Distributed Observer for Linear Systems with Multirate Sampled Outputs Involving Multiple Delays

Abstract

This paper focuses on the design of a continuous distributed observer for linear systems under multirate sampled output measurements involving multiple delays. It is mathematically proved that the continuous distributed observer can achieve estimation in a sensor network environment, where output measurements from each sensor are available at different sampling instants, whether these times are periodic or aperiodic, and despite the presence of multiple time-varying delays. Each sampled and delayed measurement represents a node of the network, necessitating a dedicated observer for each node, which has access to only part of the system’s output and communicates with its neighbors according to a given network graph. The exponential convergence of the error dynamics is ensured by Lyapunov stability analysis, which accounts for the influence of the sampled and delayed measurements at each node. To demonstrate the effectiveness of the proposed observer, simulation tests were conducted on the tracking control of chasing satellites in low Earth orbit (LEO), encompassing both small and large sampling rates and delays. The continuous distributed observer with sampled output measurements exhibited convergence in scenarios with different sampling intervals, even in the presence of time-varying delays, achieving asymptotic omniscience, as demonstrated in the convergence analysis.

1. Introduction

Distributed state estimation is a problem that is increasingly being studied due to the growing application of sensors in spatially located networks [1,2], such as sensor networks used in aeronautics and astronautics [3], localization [4], the operation of electrical networks [5], and other areas. In the formation control of linear multi-agent systems, a distributed observer enables each agent to estimate its state and coordinate to achieve stable formation [6]. The algorithms used for spatially located sensor networks can be classified into two main strategies: distributed Kalman filters [7,8] and distributed state observers [2,9]. Unlike distributed Kalman filters, distributed state observers have been able to solve the problem of computational overhead and facilitate system expansion [10].

A distributed observer is formed by a collection of individual observers that are spatially distributed. Each individual or local observer lacks sufficient information to fully estimate the state on its own. However, by combining its own measurements with the estimations of its neighbors, each local observer can achieve a complete estimation of the state [11]. This process requires consensus among the observers, facilitated by a communication network.

Early work on distributed observers used augmented states with discrete representations of the systems, which led to a computational load problem due to the augmented model [12,13]. Later, ref. [14] addressed this issue by splitting the system and computing two gains: one for error adjustment and one for the consensus term. Building on this idea, ref. [9] employed linear matrix inequalities (LMIs) for gain calculation, offering a more efficient methodology. Their algorithm allows for any desired error decay rate and decouples the network’s topological information using an auxiliary graph. In [15], a fully distributed design was implemented, refining the original methodology to provide greater flexibility and adaptability to network changes, using an adaptive scheme to adjust the coupling gain among the agents. Distributed observers have also been designed under the assumption that the outputs are continuously available [9,14,15]. However, in practice this is not always the case, as outputs are sampled when implemented on computers or microcontrollers. Moreover, sampling times can often be periodic or aperiodic [16,17]. Alternatively, when fully discrete methods are considered [13,18,19,20], these methods rely on very high sampling rates, which are often impractical in real-world applications.

The majority of studies on distributed observers concentrate on the challenge of communication [21,22,23], leaving aside the problem that each measurement output of the systems can be sampled at different sampling times. This issue arises because the output measurements can be provided by different types of sensors, each with its own sampling rate, whether periodic or aperiodic. Among the more significant works focused on communication between observers, ref. [21], inspired by the works of [9,14], performs event-driven communication between neighboring observers, i.e., neighboring observers only communicate and share their estimate when the corresponding event activation condition is triggered. In [22], the communication between distributed observers is intermittent, while the sensor measurements are continuous. This approach considers asynchronous communication based on the standard gossip protocol, with the communication interval being randomly selected. Conversely, ref. [23] employs a hybrid observer at each node, where inter-node communication is sampled using the multi-hop decomposition approach presented in [10].

However, the mentioned works assume that each output measurement is performed in continuous time and that these measurements are ideal, without delay effects. Delays are unavoidable and negatively impact the performance and stability of the system. Interesting work has been reported on this topic [11,24,25]. In [24], the authors assume that measurements are nearly instantaneous but consider delays in communication with variable sampling intervals. In [11], communication delays are considered constant and uniform. On the other hand, in [25], an observer is designed to converge in a fixed time, allowing it to compensate for delays in data transmission after they occur. Several studies have explored sampling transmission signals to enhance the feasibility of achieving more realistic scenarios, as in the work of [6], where formation control is based on a distributed observer with asynchronously sampled signals. However, these studies overlook the impact of signal delays. In contrast, in [26], the issue of delays in information transfer is addressed, rejecting them robustly through using a special Lyapunov function. However, this approach assumes that both the information transfer and the system operate in a continuous-time framework. In this regard, it is interesting to prove a novel distributed observer approach for the output measurements problem in the presence of multiple delays and multiple sampling periods.

In networked control systems, distributed observers are crucial for coordinating sensors, actuators, and state estimators that operate asynchronously. In this regard, the distributed observers have been designed for networked sensors as [9,17,20,27,28]. Another problem has been addressed in [6], it is applied in the formation control problem of multi-agent systems, where local state estimation with asynchronously sampled data ensures the system’s stability and performance. This relationship illustrates how the concepts of robustness, adaptability, and distributed control are relevant not only in multi-agent systems with distributed observers but also in systems like a USV-UAV cooperative system for maritime search missions, where robust adaptive event-triggered control strategies are employed to manage sensor faults and communication burdens effectively [29]. On the other hand, ref. [30] proposed a distributed control law based on a distributed observer for Lipschitz nonlinear systems with model mismatch, considering factors such as the coexistence of control inputs, nonlinearity, and uncertainty.

It should be noted that the most realistic scenario is that each observer receives sampled measurements with multirate sampling, depending on the type of sensor. Motivated by the above discussion, this work deals with the problem of continuous distributed observer design in the presence of sampled output measurements under multirate sampling conditions, whether asynchronous or synchronous and in the face of multiple delays. This approach allows maintaining continuous communication between each interconnected node in the observer to achieve continuous estimation of the complete state of a linear system. Very little work has been conducted on the delays that occur in both measurements and communication. For instance, refs. [24,25] proposed distributed observers that converge in finite time despite communication delays, considering continuous output. However, despite these contributions, it is important to address the output measurement problem under multirate samplings and multiple time-varying delays given the nature of each sensor.

Therefore, in this work, an analysis is conducted using multi-sampling output measurements under multiple delays. The study focuses comprehensively on estimating the state of a linear system by measurements of sampled outputs at different sampling times in a distributed sensor network. Figure 1 illustrates the concept of the continuous distributed observer in the presence of the delayed and sampled output measurements. Thus, local observers have access to a portion of the observed linear system and communicate with neighboring nodes according to the communication graph. Consequently, each local observer in the distributed network provides an accurate estimate of all internal variables. The originality of the distributed observer proposed approach lies in the use of the output error predictors, which are updated every time a new measurement is received. In this sense, each node sensor i of the sensor network corresponds to a local observer, which is independent of each measurement of the node sensor, i.e., regardless of the nature of the time delay or sampling interval of each node sensor, which is very practical, the only restriction is the maximum delay or the maximum sampling or the combination of both permitted. Each predictor is directly incorporated into the observer to minimize the state estimation error compared to traditional approaches, which usually use sampled-data mechanisms [1,6].

The structure of the proposed observer is very similar to the continuous-time observer described in [9], with the main difference being that the observation error term is replaced by a new variable, which predicts the estimated state in the no measurement interval and it is initialized with the estimation error at the sampling instant corresponding to the sensor node i. This approach has been previously applied in multi-output systems, as mentioned in [31,32]. However, few studies have explored this approach in the context of distributed systems. A related study can be found in [33], where two different predictors are introduced depending on the system’s observability for the undelayed output measurements. Thus, the present work provides an alternative solution to facilitate digital implementation in distributed systems. Additionally, our approach takes into account the problem of potential measurement delays, making the digital implementation more realistic. In this way, accurate and reliable estimation is achieved despite the asynchrony, the multirate nature of the measurements, and potential delays.

