Time-Varying Formation Tracking Control of Linear Multi-Agent Systems with Magnitude and Rate Saturation and Unknown Disturbances

Abstract

In this paper, we study the leader-following time-varying formation (TVF) tracking control of general linear multi-agent systems (MASs) with nonzero control input of the leader, and the followers which have magnitude and rate saturation (MRS) and unknown disturbances. Under the assumption that only the followers connecting to the leader have access to the leader’s input and state, an output feedback controller incorporating a distributed extended state observer (ESO) is developed to ensure the asymptotic convergence of the formation errors without input saturation. Then, a saturation model is inserted to each follower’s dynamics to constrain the magnitude and rate of the control input, with consideration of MRS. Anti-windup protection loops are applied to compensate for the saturated signals to improve the closed-loop performance. Finally, the theoretical findings are demonstrated via a series of numerical simulations.

1. Introduction

Over the past few decades, the cooperative control of multi-agent systems (MASs) has received widespread attention and is increasingly being applied in diverse fields including intelligent transportation, maritime patrol, aerospace, and smart factories [1,2,3,4,5]. Its objective is to guide all agents in achieving and maintaining a desired configuration according to task requirements. In Ref. [6], time-varying formation (TVF) tracking for second-order systems is addressed by using the relative information of the neighboring agents under switching topologies. In Ref. [7], a distributed controller is designed for high-order MASs to achieve the desired formation. In Ref. [8], the analysis and design of formation–containment control are investigated for high-order linear time-invariant MASs with time delays under directed interaction topologies. Reference [9] introduces a novel consensus control strategy for MASs comprised of electromechanical actuators. In Ref. [10], a general TVF tracking is proposed for general linear MASs with time delays. Under switching directed topology, the TVF of general linear MASs is solved in [11]. In Ref. [12], TVF with multiple leaders is discussed for general linear MASs. The TVF tracking problem is investigated for multi-leader MASs under sampled-data control in [13]. In Ref. [14], the output TVF tracking problem of heterogeneous MASs under model uncertainties and external disturbances is explored by an adaptive event-triggered mechanism. In Ref. [15], the TVF problem is studied for general linear MASs with unknown tracking leader inputs. The adaptive TVF containment tracking problem is discussed based on state observers for general linear MASs with unknown and bounded leaders’ inputs in [16]. For more recent works on TVF, please see refs. [17,18,19,20] and the references therein.

In practical implementations, MASs inevitably face external disturbances, which can weaken their ability to achieve predetermined configurations. One way to deal with the disturbances or uncertainties in solving formation control is to combine the consensus theory of MASs and the traditional robust control theory of single systems. Considerable efforts have extended robust control methods from single systems to MASs for consensus control. In Ref. [21], adaptive robust TVF tracking control for high-order linear MASs with uncertainties and external disturbances is investigated, where the leader is subject to a nonzero control input. In Ref. [22], the robust $H_2$ consensus problem of linear MASs is solved with parametric uncertainties and noise signals. In Ref. [23], a distributed $L_2$ robust control strategy is proposed for heterogeneous MASs subject to uncertainties and disturbances to achieve TVF under static and switching topologies. The robust TVF tracking for modified high-order linear MASs with heterogeneous uncertainties, external disturbances and unknown leader input is studied in Ref. [24]. The TVF $H_{\infty }$ control problem of heterogeneous MASs subject to uncertainties and external disturbances under static and switching topologies is studied by the internal model-based robust formation controllers in [25]. In Ref. [26], the robust $H_{\infty }$ output TVF problem for heterogeneous nonlinear MASs is investigated, where the actuator faults and external disturbances are considered. The TVF for general linear MASs with external disturbances and time delays is addressed in [27], where the influence of the disturbances is minimized by optimizing the $H_{\infty }$ performance index. Another method is based on an extended state observer (ESO), where the disturbances or uncertainties are treated as additional states and compensated in the control design process. Compared with the robust control method, the ESO-based control shows excellent performance in estimating the disturbances or uncertainties. In Ref. [28], the leader-following tracking problem for general linear MASs under unknown disturbances is addressed using a distributed ESO. In Ref. [29], the formation control problem is studied for second-order nonlinear MASs with ESO. In Ref. [30], an ESO-based distributed model predictive control method is proposed for multi-robot formation under unknown disturbances. In Ref. [31], practical TVF tracking for high-order nonlinear MASs is studied using a distributed disturbance observer under directed topologies, ensuring arbitrarily small tracking errors. In Ref. [32], a robust TVF problem for second-order MASs subjected to external disturbances is investigated by ESO. In Ref. [33], the TVF is solved for second-order nonlinear MASs subject to uncertainties by event-triggered fuzzy ESO. Finite-time formation control for multi-UAVs with actuator faults and unmeasurable states is achieved using an ESO that estimates uncertainties, disturbances, and faults in [34]. A prescribed-time ESO is developed to compensate for external disturbances and internal unknown dynamics for vehicles to solve formation control strategy in [35]. Based on the aforementioned articles, few works have considered the TVF tracking control of high-order linear MASs with unknown disturbances and nonzero leader inputs by using ESO, which is one of the motivations of this study.

Actuator saturation arising from physical constraints and safety requirements may also degrade system performance or destabilize the system. In Ref. [36], a saturated formation tracking controller is designed to address the three-dimensional formation tracking control of multiple AUVs subject to input saturation. In Ref. [37], the adaptive TVF for general linear MASs subject to actuator failure and input saturation is investigated by using the hyperbolic tangent function in the actuator model. The distributed adaptive control schemes are proposed to achieve TVF for general linear MASs with input saturation in [38,39] via the low-gain feedback approach, where multiple leaders are considered in [39]. In Ref. [40], the adaptive TVF of general linear MASs with input saturation and unknown leader inputs is further studied. In Ref. [41], practical output TVF tracking control for nonlinear strict-feedback MASs with input saturation and unknown leader inputs is addressed based on a distributed ESO. In Ref. [42], the TVF control for multiple quadrotors under input saturation and unknown disturbances is studied via an ESO-based approach.

Actuator magnitude saturation is often accompanied with rate saturation, such as limitations on the position or speed of mechanical actuators. For single systems, the anti-windup(AW) active disturbance rejection control (ADRC) is designed for uncertain nonlinear systems with MRS in [43,44]. In Ref. [45], an observer-based controller for linear heterogeneous leader–follower systems with position and rate saturation is designed to address the output consensus problem. An AW compensator is designed to address the leader–follower consensus control for general linear MASs with MRS in [46]. The TVF tracking problem for underactuated surface vehicles with MRS in actuators is investigated by using a nonlinear extended observer in [47]. In Ref. [48], a minimal learning parameters-based adaptive control is designed for dynamic positioning to deal with the input MRS, unknown disturbance and dynamic uncertainties. To the best of the authors’ knowledge, the TVF for high-order MASs with unknown disturbances and MRS has not been well addressed, which motivates us to study this problem.

