Event-Triggered Disturbance Estimation and Output Feedback Control Design for Inner-Formation Systems

Abstract

This study investigates an event-triggered disturbance estimation approach for the inner-formation system. An extended state observer is designed using an event-based sampling scheme, which offers advantages over traditional estimation methods by reducing information transmission and unnecessary output information exchange while ensuring accurate system estimation performance. Additionally, a method for designing output-feedback control is proposed. The separation of feedback control and event-based observation in the design of output feedback allows us to apply existing optimal control algorithms to the targeted plant without compromising our established event-triggered sampling methods. A numerical simulation is presented, and we demonstrate the effectiveness of our proposed approach for the inner-formation system.

1. Introduction

The inner-formation technology, also known as drag-free control technology [1,2], enables the achievement of pure gravitational orbits crucial in space physics and other domains by effectively compensating for non-conservative forces acting on spacecraft, making it extensively utilized in space science experimental platforms [3,4,5,6,7,8,9,10,11,12,13]. This control method utilizes external actuation to counteract relative motion deviations between the external and internal spacecraft, thereby achieving compensation for interference from the external spacecraft. Due to its significant developmental potential, scholars have conducted extensive research on the inner-formation system [14,15,16,17,18] and drag-free spacecraft [19,20,21,22,23,24,25,26,27,28,29,30,31,32].

In the inner-formation control system, the research focus lies in investigating the dynamics of relative motion between internal and external spacecraft, followed by designing control strategies. However, when studying relative motion characteristics, it is often challenging to directly measure the velocity and acceleration of relative motion as well as external interference sources and uncertainties due to limited sensor capabilities. To address this issue, a modified Unscented Kalman Filter algorithm is employed for estimating the relative motion between the inner satellite and outer satellite in [14]. Additionally, [33] proposes an estimation method for the relative motion state and disturbance of test masses in drag-free satellites based on the self-recurrent wavelet neural network. Moreover, observers can be utilized to analyze the dynamic characteristics of the system. An extended state observer not only estimates the system state but also quantifies additional interference sources and uncertainties. The extended state observer was initially introduced in reference [34], and subsequently developed further with a design approach that incorporates high gain parameters for system gain configuration. Reference [35] provides a comprehensive explanation on designing high-gain extended state observers, which have gained significant attention due to their superior observation efficiency. For a two-test-mass inner-formation drag-free system, an observer-based Model Predictive Control (MPC) is proposed, demonstrating its feasibility for deep space exploration according to simulation results [36]. Furthermore, active disturbance rejection control incorporating an extended state observer is investigated for achieving frequency domain performance requirements in gravitational detection missions [37].

However, these estimation algorithms encounter a limitation in that they fail to consider the constraints imposed by system information transmission resources when utilizing system output information for state estimation. The system acquires such information through continuous or periodic sampling methods. By adopting event-driven estimation, this issue can be addressed and improved upon. Event-triggered method determines the sampling time of the system by establishing specific triggering conditions, thereby enabling high-frequency sampling during periods requiring more frequent measurements and low-frequency sampling during more stable stages, thus optimizing resource allocation for sampling [38]. Since its inception, event-based estimation has made significant advancements and found applications across diverse domains. In [39], the event-triggered method is considered for a state feedback controller with an input-to-state stability (ISS) result, and the minimum sampling interval is obtained by a Comparison Lemma [40]. The event-triggered method is introduced to spacecraft systems and underwater systems in [41] and [42], respectively; the event-based controller shows its advantage with non-periodic sampling. Additionally, the event-triggered approach is also introduced into estimation design, i.e., event-triggered observers for delay affine systems in [43], event-triggered cascade high-gain observers in [44], and event-triggered observers for one-side Lipschitz systems in [45].