The key contributions can be summarized in the following points:

• The proposed observer approach is suited for measurements sampled over extended periods. Longer sampling periods, whether periodic or aperiodic, present significant challenges as they are necessary in contexts of slow change or limited resources but may result in the loss of important details. This contrasts with the approach in [6,17], which focuses on very small sampling intervals, this work emphasizes increasing the sampling periods to ensure observer stability, even in the presence of extended sampling intervals.
• Measurement delays are common and can arise from different factors, including technological limitations, data transmission processes, and signal processing. Despite its prevalence, this phenomenon has received insufficient attention in the scientific literature. The proposed approach tackles these delays by implementing precise models to accurately align the measurements. There are few works that address this problem, considering delay problems between the communication of each node [34], which has a major restriction in the suggested delay values, which makes it impractical.
• In contrast to many studies that use sampled communication, the proposed approach allows continuous estimation between sampling points, thereby eliminating the need for frequent communication. This capability allows for longer sampling intervals without sacrificing precision, thereby avoiding the need for hybrid observers and enhancing system flexibility and efficiency. Thus, our observer presents a smoother estimation, mainly due to the avoidance of using a sampled-date mechanism, such as [6,34].

The rest of the paper is organized as follows. Section 2 presents the problem statement and the mathematical model of a linear physical system whose output is partitioned with sampled outputs. Section 3 introduces the continuous distributed observer with sampled outputs, where state estimation is carried out continuously at a known time interval and updated at each sampled output, whether at a periodic or aperiodic sampling rate. Section 4 demonstrates the effectiveness of the method through simulations, where simulation tests were conducted on the tracking control of chasing satellites in low Earth orbit (LEO), encompassing both small and large sampling rates and delays. Finally, Section 5 provides the conclusions.

Preliminaries: Consider a directed or weighted digraph $\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})$, of order n, with node set $\mathcal{V}=\{v_1,v_2,\dots ,v_n\}$, edge set $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$, and weighted adjacency matrix $\mathcal{A}=[aij]$, where $a_{ij}$ represents the weight of the edge from node i to node j. The adjacency matrix elements $a_{ij}>0$, and $\mathcal{A}\in {\mathbb{R}}^{N\times N}$. Let $\mathcal{L}=[l_{ij}]$ be the Laplacian matrix associated with $\mathcal{G}$, where $\mathcal{L}\in {\mathbb{R}}^{N\times N}$. Thus, there is a direct relationship between the property of being strongly connected and the rank of its Laplacian matrix. Specifically, the Laplacian matrix of a strongly connected digraph has a unique eigenvalue at zero.

2. Problem Statement

Consider a physical system that is monitored by N sensors, represented by the following linear mathematical model under multirate sampled output measurements involving multiple delays, which is more general than the one focused on [9,17,27]:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{x}(t)=Ax(t)\hfill \\ \overline{y}(t)=y(t-\tau (t))=Cx(t-\tau (t))\hfill \end{array}\right.\end{array}$$

(1)

where $x(t)\in {\mathbb{R}}^n$ is the state vector, $A\in {\mathbb{R}}^{n\times n}$ is the system matrix, $C\in {\mathbb{R}}^{p\times n}$ is the output matrix, and $y(t-\tau (t))\in {\mathbb{R}}^p$ denotes the delayed output measurements acquired at different moments in time; $\tau (t):{\mathbb{R}}^{+}\to [0,{\tau }_{max}+{\mathrm{\Delta }}_{max}]$ is a piecewise continuous function, where ${\tau }_{max}$ is a positive constant representing the maximum delay and ${\mathrm{\Delta }}_{max}$ is the upper bound of the maximum sampling time.

Since each sampled output measurement is available at the time instants $t_k$, corresponding to the sampled output measurement at the time instant $t_k-\tau (t_k)$, thus, the sampled and delayed output is represented as follows:

$$\overline{y}(t)=y(t-(t-t_k+\tau (t_k))=y(t_k-\tau (t_k))=Cx(t_k-\tau (t_k))$$

(2)

since the continuous function $\tau (t)$ is denoted as $\tau (t)=t-t_k+\tau (t_k)$.

Thus, the sampled and delayed output can be partitioned as:

$$\overline{y}(t)=y_i(t_k^i-{\tau }_i(t_k^i))=\left[\begin{array}{c}{y}_1(t_k^1-{\tau }_1(t_k^1))\\ y_2(t_k^2-{\tau }_2(t_k^2))\\ ⋮\\ y_N(t_k^N-{\tau }_N(t_k^i))\end{array}\right]=\left[\begin{array}{c}{C}_1{x}_1(t_k^1-{\tau }_1(t_k^1))\\ C_2{x}_2(t_k^2-{\tau }_2(t_k^2))\\ ⋮\\ C_N{x}_N(t_k^N-{\tau }_N(t_k^i))\end{array}\right]=C_{i}x(t_k^i-{\tau }_i(t_k^i))$$

(3)

where $y_i(t_k^i-{\tau }_i(t_k^i))\in {\mathbb{R}}^{p_i}$ and ${\sum }_{i=1}^N{p}_i=p$, with $C_i\in {\mathbb{R}}^{p_i\times n}$. This means that the proportion $y_i(t_k^i-{\tau }_i(t_k^i))=C_{i}x(t_k^i-{\tau }_i(t_k^i))\in {\mathbb{R}}^{p_i}$ is the delayed measurement available to sensor node i. The delayed system output is available only at the sampling instants that satisfy $0\le t_0^i-{\tau }_i(t_0^i)<\dots <t_k^i-{\tau }_i(t_k^i)<t_{k+1}^i-{\tau }_i(t_{k+1}^i)<\dots$. Thus, one has ${\tau }_k^i=(t_{k+1}^i-{\tau }_i(t_{k+1}^i))-(t_k^i-{\tau }_i(t_k^i))$ and $\underset{k\to +\infty }{lim}({\tau }_k^i)=+\infty ,i=1,\dots ,p$. It is assumed that there exists an upper bound of the sampling partition diameter of each node sensor i${\mathrm{\Delta }}_{max}^i>0$, and an upper bound of the sampling partition diameter ${\mathrm{\Delta }}_{max}>0$ of the sampled and delayed output $\overline{y}(t)$, i.e.,

$$0<t_{k+1}^i-t_k^i={\mathrm{\Delta }}_i\le {\mathrm{\Delta }}_{max}^i\le {\mathrm{\Delta }}_{max},\forall k\ge 0,i=1,\dots ,p$$

(4)

In addition, one assumes that each delay at the sampling instants $t_k^i$, i.e., ${\tau }_i(t_k^i)$ are bounded functions as follows:

$$|{\tau }_i(t_k^i)|\le {\tau }_i\le {\tau }_{max}^i\le {\tau }_{max},\forall k\ge 0,i=1,\dots ,p$$

(5)

where ${\tau }_{max}^i$ is the upper bound of the time-varying delay of each sensor node i, and ${\tau }_{max}$ represents the maximum delay.

Therefore, one obtains, as in [32,35]:

$$0<{\mathrm{\Delta }}_{max}^i+{\tau }_{max}^i<{\chi }_i<{\mathrm{\Delta }}_{max}+{\tau }_{max}<\chi .$$

(6)

The following assumptions shall be considered throughout the design of the observer.

Assumption 1. 

The pair $(A,C)$ is observable; however, a submatrix $C_i$ of C may not guarantee the observability of the system.