Motivated by the aforementioned analysis, this paper investigates the TVF tracking control of general linear MASs with nonzero leader inputs, where the followers are subject to unknown disturbances and MRS in the control inputs. First, a distributed estimator is designed to estimate the leader’s state and control input, and an ESO including a disturbance observer is employed to estimate the system state and unknown disturbance of each follower. A distributed formation controller is designed incorporating the estimates, which ensures that the formation tracking errors asymptotically converge. Second, a MRS model with protection loops is inserted into each follower’s dynamics such that the original system is transformed into a linear system with magnitude-only saturation. The formation tracking error is minimized by formulating an optimization problem when MRS is encountered. Finally, the control gains, observer gains and the anti-windup gains are derived in the form of linear matrix inequalities (LMIs).

The rest of this paper is organized as follows. Section 2 presents the preliminaries and problem formulation. Section 3 details the controller design and provides the main stability theorem with its proof. Section 4 addresses the controller design under saturation constraints and presents the corresponding stability analysis. Section 5 validates the effectiveness of the proposed method through numerical simulations. Finally, Section 6 concludes the paper.

$Notations$. Let $I_n$ denote the $n\times n$ identity matrix, and ${\mathbf{1}}_N\in R^n$ represent a column vector with each entry being one. For a symmetric matrix P, $P>0$ and $P<0$ indicate that P is positive definite and negative definite, respectively. $A^T$ denotes the transpose of A and $\mathrm{H}e\left\{A\right\}=A+A^T$. The Kronecker product is expressed as ⊗. $\mathrm{diag}\{A_1,\dots ,A_n\}$ represents a block diagonal matrix with matrices $A_i$ being the diagonal elements. $R^n$ denotes the n-dimensional Euclidean space. For a vector $x\in R^n$, the norm of x is defined as $\parallel x\parallel =\sqrt{x^{T}x}$. The space of square integrable functions is denoted by $L_2$, that is, for any $x\left(t\right)\in L_2$, ${\int }_0^{\infty }{x}^T\left(t\right)x\left(t\right)dt<\infty$. For $x\left(t\right)\in L_2$, ${\parallel x\left(t\right)\parallel }_2={\left({\int }_0^{\infty }{x}^T\left(t\right)x\left(t\right)dt\right)}^{1/2}$. The notation ‘∗’ in symmetric matrices represents the off-diagonal terms.

2. Preliminaries and Problem Description

2.1. Graph Theory Basics

For a group of $N+1$ agents, a directed graph $\mathcal{G}$ consists of a pair ($\mathcal{V}$, $\mathcal{E}$), where $\mathcal{V}=\{v_0,v_1,\dots ,v_N\}$ is a nonempty set of nodes, and $\mathcal{E}\subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges whose element is denoted by $e_{ij}=(v_i,v_j)$. Node $v_0$ denotes the leader, and nodes $i=1,\dots ,N$ denote the followers. We consider that $e_{ij}=(v_i,v_j)\in \mathcal{E}$ if and only if node $v_i$ can send its information to node $v_j$. The leader cannot receive information from any followers. The set of neighbors of node $v_i$ is denoted by ${\mathcal{N}}_i=\{v_j\mid (v_j,v_i)\in \mathcal{E}\}$. Denote the adjacency matrix as $\mathcal{A}=\left(a_{ij}\right)$ with $a_{ij}>0$ if $(v_j,v_i)\in \mathcal{E}$, else $a_{ij}=0$ and $a_{ii}=0$. $D=\mathrm{diag}\left\{d_i\right\}$ is called the in-degree matrix, where $d_i={\sum }_{j=0}^N{a}_{ij}$ is the weighted in-degree of node $v_i$, and the Laplacian matrix is defined as $\mathcal{L}=\mathcal{D}-\mathcal{A}$.

2.2. Problem Description

Consider an MAS consisting of $N+1$ linear dynamic equations, where node $v_0$ represents the leader, and node $v_i,i=1,2,\dots ,N$ represent the followers. The leader’s dynamic model is given by

$$\left\{\begin{array}{c}{\dot{x}}_0\left(t\right)=A{x}_0\left(t\right)+B{u}_0\left(t\right)\hfill \\ y_0\left(t\right)=C{x}_0\left(t\right)\hfill \end{array}\right.$$

(1)

where $x_0\left(t\right)\in R^n$ is the leader state vector, $u_0\left(t\right)\in R^q$ is the leader’s control input, and $y_0\left(t\right)\in R^r$ is the leader output. Note that the nonzero control input $u_0\left(t\right)$ generates more reference signals to guide the followers along the desired trajectory, such as to avoid obstacles.

The dynamics of follower i is described by

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A{x}_i\left(t\right)+B{u}_{pi}\left(t\right)+B{w}_i\left(t\right)\hfill \\ y_i\left(t\right)=C{x}_i\left(t\right),i=1,\dots ,N\hfill \end{array}\right.$$

(2)

where $x_i\left(t\right)\in R^n$ is the state vector, $w_i\left(t\right)\in R^q$ is the unknown disturbance, $u_{pi}\left(t\right)\in R^q$ is the control input, and $y_i\left(t\right)\in R^r$ is the output for the ith follower.

A distributed output feedback controller is designed for follower tracking and formation achievement. In order to specify the expected TVF for the followers, a vector $f\left(t\right)={\left[\begin{array}{c}{f}_1^T\left(t\right), f_2^T\left(t\right),\dots , f_N^T\left(t\right)\end{array}\right]}^T\in R^{Nn}$ is introduced, where $f_i\left(t\right)$ is the offset vector with respect to the TVF reference.

Assumption 1.

Vectors $f_i\left(t\right)$, $i=1,\dots ,N$ are piecewise continuously differentiable, and $f_i\left(t\right)$ and its derivative ${\dot{f}}_i\left(t\right)$ for each $i=1,\dots ,N$ are bounded.

Note that the leader state $x_0\left(t\right)$ can be treated as a reference trajectory of each follower for the leader-following consensus problem, and $x_0\left(t\right)+f_i\left(t\right)$ as the trajectory for TVF tracking control. To avoid the reference trajectory diverging, we assume that the leader’s control input $u_0\left(t\right)$ leads to a stable closed-loop system, and that the offset vector $f_i\left(t\right)$ and ${\dot{f}}_i\left(t\right)$ are bounded. Specifically, Assumption 1 is naturally satisfied for the time-invariant formation control due to ${\dot{f}}_i\left(t\right)\equiv 0$.

Definition 1.

Consider the MASs (1) and (2) on a directed graph $\mathcal{G}$. For any bounded initial states, if the following holds

$$\underset{t\to \infty }{lim}(x_i\left(t\right)-f_i\left(t\right)-x_0\left(t\right))=0,i=1,\dots ,N,$$

(3)

then the MASs (1) and (2) achieve TVF tracking.

We assume the followers connecting to the leader have access to its state and control input signal, and make the following mild assumptions regarding the communication topology and system matrices.

Assumption 2.

Graph $\mathcal{G}$ contains a directed spanning tree with the leader as the root.

Under Assumption 2, the Laplacian matrix associated with $\mathcal{G}$ can be partitioned as $\mathcal{L}=\left[\begin{array}{cc}0& 0_{1\times N}\\ {\mathcal{L}}_2& {\mathcal{L}}_1\end{array}\right]$, with ${\mathcal{L}}_1\in R^{N\times N}$ and ${\mathcal{L}}_2\in R^{N\times 1}$.