Motivated by this approach, this article proposes an event-driven estimation and output feedback control method for the relative motion equation of internal formation formations. The advantages of the proposed algorithm are as follows: (1) Compared to traditional estimation methods, the process of event-triggered estimation can be more efficiently sampled to minimize the acquisition of unnecessary sampling information by implementing a well-designed triggering mechanism, which initiates sampling only when deemed necessary. (2) In the methodologies proposed in this paper, feedback control and event-triggered observation can be separated. It means that we can leverage existing feedback control algorithms by introducing our event-based extended state observers extensively. The design presented herein is partially similar to the separation principle in continuous design, which is also observed in other works in the literature on observer design, such as in [46]. The event-triggered control method proposed in this paper can theoretically replicate the convergent upper bound achievable through continuous sampling, thereby facilitating more effective utilization of system output information without compromising system performance. This holds significant research implications, as it offers potential improvements and enhancements to existing continuous output feedback control rates based on interference observation under this novel sampling mechanism.

The organizational structure of this article is as follows. Section 2 introduces the plant under consideration and formulates the problem that needs to be addressed. Building upon this, Section 3 presents the main results concerning event-triggered observer for inner-formation dynamics, followed by a proposal for an output feedback control based on the observer. In Section 4, a numerical example is provided to demonstrate the application of our method to a flying plant through numerical simulation. Finally, Section 5 concludes with key findings and outlines future research directions. Additionally, the proofs of the relevant theorems in this paper are provided in the Appendix A.

2. Problem Formulation

In this section, the dynamics we intend to investigate will be presented, while the primary issue that requires resolution is articulated herein. Firstly, let us consider the following relative motion dynamics of inner formation systems:

$${\ddot{r}}_{12}+2n\times {\dot{r}}_{12}+n\times \left(n\times r_{12}\right)+\dot{n}\times r=\frac{f_2}{m_2}-\frac{f_1}{m_1}+u$$

(1)

where SC1 and SC2 denote the outer spacecraft and inner spacecraft, respectively, with $r_{12}=\left[\begin{array}{ccc}{r}_1& r_2& r_3\end{array}\right]$ representing the vector from SC1 to SC2, $n=\left[\begin{array}{ccc}{n}_1& n_2& n_3\end{array}\right]$ representing the angular velocity of SC1, $f_1$ and $f_2$ representing external disturbances acting on SC1 and SC2, respectively, $m_1$ and $m_2$ corresponding to the masses of SC1 and SC2, respectively, and u representing the control input for the outer spacecraft. It can be observed that ${\ddot{r}}_{12}\left(t\right)$ comprises both external disturbances and uncertainties, which are collectively represented as $h_i\left(t\right)$:

$$\left\{\begin{array}{c}{h}_1\left(t\right)=\Delta a_1+2n_3{\dot{r}}_2-2n_2{\dot{r}}_3+{n_3}^2{r}_1+{n_2}^2{r}_1-n_2{n}_1{r}_2+{\dot{n}}_3{r}_2-n_3{n}_1{r}_3-{\dot{n}}_2{r}_3\\ h_2\left(t\right)=\Delta a_2-2n_3{\dot{r}}_1+2n_1{\dot{r}}_3-n_2{n}_1{r}_1-{\dot{n}}_3{r}_1+{n_3}^2{r}_2+{n_1}^2{r}_2-n_3{n}_2{r}_3+{\dot{n}}_1{r}_3\\ h_3\left(t\right)=\Delta a_3+2n_2{\dot{r}}_1-2n_1{\dot{r}}_2-n_3{n}_1{r}_1+{\dot{n}}_2{r}_1-n_3{n}_2{r}_2-{\dot{n}}_1{r}_2+{n_2}^2{r}_3+{n_1}^2{r}_3\end{array}\right.$$

(2)

where $\Delta a={\left[\begin{array}{ccc}\Delta a_1& \Delta a_2& \Delta a_3\end{array}\right]}^T$ represents the term $\frac{f_2}{m_2}-\frac{f_1}{m_1}$, which means the disturbance and uncertainty of the system. Then, the following assumption is proposed.

