# High gain observer design for some networked control systems

Tarek Ahmed-Ali, Françoise Lamnabhi-Lagarrigue

► To cite this version:

Tarek Ahmed-Ali, Françoise Lamnabhi-Lagarrigue. High gain observer design for some networked control systems. IEEE Transactions on Automatic Control, 2012, 57 (4), pp.995-1000. 10.1109/TAC.2011.2168049 . hal-00648730

HAL Id: hal-00648730

https://centralesupelec.hal.science/hal-00648730v1

Submitted on 6 Jul 2021

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
High Gain Observer Design for Some
Networked Control Systems

Tarek Ahmed-Ali and Françoise Lamnabhi-Lagarrigue

*Abstract*—New results on high gain observer design for networked con-
trol systems via an emulation-like approach are presented. By using a gen-
eral framework and a Lyapunov approach, we derive some explicit condi-
tions on the maximum allowable transmission interval that ensure an ex-
ponential convergence of the observation error for a large class of network
protocols.

*Index Terms*—High gain observer, networked control systems (NCSs), time-delay systems.

I. INTRODUCTION

The control and observation of systems called Networked control
systems (NCSs) are currently attracting a lot of attention in the con-
trol community. In many applications, the interest for NCSs is moti-
vated by many advantages they offer such as the ease of maintenance
and installation, the greater flexibility and the low cost. For these rea-
sons, many industrial control applications use a serial communication
channel to connect sensors and controllers. In NCSs, the serial com-
munication channel has many nodes (sensors and actuators) but the
signals of these nodes cannot be transmitted at the same time. The rule
that selects which node will use the network to transmit its data, is

called scheduling network protocol. In the present letter, only commu-
nication constraints on sensors are considered. It is well known that
the stability of these systems is largely determined by the transmis-
sion protocol used and by the so-called maximum allowable transfer
interval (MATI), i.e., the maximum allowable time between any two
successive transmissions in the network. There exist several families
of protocols in the literature, but in this study, only Uniformly Glob-
ally Exponentially Stable (UGES) protocols are considered. This class
includes several protocols, such Round Robin (RR), Try-Once-Discard
(TOD) and sampled data systems. For further details on NCSs, proto-
cols and their properties, see the overview papers [1]–[4].

The present note is focused on the implementation of a class of non-
linear observers on NCSs. Although the control of these systems has re-
ceived considerable attention in the recent years, only few papers which
deal with the observer design problem exist in the literature. For ex-
ample, in deterministic case, the authors in [5] derived some conditions
in order to maintain the observability of discrete-time linear systems
under network communications. In [6], the authors proposed a method
for the mutual design of protocols and observers for linear systems.
The considered protocol gives the priority to the node where the error,
between the measure and the estimate is the biggest, according to some
weighted norms. Sufficient conditions for quadratic stability properties
are then given using matrix inequalities. In [7], the authors consider the
design of a linear observer for NCSs in the presence of time delay and
sampling phenomenons without considering the constraints introduced
by the protocols. They give sufficient conditions guaranteeing asymp-
totic stability, by using a *Lyapunov-Krasovskii* approach. Recently, a
small gain approach was used in [8] and [9] for the observers design via
an emulation-like approach for some classes of networked control sys-
tems. The results contained in [8] and [9] state that if the observer has
some robustness properties with respect to measurements errors then,
for sufficiently small MATI, the stability in an appropriate sense of the
observation error is guaranteed for several classes of network protocols.
On the other hand, even without disturbances, the framework devel-
oped in [9] does not allow asymptotic stability of the observation error
when the network is in zero order hold (ZOH) fashion. In the present
study, we focus on a class of nonlinear triangular systems without con-
sidering disturbances. More precisely, we provide a framework and use
a Lyapunov approach to guarantee an exponential stability of the ob-
servation error and improve the bounds of MATI compared to small
gain approaches. We specially show that by using the framework pre-
sented below and a suitable *Lyapunov-Krasovskii functional*, we can
derive an exponential stability of the observation error when the net-
work is in ZOH fashion. Note that the results presented in the present
letter can be easily extended to several classes of observers such, for
example, linear observers and the class of observers described in [10].
The present note is organized as follows:

In Section II, we present the notations which will be used throughout
the letter. The framework that we propose is formulated in Section III.
In Section IV, we present our results on the emulation of high gain
observers on a class of networks.

II. NOTATIONS AND PRELIMINARIES

First some mathematical notations are introduced. Let $\mathbf{R} =$
$(-\infty, \infty)$, $\mathbf{R}_+ = [0, \infty)$ and $\mathbf{N}$ is the set of natural numbers. The
notation $\mathbf{N}^*$ denotes the set of strictly positive integers. The usual
euclidian norm of any vector $v$ will be noted by $|v|$ and the identity
matrix of size $p$ is denoted by $I_p$. The matrix $0_p$ is a square matrix
of size $p$ with all values equal to zero. The notations $\lambda_{min}(S)$ and
$\lambda_{max}(S)$ denote respectively the minimum and maximum eigenvalues
of the square matrix $S$. A continuous function $\Gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+$