Lemma 1

([5]). All eigenvalues of ${\mathcal{L}}_1$ have positive real parts, and each term of $-{\mathcal{L}}_1^{-1}{\mathcal{L}}_2$ is non-negative, and the sum of the rows of $-{\mathcal{L}}_1^{-1}{\mathcal{L}}_2$ equals to $1_N$.

Assumption 3.

The external disturbance of each follower $w_i\left(t\right)$ and the control input of the leader $u_0\left(t\right)$ satisfy:

(i) 

${\dot{w}}_i\left(t\right)=h_i\left(t\right)$, $\underset{t\to \infty }{lim}{h}_i\left(t\right)=0$, and $\underset{t\to \infty }{lim}{w}_i\left(t\right)=w_0$, where $w_0$ is an unknown constant vector;

(ii) 

${\dot{u}}_0\left(t\right)=h_0\left(t\right)$, $\underset{t\to \infty }{lim}{h}_0\left(t\right)=0$, and $\underset{t\to \infty }{lim}{u}_0\left(t\right)=r_0$, where $r_0$ is an unknown constant vector;

(iii) 

$w_i\left(t\right)$, $h_i\left(t\right)$, $u_0\left(t\right)$ and $h_0\left(t\right)$ are bounded for any $t\ge 0$, and the compact forms $w=[w_1,\dots ,w_N]$, $h=[h_1,\dots ,h_N]$, ${\overline{u}}_0=1_N\otimes u_0$ and ${\overline{h}}_0=1_N\otimes h_0$ satisfy $w^{T}w\le {\overline{w}}^2$, ${\overline{u}}_0^T{\overline{u}}_0\le {\overline{w}}^2$, $h^{T}h\le {\overline{h}}^2$, and ${\overline{h}}_0^T{\overline{h}}_0\le {\overline{h}}^2$, for some constants $\overline{w}>0$ and $\overline{h}>0$.

Assumption 4.

The triple $(A,B,C)$ is controllable and observable, and satisfies

$$\mathit{rank}\left[\begin{array}{cc}C& 0\\ A& B\end{array}\right]=n+q,\mathrm{rank}\left(B\right)=q.$$

In light of $\mathrm{rank}\left(B\right)=q$, there exists a nonsingular matrix $\mathrm{\Gamma }={\left[{\widehat{B}}^T{\tilde{B}}^T\right]}^T$ with $\widehat{B}\in R^{q\times n}$, $\tilde{B}\in R^{(n-q)\times n}$ such that $\widehat{B}B=I_q$ and $\tilde{B}B=0$ from [21].

We consider that the control input of each follower $u_{pi}$ is subject to both MASs, i.e., for each $i=1,\dots ,N$.

$$\begin{array}{c}\hfill \left|u_{pi,\left(k\right)}\right|\le m,\left|{\dot{u}}_{pi,\left(k\right)}\right|\le r,k=1,\dots ,q\end{array}$$

(4)

where $m>0$ and $r>0$ are the actuator MRS limits, respectively. We employ the anti-windup approach to the MRS impact on the closed-loop system, i.e.,

(i)

Within the closed-loop control system under constraints, design a distributed dynamic output feedback controller to guarantee that (3) holds;

(ii)

For a given magnitude bound $m>0$ and rate bound $r>0$, ensure the boundedness of the TVF error and minimize by an AW compensation.

3. Distributed Extended State Observer-Based Formation Control

In the first step, we intend to design an observer-based distributed controller for the unconstrained agents. Here, we take the unknown disturbance $w_i\left(t\right)$ as an extended state; thus, the dynamics of the follower in (2) can be written in an augmented state space form

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A{x}_i\left(t\right)+B{u}_{pi}\left(t\right)+B{w}_i\left(t\right)\hfill \\ {\dot{w}}_i\left(t\right)=h_i\left(t\right)\hfill \\ y_i\left(t\right)=C{x}_i\left(t\right),i=1,\dots ,N\hfill \end{array}\right.$$

(5)

Inspired by the works in [3,46], we designing the following distributed estimators for each follower to respectively estimate the leader’s state and input.

$$\left\{\begin{array}{c}{\dot{r}}_i=-\mu ({\displaystyle \sum _{j=1}^N}{a}_{ij}(r_i-r_j)+a_{i0}(r_i-u_0))\hfill \\ \dot{{\eta }_i}=A{\eta }_i-\beta ({\displaystyle \sum _{j=1}^N}{a}_{ij}({\eta }_i-{\eta }_j)+a_{i0}({\eta }_i-x_0))+B{r}_i\hfill \end{array}\right.$$

(6)

where $r_i\in R^q$, ${\eta }_i\in R^n$, and $\beta >0$, $\mu >0$ are the coupling gains to be tuned.

Also, we design an observer to estimate the augmented state ${[x_i^T,w_i^T]}^T$ of each follower i

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_i=A{\widehat{x}}_i+B{u}_i+B{\widehat{w}}_i+L_1(y_i-{\widehat{y}}_i)\hfill \\ {\dot{\widehat{w}}}_i=L_2(y_i-{\widehat{y}}_i)\hfill \end{array}\right.$$

(7)

where ${\widehat{y}}_i=C{\widehat{x}}_i$ is the observer output, $L_1$ and $L_2$ are observer gains with appropriate dimensions to be determined.

With the help of (6) and (7), the controller output $u_i$ is then designed as

$$u_i=K_1({\widehat{x}}_i-{\eta }_i-f_i)-{\widehat{w}}_i+r_i+z_i$$

(8)

where $K_1$, with an appropriate dimension, is the control gain, $z_i$ is the TVF tracking compensation signal given by $z_i=-\widehat{B}A{f}_i+\widehat{B}{\dot{f}}_i$.

Combining systems (1), (5) and the controller (6)–(8), the closed-loop dynamics of follower i with $u_{pi}=u_i$ is obtained as follows.

$$\left\{\begin{array}{c}{\dot{\zeta }}_i=(A+B{K}_1){\zeta }_i-B{K}_1(e_{xi}+e_{\eta i})+B{e}_{wi}+B{e}_{ri}+d_i\hfill \\ {\dot{e}}_{xi}=(A-L_{1}C)e_{xi}+B{e}_{wi}\hfill \\ {\dot{e}}_{wi}=-L_{2}C{e}_{xi}+h_i\hfill \\ {\dot{e}}_{\eta i}=A{e}_{\eta i}-\beta ({\displaystyle \sum _{j=1}^N}{a}_{ij}(e_{\eta i}-e_{\eta j})+a_{i0}{e}_{\eta i})+B{e}_{ri}\hfill \\ {\dot{e}}_{ri}=-\mu ({\displaystyle \sum _{j=1}^N}{a}_{ij}(e_{ri}-e_{rj})+a_{i0}{e}_{ri})-h_0\hfill \end{array}\right.$$

(9)

where the closed-loop variables are defined by ${\zeta }_i=x_i-x_0-f_i$, $e_{xi}=x_i-{\widehat{x}}_i$, $e_{wi}=w_i-{\widehat{w}}_i$, $e_{ri}=r_i-u_0$, $e_{\eta i}={\eta }_i-x_0$, $e_{si}={[e_{xi}^T,e_{wi}^T]}^T$, and $d_i=B{z}_i+A{f}_i-{\dot{f}}_i$.