Assumption 1.

$h_i\left(t\right)$ ($i=1,2,3$) is derivable, and its derivative is bounded by some positive constant $M_i$, which means that $\frac{d|h_i\left(t\right)|}{dt}\le M_i$.

This assumption serves as a fundamental premise in the design of extended observers, enabling us to conduct differential calculations and estimations of the extended state. Building upon this assumption, we elaborate on the key issues that necessitate attention in this article:

• To propose an event-triggered extended observer for system (1), based on the output information obtained through event-based sampling, providing estimation for all states as well as external disturbances and uncertainties;
• To design an output feedback controller using the proposed event-triggered observer, ensuring practical applicability in the presence of bounded disturbances and uncertainties;
• To determine a minimum interval to ensure the absence of Zeno behavior under the proposed sampling scheme.

3. Main Results

This section states the main results in this work, the event-based estimation design is given, and a output feedback is proposed based on the observer.

3.1. Event-Triggered Disturbance Estimation for Relative Motion Dynamics

Considering the dynamics (1), ${\varsigma }_1\left(t\right)$, ${\varsigma }_2\left(t\right)$${\varsigma }_3\left(t\right)$ represent the estimation to $r_1$, $r_2$ and $r_3$ respectively, ${\psi }_1\left(t\right)$, ${\psi }_2\left(t\right)$${\psi }_3\left(t\right)$ represent the estimation to ${\dot{r}}_1$, ${\dot{r}}_2$ and ${\dot{r}}_3$ respectively, and ${\varphi }_1\left(t\right)$, ${\varphi }_2\left(t\right)$${\varphi }_3\left(t\right)$ represent the estimation to ${\ddot{r}}_1$, ${\ddot{r}}_2$ and ${\ddot{r}}_3$, respectively. Since the event-triggered extended observer is proposed as:

$$\begin{array}{ccc}\hfill & & {\dot{\xi }}_i\left(t\right)={\psi }_i\left(t\right)+{\ell }_i{\beta }_{i1}\left(r_i\left(t_k\right)-{\xi }_i\left(t\right)\right)\hfill \\ & & {\dot{\psi }}_i\left(t\right)={\varphi }_i\left(t\right)+{\ell }_i^2{\beta }_{i2}\left(r_i\left(t_k\right)-{\xi }_i\left(t\right)\right)+u_i\left(t\right)\hfill \\ & & {\dot{\varphi }}_i\left(t\right)={\ell }_i^3{\beta }_{i3}\left(r_i\left(t_k\right)-{\xi }_i\left(t\right)\right),i=1,2,3\hfill \end{array}$$

(3)

for any $t\in [t_k,t_{k+1})$, where $u_i\left(t\right)$ represent the control input for each axis, $K_i\left(\beta \right)={\left[\begin{array}{ccc}{\beta }_{i1}& {\beta }_{i2}& {\beta }_{i3}\end{array}\right]}^T$ represents the gain of the observer, ${\ell }_i$ is the high-gain parameter, $t_k$ ($k\in \mathbb{N}$) denotes the sampling moment, and the first sampling happens at the initial time $t_0$. The event-triggered mechanism selectively transmits output information by monitoring at the sensor end, allowing for the separate monitoring and transmission of three-axis output information. The transmission mechanism is designed to trigger sampling based on some positive constants ${\Pi }_1$, ${\Pi }_2$, and ${\Pi }_3$ by design:

$$\left|r_1\left(t\right)-r_1\left(t_k\right)\right|\le {\Pi }_1$$

(4)

$$\left|r_2\left(t\right)-r_2\left(t_k\right)\right|\le {\Pi }_2$$

(5)

$$\left|r_3\left(t\right)-r_3\left(t_k\right)\right|\le {\Pi }_3$$

(6)

The estimation errors are defined as $\tilde{\xi }\left(t\right)=r_i\left(t\right)-\xi \left(t\right)$, $\tilde{\psi }\left(t\right)={\dot{r}}_i\left(t\right)-\psi \left(t\right)$ and $\tilde{\varphi }\left(t\right)={\ddot{r}}_i\left(t\right)-\varphi \left(t\right)$. Then, the following theorem is obtained.