T. Ahmed-Ali is with GREYC UMR CNRS 6072, Caen Cedex 14050, France
(e-mail: tarek.ahmed-ali@greyc.ensicaen.fr).

F. Lamnabhi-Lagarrigue is with CNRS, Laboratoire des Signaux et Systemes (INSIS-INS2I), European Embedded Control Institute (EECI) SUPELEC, Gif-sur-Yvette 91192, France (e-mail: lamnabhi@lss.supelec.fr).
is said to be of class $\mathcal{K}$, if it is zero at zero, strictly increasing. It is of class $\mathcal{K}_\infty$, if it is of class $\mathcal{K}$ and unbounded. A continuous function $\varphi : \mathbf{R}_+ \times \mathbf{R}_+ \to \mathbf{R}_+$ is said to be of class $-\mathcal{K}\mathcal{L}$, if $\varphi(\cdot, t)$ is of class $\mathcal{K}$ for each $t \ge 0$, and $\varphi(s, \cdot)$ is non increasing and satisfies $\lim_{t \to -\infty} \varphi(s, t) = 0$. The notation $e_y(a^+) = 0$ represents $\lim_{t \to -\infty} e_y(t)$ when it exists. The term $W_{[t_0, t)}$ represents ess. $\sup_{s \in [t_0, t]} |W(s)|$. Throughout this letter the vectors $x \in \mathbf{R}^n$ and $y \in \mathbf{R}^p$ represent respectively the state vector and the outputs of the considered systems.

We consider the following class of nonlinear systems:

$$ \begin{cases} \dot{x} = Ax + f(x) \\ y = Cx = x^1 \end{cases} \tag{1} $$

where $x = (x^1, x^2, \dots, x^q)^T \in \mathbf{R}^n$ with $x^k \in \mathbf{R}^p$ and $p, q = n$

$$ A = \begin{pmatrix} 0_p & I_p & 0_p & \cdots & 0_p \\ 0_p & 0_p & I_p & 0_p & \vdots \\ \vdots & 0_p & \cdots & I_p & 0_p \\ 0_p & \vdots & \cdots & 0_p & I_p \\ 0_p & \cdots & \cdots & \cdots & 0_p \end{pmatrix} \tag{2} $$

and

$$ C = (I_p \quad 0_p \quad \dots \quad 0_p). \tag{3} $$

Throughout this letter, we assume that the following hypotheses are satisfied:

**Hypothesis 1:** The vector $f(x)$ has the following form:

$$ f(x) = (f_1(x^1), \dots, f_s(x^1, \dots, x^s), \dots, f_q(x))^T. \tag{4} $$

**Hypothesis 2:** The functions $f_s$ are globally Lipschitz, i.e.:

$$ \exists \beta_0 > 0 \text{ such that } \forall (x_1, x_2) \in \mathbf{R}^{p, s} \times \mathbf{R}^{p, s} $$

$$ |f_s(x_1) - f_s(x_2)| \le \beta_0 |x_1 - x_2|, \quad (s = 1, \dots, q). \tag{5} $$

Using these hypotheses, the following observer has been proposed in [11] for the class of systems (1) without considering network:

$$ \dot{x} = A\hat{x} + f(\hat{x}) - \theta \Delta^{-1} S^{-1} C^T (C\hat{x}(t) - y) \tag{6} $$

where $\theta$ is a positive constant satisfying $\theta \ge 1$. $S$ is a symmetric positive definite matrix, solution of the algebraic Lyapunov equation

$$ SA + A^T S - C^T C = -S \tag{7} $$

and $\Delta$ is a diagonal matrix which has the following form:

$$ \Delta = Diag \left( I_p, \dots, \frac{1}{\theta} I_p, \dots, \frac{1}{\theta^{q-1}} I_p \right). \tag{8} $$

The exponential convergence of this observer has been derived by using the Lyapunov function $V = \bar{x}^T S \bar{x}$, where $\bar{x} = \Delta \hat{x}$. The authors prove in [11] that its time derivative satisfies

$$ \dot{V} \le -(\theta - \beta)V - \theta \bar{x}^T C^T C \bar{x} \quad \text{where} \quad \beta = 2\sqrt{q} \frac{\lambda_{max}(S)}{\lambda_{min}(S)} \beta_0. \tag{9} $$

### III. OBSERVERS FOR NETWORKED CONTROL SYSTEMS

In this section, we present a framework dedicated to implementation of a class of nonlinear observers on a class of networks. Let us consider the nonlinear systems

$$ \begin{cases} \dot{x} = f_M(x) \\ y = h_M(x) \end{cases} \tag{10} $$

and suppose that there exists an observer described by the dynamics

$$ \dot{x} = f_o(\hat{x}, (h_M(\hat{x}) - y)) \tag{11} $$

observation error $\hat{x} = \hat{x} - x$ which is represented by the system

$$ \dot{\hat{x}} = f_o(\hat{x}, (h_M(\hat{x}) - y)) - f_M(x) \tag{12} $$

is globally asymptotically stable. Now, let us consider the case where all outputs of system (10) are connected to observer (11) via a serial communication channel. We suppose that the signals of these sensors cannot be transmitted to observer (11) at the same time, then this problem can be modeled by the following framework:

$$ \left\{ \begin{aligned} \dot{x} &= f_M(x) \quad t \in [t_{i-1}, t_i] \\ x(t_i^+) &= x(t_i) \\ y &= h_M(x) \\ \dot{\hat{x}} &= f_o(\hat{x}, \hat{w} - w) \quad t \in [t_{i-1}, t_i] \\ \hat{x}(t_i^+) &= \hat{x}(t_i) \\ \dot{\hat{w}} &= g_w(w, \hat{w}, \hat{x}) \quad t \in [t_{i-1}, t_i] \\ w(t_i^+) &= y_s(t_i) + h_y(i, e_y(t_i)) \\ \dot{\hat{w}} &= \hat{g}_{\hat{w}}(\hat{w}, w, \hat{x}) \quad t \in [t_{i-1}, t_i] \\ \hat{w}(t_i^+) &= h_M(\hat{x}(t_i)) \\ \dot{y}_s &= g_{y_s}(y_s, x) \quad t \in [t_{i-1}, t_i] \\ y_s(t_i) &= y(t_i) \quad i \in \mathbf{N}^* \\ e_y &= w - y_s \end{aligned} \right. \tag{13} $$

