta }_n x + CA^{n-1}Dw$, $\xi =x\in \mathbb {R} ^{n}$ and $\varXi = (\bar{\varDelta }_1^T,\ldots ,\bar{\varDelta }_{n-1}^T,0)^T \in \mathbb {R}^{n \times n}$. It is given that the extended discontinuous vector field F, with $\varphi \equiv 0$, is r-homogeneous with degree $d = -1$ for vectors of weights $r=(n,n-1,\ldots ,1)>0$ and $\tilde{r}=(n-1,\ldots ,1,1) > 0$, i.e., $f(\varLambda _r(\lambda )\varepsilon ,\varLambda _{\tilde{r}}(\lambda )\xi ) = \lambda ^d \varLambda _r f(\varepsilon ,\xi )$ holds.

Hence, according to Theorem 5, since $\min \tilde{r}_j = 1 > 0$, with $j=\overline{1,n+1}$, and system (48)–(49), with $\bar{\varDelta }_l = 0$, $l=\overline{l,n-1}$, is GFTS for any $x\in \mathscr {X}$ and $w\in \mathscr {W}$; then the system (9), when $\mu =1$, is ISS with respect to x. Moreover, since $d = -1$, system (9), when $\mu =1$, is FT-ISS.

1.8 Proof of Theorem 1 with $\mu \in (0,1)$

The error dynamics (9), with $\mu \in (0,1)$ and $\bar{\varDelta } x + \bar{D}w \equiv 0$, can be written as

$$\begin{aligned} \dot{\varepsilon }_{l} &=\varepsilon _{l+1} - k_{i}p_{11}^{\frac{-l\mu }{2+2\mu (n-1)}} \lceil \varepsilon _{1} \rfloor ^{\frac{1+\mu (n-1-l)}{1+ \mu (n-1)}}, \forall l=\overline{1,n-1}, \end{aligned}$$

(50)

$$\begin{aligned} \dot{\varepsilon }_{n} &= - k_{n} p_{11}^{\frac{-n\mu }{2+2\mu (n-1)}}\lceil \varepsilon _{1}\rfloor ^{\frac{1-\mu }{1+ \mu (n-1)}}, \end{aligned}$$

(51)

This system is r-homogeneous with degree $d = -\mu \in (-1,0)$ for a vector of weights $r=(1+\mu (n-1),1+\mu (n-2),\ldots ,1)$, and its dynamics is nonlinear but continuous. Note that system (50)–(51) also admits the strong, proper, smooth and r-homogeneous of degree m Lyapunov function, given by Theorem 8. Therefore, based on [9], if $\bar{\varDelta } x + \bar{D}w \equiv 0$, there exist some positive constants $k_{i}$, $i=\overline{1,n}$ and $p_{11}$ such that system (50)–(51) is GFTS.

Define an extended auxiliary vector field F for system (9), when $\mu \in (0,1)$, as

$$\begin{aligned} F(\varepsilon ,\xi ) &= \left( \begin{array}{cc} f^T(\varepsilon ,\xi ) & 0_p^T \end{array}\right) ^T \\ &= \left( \begin{array}{cc} \varepsilon ^TA_{0}^T - \chi \vert ^T_{\mu \in (0,1)}(\varepsilon _1) + \xi ^T\varXi ^T & 0_{n+1}^T \end{array}\right) ^T \in \mathbb {R}^{n+(n+1)}, \end{aligned}$$

where $\xi =(x^T,w^T)^T \in \mathbb {R} ^{n+1}$ and $\varXi = (\bar{\varDelta },\bar{D}) = (\varXi _1^T,\ldots ,\varXi _n^T)^T\in \mathbb {R}^{n \times (n+1)}$, with $\varXi _i \in \mathbb {R}^{1 \times (n+1)}$, $i=\overline{1,n}$, the rows of matrix $\varXi$. It is given that the extended continuous vector field F is r-homogeneous with degree $d \in (-1,0)$ for vectors of weights $r=(1+\mu (n-1),1+\mu (n-2),\ldots ,1)$ and $\tilde{r}=(1+\mu (n-2),\ldots ,1,1,1) > 0$, i.e.,  $f(\varLambda _r(\lambda )\varepsilon ,\varLambda _{\tilde{r}}(\lambda )\xi ) = \lambda ^d \varLambda _r f(\varepsilon ,\xi )$ holds.