Theorem 1.

Let us consider plant (1), observer (14) and the driven conditions (4)–(6) if there exist some positive definite matrices $P_1$, $P_2$, $P_3$, and some constants $\theta >0$, $0<\sigma <1$ such that

$$0<∥P_i{K}_i\left(\beta \right)∥{\Pi }_i\le \frac{1}{{{\ell }_i}^3}\left(\frac{\lambda \sigma }{2}\sqrt{\frac{\underline{\varpi }}{\overline{\varpi }}}\theta -∥P_i∥M_i\right),i=1,2,3$$

(7)

where $\overline{\varpi }$ and $\underline{\varpi }$ represent the maximum and minimum eigenvalues among $P_1$, $P_2$, and $P_3$, respectively. And then, there exists some ${\beta }_{i1}$, ${\beta }_{i2}$ and ${\beta }_{i3}$ such that the estimation error will meet the bound

$$\left|{\tilde{\xi }}_i\left(t\right)\right|\le \frac{\theta }{{\ell }_i^3}$$

(8)

$$\left|{\tilde{\psi }}_i\left(t\right)\right|\le \frac{\theta }{{\ell }_i^2}$$

(9)

$$\left|{\tilde{\varphi }}_i\left(t\right)\right|\le \frac{\theta }{{\ell }_i}$$

(10)

for any $t\ge t_0+T(\ell )$, where $\ell =min\{{\ell }_1,{\ell }_2,{\ell }_3\}$, and $T(\ell )$ is some positive constant which satisfies $\underset{\ell \to \infty }{limsup}T(\ell )=0$.

Remark 1.

It is a common conclusion to obtain practical convergence in estimates under bounded disturbances. From the fact (A10), we show that the ultimate bound of $\varsigma \left(t\right)$

$$\underset{t\to \infty }{limsup}\left|{\tilde{\xi }}_i\left(t\right)\right|\le \frac{\theta }{{\ell }^3},1\le i\le 3$$

(11)

$$\underset{t\to \infty }{limsup}\left|{\tilde{\psi }}_i\left(t\right)\right|\le \frac{\theta }{{\ell }^2},1\le i\le 3$$

(12)

$$\underset{t\to \infty }{limsup}\left|{\tilde{\varphi }}_i\left(t\right)\right|\le \frac{\theta }{\ell },1\le i\le 3$$

(13)

with some constant θ, and it is obvious to obtain same results by continuous extended observer:

$$\begin{array}{ccc}\hfill & & {\dot{\xi }}_i\left(t\right)={\psi }_i\left(t\right)+{\ell }_i{\beta }_{i1}\left(r_i\left(t\right)-{\xi }_i\left(t\right)\right)\hfill \\ & & {\dot{\psi }}_i\left(t\right)={\varphi }_i\left(t\right)+{\ell }_i^2{\beta }_{i2}\left(r_i\left(t\right)-{\xi }_i\left(t\right)\right)+u_i\left(t\right)\hfill \\ & & {\dot{\varphi }}_i\left(t\right)={\ell }_i^3{\beta }_{i3}\left(r_i\left(t\right)-{\xi }_i\left(t\right)\right),i=1,2,3\hfill \end{array}$$

(14)

Remark 2.

From the fact (A10), we obtain the ultimate bound $\underset{\ell \to \infty }{limsup}∥{\tilde{\xi }}_i\left(t\right)∥=\underset{\ell \to \infty }{limsup}∥{\tilde{\psi }}_i\left(t\right)∥=\underset{\ell \to \infty }{limsup}∥{\tilde{\varphi }}_i\left(t\right)∥=0$ for any $t\ge t_0$, and this obtained result is in line with the design outcomes of the extended state observer for continuous time sampling. We can indicate that the system convergence upper bound can be enhanced by adjusting parameter ℓ. This advantage remains unchanged even with the introduction of event-triggered mechanisms.