The vector $\hat{x}$ is the continuous-time estimate of the system state $x$. The monotonically increasing sequence $t_j, j \in \mathbf{N}$ represents the transmission instants. The vectors $w$ and $\hat{x}$ (which are re-initialized at each instant $t_j$) represent respectively most recently transmitted output values via network and the prediction of the observer output. The vector $y_s$ (which is also re-initialized at $t_j$) allows us the possibility to describe the sampling phenomena. More precisely, $y_s$ can be used to represent the output of the zero order hold (ZOH) device. This variable is specially useful when the network operates in ZOH fashion. In this case we replace conjointly the outputs of the system and the observer by their sampled signals before considering the constraints introduced by the network. As, we will see in the sequel, this framework allows us to derive an asymptotic stability of the observation error when the network operates in ZOH fashion. The functions $g_{y_s}, g_w$ and $\hat{g}_{\hat{x}}$ represent prediction functions between two transmission instants. The error induced by the network is represented by the vector $e_y$. The protocol $h_y$ is the algorithm by which the access to network of each node is determined. At each $t_j$, the protocol $h_y$ selects which nodes $k \in \{1, \dots, l\}$ can transmit its data throughout the network. This algorithm is represented by the discrete time system: $e_y(t_j^+) = h_y(j, e_y(t_j)) \quad j \in \mathbf{N}$. Throughout this note, the variable $\tau = \max(t_{j+1} - t_j)$ represents the maximum allowable transfer interval (MATI) and $\tau_{MATI}$ represents an upper bound of $\tau$. To prevent zero solution we suppose that the MATI satisfies $\tau > t_{j+1} - t_j > \mu$, where $\mu$ is an arbitrary positive constant. Note that the sampled-data systems are a special case of NCSs since all sensor and control signals are transmitted at each transmission instant. In this case, the protocol $h_y$ is equal to zero, and the maximum allowable time between any two successive transmissions is called maximum allowable sampling period MASP. Compared to [12] where a framework based on discrete-time approximate models is developed for the observer design for sampled-data systems, we can say that the framework (13) is an alternative approach to [12].

**Hypothesis 3:** We suppose that the protocol $h_y$ is UGES. This means that there exists a positive function, $W: \mathbf{N} \times \mathbf{R}^n \to \mathbf{R}_+$ and some
positive constants $\lambda \in [0, 1)$, $a_1, a_2$ such that, for all $j \in \mathbf{N}$ and for all $e_y \in \mathbf{R}^{n_{ey}}$

$$a_1|e_y| \le W(j, e_y) \le a_2|e_y| \quad (14)$$

$$W(j+1, h_y(j, e_y)) \le \lambda W(j, e_y). \quad (15)$$

**Hypothesis 4:** We also assume that there exists a positive constant $M_0$ so that for all $j \in \mathbf{N}$ and for all $e_y \in \mathbf{R}^{n_{ey}}$

$$\left| \frac{\partial W(j, e_y)}{\partial e_y} \right| \le M_0. \quad (16)$$

This condition is satisfied by many protocols such Round Robin (RR) and Try-Once-Discard (TOD).

## IV. HIGH GAIN OBSERVER FOR NCSs

In this section we focus on the implementation of high gain observer (6) over a network with UGES protocols. We present two examples: The first one is an implementation in ZOH fashion, whereas the second one, is based on the introduction of an output predictor between two transmission instants. In both cases we prove exponential convergence of the high gain observer for sufficiently small MATI.

### A. High Gain Observer in ZOH Fashion

When the network operates in ZOH fashion, we will consider that all prediction functions in (13) are equal to zero. This means that the corrector term $\hat{w} - w$ is held constant between two transmission instants. Following framework (13), we propose this observer:

$$\left\{
\begin{aligned}
\dot{\hat{x}} &= A\hat{x} + f(\hat{x}) - \theta\Delta^{-1}S^{-1}C^T(\hat{w} - w) \\
\dot{w} &= 0 \quad t \in [t_{i-1}, t_i] \\
w(t_i^+) &= y_s(t_i) + h_y(i, e_y(t_i)) \\
\dot{w} &= 0 \quad t \in [t_{i-1}, t_i] \\
\dot{w}(t_i^+) &= C\hat{x}(t_i) \\
\dot{y}_s &= 0 \quad t \in [t_{i-1}, t_i] \\
y_s(t_i) &= y(t_i) \\
e_y &= w - y_s
\end{aligned}
\right.
\quad (17)$$

if we consider the observation error $\tilde{x} = \hat{x} - x$, then we have

$$\left\{
\begin{aligned}
\dot{\tilde{x}} &= (A - \theta \Delta^{-1} S^{-1} C^T C) \tilde{x} + f(\tilde{x} + x) - f(x) \\
&- \theta \Delta^{-1} S^{-1} C^T (C \tilde{x} (t_{i-1}^+) - C \tilde{x}(t) - e_y) \quad t \in [t_{i-1}, t_i] \\
\dot{e}_y &= 0 \quad t \in [t_{i-1}, t_i] \\
\tilde{x}(t_i^+) &= \tilde{x}(t_i) \\
e_y(t_i^+) &= h_y(i, e_y(t_i))
\end{aligned}
\right.
\quad (18)$$