Assumption 2. 

The graph $\mathcal{G}$ is directed and has the property of strong connectivity.

Given the system to be estimated of Equations (1)–(3) and a communication structure $\mathcal{G}$ among each observer, the distributed observer structure under multirate sampling and multiple delays are expected to achieve omniscience asymptotically for all nodes $i\in \mathcal{V}$ [13]. This indicates that

$$\underset{t\to \infty }{lim}\parallel {\widehat{x}}_i(t)-x(t)\parallel =0.$$

(7)

Consequently, to address the challenges posed by sampled and delayed output measurements affected by multirate sampling and multiple delays, the design of the continuous-time distributed observer structure is proposed. This redesign aims to ensure the asymptotic omniscience of the network, allowing for continuous-time estimation, as presented in [9].

3. Distributed Observer with Sampled and Delayed Output Measurements

This section presents the distributed observer with sampled and delayed output measurements under multirate sampling and multiple delays for the linear system of Equations (1) and (3). The distributed observer structure is described by the following equations:

$$\left\{\begin{array}{cc}\hfill {\dot{\widehat{x}}}_i(t)& =A{\widehat{x}}_i(t)-L_i{\eta }_i(t)+\gamma M_i\sum _{j=1}^N{a}_{ij}\left({\widehat{x}}_j(t)-{\widehat{x}}_i(t)\right),i\in \mathcal{V}\hfill \\ \hfill {\dot{\eta }}_i(t)& =-C_i{L}_i{\eta }_i(t),t\in \left[t_k^i-{\tau }_i(t_k^i),t_{k+1}^i-{\tau }_i(t_{k+1}^i)\right),k\in \mathbb{N}\hfill \\ \hfill {\eta }_i\left(t_k^i-{\tau }_i(t_k^i)\right)& =C_{i}x\left(t_k^i-{\tau }_i(t_k^i)\right)-C_i{\widehat{x}}_i\left(t_k^i-{\tau }_i(t_k^i)\right),t=t_k^i-{\tau }_i(t_k^i)\hfill \end{array}\right.$$

(8)

where the value of ${\widehat{x}}_i\in {\mathbb{R}}^n$ is the estimated state vector of the observer node i; $a_{ij}$ is the $(i,j)$-th item of the adjacency matrix $\mathcal{A}$, which relates to a communication network; $\gamma \in \mathbb{R}$ is a coupling gain to be designed and $L_i\in {\mathbb{R}}^{n\times p_i}$ and $M_i\in {\mathbb{R}}^{n\times n}$ are gain matrices to be designed. The function ${\eta }_i(t)$ is continuous over the time interval $[t_k^i-{\tau }_i(t_k^i),t_{k+1}^i-{\tau }_i(t_{k+1}^i))$, which is set at each sampling instant $t_k^i-{\tau }_i(t_k^i)$ using only the sampled output measurement $y_i((t_k^i)-{\tau }_i(t_k^i))$.

For a comprehensive analysis of the observers, the estimation error of the i-th node observer is defined as follows ${\tilde{x}}_i(t):={\widehat{x}}_i(t)-x(t)$, combining Equation (1) with Equation (8), we obtain:

$${\dot{\tilde{x}}}_i(t)=A{\tilde{x}}_i(t)-L_i{\eta }_i(t)+\gamma M_i\sum _{j=1}^N{a}_{ij}\left({\tilde{x}}_j(t)-{\tilde{x}}_i(t)\right)$$

(9)

Now, starting from ${\dot{\eta }}_i(t)$ in Equation (8), yields

$${\eta }_i(t)=e^{-C_i{L}_i(t-(t_k-{\tau }_i(t_k^i))}{C}_i\tilde{x}(t_k^i-{\tau }_i(t_k^i))$$

(10)

Defining $z_i(t)$ as an auxiliary variable,

$$\begin{array}{cc}\hfill z_i(t)& =C_i\tilde{x}(t)-{\eta }_i(t)\hfill \end{array}$$

(11)

Substituting Equation (11) in Equation (9),

$${\dot{\tilde{x}}}_i(t)=\left(A-L_i{C}_i\right){\tilde{x}}_i(t)+L_i{z}_i(t)+\gamma M_i\sum _{j=1}^N{a}_{ij}\left({\tilde{x}}_j(t)-{\tilde{x}}_i(t)\right)$$

where $\tilde{x}(t):=col({\tilde{x}}_1^T(t),{\tilde{x}}_2^T(t),\dots ,{\tilde{x}}_N^T(t))$ is the joint vector of errors. Therefore, the following global error can be written:

$$\dot{\tilde{x}}(t)=\mathrm{\Lambda }\tilde{x}(t)+Lz(t)-\gamma \overline{M}\left(\mathcal{L}\otimes I_n\right)\tilde{x}(t)$$

(12)

where

$$\begin{array}{c}\mathrm{\Lambda }=diag\left\{A-L_1{C}_1,\dots ,A-L_N{C}_N\right\},\\ L=diag\left\{L_1,\dots ,L_N\right\},\\ z(t)=diag\left\{z_1(t),\dots ,z_N(t)\right\},\\ \overline{M}=diag\left\{M_1,\dots ,M_N\right\}.\end{array}$$

Remark 1. 

Since, $\left(A,C_i\right)$ may not be observable or detectable, an orthogonal transformation is required in order to achieve the observability of the pair $\left(A,C_i\right)$. Following the approach in [9] and [14], an orthogonal matrix $T_i$ (i.e., $T_i{T}_i^T=I_n$) for $i\in \mathcal{V}$ is defined. Then, matrices A and $C_i$ are transformed by the state space transformation $T_i$:

$$T_i^{T}A{T}_i=\left[\begin{array}{cc}{A}_{io}& 0\\ A_{ir}& A_{iu}\end{array}\right],C_i{T}_i=\left[\begin{array}{cc}{C}_{io}\hfill & 0\hfill \end{array}\right]$$

(13)

where $C_{io}\in {\mathbb{R}}^{p_i\times v_i},A_{io}\in {\mathbb{R}}^{v_i\times v_i},A_{ir}\in {\mathbb{R}}^{\left(n-v_i\right)\times v_i}$, $A_{iu}\in {\mathbb{R}}^{\left(n-v_i\right)\times \left(n-v_i\right)}$, $n-v_i$ is the dimension of the unobservable subspace of the pair $\left(A,C_i\right)$, and $v_i$ is the dimension of the observable subspace of the pair $\left(A,C_i\right)$.

The following set of lemmas will be used to ensure the convergence of the proposed observer structure.

Lemma 1 

([9]). If $\mathcal{G}$ is a strongly connected directed graph, there exists a unique positive row vector h such that $h\mathcal{L}=0$ and $h{\mathbf{1}}_N=N$. Defining $\mathcal{H}$ as the diagonal matrix with the elements of h on its diagonal, the matrix $\mathcal{T}:=\mathcal{H}\mathcal{L}+{\mathcal{L}}^T\mathcal{H}$ is positive semi-definite and satisfies ${\mathbf{1}}_N^T\mathcal{T}=0$ and $\mathcal{T}{\mathbf{1}}_N=0$.

Lemma 2 

([9]). Linking the Laplacian matrix $\mathcal{L}$ to the strongly connected directed graph $\mathcal{G}$, for all $g_i>0,i\in \mathcal{V}$, it is necessary to have $ϵ>0$ such that

$$T^T\left(\mathcal{T}\otimes I_n\right)T+\widehat{G}>ϵI_{nN}$$

(14)

where $\mathcal{T}$ is specified as described in Lemma 1, T is a diagonal matrix with entries $T_1,\dots ,T_N$, and $\widehat{G}$ is a diagonal matrix whose elements are determined by $g_i$.