3.2. Observer-Based Output Feedback Control with Event-Triggered Sampling

Considering the plant (1), our objective is to design an observer-based output-feedback control law using event-based sampling in this subsection. Our analysis approach aims to identify a broad range of output feedback control rates that satisfy the conditions and achieve practical convergence of the system under existing event-triggered estimation, while incorporating control design and event-driven separation. An output feedback control signal is provided by:

$${\widehat{u}}_i\left(t\right)={\gamma }_i\left({\varsigma }_i\right)$$

(15)

The function ${\gamma }_i$ presented here is a universally applicable output feedback expression that relies on the system state. Our original design intention was to leverage these states. As long as the control rate of Proposition 1 is satisfied, it can be seamlessly integrated with our event-driven approach to achieve control convergence. Further elucidation regarding the properties of ${\gamma }_i$ will be provided in subsequent context. Based on the active disturbance rejection control approach, the controller we use, in fact, is:

$$u_i\left(t\right)={\widehat{u}}_i\left(t\right)-{\varphi }_i\left(t\right)$$

(16)

The following dynamics is given:

$$\left[\begin{array}{c}{\dot{r}}_i\left(t\right)\\ {\ddot{r}}_i\left(t\right)\\ {\dot{\xi }}_i\left(t\right)\\ {\dot{\psi }}_i\left(t\right)\\ {\dot{\varphi }}_i\left(t\right)\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ {\ell }_i{\beta }_{i1}& 0& -{\ell }_i{\beta }_{i1}& 1& 0\\ {{\ell }_i}^2{\beta }_{i2}& 0& -{{\ell }_i}^2{\beta }_{i2}& 0& 1\\ {{\ell }_i}^3{\beta }_{i3}& 0& -{{\ell }_i}^3{\beta }_{i3}& 0& 0\end{array}\right]\left[\begin{array}{c}{r}_i\left(t\right)\\ {\dot{r}}_i\left(t\right)\\ {\xi }_i\left(t\right)\\ {\psi }_i\left(t\right)\\ {\varphi }_i\left(t\right)\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 1& 0& 1\\ 0& {\ell }_i{\beta }_{i1}& 0\\ 0& {{\ell }_i}^2{\beta }_{i2}& 1\\ 0& {{\ell }_i}^3{\beta }_{i3}& 0\end{array}\right]\left[\begin{array}{c}{h}_i\left(t\right)-{\varphi }_i\left(t\right)\\ r_i\left(t_k\right)-r_i\left(t\right)\\ {\widehat{u}}_i\left(t\right)\end{array}\right]$$

(17)

Here, we make the following transformation:

$$\left[\begin{array}{c}\frac{{\dot{r}}_i\left(t\right)}{{\ell }_i}\\ \frac{{\ddot{r}}_i\left(t\right)}{{{\ell }_i}^2}\\ \frac{{\dot{\xi }}_i\left(t\right)}{{\ell }_i}\\ \frac{{\dot{\psi }}_i\left(t\right)}{{{\ell }_i}^2}\\ \frac{{\dot{\varphi }}_i\left(t\right)}{{{\ell }_i}^3}\end{array}\right]=\left[\begin{array}{ccccc}0& {\ell }_i& 0& 0& 0\\ 0& 0& 0& 0& 0\\ {\ell }_i{\beta }_{i1}& 0& -{\ell }_i{\beta }_{i1}& {\ell }_i& 0\\ {\ell }_i{\beta }_{i2}& 0& -{\ell }_i{\beta }_{i2}& 0& {\ell }_i\\ {\ell }_i{\beta }_{i3}& 0& -{\ell }_i{\beta }_{i3}& 0& 0\end{array}\right]\left[\begin{array}{c}\frac{r_i\left(t\right)}{{\ell }_i}\\ \frac{{\dot{r}}_i\left(t\right)}{{{\ell }_i}^2}\\ \frac{{\xi }_i\left(t\right)}{{\ell }_i}\\ \frac{{\psi }_i\left(t\right)}{{{\ell }_i}^2}\\ \frac{{\varphi }_i\left(t\right)}{{{\ell }_i}^3}\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ \frac{1}{{{\ell }_i}^2}& 0& \frac{1}{{{\ell }_i}^2}\\ 0& {\beta }_{i1}& 0\\ 0& {\beta }_{i2}& \frac{1}{{{\ell }_i}^2}\\ 0& {\beta }_{i3}& 0\end{array}\right]\left[\begin{array}{c}{h}_i\left(t\right)-{\varphi }_i\left(t\right)\\ r_i\left(t_k\right)-r_i\left(t\right)\\ {\widehat{u}}_i\left(t\right)\end{array}\right]$$

(18)

For the sake of convenience, we define a new state ${\mathsf{\Theta }}_i={\left[\begin{array}{ccccc}\frac{r_i}{{\ell }_i}& \frac{{\dot{r}}_i}{{{\ell }_i}^2}& \frac{{\xi }_i}{{\ell }_i}& \frac{{\psi }_i}{{{\ell }_i}^2}& \frac{{\varphi }_i}{{{\ell }_i}^3}\end{array}\right]}^T$ and a new function $F\left({\mathsf{\Theta }}_i\right)$:

$$F\left({\mathsf{\Theta }}_i\right)=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ {\beta }_{i1}& 0& -{\beta }_{i1}& 1& 0\\ {\beta }_{i2}& 0& -{\beta }_{i2}& 0& 1\\ {\beta }_{i3}& 0& -{\beta }_{i3}& 0& 0\end{array}\right]{\mathsf{\Theta }}_i\left(t\right)+\left[\begin{array}{c}0\\ \frac{1}{{{\ell }_i}^3}\\ 0\\ \frac{1}{{{\ell }_i}^3}\\ 0\end{array}\right]{\widehat{u}}_i\left(t\right)$$

(19)

The introduced controller (15) satisfies the following proposition.

Proposition 1.

There exists a Lyapunov function $W\left({\mathsf{\Theta }}_i\right)$ which satisfies the following conditions:

$$a_1{∥{\mathsf{\Theta }}_i∥}^2\le W\left({\mathsf{\Theta }}_i\right)\le a_2{∥{\mathsf{\Theta }}_i∥}^2$$

(20)

$$\frac{\partial W}{\partial {\mathsf{\Theta }}_i}F\left({\mathsf{\Theta }}_i\right)\le -a_3{∥{\mathsf{\Theta }}_i∥}^2$$

(21)

$$∥\frac{\partial W}{\partial {\mathsf{\Theta }}_i}∥\le a_4∥{\mathsf{\Theta }}_i∥$$

(22)

where $a_1$, $a_2$, $a_3$, and $a_4$ are some positive constants.

It is evident that the output feedback control algorithm (15), which satisfies Proposition (1), can achieve exponential convergence of the system without considering estimation errors associated with interference and event-driven introduction. Although the specific form of the output feedback algorithm employed here remains undefined, various optimization control methods can ultimately be applied. The analysis approach adopted in this study draws inspiration from the continuous-time high-gain observer analysis approach on the separation principle [46]. With bounded interference introduced, the observer does not impose any additional changes to the calculation of control feedback gain for the system. Furthermore, incorporating an event-triggered mechanism does not introduce any extra design burden to the system.

Theorem 2.

Let us consider that plant (1) observer-based output feedback control law (16), states $r\left(t\right)$, and $\dot{r}\left(t\right)$ will be practically convergent; this means that the states of (1) meet a certain limit bound by the controller (16).