As we can see, system (18) is disturbed by the term $C\tilde{x}(t_{i-1}^+) - C\tilde{x}(t)$. The Leibniz–Newton formula provides

$$C\tilde{x}(t_{i-1}^+) - C\tilde{x}(t) = -C \int_{t-\tau_1(t)}^{t} \dot{\tilde{x}}(s_1)ds_1 \quad (19)$$

with $\tau_1(t) = t - t_{i-1}^+$ for all $t \in (t_{i-1}, t_i]$. Note that we have $\tau_1(t) \in [0, \tau]$ and $\dot{\tau}_1 = 1$. This means that system (18) can be viewed as a time delay system with bounded time varying delay. This allows us to extend the approach developed in [4], [13] and [14] to the problem considered in the present note.

**Theorem 1:** Let us consider system (1) and suppose that hypotheses 1, 2, 3, 4 hold. Then, there exists a positive constant $\theta_0$, such that for all $\theta > \theta_0$ system (17) is a global exponential observer for system

(1) if $\tau \in [\mu, \tau_{MATI})$ where $\tau_{MATI} = \min(\lambda_{min}(S)/6sup\{[A - S^{-1}C^T C] + \sqrt{q}\beta_0], \lambda_{min}(S)\lambda_{max}(S^{-1})\}^2\theta, 1 - \lambda)$.

*Proof:* Consider the change of coordinates $\tilde{x} = \Delta\tilde{x}$, and from the fact that $\Delta A\Delta^{-1} = \theta A$ and $C^T C\Delta^{-1} = C^T C$, then (18) will be

$$\left\{
\begin{aligned}
\dot{\tilde{x}} &= \theta(A - S^{-1}C^T C)\tilde{x} + \Delta(f(\tilde{x} + x) - f(x)) \\
&\quad + \theta S^{-1}C^T \left( e_y + C \int_{t-\tau_1(t)}^{t} \dot{\tilde{x}}(s_1)ds_1 \right) \quad t \in [t_{i-1}, t_i] \\
\dot{e}_y &= 0 \quad t \in [t_{i-1}, t_i] \\
e_y(t_i^+) &= h_y(i, e_y(t_i))
\end{aligned}
\right.
\quad (20)$$

In order to prove this theorem, let us consider the following candidate Lyapunov-Krasovskii functional:

$$U_1 = \bar{x}^T S \bar{x} + \int_{t-\tau_{MATI}}^{t} \int_{s}^{t} |\dot{\bar{x}}(s_1)|^2 ds_1 ds + \gamma_1 \phi_1(t) W^2(j, e_y) \quad (21)$$

where $\gamma_1 > 0, j \in \mathbf{N}$ and $\phi_1(t)$ is a bounded, positive function and decreasing between $(t_{i-1}^+, t_i]$, with appropriate fixed values at instants $t_{i-1}^+$ and $t_i$. The functional $U_1$ has been used in [4], [13] and [14] for observer design with sampled and delayed measurements without network constraints communications by considering $W^2(j, e_y) = 0$. We will show below that this approach can be used here. More precisely, to prove exponential stability, it is sufficient to find $\tau_{MATI}$ and $\theta_0$ so that for $\theta > \theta_0$ the following inequalities hold:

$$\left\{
\begin{aligned}
\dot{U}_1 + \varepsilon_1 U_1 &\le 0 \quad t \in (t_{i-1}, t_i] \\
U_1(t_i^+) &< U_1(t_i) \quad \forall i \in \mathbf{N}^*
\end{aligned}
\right.
\quad (22)$$

where $\varepsilon_1 > 0$. We can easily see that the second inequality of (22) can be satisfied by choosing $\phi_1(t_{i-1}^+) = 1$ and $\phi_1(t_{i-1}^+ + \tau_{MATI}) = \lambda \forall i \in \mathbf{N}^*$ where $\lambda \in (0, 1)$.

Let us compute the derivative of $U_1$ for $t \in (t_{i-1}, t_i]$, then we have

$$\dot{U}_1 = 2\bar{x}^T S \dot{\bar{x}} + \tau_{MATI} |\dot{\bar{x}}|^2 - \int_{t-\tau_{MATI}}^{t} |\dot{\bar{x}}(s)|^2 ds \\
+ \gamma_1 \dot{\phi}_1(t) W^2(i-1, e_y). \quad (23)$$

Using (20), and Hölder inequality we can deduce that

$$|\dot{\bar{x}}|^2 \le 2k_2\theta^2 \left[ V + |I|^2 + \frac{W^2(i-1, e_y)}{a_1^2} \right] \quad (24)$$

where

$$k_2 = \frac{3sup\{[A - S^{-1}C^T C] + \sqrt{q}\beta_0], \lambda_{min}(S)\lambda_{max}(S^{-1})\}^2}{2\lambda_{min}(S)}. \quad (25)$$