Lemma 3 

([9]). The distributed observer structure of Equation (8) guarantees omniscience asymptotically. Moreover, all error dynamics of Equation (12) converge to zero with a decay rate of at least μ, considering that there exist positive definite matrices $P_{io}\in {\mathbb{R}}^{v_i\times v_i},P_{iu}\in {\mathbb{R}}^{\left(n-v_i\right)\times \left(n-v_i\right)}$, and a matrix $R_i\in {\mathbb{R}}^{v_i\times p_i}$ such that

$$\begin{array}{cc}\hfill \left[\begin{array}{cc}{\Psi }_i+{\displaystyle \frac{\gamma }{h_i}}{g}_i{I}_{v_i}& A_{ir}^T{P}_{iu}\\ P_{iu}{A}_{ir}& P_{iu}{A}_{iu}+{(P_{iu}{A}_{iu})}^T+2\mu P_{iu}\end{array}\right]-{\displaystyle \frac{\gamma }{h_i}}ϵI_n& <0,\forall i\in \mathcal{V}\hfill \end{array}$$

(15)

where ${\Psi }_i:=P_{io}{A}_{io}+A_{io}^T{P}_{io}-R_i{C}_{io}-C_{io}^T{R}_i^T+2\mu P_{io}$, thus, the gain matrices of the distributed observer of Equation (8) are

$$L_i:=T_i\left[\begin{array}{c}{L}_{io}\\ 0\end{array}\right],M_i:=T_i\left[\begin{array}{cc}{P}_{io}^{-1}& 0\\ 0& P_{iu}^{-1}\end{array}\right]T_i^T$$

(16)

where $L_{io}=P_{io}^{-1}{R}_i,i\in \mathcal{V}$.

Lemma 4 

([32]). Consider a differentiable function $v:t\in \left[t_0-\delta (t),+\infty \right[\mapsto v(t)\in {\mathbb{R}}^{+}$, where $t_0,\delta (t)\ge 0$, thus, the function $v(t)$ satisfies the following inequality

$$\dot{v}(t)\le -\alpha v(t)+\beta {\int }_{t-\delta (t)}^{t}v(s)ds,\forall t\ge t_0$$

(17)

where $\alpha >0$, $\beta >0$ and if

$${\displaystyle \frac{\beta {\tau }_M}{\alpha }}<1.$$

(18)

then, the function v has an exponential convergence to zero, i.e.,

$$v(t)\le (1+\delta (\alpha -\sigma (\delta (t))))e^{-\sigma (\delta (t))\left(t-t_0\right)}v(t_0)$$

(19)

with

$$0<\sigma (\delta (t))=(\alpha -\beta \delta (t))e^{-\alpha \delta (t)}\le \alpha .$$

(20)

On the basis of the above, the following theorem is proposed:

Theorem 1. 

Given the observable system of Equations (1)–(3) and a strongly connected directed graph $\mathcal{G}$, there exists a positive constant $\chi >0$ such that the upper bound $({\tau }_{max}^i+{\mathrm{\Delta }}_{max}^i)$ of each observer i satisfies that

$${\tau }_{max}+{\mathrm{\Delta }}_{max}<\chi$$

(21)

and $\mu >0$ is also satisfied, then the distributed observer with sampled and delayed output measurements under multirate sampling and multiple delays of Equation (8) achieves omniscience asymptotically, i.e., Equation (7), and the error system is stable, which is given:

$$\parallel \tilde{x}(t)\parallel \le \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}{e}^{-\sigma (d_{max})(t-t_0)}(1+d_{max}(\alpha -\sigma (d_{max}))\parallel \tilde{x}(0)\parallel$$

(22)

where $d_{max}={\mathrm{\Delta }}_{max}+{\tau }_{max}$, $\alpha =\mu$, $\beta =\left(\mathcal{H}\otimes I_n\right)\parallel L\parallel \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}\xi$ and

$$\sigma (d_{max})=\left(\alpha -\beta d_{max}\right)e^{-\alpha d_{max}},$$

also, by performing the following steps:

1.

Find an orthogonal matrix $T_i$ for all $i\in \mathcal{N}$ such that the pair $(A_{io},C_{io})$ is observable:

$${\overline{A}}_i=T_i^{T}A{T}_i=\left[\begin{array}{cc}{A}_{io}& 0\\ A_{ir}& A_{iu}\end{array}\right],{\overline{C}}_i=C_i{T}_i=\left[\begin{array}{cc}{C}_{io}\hfill & 0\hfill \end{array}\right]$$

(23)

2.

Compute the positive row vector $h=[h_1,\dots ,h_N]$ such that, $h\mathcal{L}=0$ and $h{\mathbf{1}}_N=N$.

3.

Calculate $ϵ>0$, proposing $g_i=1$. Such that

$$T^T(\mathcal{T}\otimes I_n)T+\widehat{G}>ϵI_{nN}$$

(24)

holds.

4.

Calculate $\gamma >0$ for all $i\in \mathcal{N}$

$$A_{iu}+A_{iu}^T-\left({\displaystyle \frac{\gamma }{h_i}}ϵ-2\mu \right)I_{n-v_i}+{\displaystyle \frac{1}{\frac{\gamma }{h_i}ϵ-2\mu }}{A}_{ir}{A}_{ir}^T<0$$

(25)

5.

Calculate $L_{io}$ such that the eigenvalues of $A_{io}-L_{io}{C}_{io}$ converge to the region $\{s\in \mathbb{C}\mid Re(s)<-\mu \}$.

6.

Compute the gain matrices $L_i$ and $M_i$ for $i\in \mathcal{N}$ such that

$$\begin{array}{cc}\hfill \left[\begin{array}{cc}{\Psi }_i+{\displaystyle \frac{\gamma }{h_i}}{g}_i{I}_{v_i}& A_{ir}^T{P}_{iu}\\ P_{iu}{A}_{ir}& P_{iu}{A}_{iu}+{(P_{iu}{A}_{iu})}^T+2\mu P_{iu}\end{array}\right]-{\displaystyle \frac{\gamma }{h_i}}ϵI_n& <0,\forall i\in \mathcal{V}\hfill \end{array}$$

(26)

where ${\Psi }_i:=P_{io}{A}_{io}+A_{io}^T{P}_{io}-R_i{C}_{io}-C_{io}^T{R}_i^T+2\mu P_{io}$

$$L_i:=T_i\left[\begin{array}{c}{L}_{io}\\ 0\end{array}\right],M_i:=T_i\left[\begin{array}{cc}{P}_{io}^{-1}& 0\\ 0& P_{iu}^{-1}\end{array}\right]T_i^T$$

(27)

where $L_{io}=P_{io}^{-1}{R}_i,i\in \mathcal{N}$. The following is obtained:

$$P_i:=T_i\left[\begin{array}{cc}{P}_{io}& 0\\ 0& P_{iu}\end{array}\right]T_i^T,P_i>0$$

(28)

thus, the observer of Equation (8) converges asymptotically with a decay rate of at least μ.

Proof. 

Consider the Lyapunov candidate function for the error system of Equation (12):

$$V\left({\tilde{x}}_1,\dots ,{\tilde{x}}_N,t\right):=\sum _{i=1}^N{h}_i{\tilde{x}}_i^T(t)P_i{\tilde{x}}_i(t)$$

(29)

where the matrix $P_i$ is given by Equation (28).