Theorem 2 proposes the performance of the output-feedback control law, and the proof is presented in the appendix.

Remark 3.

The determination of a convergence interval is an inherent outcome in the design of output feedback control under bounded disturbances, and this outcome remains valid even with the incorporation of event-triggered estimation design.

Remark 4.

It is obvious that $\underset{{\ell }_i\to \infty }{limsup}∥e^{-\frac{a_3{\ell }_i}{2}\left(t-t_0\right)}∥=\underset{{\ell }_i\to \infty }{limsup}∥e^{-\frac{\lambda \left(1-\sigma \right){\ell }_i}{2\overline{\varpi }}\left(t-t_0\right)}∥=0$ holds on, since that we get $\underset{{\ell }_i\to \infty }{limsup}∥W_i\left(t\right)∥=0$ from (A19). It means that we obtain a ultimate boundedness:

$$\underset{{\ell }_i\to \infty }{limsup}∥{\mathsf{\Theta }}_i\left(t\right)∥=0$$

(23)

Remark 5.

Here, we propose a general condition for designing output feedback control based on the aforementioned event-triggered extended observer that can be applied to the system. Specifically, after extracting disturbances and event-driven effects, the estimated state information is utilized to design output feedback control, satisfying conditions (20)–(22), thereby ensuring the practicality and convergence of the system. The employed output feedback control method can be a commonly used design approach, and in subsequent design examples, we will also present a simplified design criterion.

Remark 6.

Decoupling the system is a commonly employed technique in spacecraft control system design. In this study, we propose an observer-based approach to address this issue, which effectively mitigates decoupling-induced errors by estimating and compensating for them using the extended state. The design parameters of each axis can be adjusted based on the specific control requirements of that axis. This article presents a general conclusion that convergence can be achieved by satisfying these requirements.

3.3. Zeno Behavior Analysis

In the preceding content, we presented evidence of convergence in the observation and control within the system design. Subsequently, we will conduct an analysis on Zeno behavior. The primary objective of event-driven sampling is to ensure a specific time interval between consecutive samples, thereby preventing the occurrence of Zeno behavior. Subsequently, we will delve into an analysis of this crucial aspect. Consequently, the ensuing theorem is presented. The proof of this theorem is presented in the appendix.

Theorem 3.

Let us consider the event-based estimation (14) for system (1); there will exist a minimum interval between every two samples with driven conditions (4)–(6).

Remark 7.

Theorem 3 establishes the existence of a minimum sampling interval within our proposed driving condition, ensuring that sampling events do not occur continuously. This absence of Zeno behavior is a fundamental prerequisite for event-triggered design, and the event-based design is only meaningful under this premise. The introduction of this requirement did not impose any additional parameter constraints, as they were already ensured by the previous design.

4. Numerical Examples

In this section, we present the practical implementation of the algorithm based on the aforementioned design, demonstrated through a specific numerical simulation example. A inner-formation background with specific parameters is given, and its dynamics is with the form of (1). The simulation parameters are selected as follows: the angular velocity vector of the out-spacecraft, denoted as $n^T=\left[\begin{array}{ccc}2& 5& 3\end{array}\right]\times {10}^{-3}$ rad/s, the mass of the out-spacecraft $m_{out}=250$ kg, and the mass of the in-spacecraft $m_{in}=1.5$ kg. Here, $m_{out}$ and $m_{in}$ correspond to $m_1$ and $m_2$ in (1), respectively. The external disturbance is considered $\Delta a={\left[\begin{array}{ccc}0.001sin(2t+\frac{\pi }{4})& -0.003sin(t+\frac{\pi }{8})& 0.002sin(4t+\frac{\pi }{3})\end{array}\right]}^{T}N$. Considering the distinct operational scenarios encountered by each axis in practical spacecraft systems, diverse axis parameters were selected herein, and the disturbances experienced during simulation of each axis also vary. Consequently, these disparities may give rise to variations in the simulation specifics of the three-axis system due to the influence of nonlinear dynamics.