Now, let us set $V = \bar{x}^T S \bar{x}$, after some computations we deduce that for $t \in (t_{i-1}, t_i]$, we have

$$\begin{aligned}
\dot{U}_1 + \varepsilon_1 U_1 &\le -\theta V + \beta V + \varepsilon_1 V + 2\theta \left( |I|^2 + \frac{W^2(i-1, e_y)}{a_1^2} \right) \\
&\quad + 2\tau_{MATI} \theta^2 k_2 \left[ V + |I|^2 + \frac{W^2(i-1, e_y)}{a_1^2} \right] \\
&\quad + \varepsilon_1 \tau_{MATI} \int_{t-\tau_{MATI}}^{t} |\dot{\bar{x}}(s)|^2 ds \\
&\quad - \int_{t-\tau_{MATI}}^{t} |\dot{\bar{x}}(s)|^2 ds \\
&\quad + [\gamma_1 \dot{\phi}_1(t) + \varepsilon_1 \gamma_1 \phi_1(t)] W^2(i-1, e_y)
\end{aligned}$$
where $I = \int_{t-\tau_{\text{MATI}}(t)}^{t} \dot{x}(s)ds$. From Jensen's inequality, we can derive

$$|I|^2 \le \tau_{\text{MATI}} \int_{t-\tau_{\text{MATI}}}^{t} |\dot{\bar{x}}(s)|^2 ds. \quad (26)$$

By choosing

$$\dot{\phi}_1 = -\epsilon_1\phi_1 - 1 \quad \forall t \in (t_{i-1}, t_i] \quad \text{and} \quad \gamma_1 = \frac{a_1^2}{2\theta + 2\tau_{\text{MATI}}\theta^2 k_2} \quad (27)$$

we can say that the inequality $\dot{U}_1 + \epsilon_1 U_1 \le 0$ is ensured, if $\tau_{\text{MATI}}$ satisfies both (27) and the following inequalities:

$$\left\{ \begin{array}{l} \theta - \beta - \epsilon_1 - 2\tau_{\text{MATI}}\theta^2 k_2 > 0 \\ 1 - 2\theta^2 k_2\tau_{\text{MATI}}^2 - \epsilon_1\tau_{\text{MATI}} - 2\tau_{\text{MATI}}\theta > 0 \end{array} \right. . \quad (28)$$

From this, and by integrating (27) on $(t_{i-1}, t_{i-1} + \tau_{\text{MATI}}]$ and setting $\epsilon_1 \to 0$, we derive the following conditions:

$$\left\{ \begin{array}{l} \tau_{\text{MATI}} = \min \left\{ \frac{1}{4k_2\theta}, 1 - \lambda \right\} \\ \theta > \sup\{2\beta, 1\} \end{array} \right. \quad (29)$$

using the fact that $|\bar{x}| \le (1/\sqrt{\lambda_{\min}(S)})\sqrt{\bar{U}_1(t)}$ and from (22) then we deduce that under conditions (29) there exist $c_1 > 0$ and $c_2 > 0$ such that $|\bar{x}| \le c_1 \exp^{-c_2(t-t_0)} \forall t \ge t_0$. From this we can say that $|\bar{x}|$ converges exponentially towards zero.

## B. High Gain Observer With an Output Predictor

The idea of using a predictor of the output was introduced in [15] for the design of sampled-data observers. More precisely, this predictor is used by the observer between two transmission instants and its model is a copy of the output system model. This gives this differential equation $\dot{w} = (\partial h_M(\hat{x})/\partial \hat{x})f_M(\hat{x})$ where the initial condition $w(t_0)$ is arbitrary and at each transmission instant $t_i$ the value of $w$ is reseated like this: $w(t_i^+) = y(t_i)$. This idea has been extended to networked systems in [8] and [9] and some bounds of MATI have been derived using small gain approach. The aim of this section is to improve the bound of MATI for high gain observer when an output predictor is used. Following the framework (13), we can write the following observer:

$$\left\{ \begin{array}{l} \dot{\hat{x}} = A\hat{x} + f(\hat{x}) - \theta\Delta^{-1}S^{-1}C^T(\hat{w} - w) \\ \dot{w} = \hat{x}^2 + f_1(\hat{x}^1) \quad t \in [t_{i-1}, t_i] \\ w(t_i^+) = y_s(t_i) + h_y(i, e_y(t_i)) \\ y_s = y \\ \hat{w} = C\hat{x}(t) \end{array} \right. \quad (30)$$

Let us consider the observation error $\tilde{x} = \hat{x} - x$, then we have

$$\left\{ \begin{array}{l} \dot{\tilde{x}} = (A - \theta \Delta^{-1} S^{-1} C^T C) \tilde{x} + f(\tilde{x} + x) - f(x) \\ + \theta \Delta^{-1} S^{-1} C^T e_y \quad t \in [t_{i-1}, t_i] \\ \dot{e}_y = \tilde{x}^2 + f_1(\tilde{x}^1 + x^1) - f_1(x^1) \quad t \in [t_{i-1}, t_i] \\ \tilde{x}(t_i^+) = \tilde{x}(t_i) \\ e_y(t_i^+) = h_y(i, e_y(t_i)) \end{array} \right. \quad (31)$$

**Theorem 2:** Let us consider system (1) and suppose that hypotheses 1, 2, 3, 4 hold. Then for all $\theta > \max\{\beta, 1\}$, system (30) is a global exponential observer for system (1) if $\tau \in [\mu, \tau_{\text{MATI}})$ where $\tau_{\text{MATI}} = a_1((\pi/2) - 2\operatorname{atan}(\lambda))\sqrt{\lambda_{\min}(S)(\theta - \beta)}/M_0(\theta + \beta_0)\sqrt{\theta}$.