The time-derivative of $V(\tilde{x},t)=V({\tilde{x}}_1,\dots ,{\tilde{x}}_N,t)$ along the trajectories of $\tilde{x}(t)$ is obtained as

$$\begin{array}{cc}\hfill \dot{V}\left(\tilde{x}(t),t\right)=& {\tilde{x}}^T(t)\left(\left(\mathcal{H}\otimes I_n\right)\left(P\mathrm{\Lambda }+{\mathrm{\Lambda }}^{T}P\right)\right)\tilde{x}(t)\hfill \\ \hfill & -\gamma {\tilde{x}}^T(t)\left(P\overline{M}\left(\mathcal{H}\mathcal{L}\otimes I_n\right)+\left({\mathcal{L}}^T\mathcal{H}\otimes I_n\right){\overline{M}}^{T}P\right)\tilde{x}(t)\hfill \\ \hfill & +\left(\mathcal{H}\otimes I_n\right)\left(\tilde{x}{(t)}^{T}PLz(t)+z^T(t)L^{T}P\tilde{x}(t)\right)\hfill \end{array}$$

(30)

with $P=diag\left\{P_1,\dots ,P_N\right\}$. Given that the matrix $M_i$ in Equation (16) is defined as $M_i=P_i^{-1}$, it follows that $\overline{M}=P^{-1}$. Consequently, Equation (30) becomes

$$\begin{array}{cc}\hfill \dot{V}\left(\tilde{x}(t),t\right)& ={\tilde{x}}^T(t)\left(\left(\mathcal{H}\otimes I_n\right)\left(P\mathrm{\Lambda }+{\mathrm{\Lambda }}^{T}P\right)-\gamma \left(\mathcal{T}\otimes I_n\right)\right)\tilde{x}(t)+\left(\mathcal{H}\otimes I_n\right)\left(2{\tilde{x}}^T(t)PLz(t)\right)\hfill \end{array}$$

(31)

where $\mathcal{T}=\mathcal{H}\mathcal{L}+{\mathcal{L}}^T\mathcal{H}$. On the other hand, the auxiliary function $z(t)$ given in Equation (11) is analyzed. Note, that $z(t_k^i-{\tau }_i(t_k^i))=0$. Moreover, the time-derivative of $z(t)$ can be expressed as

$$\begin{array}{cc}\hfill \dot{z}(t)& =C\left(\left(A-LC\right)\tilde{x}(t)+Lz(t)+\gamma \overline{M}\left(\mathcal{L}\otimes In\right)\tilde{x}(t)\right)+CL{e}^{-CL(t-(t_k^i-{\tau }_i(t_k^i))}C\tilde{x}(t_k^i-{\tau }_i(t_k^i))\hfill \\ \hfill & =CA\tilde{x}(t)+C\gamma \overline{M}\left(\mathcal{L}\otimes In\right)\tilde{x}(t)\hfill \end{array}$$

(32)

Integrating Equation (32) from ${\delta }_i(t)=t-(t_k^i-{\tau }_i(t_k^i))$ to t, further considering that $z(t_k^i-{\tau }_i(t_k^i))=0$, we obtain

$$\begin{array}{cc}\hfill z(t)& ={\int }_{t-{\delta }_i(t)}^t\left(CA\tilde{x}(s)+C\gamma P^{-1}\left(\mathcal{L}\otimes In\right)\tilde{x}(s)\right)ds\hfill \\ \hfill & =\left(CA+C\gamma P^{-1}\left(\mathcal{L}\otimes In\right)\right){\int }_{t-{\delta }_i(t)}^t\tilde{x}(s)ds\hfill \end{array}$$

(33)

and hence

$$\begin{array}{cc}\hfill \parallel z(t)\parallel & \le \parallel \left(CA+C\gamma P^{-1}\left(\mathcal{L}\otimes In\right)\right){\int }_{t-{\delta }_i(t)}^t\tilde{x}(s)\parallel ds\hfill \\ \hfill & \le \parallel C\parallel \parallel A\parallel +\parallel C\parallel \gamma \parallel P^{-1}\parallel \parallel \left(\mathcal{L}\otimes In\right)\parallel {\int }_{t-{\delta }_i(t)}^t\parallel \tilde{x}(s)\parallel ds\hfill \\ \hfill & \le \xi {\int }_{t-{\delta }_i(t)}^t\parallel \tilde{x}(s)\parallel ds\hfill \end{array}$$

(34)

where $\xi =\parallel C\parallel \parallel A\parallel +\parallel C\parallel \gamma \parallel P^{-1}\parallel \parallel \left(\mathcal{L}\otimes In\right)\parallel$. Thus, from Equation (31)

$$\parallel 2{\tilde{x}}^T(t)PLz(t)\parallel \le 2\parallel L\parallel \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}\xi {\int }_{t-{\delta }_i(t)}^t\sqrt{V(\tilde{x}(s),s)}ds\times \sqrt{V(\tilde{x}(t),t)}$$

(35)

Hence,

$$\begin{array}{cc}\hfill \dot{V}\left(\tilde{x}(t),t\right)& \le {\tilde{x}}^T(t)\left(\left(\mathcal{H}\otimes I_n\right)\left(P\mathrm{\Lambda }+{\mathrm{\Lambda }}^{T}P\right)-\gamma \left(\mathcal{T}\otimes I_n\right)\right)\tilde{x}(t)\hfill \\ \hfill & +2\left(\mathcal{H}\otimes I_n\right)\parallel L\parallel \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}\xi {\int }_{t-{\delta }_i(t)}^t\sqrt{V(\tilde{x}(s),s)}ds\times \sqrt{V(\tilde{x}(t),t)}\hfill \end{array}$$

(36)

On the other hand, from the inequalities of Equations (14) and (15) of Lemmas 2 and 3, respectively, the following inequality is obtained

$$diag\{Q_1,\dots ,Q_N\}-T^T\gamma \left(\mathcal{T}\otimes I_n\right)T<0,$$

(37)

where $Q_i=h_i\left[\begin{array}{cc}{\Psi }_i& A_{ir}^T{P}_{iu}\\ P_{iu}{A}_{ir}& P_{iu}{A}_{iu}+{(P_{iu}{A}_{iu})}^T+2\mu P_{iu}\end{array}\right]$, $i\in \mathcal{N}$, with ${\Psi }_i$ defined in Lemma 3. Since $L_{io}=P_{io}^{-1}{R}_i$, pre- and post- multiplying the inequality of Equation (37) with T and $T^T$, this result in

$$\left(\mathcal{H}\otimes I_n\right)\left(P\mathrm{\Lambda }+{\mathrm{\Lambda }}^{T}P+2\mu P\right)-\gamma \left(\mathcal{T}\otimes I_n\right)<0,$$

(38)

Therefore, from the above analysis, it is guaranteed that

$$\left(\mathcal{H}\otimes I_n\right)\left(P\mathrm{\Lambda }+{\mathrm{\Lambda }}^{T}P+2\mu P\right)-\gamma \left(\mathcal{T}\otimes I_n\right)<-2\mu V(\tilde{x}(t),t).$$

(39)

Building on the analysis and considering Lemma 4, it follows that

$$\begin{array}{cc}\hfill \dot{V}\left(\tilde{x}(t),t\right)& \le -2\mu V(\tilde{x}(t),t)+2\left(\mathcal{H}\otimes I_n\right)\parallel L\parallel \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}\xi {\int }_{t-{\delta }_i(t)}^t\sqrt{V(\tilde{x}(s),s)}ds\times \sqrt{V(\tilde{x}(t),t)}\hfill \end{array}$$

(40)

or equivalently

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\sqrt{V\left(\tilde{x}(t),t\right)}& \le -\alpha \sqrt{V(\tilde{x}(t),t)}+\beta {\int }_{t-{\delta }_i(t)}^t\sqrt{V(\tilde{x}(s),s)}ds\hfill \end{array}$$

(41)

where $\alpha =\mu$ and

$$\beta =\left(\mathcal{H}\otimes I_n\right)\parallel L\parallel \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}\xi$$