Letting the stack form variables as $\zeta ={[{\zeta }_1^T,\dots ,{\zeta }_N^T]}^T$, $e_x={[e_{x1}^T,\dots ,e_{xN}^T]}^T$, $e_w={[e_{w1}^T,\dots ,e_{wN}^T]}^T$, $e_{\eta }={[e_{\eta 1},\dots ,e_{\eta N}]}^T$, $e_r={[e_{r1}^T,\dots ,e_{rN}^T]}^T$, $e_s={[e_x^T,e_w^T]}^T$, and $e_o={[e_{\eta }^T,e_r^T]}^T$, we can obtain the global closed-loop system as follows

$$\left\{\begin{array}{c}\dot{\zeta }={\overline{A}}_{\zeta }\zeta +{\overline{B}}_s{e}_s+{\overline{B}}_o{e}_o+d\hfill \\ {\dot{e}}_s={\overline{A}}_{es}{e}_s+{\overline{E}}_{1}h\hfill \\ {\dot{e}}_o={\overline{A}}_{eo}{e}_o+{\overline{E}}_2{h}_0\hfill \end{array}\right.$$

(10)

where $d=(I_N\otimes B)z+(I_N\otimes A)f-(I_N\otimes I_n)\dot{f}$, $h={[h_1^T,\dots ,h_N^T]}^T$, ${\overline{h}}_0=1_N\otimes h_0$, $z={[z_1^T,\dots ,z_N^T]}^T$, and the system matrices are

$${\overline{A}}_{\zeta }=I_N\otimes (A+B{K}_1),{\overline{A}}_{es}=\left[\begin{array}{cc}{I}_N\otimes (A-L_{1}C)& I_N\otimes B\\ -I_N\otimes L_{2}C& 0\end{array}\right],{\overline{A}}_{eo}=\left[\begin{array}{cc}{I}_N\otimes A-\beta {\mathcal{L}}_1\otimes I_n& I_N\otimes B\\ 0& -\mu {\mathcal{L}}_1\otimes I_n\end{array}\right]$$

$${\overline{B}}_s=\left[\begin{array}{cc}-I_N\times B{K}_1& I_N\otimes B\end{array}\right],{\overline{B}}_o=\left[\begin{array}{cc}{I}_N\otimes B& -I_N\otimes B{K}_1\end{array}\right],{\overline{E}}_1=\left[\begin{array}{c}0\\ I\end{array}\right]{\overline{E}}_2=\left[\begin{array}{c}0\\ -I\end{array}\right].$$

From system (10), it is obvious that the TVF tracking is achieved if $\zeta \left(t\right)\to 0$ as time $t\to \infty$. Next, we give the sufficient condition to guarantee that system (10) is asymptotically stable. We first introduce two indispensable lemmas below.

Lemma 2

([28]). $(\overline{A},\overline{C})$ is observable if and only if $(A,C)$ is observable and Assumption 4 is satisfied, where $\overline{A}=\left[\begin{array}{cc}A& B\\ 0& 0\end{array}\right]$, $\overline{C}=\left[\begin{array}{cc}C& 0\end{array}\right]$.

Lemma 3

([28]). Consider a linear system $\dot{x}\left(t\right)=Ax\left(t\right)+Bu\left(t\right)$. If A is a Hurwitz matrix, $u\left(t\right)$ is bounded, and $\underset{t\to \infty }{lim}u\left(t\right)=0$, then $\underset{t\to \infty }{lim}x\left(t\right)=0$.

Theorem 1.

Consider the closed-loop MASs (10). Under Assumptions 1–4, if there exist positive definite matrices $P_1\in R^{n\times n}$, $P_2\in R^{(n+q)\times (n+q)}$, $Q_3\in R^{n\times n}$, matrices G, F with suitable dimensions, a positive scalar $\mu >0$, and $\beta >1/\mathrm{Re}\left\{{\lambda }_1\right\}$, such that the subsequent matrix inequality is satisfied:

$$\begin{array}{cc}\hfill & A^T{P}_1+P_{1}A-2P_1<0\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {\overline{A}}^T{P}_2-\overline{C}{G}^T+P_2\overline{A}-G\overline{C}<0\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & Q_3{A}^T+F^T{B}^T+A{Q}_3+BF<0\hfill \end{array}$$

(13)

and the TVF tracking feasibility condition is satisfied

$$\underset{t\to \infty }{lim}\tilde{B}A{f}_i\left(t\right)-\tilde{B}{\dot{f}}_i\left(t\right)=0$$

(14)

then the system is asymptotically stable, and the control gain and observer gain can be obtained by $K_1=F{Q}_3^{-1}$ and $L=\left[\begin{array}{c}{L}_1\\ L_2\end{array}\right]=P_2^{-1}G$.

Proof. 

First, we prove that the estimation error $e_o\left(t\right)$ goes to zero as $t\to \infty$, under condition (11). Assumption 3 implies that ${\overline{h}}_0\to 0$ as $t\to \infty$. From Lemma 3, it can be obtained that $e_o\left(t\right)\to 0$ as long as the matrix ${\overline{A}}_{eo}$ is Hurwitz. Now choose a positive definite matrix $P_1\in R^{n\times n}$, $P_1>0$ and construct ${\mathcal{P}}_1=I_N\otimes P_1$; then we have

$${\mathcal{P}}_1{\overline{A}}_{eo}+{\overline{A}}_{eo}^T{\mathcal{P}}_1=I_N\otimes (P_{1}A+A^T{P}_1)-\beta ({\mathcal{L}}_1+{\mathcal{L}}_1^T)\otimes P_1$$

Let $U\in R^{N\times N}$ be a unitary matrix that satisfies $U^T{\mathcal{L}}_{1}U=\mathrm{\Lambda }=\mathrm{diag}\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_N\}$. From Lemma 1, we have that all the eigenvalues of ${\mathcal{L}}_1$ have positive real parts, and $\mathrm{Re}\{{\lambda }_1\}\le \mathrm{Re}\{{\lambda }_2\}\le \cdots \le \mathrm{Re}\{{\lambda }_N\}$ without loss of generality. Pre- and post-multiplying $U^T\otimes I_n$ and its transpose, we obtain that (11) guarantees that ${\mathcal{P}}_1{\overline{A}}_{eo}+{\overline{A}}_{eo}^T{\mathcal{P}}_1<0$, by setting $\beta >1/\mathrm{Re}\left\{{\lambda }_1\right\}$, which implies ${\overline{A}}_{eo}$ is the Hurwitz.