*Proof:* As in the above section, using the same change of coordinates then (31) will be

$$\left\{ \begin{array}{l} \dot{\bar{x}} = \theta(A - S^{-1}C^T C)\bar{x} + \Delta(f(\tilde{x} + x) - f(x)) \\ + \theta S^{-1}C^T e_y \quad t \in [t_{i-1}, t_i] \\ \dot{e}_y = \theta \bar{x}^2 + f_1(\tilde{x}^1 + x^1) - f_1(x^1) \quad t \in [t_{i-1}, t_i] \\ e_y(t_i^+) = h_y(i, e_y(t_i)) \end{array} \right. \quad (32)$$

Let us consider the following candidate Lyapunov function as in [16]:

$$U(t) = \sigma \bar{x}^T(t) S \bar{x}(t) + \gamma \phi(t) W^2 (j, e_y(t)) \quad (33)$$

where $j \in \mathbf{N}$, $\sigma$ and $\gamma$ are two positive constants which will be computed below. The positive function $\phi(t)$ is bounded and decreasing on the transmission interval $(t_{i-1}, t_i] \in \mathbf{N}^*$. This function satisfies the following conditions:

$$\phi(t_{i-1}^+) = \lambda^{-1} \quad \text{and} \quad \phi(t_{i-1}^+ + \tau_{\text{MATI}}) = \lambda \quad \forall i \in \mathbf{N}^*$$

where $\lambda \in (0, 1)$, to guarantee that $U(t_i^+) \le U(t_i)$. Now, let us compute the time derivative of $U$ between $(t_{i-1}, t_i]$, then we have

$$\dot{U} = 2\sigma\bar{x}^T S\dot{\bar{x}} + \gamma\dot{\phi}W^2(i-1, e_y) + 2\gamma\phi W(i-1, e_y)\dot{W}(i-1, e_y) \quad (34)$$

after some computations, we deduce that

$$\dot{U} \le -\sigma(\theta - \beta)\lambda_{\min}(S)|\bar{x}|^2 + \sigma\theta|e_y|^2 \\ + \gamma\dot{\phi}W^2(i-1, e_y) + 2\gamma\phi W(i-1, e_y)\left|\frac{\partial W(i-1, e_y)}{\partial e_y}\right| |\dot{e}_y|.$$

Let us remark that

$$|\dot{e}_y| \le \theta|\bar{x}|^2 + \left|f_1(\hat{x}^1) - f_1(x^1)\right| \le (\theta + \beta_0)|\bar{x}| \quad (35)$$

using the property (16), we derive the following inequalities $|\partial W(i-1, e_y)/\partial e_y||\dot{e}_y| \le M_0(\theta + \beta_0)|\bar{x}|$ and

$$\dot{U} \le -\sigma(\theta - \beta)\lambda_{\min}(S)|\bar{x}|^2 + \sigma\theta|e_y|^2 \\ + \gamma\dot{\phi}W^2(i-1, e_y) + 2\gamma\phi W(i-1, e_y)M_0(\theta + \beta_0)|\bar{x}| + \epsilon_0 W^2(i-1, e_y)$$

Using the fact that the protocol $h_y$ is UGES, and by adding and subtracting the term $\epsilon_0 W^2(i-1, e_y)$, where $\epsilon_0 > 0$, thus we have

$$\begin{aligned} \dot{U} \le & -\sigma(\theta - \beta)\lambda_{\min}(S)|\bar{x}|^2 - \epsilon_0 W^2(i-1, e_y) \\ & + \frac{\sigma\theta}{a_1^2}W^2(i-1, e_y) + \gamma\dot{\phi}W^2(i-1, e_y) \\ & + 2\gamma\phi W(i-1, e_y)M_0(\theta + \beta_0)|\bar{x}| + \epsilon_0 W^2(i-1, e_y) \end{aligned}$$

If we choose

$$\dot{\phi} = -\gamma(\phi^2 + 1) \quad \forall t \in (t_{i-1}, t_i] \quad (36)$$

thus, we will have

$$\begin{aligned} \dot{U} \le & -\sigma(\theta - \beta)\lambda_{\min}(S)|\bar{x}|^2 + M_0^2(\theta + \beta_0)^2|\bar{x}|^2 \\ & - \epsilon_0 W^2(i-1, e_y) - \left[ \gamma^2 - \epsilon_0 - \frac{\sigma\theta}{a_1^2} \right] W^2(i-1, e_y) \end{aligned}$$
where $\epsilon_0 > 0$. Note that the term $\epsilon_0 W^2(i-1, e_y)$ is introduced to guarantee the exponential convergence. Choosing

$$ \left\{ \begin{aligned} \sigma &= \frac{M_0^2(\theta+\beta_0)^2}{(\theta-\beta)\lambda_{min}(S)} + \frac{\epsilon_0}{(\theta-\beta)\lambda_{min}(S)} \\ \gamma &= \sqrt{\frac{\sigma\theta}{a_1^2} + \epsilon_0} \end{aligned} \right. \quad (37) $$