Based on these conditions, we propose a parameter turning in detail here. The parameter of the event-triggered observer is considered to be ${\ell }_1={\ell }_2={\ell }_3=6$, $K_i\left(\beta \right)={\left[\begin{array}{ccc}1.2252& 2.2267& 0.7633\end{array}\right]}^T$.

Remark 8.

Here, $K_i\left(\beta \right)$ is obtained by condition (A4). The calculation method can be obtained through feature root configuration methods or appropriate LMI (Linear Matrix Inequality) calculation methods. In this paper, we present an LMI-based approach for calculating the method. Specifically, we consider the following LMI conditions: $Q_1^{T}P+P{Q}_1-Q_2^T{R}^T-R{Q}_2<0$, where $Q_1=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$, $Q_2=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$, P is a positive definite matrix, and R is a matrix of appropriate order. Calculating P and R through the LMI condition, we obtain $K_i\left(\beta \right)=P^{-1}R$.

And a normal output feedback control is considered here. We use

$$u_1\left(t\right)=-\frac{1}{50}\left[\begin{array}{ccc}5.2& 5& -2\end{array}\right]\left[\begin{array}{ccc}{\xi }_1& {\psi }_1& {\varphi }_1\end{array}\right]$$

$$u_2\left(t\right)=-\frac{1}{50}\left[\begin{array}{ccc}2.5& 3.5& 0\end{array}\right]\left[\begin{array}{ccc}{\xi }_1& {\psi }_1& {\varphi }_1\end{array}\right]$$

$$u_3\left(t\right)=-\frac{1}{50}\left[\begin{array}{ccc}6& 5& -2\end{array}\right]\left[\begin{array}{ccc}{\xi }_1& {\psi }_1& {\varphi }_1\end{array}\right]$$

It is obvious to find that the controllers meet the conditions (20)–(22).

The control results are depicted in Figure 1. As evident from the figure, employing an event-driven sampling design, the system converges to a specific range under the output feedback control rate.

The transmission of sampling information is illustrated in Figure 2. It can be observed that during the initial stages of the system, when additional data are required to support control instructions, sampling occurs at a high frequency. However, once the system achieves stability, there is a significant decrease in the sampling rate, thereby indicating the rationality of this resource allocation design.

The estimation results of the three axes in Figure 3, Figure 4 and Figure 5 demonstrate convergence patterns consistent with the theoretical findings. By ensuring accurate estimations of displacement and velocity, the system disturbances and uncertainties are also estimated within a certain range.

5. Conclusions

The present work addresses an event-based disturbance estimation and output feedback control for inner-formation systems. In this study, we employ an event-triggered extended state observer to estimate the states of the inner-formation system, along with a corresponding output feedback control. By utilizing event-based sampling and avoiding the collection of invalid output information, our designed method achieves bounded estimation errors for the system state in terms of performance. Furthermore, we design output feedback control rates based on the proposed observer, ensuring a separation between the controller and observer that converges to an exact interval under the influence of the control law. We conduct simulations and analyses under specific numerical backgrounds that conform to relative motion dynamics in an inner formation context, demonstrating the successful application of our design algorithm.

However, it is important to note that certain conservative aspects still exist in these algorithms, such as high sampling frequency during early stages and noise during the sampling process; addressing these issues will be a focus for future solutions. As this study represents a preliminary theoretical investigation into inner-formation systems, our proposed event-triggered control method can theoretically reproduce the convergent upper bound achievable by continuous sampling without affecting system performance, thus offering significant research significance. This implies that existing continuous output feedback control rates based on interference observation can be improved and enhanced under this new sampling mechanism; detailed discussions regarding its application in engineering practice will be presented in future research.