Next, we prove that (12) guarantees that the estimation error $e_s\left(t\right)$ asymptotically converges. Assumption 3 yields that $h\to 0$. Using Lemma 3, we can obtain that $e_s\left(t\right)\to 0$ as time $t\to \infty$ as long as ${\overline{A}}_{es}$ is Hurwitz. Choose a positive definite matrix $P_2\in R^{(n+q)\times (n+q)}$, $P_2=\left[\begin{array}{cc}{P}_2^{11}& P_2^{12}\\ P_2^{12}& P_2^{22}\end{array}\right]$ and construct ${\mathcal{P}}_2=\left[\begin{array}{cc}{I}_N\otimes P_2^{11}& I_N\otimes P_2^{12}\\ I_N\otimes P_2^{12}& I_N\otimes P_2^{22}\end{array}\right]$. Then, we have

$${\mathcal{P}}_2{\overline{A}}_{es}+{\overline{A}}_{es}^T{\mathcal{P}}_2=\mathrm{He}\left[\begin{array}{cc}{I}_N\otimes (P_2^{11}(A-L_{1}C)-P_2^{12}{L}_{2}C)& I_N\otimes P_2^{11}B\\ I_N\otimes (P_2^{12}(A-L_{1}C)-P_2^{22}{L}_{2}C)& I_N\otimes P_2^{12}B\end{array}\right]$$

After congruence transformations, we obtain that ${\mathcal{P}}_2{\overline{A}}_{es}+{\overline{A}}_{es}^T{\mathcal{P}}_2<0$ is equivalent to the following inequality

$$I_N\otimes \mathrm{He}\left[\begin{array}{cc}(P_2^{11}(A-L_{1}C)-P_2^{12}{L}_{2}C)& P_2^{11}B\\ (P_1^{12}(A-L_{1}C)-P_2^{22}{L}_{2}C)& P_2^{12}B\end{array}\right]=I_N\otimes (P_2(\overline{A}-L\overline{C})+{(\overline{A}-L\overline{C})}^T{P}_2)<0$$

which can be guaranteed by (12) by setting $G=P_{2}L$. Note that Lemma 2 ensures the feasibility of (12).

Last but not the least we prove that the TVF tracking error $\zeta \left(t\right)$ asymptotically converges. The TVF tracking feasibility condition (14) and the property that $\tilde{B}B=0$ can guarantee that $\underset{t\to \infty }{lim}\tilde{B}A{f}_i\left(t\right)-\tilde{B}{\dot{f}}_i\left(t\right)+\tilde{B}B{z}_i\left(t\right)=0.$

The TVF tracking compensation signal $z_i$ is set as $z_i=-\widehat{B}A{f}_i+\widehat{B}{\dot{f}}_i$; it can then be obtained from the property $\widehat{B}B=I$ that $\widehat{B}A{f}_i\left(t\right)-\widehat{B}{\dot{f}}_i\left(t\right)+\widehat{B}B{z}_i\left(t\right)=0.$ Due to the fact that $\mathrm{\Gamma }={\left[{\widehat{B}}^T{\tilde{B}}^T\right]}^T$ and $\mathrm{\Gamma }$ is nonsingular, we have that

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}\mathrm{\Gamma }(B{z}_i\left(t\right)+A{f}_i\left(t\right)-{\dot{f}}_i\left(t\right))=\underset{t\to \infty }{lim}\left[\begin{array}{c}\widehat{B}A{f}_i\left(t\right)-\widehat{B}{\dot{f}}_i\left(t\right)+\widehat{B}B{z}_i\left(t\right)\\ \tilde{B}A{f}_i\left(t\right)-\tilde{B}{\dot{f}}_i\left(t\right)+\tilde{B}B{z}_i\left(t\right)\end{array}\right]=0\end{array}$$

(15)

and then

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}B{z}_i\left(t\right)+A{f}_i\left(t\right)-{\dot{f}}_i\left(t\right)=0\end{array}$$

(16)

that is, $\underset{t\to \infty }{lim}d\left(t\right)=0$. Choose a positive definite matrix ${\mathcal{P}}_3=I_N\otimes P_3$, where $P_3\in R^{n\times n}$ and $P_3>0$. Then, the inequality ${\mathcal{P}}_3{\overline{A}}_{\zeta }+{\overline{A}}_{\zeta }^T{\mathcal{P}}_3<0$ can be guaranteed by $\mathrm{He}\{P_{3}A+P_{3}A{K}_1\}<0$, which is equivalent to (13) by setting $Q_3=P_3^{-1}$ and $F=K_1{Q}_3$. Thus it can be confirmed that the TVF tracking error $\zeta \left(t\right)\to 0$ by Lemma 3. This completes the proof. □

Remark 1.

Under Assumption 3 that the external disturbances $w_i\left(t\right)$ and the control input of the leader $u_0\left(t\right)$ have steady constant values, the estimates ${\widehat{w}}_i$ and $r_i$ are introduced into $u_i\left(t\right)$ in (8) to eliminate the influence, which guarantees that the tracking errors asymptotically converge. In Ref. [28], a similar assumption is given for the unknown external disturbances, whereas an autonomous leader agent is considered. For more general systems with time-varying disturbances, the robust control method can be applied to minimize the influence of the disturbances—see Ref. [46] for an example.

4. Anti-Windup Formation Control with Magnitude and Rate Saturation

To address actuator saturation, an AW compensator based on (7) and (8) is proposed. Regional conditions are introduced to handle open-loop unstable systems, while global results apply to open-loop stable ones.

To satisfy the input constraint given by (4), the dynamics of the follower in (5) is reformulated to the following, which is also shown in Figure 1.

$$\left\{\begin{array}{c}{\dot{x}}_i\left(t\right)=A{x}_i\left(t\right)+B{\mathrm{sat}}_m\left({\delta }_i\left(t\right)\right)+B{w}_i\left(t\right)\hfill \\ {\dot{w}}_i\left(t\right)=h_i\left(t\right)\hfill \\ \dot{{\delta }_i}\left(t\right)={\mathrm{sat}}_r(K(u_i\left(t\right)-{\delta }_i\left(t\right))+{\Delta }_r{q}_{ri}\left(t\right))\hfill \\ y_i\left(t\right)=C{x}_i\left(t\right)\hfill \end{array}\right.$$

(17)

where $q_{ri}$ is the rate deadzone signal defined by $q_{ri}={\eta }_{ci}-{\mathrm{sat}}_r\left({\eta }_{ci}\right)$, ${\Delta }_r$ is the AW gain to be determined, and ${\mathrm{sat}}_m\left({\delta }_i\right)$ and ${\mathrm{sat}}_r\left({\delta }_i\right)$ are MRS functions.

Remark 2.

Due to the limitations on the MRS of the control input $u_{pi}$, the controller output $u_i$ fails to compensate the unknown disturbances $w_i$’s, the TVF signals $f_i$’s and the unknown leader’s input $u_0$ effectively. The loss of control effect is mitigated by adding AW signals $v_{ri}:={\Delta }_r{q}_{ri}\left(t\right)$ to (17), and $v_{mi}:=\left[v_{i1}{v}_{i2}{v}_{i3}\right]$ to the observer-based controller in (18). Moreover, the TVF tracking error due to control loss can be minimized by formulating an optimization problem in Theorem 2.

Then, the observer (7) and controller (8) are augmented with static AW loops

$$\left\{\begin{array}{c}{\dot{\widehat{x}}}_i=A{\widehat{x}}_i+B{\mathrm{sat}}_m\left({\delta }_i\right)+B{\widehat{w}}_i+L_1(y_i-{\widehat{y}}_i)+v_{1i}\hfill \\ {\dot{\widehat{w}}}_i=L_2(y_i-{\widehat{y}}_i)+v_{2i}\hfill \\ u_i=K_1({\widehat{x}}_i-{\eta }_i-f_i)-{\widehat{w}}_i+z_i+r_i+v_{3i}\hfill \end{array}\right.$$

(18)

The additive terms $v_{1i}$, $v_{2i}$ and $v_{3i}$ are introduced as $v_{1i}=\left[I_{n}00\right]{\Delta }_m{q}_{mi}$, $v_{2i}=\left[0I_{q}0\right]{\Delta }_m{q}_{mi}$, and $v_{3i}=\left[00I_q\right]{\Delta }_m{q}_{mi}$, respectively, where $q_{mi}$ is the deadzone signal, defined by $q_{mi}={\delta }_i-{\mathrm{sat}}_m\left({\delta }_i\right)$, and ${\Delta }_m$ is the AW gain to be determined.