Therefore, according to Lemma 4, it follows that

$$\begin{array}{cc}\hfill \sqrt{V\left(\tilde{x}(t)\right)}& \le (1+\delta (t)(\alpha -\sigma (\delta (t))))e^{-\sigma (\delta (t))(t-t_0)}\sqrt{V(\tilde{x}(0),t)}\hfill \\ \hfill & \le (1+d_{max}(\alpha -\sigma (d_{max}))e^{-\sigma (d_{max})(t-t_0)}\sqrt{V(\tilde{x}(0),t)}\hfill \end{array}$$

(42)

where $d_{max}={\mathrm{\Delta }}_{max}+{\tau }_{max}$ and

$$0<\sigma (d_{max})=\left(\alpha -\beta d_{max}\right)e^{-\alpha d_{max}}.$$

(43)

and suppose that the maximum values of the sampling time ${\mathrm{\Delta }}_{max}$ and delay ${\tau }_{max}$ satisfy the following condition ${\mathrm{\Delta }}_{max}+{\tau }_{max}=d_{max}<{\displaystyle \frac{\alpha }{\beta }}=\chi$, i.e., ${\displaystyle \frac{\beta d_{max}}{\alpha }}<1$ or explicitly

$$d_{max}{\displaystyle \frac{\left(\mathcal{H}\otimes I_n\right)\parallel L\parallel \sqrt{\frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}\xi }{\mu }}<1$$

(44)

where the maximum value $d_{max}$ satisfies the following condition

$$d_{max}<{\displaystyle \frac{\alpha }{\beta }}={\displaystyle \frac{\mu }{\left(\mathcal{H}\otimes I_n\right)\parallel L\parallel \sqrt{\frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}\xi }}\triangleq \chi$$

(45)

Back to terms of the estimation error $\tilde{x}(t)$, it can be stated that

$$\parallel \tilde{x}(t)\parallel \le \sqrt{{\displaystyle \frac{{\lambda }_{max}(P)}{{\lambda }_{min}(P)}}}{e}^{-\sigma (d_{max})(t-t_0)}(1+d_{max}(\alpha -\sigma (d_{max}))\parallel \tilde{x}(0)\parallel$$

(46)

From Equation (46), it is noticed that the estimation error $\tilde{x}(t)$ decreases asymptotically to zero at an exponential rate. Then, with the parameters calculated as $ϵ$, $\gamma$, $P_{iu}$, $P_{io}$ and those calculated in steps 3 and 6 of Theorem 1, the inequality of Equation (26) is guaranteed to be satisfied. This way, the distributed observer with sampled and delayed output measurements of Equation (8) with the computed gain matrices $L_i$, $M_i$ and $\gamma$ achieves omniscience asymptotically, i.e., Equation (7), with a decay rate of at least $\mu$, as expressed in Equation (39). □

Remark 2. 

It is important to state that the variable $\delta (t)$ corresponds to the upper bound either of the output delay, the sampling partition diameter, or the sum of these bounds if the outputs are sampled and delayed. The case $\delta (t)=0$ corresponds to the case where the outputs are available in a continuous form. According to Lemma 4, one can check that the function $\delta (t)\mapsto \sigma (\delta )$ is decreasing and that $\sigma (0)=\alpha$. Hence, the fastest rate of convergence of the observation error $\overline{x}(t)$ (to zero or the ultimate ball), as well as the smallest value of the asymptotic ultimate bound, are obtained with $\delta (t)=0$. When $d_{max}\ge \delta (t)>0$, the behavior of the observation error $\overline{x}(t)$ will still be similar to that of $\delta (t)=0$, but the rate of convergence decreases, and the magnitude of the asymptotic ultimate bound increases.

Remark 3. 

It is essential to bear in mind that high-dimensional structures generate results in large system matrices and correspondingly high-dimensional LMIs, as described in Theorem 1 (points 4–6). The proposed LMIs, however, can offer greater flexibility in problem-solving, potentially reducing the computational complexity associated with calculating observer gains, regardless of the system order or sensor network size. The main limitation lies in the maximum delay sizes and sampling intervals. This complexity reduction is achieved by decoupling the network’s topological information from the local gain matrices of the observer through the introduction of an auxiliary undirected graph. Moreover, the scalability of this approach primarily depends on the sensor network size, as the distributed observer structure consists of N local observers. Each sensor node i corresponds to a local observer, and to address the scalability issue, the distributed observer is represented by a simple structure with only two equations. These equations operate independently of the measurements from each sensor node, making the approach practical regardless of variations in time delays or sampling intervals across the sensor network.

4. Numerical Results

In this section, the main findings of the proposed distributed observer with sampled delayed output measurements under multirate sampling and multiple delays are examined. A numerical example involving the tracking control of chasing satellites in low Earth orbit (LEO) is provided to demonstrate its effectiveness. The operation of LEO satellites requires precise real-time orbits. The precise orbit determination (POD) of LEO satellites mainly depends on the Global Positioning System (GPS). However, the trend towards highly autonomous spacecraft provides a strong motivation for accurate real-time navigation of LEO satellites. Unfortunately, it is not simple to retrieve stable and high-accuracy real-time orbit information of LEO satellites at all times since LEO satellites absorb different code delays when receiving and transmitting signals [36]. Thus, the performance of onboard real-time POD needs to consider the balance between computational efficiency, in-orbit processor resources, and accuracy. Usually, the errors appear more frequently in space-borne measurements than those from the ground, which are often discontinuous; a large number of outliers may occur. In this sense, the dynamic model and the estimation technique are generally applied for real-time POD to improve the reliability and stability of orbit solutions; for example, in [37], a Kalman Filter for Real-Time Precise Orbit Determination has been proposed. The limited capability of the onboard processor restricts the resolution of many measurements. Moreover, in the case of insufficient measurements or frequent loss of satellite tracking, estimating a large number of measurements together with the dynamic model parameters would also lead to the singularity of solutions. Considering the balance between the onboard processor’s capability and the accuracy of estimating unmeasured variables is essential.

It is thus assumed that the target satellite is the reference and moves in a circular orbit of radius ${\rho }_0$, ${\dot{\omega }}_0=0$, and ${\omega }_0=\sqrt{\mathfrak{g}/{\rho }_0^3}$, where $\mathfrak{g}$ is the standard gravity parameter. The relative dynamic of the chasing satellites regarding the target satellite is linearized with the following reference frame. A right-handed coordinate system is chosen, with the origin at the center of mass of the target satellite. The x-axis lies tangent to the orbit, the y-axis points along the position vector from the center of the Earth to the target satellite, and the z-axis is aligned with the orbital angular momentum vector, perpendicular to the plane of the target satellite’s orbit. The linearized equations of the relative dynamics of the target satellite are given by the Clohessy–Wiltshire (C-W) equations [38,39]:

$$\begin{array}{cc}\hfill u_x(t)& =\ddot{x}(t)-2{\omega }_0\dot{y}(t)\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill u_y(t)& =\ddot{y}(t)+2{\omega }_0\dot{x}(t)-3{\omega }_0^{2}y(t)\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill u_z(t)& =\ddot{z}(t)+{\omega }_0^{2}z(t)\hfill \end{array}$$

(49)