thus we can say that there exists a positive constant $\xi$, so that $\dot{U}(t) \le -\xi U(t) \in (t_{i-1}, t_i]$ this means that for all $t \in (t_{i-1}, t_i]$

$$ U(t) \le U(t_{i-1}^+) \exp^{-\xi(t-t_{i-1}^+)} \quad (38) $$

and

$$ U(t_i) \le U(t_{i-1}^+) \exp^{-\xi(t_i-t_{i-1}^+)} \quad (39) $$

on the other hand, we have

$$ U(t_i^+) = V(\bar{x}(t_i^+)) + \gamma \lambda^{-1} W^2(c_y(t_i^+)) $$

using the fact that the protocol is UGES, and $\bar{x}(t_i^+) = \bar{x}(t_i)$, then we can write

$$ U(t_i^+) \le V(\bar{x}(t_i)) + \gamma \lambda W^2(c_y(t_i)) \le U(t_i) \quad i \in \mathbf{N}^* \quad (40) $$

thus, we derive

$$ \begin{aligned} U(t) &\le U(t_{i-1}^+) \exp^{-\xi(t-t_{i-1}^+)} \quad \forall t \in (t_{i-1}, t_i] \\ U(t_i^+) &\le U(t_{i-1}^+) \exp^{-\xi(t_i-t_{i-1}^+)} \quad i \in \mathbf{N}^*. \end{aligned} \quad (41) $$

Using the fact that $|\bar{x}| \le (1/\sqrt{\sigma\lambda_{min}(S)})\sqrt{U(t)}$ and from (41), then we deduce that there exist $\mu_1 > 0$ and $\mu_2 > 0$ such that $|\bar{x}| \le \mu_1\exp^{-\mu_2(t-t_0)} \forall t \ge t_0$. This means that $\bar{x}$ converges exponentially towards zero, and the value of $\tau_{MATI}$ is then derived by integrating (36) on $(t_{i-1}, t_{i-1} + \tau_{MATI}]$, with $\epsilon_0 \to 0$. This leads to

$$ \begin{aligned} \tau_{MATI} &= \lim_{\epsilon_0 \to 0} \left( \frac{1}{\gamma} \left[ \operatorname{atan}(\lambda^{-1}) - \operatorname{atan}(\lambda) \right] \right) \\ &= \frac{a_1 \left( \frac{\pi}{2} - 2\operatorname{atan}(\lambda) \right) \sqrt{\lambda_{min}(S)(\theta - \beta)}}{M_0(\theta + \beta_0)\sqrt{\theta}}. \end{aligned} \quad (42) $$

1) Comparison With Small Gain Approach: Let us consider the function $V = \bar{x}^T S \bar{x}$, then after simple computations, we can easily derive the following inequality:

$$ \dot{V} \le -(\theta - \beta)V + \theta \frac{W^2}{a_1^2} \quad (43) $$

using the comparison lemma, then we will have

$$ |\bar{x}| \le \sqrt{\frac{V(t_0)}{\lambda_{min}(S)} \exp\left(\frac{-(\theta-\beta)}{2}(t-t_0)\right)} + \sqrt{\frac{\theta}{\lambda_{min}(S)(\theta-\beta)}} \frac{W_{[t_0,t)}}{a_1} \quad (44) $$

where $\sqrt{V(t_0)/\lambda_{min}(S)} \exp((-(\theta-\beta))/2)(t-t_0)$ is a function of class $KL$.

Now, let us define, $\tilde{y} = M_0(\theta + \beta_0)\bar{x}$, then we derive the gain $\gamma_0$ from $W$ to $\tilde{y}$

$$ \gamma_0 = \frac{M_0(\theta + \beta_0)}{a_1} \sqrt{\frac{\theta}{\lambda_{min}(S)(\theta - \beta)}}. \quad (45) $$

Using (44) and the fact that

$$ \left| \frac{\partial W(i-1, e_y)}{\partial e_y} \right| |\dot{e}_y| \le M_0(\theta + \beta_0) |\bar{x}| \quad (46) $$

then from Theorem 4 in [2] we can say that $\tau_{MATI}$ is given by

$$ \tau_{MATI} = \frac{1 - \lambda}{\gamma_0} = \frac{a_1 (1 - \lambda) \sqrt{\lambda_{min}(S)(\theta - \beta)}}{M_0(\theta + \beta_0) \sqrt{\theta}}. \quad (47) $$

<figure><img src="image_2.png" /><figcaption>Fig. 1. Evolution of the first and second states for $\tau_{MATI} = 0.01$.</figcaption></figure>

Now let us compute the ratio $R$ between (42) and (47), then we have $R = ((\pi/2) - 2\operatorname{atan}(\lambda))/(1 - \lambda)$. Since $1 - \lambda \le (\pi/2) - 2\operatorname{atan}(\lambda)$ for all $\lambda \in [0, 1)$, then we deduce that $R \ge 1$. This means that the bound derived from Lyapunov approach is larger than the one derived from small gain approach. This result means that we have also enlarged the interval of admissible values of the gain $\theta$ for any fixed $\tau_{MATI}$ and therefore we have also improved the speediness of convergence. Note that in all formulas of $\tau_{MATI}$, it is clear that $\tau_{MATI}$ depends inversely on $\theta$. This obviously leads to small values of $\tau_{MATI}$ for high values of the gain $\theta$.