As shown in Figure 1, we have

$$K(u_i-{\delta }_i)+{\Delta }_r{q}_{ri}={\eta }_{ci}$$

(19)

With the variables defined in (9), the closed-loop system under saturated input $u_{pi}={\mathrm{sat}}_m\left({\delta }_i\right)$ is given by combining (6), (17) and (18)

$$\left\{\begin{array}{c}{\dot{\zeta }}_i=A{\zeta }_i+B{\delta }_i+B{w}_i-B{u}_0-B{q}_{mi}+{\tilde{f}}_i\hfill \\ \dot{{\delta }_i}=K{K}_1({\zeta }_i-e_{xi}-e_{{\eta }_i})-K{\delta }_i-K{e}_{wi}+K{e}_{ri}-K{w}_i+K{u}_0-K\widehat{B}{\tilde{f}}_i+K\left[00I_q\right]{\Delta }_m{q}_{mi}+{\Delta }_r{q}_{ri}-q_{ri}\hfill \\ {\dot{e}}_{xi}=(A-L_{1}C)e_{xi}+B{e}_{wi}-\left[I_{n}00\right]{\Delta }_m{q}_{mi}\hfill \\ {\dot{e}}_{wi}=-L_{2}C{e}_{xi}-\left[0I_{q}0\right]{\Delta }_m{q}_{mi}\hfill \\ {\dot{e}}_{{\eta }_i}=A{e}_{{\eta }_i}-\beta ({\displaystyle \sum _{j=1}^N}{a}_{ij}(e_{{\eta }_i}-e_{{\eta }_i})+a_{i0}{e}_{{\eta }_i}+B{e}_{ri}\hfill \\ {\dot{e}}_{ri}=-\mu ({\displaystyle \sum _{j=1}^N}{a}_{ij}(e_{ri}-e_{rj})+a_{i0}{e}_{ri})-h_0\hfill \end{array}\right.$$

(20)

where we set ${\tilde{f}}_i:=A{f}_i-{\dot{f}}_i$.

Building on AW methods for single systems, we propose a network-level AW design. We define the augmented system state corresponding to agent i as $x_{cl,i}={\left[{\zeta }_i^T{\delta }_i^T{e}_{xi}^T{e}_{wi}^T{e}_{{\eta }_i}^T{e}_{ri}^T\right]}^T$, and the compact form vectors as $\delta ={[{\delta }_1^T\dots {\delta }_N^T]}^T$, ${\eta }_c={[{\eta }_{c1}^T\dots {\eta }_{cN}^T]}^T$, $\tilde{f}={[f_1^T\dots f_N^T]}^T$, $q_m={[q_{m1}^T\dots q_{mN}^T]}^T$, $q_r={[q_{r1}^T\dots q_{rN}^T]}^T$, and $x_{cl}={\left[{\zeta }^T{\delta }^T{e}_x^T{e}_w^T{e}_{\eta }^T{e}_r^T\right]}^T$. In the following, we take $\tilde{w}:={\left[w^T{\tilde{f}}^T{\overline{u}}_0^T{h}^T{\overline{h}}_0^T\right]}^T$ as an augmented global disturbance, which is bounded from Assumption 1 and Assumption 3, and satisfies

$$\begin{array}{c}\hfill {\tilde{w}}^T\tilde{w}\le w_{max}^2\end{array}$$

(21)

where $w_{max}>0$ is a predetermined constant. Then, the global TVF tracking error system is

$$\left\{\begin{array}{c}{\dot{x}}_{cl}={\overline{A}}_{cl}{x}_{cl}+{\overline{B}}_{x\tilde{w}}\tilde{w}+({\overline{B}}_r+{\overline{B}}_{\delta r}{\overline{\Delta }}_r)q_r+({\overline{B}}_m+{\overline{B}}_{\delta m}{\overline{\Delta }}_m)q_m\hfill \\ u={\overline{K}}_{cl}{x}_{cl}+{\overline{B}}_{u\tilde{w}}\tilde{w}+(I_N\otimes \left[00I_q\right]){\overline{\Delta }}_m{q}_m\hfill \\ \delta ={\overline{C}}_m{x}_{cl}\hfill \\ {\eta }_c={\overline{C}}_r{x}_{cl}+{\overline{B}}_{\eta \tilde{w}}\tilde{w}+(I_N\otimes K\left[00I_q\right]){\overline{\Delta }}_m{q}_m+{\overline{\Delta }}_r{q}_r\hfill \\ \zeta ={\overline{C}}_z{x}_{cl}\hfill \end{array}\right.$$

(22)

where ${\overline{\Delta }}_m=I_N\otimes {\Delta }_m$, ${\overline{\Delta }}_r=I_N\otimes {\Delta }_r$, and system matrices are given in (23) on the next page. For later convenience, we also list system matrices associated with agent i.