The position components of the target satellite are represented by $x(t)$, $y(t)$, and $z(t)$; $u_x(t)$, $u_y(t)$, $u_z(t)$ represent the control inputs and ${\omega }_0$ denotes the angular rate of the target satellite. Expressing the linear differential Equations (47)–(49) in state-space form is convenient [34]. The state vector $\zeta (t)={[{\overline{\zeta }}_1(t){\overline{\zeta }}_2(t)]}^T$ is defined, where ${\overline{\zeta }}_1(t)$ contains the position components ${\overline{\zeta }}_1(t)={[x(t)y(t)z(t)]}^T$, and ${\overline{\zeta }}_2(t)$ contains the velocity components ${\overline{\zeta }}_2(t)={[\dot{x}(t)\dot{y}(t)\dot{z}(t)]}^T$, and $u(t)$ = ${[u_x(t)u_y(t)u_z(t)]}^T$

$$\dot{\zeta }(t)=A\zeta (t)+Bu(t)$$

(50)

with

$$\begin{array}{cc}\hfill A=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 2{\omega }_0& 0\\ 0& 3{\omega }_0^2& 0& -2{\omega }_0& 0& 0\\ 0& 0& -{\omega }_0^2& 0& 0& 0\end{array}\right],& B=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\hfill \end{array}$$

(51)

where the angular velocity of the target satellite ${\omega }_0$ is fixed at $0.001$ rad/s. In addition, the control input is defined as

$$u(t)=[-F_1-F_2]\left[\begin{array}{c}{\overline{\zeta }}_1(t)-{\varphi }_1(t)\\ {\overline{\zeta }}_2(t)-{\varphi }_2(t)\end{array}\right].$$

(52)

where ${\varphi }_1{(t)=[sin(2\pi t),cos(2\pi t),sin(2\pi t)+8)]}^T$, and ${\varphi }_2(t)={\dot{\varphi }}_1(t)$ are the desired trajectory. The matrices $F_1$ and $F_2$ represent the feedback gain matrices and are defined as follows:

$$\begin{array}{c}{F}_1=\left[\begin{array}{ccc}27.0750& 0& 0\\ 0& 5.4150& 0\\ 0& 0& 3.6100\end{array}\right],F_2=\left[\begin{array}{ccc}10.4500& 0.0020& 0\\ -0.0020& 4.7500& 0\\ 0& 0& 4.7500\end{array}\right].\end{array}$$

(53)

Assume that the sensor measurements are ${\overline{y}}_1(t)=x(t_k^1-{\tau }_1(t_k^1))$, ${\overline{y}}_2(t)=y(t_k^2-{\tau }_2(t_k^2))$ and ${\overline{y}}_3(t)=z(t_k^3-{\tau }_3(t_k^3))$, which are sampled and delayed, in addition, acknowledging that velocities are typically unmeasurable; therefore, the sampled and delayed output is denoted as follows:

$$\overline{y}(t)={\left[{\overline{y}}_1(t){\overline{y}}_2(t){\overline{y}}_3(t)\right]}^T=C\zeta (t)$$

(54)

where

$$C=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right]=\left[\begin{array}{c}{C}_1\\ C_2\\ C_3\end{array}\right]$$

(55)

According to Equations (50) and (54), this system is in the form of Equations (1) and (3), respectively. Thus, the proposed approach’s effectiveness can be verified.

Now, the strongly connected digraph in Figure 2 shows that Observers 1 and 3 can obtain information from Observer 2, and Observer 2 can receive information from the two local observers.

The Laplacian matrix associated with this digraph is

$$\mathcal{L}=\left[\begin{array}{ccc}1& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right].$$

(56)

In order to calculate the gains $L_1$, $L_2$, $L_3$, $M_1$, $M_2$ and $M_3$, a series of preliminary steps must be carried out. These steps involve the calculation of values that will later be used in Equation (27). For a more detailed analysis, let $\mathcal{O}=obsv(A,C)$ and ${\mathcal{O}}_i=obsv(A,C_i)$ be the observability matrix of the pairs $(A,C)$ and $(A,C_i)$, respectively; with $rank({\mathcal{O}}_1)=4$, $rank({\mathcal{O}}_2)=3$, and $rank({\mathcal{O}}_3)=2$. It is evident that, for each local observer, the pair $(A,C_i)$ is unobservable. Nevertheless, the pair $(A,C)$ is observable because the observability matrix $\mathcal{O}$ has full rank $rank(\mathcal{O})=6$. Therefore, due to local unobservability, matrix transformations are necessary to address this constraint, considering the following orthogonal matrices $T_i$

$$T_1=\left[\begin{array}{cccccc}\hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\end{array}\right],T_2=\left[\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1\end{array}\right],T_3=\left[\begin{array}{cccccc}\hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$$

The new transformed matrices, according to Equation (23), ${\overline{A}}_i=T_i^{T}A{T}_i$ and ${\overline{C}}_i=C_i{T}_i$, are obtained

$$\begin{array}{cc}\hfill {\overline{A}}_1& =\left[\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0.0020& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill -0.0020& \hfill 0& \hfill 3\times {10}^{-6}& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1\times {10}^{-6}& \hfill 0\end{array}\right],\hfill \\ \hfill {\overline{A}}_2& =\left[\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 3\times {10}^{-6}& \hfill 0& \hfill 0.0020& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill -0.0020& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\times {10}^{-6}& \hfill 0\end{array}\right],\hfill \\ \hfill {\overline{A}}_3& =\left[\begin{array}{cccccc}\hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill -1\times {10}^{-6}& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0.0020\\ \hfill 0& \hfill 0& \hfill -3\times {10}^{-6}& \hfill 0& \hfill -0.0020& \hfill 0\end{array}\right]\hfill \end{array}$$

$${\overline{C}}_1=\left[\begin{array}{cccccc}\hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]={\overline{C}}_2={\overline{C}}_3$$

From which we obtain, $A_{io}$, $A_{iu}$,$A_{ir}$ and $C_{io}$, as shown in Equation (23). For the calculation of $\gamma$ in Equation (25), a value of $\mu =0.01$ is proposed, while $ϵ$ is determined using Equation (24), yielding $ϵ=0.0820$ and $\gamma =2.021$. These values are then used in Equation (26), from which the parameters $L_{io}$, $P_{io}$, and $P_{iu}$ are computed. Subsequently, these parameters are employed in Equation (27) to determine the gains $L_1$, $L_2$, $L_3$, $M_1$, $M_2$ and $M_3$. Then, the resulting local observer gain matrices are as follows:

$$L_1=\left[\begin{array}{c}0.3590\\ 433.8611\\ 0\\ 0.1400\\ 0.5591\\ 0\end{array}\right],L_2=\left[\begin{array}{c}0\\ 1.0007\\ 0\\ -11.2996\\ 1.5786\\ 0\end{array}\right],L_3=\left[\begin{array}{c}0\\ 0\\ 0.8952\\ 0\\ 0\\ 1.3059\end{array}\right]$$