## V. SIMULATIONS

In this part, we present some simulations on the flexible joint robotic arm used in [13]

$$ \left\{ \begin{aligned} \dot{x}_1 &= x_3 + \delta(t) \\ \dot{x}_2 &= x_4 + \delta(t) \\ \dot{x}_3 &= -48.6x_1 + 48.6x_2 - 1.25x_3 + 21.6u + \delta(t) \\ \dot{x}_4 &= 19.5x_1 - 19.5x_2 - 3.3\sin(x_3) + \delta(t) \\ y_1 &= x_1 \\ y_2 &= x_2 \end{aligned} \right. \quad (48) $$

Without unknown uncertainties $\delta(t)$, system (48) satisfies hypotheses 1, 2, 3, 4 with $p = q = 2$ and $x = (x^1, x^2)$ where $x^1 = (x_1, x_2)$ and $x^2 = (x_3, x_4)$.

Constrained by the size of the present note, we focused on the simulations of observer (30) with Round-Robin (RR) protocols, that consists in granting access to each component of the vector outputs after $p$ transmission instants where $p$ is the number of outputs [2]. In the simulations presented below, we also suppose that unknown uncertainties $\delta(t)$ are represented by random function of Matlab with variance $= 0.8$. The following simulations are performed with $u = 0.1\sin(t)$, $\theta = 40$ and $\tau_{MATI} = 0.01$ (see Figs. 1 and 2).

As we can see, even with some uncertainties, the results remain good with $\theta = 40$ and $\tau_{MATI} = 0.01$. This is not a general conclusion. Indeed the uncertainties can cause robustness problems especially for high values of $\theta$.

## VI. CONCLUSION

In this letter, some results on the design of high gain observer for networked systems with UGES protocols were presented. These results can be easily extended to other observers. In future work, we will present some results concerning other classes of protocols.
<figure><img src="image_3.png" /><figcaption>Fig. 2. Evolution of the third and fourth states for $\tau_{MATI} = 0.01$.</figcaption></figure>

REFERENCES

[1] G. C. Walsh, H. Ye, and G. Bushnell, "Stability analysis of networked control systems," *IEEE Trans. Control Sys. Technol*, vol. 10, no. 3, pp. 438-446, May 2002.

[2] D. Nešić and A. Teel, "Input-output stability properties of networked control systems," *IEEE Trans. Autom. Control*, vol. 49, no. 10, pp. 1650-1667, Oct. 2004.

[3] M. Yu, L. Wang, and T. Chu, "Sampled-data stabilization of networked control systems with nonlinearity," *Proc. Inst. Elect. Eng.*, vol. 156, no. 6, pp. 609-614, Nov. 2005.

[4] J. Hespanha, P. Naghshtabrizi, and Y. Xu, "A survey of recent results in networked control systems," *IEEE Special Issue Technol. Netw. Control Syst.*, vol. 95, no. 1, pp. 138-162, 2007.

[5] B. Sinopoli, L. Schenato, M. Franschetti, K. Poolla, M. Jordan, and S. Sastry, "Kalman filtering with intermittent observations," *IEEE Trans. Autom. Control*, vol. 49, no. 9, pp. 1453-1464, Sep. 2004.

[6] D. Dačić and D. Nešić, "Observer design for wired linear networked control systems using matrix inequalities," *Automatica*, vol. 44, pp. 2840-2848, 2008.

[7] P. Naghshtabrizi and J. P. Hespanha, "Designing an observer-based controller for a network control system," in *Proc. 44th Conf. Decision Control*, 2005, [CD ROM].

[8] R. Postoyan, T. Ahmed-Ali, and F. Lannabhi-Lagarrigue, "Observers for classes of nonlinear networked control systems," in *Proc. Conf. Signal Syst. Devices (SSD'09) Djerba-Tunisie*, 2009, [CD ROM].

[9] R. Postoyan and D. Nešić, "A framework for the observer design for networked control," in *Proc. Amer. Control Conf. (ACC'2010)*, Baltimore, MD, pp. 3678-3683.

[10] A. Zemouche, M. Boutayeb, and G. I. Bara, "Observers for a class of Lipschitz systems with extension to $H_\infty$ performance analysis," *Syst. Control Lett.*, vol. 57, no. 1, pp. 18-27, 2008.

[11] M. Farza, M. M'Saad, and L. Rossignol, "Observer design for a class of mimo nonlinear systems," *Automatica*, vol. 40, pp. 135-143, 2004.

[12] M. Arcak and D. Nešić, "A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation," *Automatica*, vol. 40, pp. 1931-1938, 2004.

[13] T. Raff, M. Kogel, and F. Allgower, "Observer with sample-and-hold updating for lipschitz nonlinear systems with nonuniformly sampled measurements," in *Proc. 48th IEEE-ACC*, Shanghai, China, 2008, [CD ROM].

[14] T. Ahmed-Ali, E. Cherrier, and M. M'Saad, "Cascade high gain observers for nonlinear systems with delayed output," in *Proc. 48th IEEE-CDC*, Shanghai, China, 2009, [CD ROM].

[15] I. Karafyllis and C. Kravaris, "From continuous-time design to sampled-data design of nonlinear observers," *IEEE Trans. Autom. Control*, vol. 54, no. 9, pp. 2169-2174, Sep. 2009.

[16] D. Carnevale, A. Teel, and D. Nešić, "A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems," *IEEE Trans. Autom. Control*, vol. 52, no. 5, pp. 892-897, May 2007.