$$\begin{array}{c}\hfill \begin{array}{c}{\overline{A}}_{cl}=\left[\begin{array}{cccccc}{I}_N\otimes A& I_N\otimes B& 0& 0& 0& 0\\ I_N\otimes K{K}_1& I_N\otimes K& -I_N\otimes K{K}_1& -I_N\otimes K& -I_N\otimes K{K}_1& I_N\otimes K\\ 0& 0& I_N\otimes (A-L_{1}C)& B& 0& 0\\ 0& 0& -I_N\otimes L_{2}C& 0& 0& 0\\ 0& 0& 0& 0& (A-\beta {\mathcal{L}}_1)\otimes I_n& I_N\otimes B\\ 0& 0& 0& 0& 0& -u{\mathcal{L}}_1\otimes I_n\end{array}\right]\hfill \\ {\overline{B}}_{x\tilde{w}}=\left[\begin{array}{ccccc}{I}_N\otimes B& I_N\otimes I_n& -I_N\otimes B& 0& 0\\ -I_N\otimes K& -I_N\otimes K\widehat{B}& I_N\otimes K& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_N\otimes I_n& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -I_N\otimes I_n\end{array}\right]\hfill \\ \left(\begin{array}{cccc}{\overline{B}}_m& {\overline{B}}_{\delta m}& {\overline{B}}_r& {\overline{B}}_{\delta r}\end{array}\right)=\left(\begin{array}{cccc}\left[\begin{array}{c}-I_N\otimes B\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]& \left[\begin{array}{c}0\\ I_N\otimes K\left[00I_q\right]\\ -I_N\otimes \left[I_{n}00\right]\\ I_N\otimes \left[0I_{n}0\right]\\ 0\\ 0\end{array}\right]& \left[\begin{array}{c}0\\ -I_N\otimes I_n\\ 0\\ 0\\ 0\\ 0\end{array}\right]& \left[\begin{array}{c}0\\ I_N\otimes I_n\\ 0\\ 0\\ 0\\ 0\end{array}\right]\end{array}\right)\hfill \\ \left(\begin{array}{c}{\overline{K}}_{cl}\\ {\overline{C}}_m\\ {\overline{C}}_r\\ {\overline{C}}_z\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{cccccc}{I}_N\otimes K_1& 0& -I_N\otimes K_1& -I_N\otimes I_n& -I_N\otimes K_1& I_N\otimes I_n\end{array}\right]\\ \left[\begin{array}{cccccc}0& I_N\otimes I_n& 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccccc}{I}_N\otimes K{K}_1& -I_N\otimes K& -I_N\otimes K{K}_1& K& -I_N\otimes K{K}_1& I_N\otimes K\end{array}\right]\\ \left[\begin{array}{cccccc}{I}_N\otimes I_n& 0& 0& 0& 0& 0\end{array}\right]\end{array}\right)\hfill \\ \left(\begin{array}{c}{\overline{B}}_{u\tilde{w}}\\ {\overline{B}}_{\eta \tilde{w}}\end{array}\right)=\left(\begin{array}{c}\left[\begin{array}{ccccc}-I_N\otimes I_n& -I_N\otimes \widehat{B}& -I_N\otimes I_n& 0& 0\end{array}\right]\\ \left[\begin{array}{ccccc}-I_N\otimes K& -I_N\otimes K\widehat{B}& I_N\otimes K& 0& 0\end{array}\right]\end{array}\right)\hfill \\ A_{cl,i}=\left[\begin{array}{cccccc}A& B& 0& 0& 0& 0\\ K{K}_1& K& -K{K}_1& -K& -K{K}_1& K\\ 0& 0& A-L_{1}C& B& 0& 0\\ 0& 0& -L_{2}C& 0& 0& 0\\ 0& 0& 0& 0& A-\beta {\lambda }_i{I}_n& B\\ 0& 0& 0& 0& 0& -\mu {\lambda }_i{I}_n\end{array}\right]\hfill \\ B_{x\tilde{w}}=\left[\begin{array}{ccccc}B& I_n& -B& 0& 0\\ -K& -K\widehat{B}& K& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& I_n& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -I_n\end{array}\right]\hfill \\ \left(\begin{array}{cc}{K}_{cl}& B_{u\tilde{w}}\\ C_m& \\ C_r& B_{\eta \tilde{w}}\\ C_z& \end{array}\right)=\left(\begin{array}{cc}\left[\begin{array}{cccccc}{K}_1& 0& -K_1& -I_n& -K_1& I_n\end{array}\right]& \left[\begin{array}{ccccc}-I_n& -\widehat{B}& -I_n& 0& 0\end{array}\right]\\ \left[\begin{array}{cccccc}0& I_n& 0& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccccc}K{K}_1& -K& -K{K}_1& K& -K{K}_1& K\end{array}\right]& \left[\begin{array}{ccccc}-K& -K\widehat{B}& K& 0& 0\end{array}\right]\\ \left[\begin{array}{cccccc}0& I_n& 0& 0& 0& 0\end{array}\right]\end{array}\right)\hfill \\ \left(\begin{array}{cccc}{B}_m& B_{\delta m}& B_r& B_{\delta r}\end{array}\right)=\left(\begin{array}{cccc}\left[\begin{array}{c}-B\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right]& \left[\begin{array}{c}0\\ K\left[00I_q\right]\\ -\left[I_{n}00\right]\\ \left[0I_{n}0\right]\\ 0\\ 0\end{array}\right]& \left[\begin{array}{c}0\\ -I_n\\ 0\\ 0\\ 0\\ 0\end{array}\right]& \left[\begin{array}{c}0\\ I_n\\ 0\\ 0\\ 0\\ 0\end{array}\right]\end{array}\right)\hfill \end{array}\end{array}$$

(23)

Following Ref. [49], the sector property for deadzone signals $q_{mi}$ and $q_{ri}$ of an agent is recalled.

Lemma 4

([49]). Given any matrices $H_m\in R^{q\times (3n+3q)}$ and $H_r\in R^{q\times (3n+3q)}$, consider the deadzone signals $q_{ri}$ and $q_{mi}$ in Figure 1. If $x_{cl,i}\in R^{3n+3q}$ satisfies $|H_{m\left(k\right)}{x}_{cl,i}|\le m$ and $|H_{r\left(k\right)}{x}_{cl,i}|\le r$, $k=1,\dots ,q$, the following inequalities hold:

$$\begin{array}{c}\hfill q_{mi}^T{M}_m(q_{mi}-{\delta }_i+H_m{x}_{cl,i})\le 0,q_{ri}^T{M}_r(q_{ri}-{\eta }_{ci}+H_r{x}_{cl,i})\le 0\end{array}$$

(24)

for $i=1,\dots ,N$ and any diagonal matrices $M_m>0$, $M_m\in R^{q\times q}$, $M_r>0$, $M_r\in R^{q\times q}$, where $H_{m\left(k\right)}$ and $H_{r\left(k\right)}$ represent the kth row of $H_m$ and $H_r$, respectively.

The following gives the network version of Lemma 4, which is indispensible for the proof of Theorem 2.

Lemma 5.

For any matrices $H_m\in R^{q\times (3n+3q)}$ and $H_r\in R^{q\times (3n+3q)}$, partitioned with six block matrices as $H_m=\left[H_{m1}{H}_{m2}{H}_{m3}{H}_{m4}{H}_{m5}{H}_{m6}\right]$ and $H_r=\left[H_{r1}{H}_{r2}{H}_{r3}{H}_{r4}{H}_{r5}{H}_{r6}\right]$, we define ${\overline{H}}_m=[I_N\otimes H_{m1}{I}_N\otimes H_{m2}{I}_N\otimes H_{m3}{I}_N\otimes H_{m4}{I}_N\otimes H_{m5}{I}_N\otimes H_{m6}]$ and ${\overline{H}}_r=[I_N\otimes H_{r1}{I}_N\otimes H_{r2}{I}_N\otimes H_{r3}{I}_N\otimes H_{r4}{I}_N\otimes H_{r5}{I}_N\otimes H_{r6}]$. If $x_{cl}\in R^{N(3n+3q)}$ satisfies $x_{cl}\in \mathcal{H}$, where

$$\begin{array}{c}\hfill \mathcal{H}:=\left\{\begin{array}{c}{x}_{cl}\in R^{N(3n+3q)};\left|{\overline{H}}_{m\left(p\right)}{x}_{cl}\right|\le m,\left|{\overline{H}}_{r\left(p\right)}{x}_{cl}\right|\le r\hfill \end{array}\right\},\end{array}$$

(25)

where $p=1,\dots ,Nq$, then we have

$$\begin{array}{c}\hfill q_m^T{\overline{M}}_m(q_m-\delta +H_m{x}_{cl})\le 0,q_r^T{\overline{M}}_r(q_r-{\eta }_c+H_r{x}_{cl})\le 0\end{array}$$

(26)

where ${q_m=[q_{m1}^T\dots q_{mN}^T]}^T$, ${q_r=[q_{r1}^T\dots q_{rN}^T]}^T$, ${\overline{M}}_m=I_N\otimes M_m$, and ${\overline{M}}_r=I_N\otimes M_r$.

Following a similar approach to Lemma 2 in [46], the proof is omitted for brevity.