Therefore, according to Theorem 4, since $\min \tilde{r}_j = 1 > 0$, with $j=\overline{1,n+1}$, and system (9), with $\mu \in (0,1)$ and $\bar{\varDelta } x + \bar{D}w \equiv 0$, is GFTS; then the system (9), when $\mu \in (0,1)$, is ISS with respect to $\xi$, and thus, it is ISS with respect to x and w.

1.9 Proof of Proposition 1

The error dynamics (9), when $\mu =0$, is given as follows

$$\begin{aligned} \dot{\varepsilon }=(A_{0}-K\bar{C})\varepsilon + \bar{\varDelta } x + \bar{D}w. \end{aligned}$$

(52)

Propose the following Lyapunov candidate function

$$ V(\varepsilon ) = \varepsilon ^T P \varepsilon , \quad P=P^T>0, $$

that satisfies the following inequalities

$$ \lambda _{\min }(P)||\varepsilon ||^2 \le V(\varepsilon ) \le \lambda _{\max }(P)||\varepsilon ||^2. $$

The time derivative of V along the trajectories of the system (52) is given by

$$ \dot{V}(\varepsilon ) = \varepsilon ^T (PA_k + A_k^TP) \varepsilon + 2\varepsilon ^T P(\bar{\varDelta } x + \bar{D}w), $$

where $A_k=A_{0}-K\bar{C}$. Then, since Assumptions 2 and 3 hold, it follows that the time derivative of V can be upper bounded as

$$ \dot{V}(\varepsilon ) \le \left( \begin{array}{c}\varepsilon \\ \bar{\varDelta }x \\ w \end{array}\right) ^T \left( \begin{array}{ccc} PA_k + A_k^TP + \beta P &{} P &{} P\bar{D} \\ \star &{} -\beta I &{} 0 \\ \star &{} \star &{} -\beta \end{array} \right) \left( \begin{array}{c}\varepsilon \\ \bar{\varDelta }x \\ w \end{array}\right) - \beta (\varepsilon ^T P \varepsilon - ||\bar{\varDelta }||^2\overline{x}^2 - \overline{w}^2), $$

for any $\beta > 0$. Therefore, if the matrix inequality

$$ \left( \begin{array}{ccc} PA_k + A_k^TP + \beta P &{} P &{} P\bar{D} \\ \star &{} -\beta I &{} 0 \\ \star &{} \star &{} -\beta \end{array} \right) \le 0, $$

is satisfied, it follows that

$$ \dot{V}(\varepsilon ) \le - \beta (\varepsilon ^T P \varepsilon - ||\bar{\varDelta }||^2\overline{x}^2 - \overline{w}^2), $$

and the time derivative of V is negative definite outside the ellipsoid $\mathscr {E}(P)=\lbrace \varepsilon \in \mathbb {R}^n \ | \ \varepsilon ^T P \varepsilon \le \bar{\varDelta }||^2\overline{x}^2 + \overline{w}^2\rbrace$, which implies that $\mathscr {E}(P)$ is an attractive ellipsoid of the error dynamics (9) for $\mu =0$.

Finally, defining the variable $Y = PK$, one obtains (11) which is an LMI in the variables $P=P^T\ge 0$ and Y for a fixed constant $\beta > 0$. Therefore, since $\varepsilon =Te$ and $\varepsilon ^T P \varepsilon \le ||\bar{\varDelta }||^2\overline{x}^2 + \overline{w}^2$, the estimation error converges exponentially to a ball around of the origin given by (12).