$$\begin{array}{cc}\hfill M_1& =\left[\begin{array}{cccccc}\hfill 1.94\times {10}^{-8}& \hfill 3.69\times {10}^{-6}& \hfill 0& \hfill 1.15\times {10}^{-9}& \hfill 4.74\times {10}^{-9}& \hfill 0\\ \hfill 3.69\times {10}^{-6}& \hfill 0.0253& \hfill 0& \hfill 7.15\times {10}^{-6}& \hfill 3.22\times {10}^{-5}& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0.1189& \hfill 0& \hfill 0& \hfill 2.61\times {10}^{-4}\\ \hfill 1.15\times {10}^{-9}& \hfill 7.15\times {10}^{-6}& \hfill 0& \hfill 2.31\times {10}^{-9}& \hfill 9.22\times {10}^{-9}& \hfill 0\\ \hfill 4.74\times {10}^{-9}& \hfill 3.29\times {10}^{-5}& \hfill 0& \hfill 9.22\times {10}^{-9}& \hfill 4.14\times {10}^{-8}& \hfill 0\\ \hfill 0& \hfill 0& \hfill 2.61\times {10}^{-4}& \hfill 0& \hfill 0& \hfill 1.04\times {10}^{-6}\end{array}\right]\hfill \\ \hfill M_2& =\left[\begin{array}{cccccc}\hfill 0.2203& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 3.36\times {10}^{-8}& \hfill 0& \hfill -9.94\times {10}^{-8}& \hfill 1.38\times {10}^{-8}& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0.9395& \hfill 0& \hfill 0& \hfill 0.0166\\ \hfill 0& \hfill -9.94\times {10}^{-8}& \hfill 0& \hfill 2.86\times {10}^{-6}& \hfill -2.84\times {10}^{-7}& \hfill 0\\ \hfill 0& \hfill 1.38\times {10}^{-9}& \hfill 0& \hfill -2.84\times {10}^{-7}& \hfill 3.97\times {10}^{-8}& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0.0166& \hfill 0& \hfill 0& \hfill 3.57\times {10}^{-4}\end{array}\right]\hfill \\ \hfill M_3& =\left[\begin{array}{cccccc}\hfill 0.9394& \hfill 0.0012& \hfill 0& \hfill 0.0166& \hfill -9.04\times {10}^{-4}& \hfill 0\\ \hfill 0.0012& \hfill 0.9599& \hfill 0& \hfill 9.93\times {10}^{-4}& \hfill 0.0172& \hfill 0\\ \hfill 0& \hfill 0& \hfill 3.28\times {10}^{-8}& \hfill 0& \hfill 0& \hfill 1.12\times {10}^{-8}\\ \hfill 0.0166& \hfill 9.93\times {10}^{-4}& \hfill 0& \hfill 3.58\times {10}^{-4}& \hfill 1.48\times {10}^{-6}& \hfill 0\\ \hfill -9.04\times {10}^{-4}& \hfill 0.0172& \hfill 0& \hfill 1.48\times {10}^{-6}& \hfill 3.73\times {10}^{-4}& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1.12\times {10}^{-8}& \hfill 0& \hfill 0& \hfill 3.30\times {10}^{-8}\end{array}\right]\hfill \end{array}$$

Sensor measurements in real environments are sampled at different frequencies and occasionally experience delays. In this scenario, two sets of experiments were designed to demonstrate the effectiveness of the distributed observer with sampled output measurements and delays: the first considering small sampling intervals and delays and subsequently increasing the sampling time and the measurement delay time.

For the simulation tests, the initial condition of the observed system is configured to $\zeta (0)=3\times {10}^{-3}{[504030000]}^T$ and each local observer ${\widehat{\zeta }}_i,i=1:3$ starts with the initial states ${\widehat{\zeta }}_1(0)=3\times {10}^{-3}{[453627000]}^T$, ${\widehat{\zeta }}_2(0)=3\times {10}^{-3}{[1008060000]}^T$, ${\widehat{\zeta }}_3(0)=3\times {10}^{-3}{[-125-100-750000]}^T$. To assess the effectiveness of the proposed method under varying conditions of multirate sampled output measurements with variable delays, the parameters in

Table 1. Experimental configurations with sampled output measurements involving multiple delays.

Scenario	Sampling Intervals for Sampled Outputs Measurements [s]	Delayed Outputs for Sampled Outputs Measurements [s]
	${\mathrm{\Delta }}_1=0.0120$	${\tau }_1=0.0120$
Small	${\mathrm{\Delta }}_2=0.0240$	${\tau }_2=0.0144$
	${\mathrm{\Delta }}_3=0.0168$	${\tau }_3=0.0168$
	${\mathrm{\Delta }}_1=0.160$	${\tau }_1=0.140$
Large	${\mathrm{\Delta }}_2=0.320$	${\tau }_2=0.168$
	${\mathrm{\Delta }}_3=0.224$	${\tau }_3=0.196$

are considered. It is important to mention that delays considered for the test are constant-time delays; for this reason, they are denoted ${\tau }_i(t_k^i)={\tau }_i$.

These configurations aim to simulate scenarios with significant delays, which are typical in practical applications of sampled measurements. Each experimental configuration produced simulation graphs illustrating the behavior of the distributed observer with multirate sampled outputs involving multiple delays. The first series of simulations are shown in Figure 3, which presents the estimation of the state variables by each observer ${\widehat{\zeta }}_i^1(t)=\widehat{x}(t)$, ${\widehat{\zeta }}_i^2(t)=\widehat{y}(t)$ and ${\widehat{\zeta }}_i^3(t)=\widehat{z}(t)$, for $i=1,2,3$. Each state component is estimated by the $\mathcal{N}$ continuous distributed observers, considering that between outputs, there are different sampling periods ${\mathrm{\Delta }}_i$ and different delays ${\tau }_i$ at each output. The values considered for this test are in Table 1, in the scenario named Small.

Continuing with the experimentation, both the sampling periods and measurement delays were increased, with the values presented in Table 1 for the Large scenario. The results are shown in Figure 4. The results demonstrate that the distributed observers with sampled output measurements and multiple delays converge to the correct state values, even though individual observability conditions are not met, as previously evaluated. Additionally, the observer achieves asymptotic omniscience, successfully estimating the state values despite outputs being sampled at different rates and subject to multiple delays.

Comparison with a Traditional Distributed Observer

In order to highlight the originality of the distributed observer proposed in the presence of output measurements under different sampling periods, the strategy was compared to a traditional distributed observer approach based on a sampled-data mechanism, as presented in [6]. Both distributed observers were applied to the position estimation problem for a chasing satellite in LEO, with the comparison made across different sampling periods.

Two tests were conducted with different sampling periods. In the first test, relatively short sampling periods were considered, with the following values: ${\mathrm{\Delta }}_1=0.5$s, ${\mathrm{\Delta }}_2=1$s, and ${\mathrm{\Delta }}_3=1.25$s. The results are shown in Figure 5 for the distributed observer based on the traditional approach [6] and in Figure 6 for the proposed distributed observer using sampled output measurements. Comparing these figures reveals that the estimation in Figure 5 exhibits greater oscillations compared to Figure 6. The distributed observer based on the traditional approach (Figure 5) is more sensitive to variations in the sampled input signals, resulting in a less smooth response. In contrast, the observer proposed with sampled output measurements (Figure 6) provides a more stable and smooth estimation, indicating that the sampling strategy implemented in this case mitigates the fluctuations in the observer’s estimation.

In the second experiment, the sampling period was increased for both the distributed observer based on the traditional approach (Figure 7) and the proposed distributed observer with sampled output measurements (Figure 8). In this case, the relative longer sampling periods were ${\mathrm{\Delta }}_1=0.77$s, ${\mathrm{\Delta }}_2=1.54$s, and ${\mathrm{\Delta }}_3=1.925$s. In this case, it can be observed that the distributed observer with sampled output measurements (Figure 8) provides a consistent estimation of the full system state, maintaining a bounded observation error despite the delay and larger sampling periods. In contrast, the distributed observer based on the traditional approach shows indeterminate behavior under these conditions, struggling to provide reliable estimates with the increased sampling intervals.

5. Conclusions

This paper proposed a continuous distributed observer with multirate sampled outputs affected by multiple time-varying delays. The distributed observer uses its own information and communicates with its neighbors in continuous time. Despite the measurements from the sensors arriving at different sampling periods and with potential delays, the observer successfully converges omniscience asymptotically. This has been demonstrated both in the convergence analysis and in the numerical example, in which various sampling and delay values were tested, showing that the observer maintains convergence, where it has shown that each sensor node i of the network sensors achieved the convergence regardless of the nature of the time delay or sampling interval of each sensor node. In addition, the disturbance observer has shown that it handles large sampling periods by efficiently integrating information from multiple sensors while maintaining accurate state estimation. This was further proven through a comparison with results available in the literature. As a possible extension for future work, it is proposed to apply this approach to multi-agent control systems, addressing the management of time-varying and unknown communication delays.