Lemma 6

([2]). Let $x\in R^n$, $y\in R^n$, $\kappa >0$ be a scalar, and $S\in R^{n\times n}$ be a positive definite matrix; then we have

$$2x^{T}y\le 1/\kappa x^{T}Sx+\kappa y^T{S}^{-1}y$$

(27)

Now, we are ready to give the following theorem to establish a sufficient condition that guarantees that the TVF tracking error is bounded and minimized.

Theorem 2.

Considering system (22), suppose there exists a positive definite matrix $Q\in R^{(3n+3q)\times (3n+3q)}$, diagonal positive definite matrices $W_m\in R^{q\times q}$, $W_r\in R^{q\times q}$, $X_m$, $X_r$ with suitable dimensions, and a positive scalar $\alpha >0$ such that the following optimization problem with LMI constraints is feasible

$$\begin{array}{cc}\hfill & \mathit{minimize}ϵ,s.t.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\mathrm{\Sigma }}_{11}& {\mathrm{\Sigma }}_{12}& {\mathrm{\Sigma }}_{13}\\ \ast & -2W_m& {\mathrm{\Sigma }}_{23}\\ \ast & \ast & {\mathrm{\Sigma }}_{33}\end{array}\right]<0\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}Q& Y_{m\left(k\right)}^T\\ Y_{m\left(k\right)}& m^2/w_{max}^2\end{array}\right]>0,\left[\begin{array}{cc}Q& Y_{r\left(k\right)}^T\\ Y_{r\left(k\right)}& r^2/w_{max}^2\end{array}\right]>0\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \left[\begin{array}{cc}Q& Q{C}_z^T\\ C_{z}Q& {ϵ}^{2}I\end{array}\right]>0\hfill \end{array}$$

(30)

where

$$\begin{array}{cc}\hfill {\mathrm{\Sigma }}_{11}& =A_{cl,1}Q+Q{A}_{cl,1}^T+\alpha Q+{\displaystyle \frac{2}{\alpha }}{\overline{B}}_{x\tilde{w}}{B}_{x\tilde{w}}^T,{\mathrm{\Sigma }}_{12}=B_m{W}_m+B_{\delta }{X}_1+Q{C}_m^T-Y_m^T\hfill \\ \hfill {\mathrm{\Sigma }}_{13}& =B_r{W}_r+B_{\delta r}{X}_2+Q{C}_r^T-Y_r^T,{\mathrm{\Sigma }}_{23}=X_1^T{\left[00I_q\right]}^{T}K,{\mathrm{\Sigma }}_{33}=-2W_r+{\displaystyle \frac{2}{\alpha }}K{B}_{\eta \tilde{w}}{B}_{\eta \tilde{w}}^{T}K+2X_2\hfill \end{array}$$

Then, if initial conditions satisfy $x_{cl}\left(0\right)\in \mathcal{E}\left(\overline{P}\right)$, then the tracking error $\zeta \left(t\right)$ is bounded by $\parallel \zeta \left(t\right)\parallel \le ϵw_{max}$, and the AW gains can be obtained by ${\Delta }_m=X_1{W}_m^{-1}$, ${\Delta }_r=X_2{W}_r^{-1}$.

Proof. 

Consider a positive definite matrix $P>0$, $P=\left[P_{ij}\right]\in R^{(3n+3q)\times (3n+3q)}$, and construct $\overline{P}=\left[{\overline{P}}_{ij}\right]$, $i,j=1,\dots ,6$, where ${\overline{P}}_{ij}=I_N\otimes P_{ij}$. We choose the Lyapunov function $V\left(t\right)=x_{cl}^T\overline{P}{x}_{cl}$, and define an ellipsoid set $\mathcal{E}\left(\overline{P}\right)=\{x_{cl}\in R^{3n+3q}:x_{cl}^T\overline{P}{x}_{cl}\le w_{max}^2$}. Next, we establish that $\mathcal{E}\left(\overline{P}\right)$ is an invariant set for system (22) by ensuring the following inequality holds:

$$\dot{V}\left(t\right)+\alpha (V\left(t\right)-{\tilde{w}}^T\tilde{w})<0,$$

(31)

From Lemma 5, Ref. (31) can be guaranteed by

$$\begin{array}{cc}\hfill & \dot{V}\left(t\right)+\alpha (V\left(t\right)-{\tilde{w}}^T\tilde{w})-2q_m^T{\overline{M}}_m(q_m-\delta +{\overline{H}}_m{x}_{cl})-2q_r^T{\overline{M}}_r(q_r-{\eta }_c+{\overline{H}}_r{x}_{cl})\le 0\hfill \end{array}$$

(32)

as long as $\mathcal{E}\left(\overline{P}\right)\subset \mathcal{H}$ and $x_{cl}\left(0\right)\in \mathcal{E}\left(\overline{P}\right)$.

Taking the derivative of $V\left(t\right)$ yields

$$\begin{array}{cc}\hfill & \dot{V}\left(t\right)+\alpha (V\left(t\right)-{\tilde{w}}^T\tilde{w})-2q_m^T{\overline{M}}_m(q_m-\delta +{\overline{H}}_m{x}_{cl})-2q_r^T{\overline{M}}_r(q_r-{\eta }_c+{\overline{H}}_r{x}_{cl})\hfill \\ \hfill =& x_{cl}^T(\overline{P}{\overline{A}}_{cl}+{\overline{A}}_{cl}^T\overline{P}+\alpha \overline{P})x_{cl}+2x_{cl}^T(\overline{P}({\overline{B}}_m+{\overline{B}}_{\delta m}{\overline{\Delta }}_m)+{\overline{C}}_m^T{\overline{M}}_m-{\overline{H}}_m^T{\overline{M}}_m)q_m\hfill \\ \hfill & +2x_{cl}^T(\overline{P}({\overline{B}}_r+{\overline{B}}_{\delta r}{\overline{\Delta }}_r)+{\overline{C}}_r^T{\overline{M}}_r-{\overline{H}}_r^T{\overline{M}}_r)q_r-2q_m^T{\overline{M}}_m{q}_m-2q_r^T({\overline{M}}_r-{\overline{M}}_r{\overline{\Delta }}_r)q_r\hfill \\ \hfill & +2q_r^T{\overline{M}}_r(I_N\otimes K\left[00I_q\right]){\overline{\Delta }}_m{q}_m+2x_{cl}^T\overline{P}{\overline{B}}_{x\tilde{w}}\tilde{w}+2q_r{\overline{M}}_r{\overline{B}}_{\eta \tilde{w}}\tilde{w}-\alpha {\tilde{w}}^T\tilde{w}\hfill \end{array}$$

(33)

According to Lemma 6, the following two inequalities hold for $\alpha >0$

$$\begin{array}{cc}\hfill & 2q_r^T{\overline{M}}_r{\overline{B}}_{\eta \tilde{w}}\tilde{w}\le {\displaystyle \frac{\alpha }{2}}{\tilde{w}}^T\tilde{w}+{\displaystyle \frac{2}{\alpha }}{q}_r^T{\overline{M}}_r{\overline{B}}_{\eta \tilde{w}}{\left(q_r^T{\overline{M}}_r{\overline{B}}_{\eta \tilde{w}}\right)}^T\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & 2x_{cl}^T\overline{P}{\overline{B}}_{x\tilde{w}}\tilde{w}\le {\displaystyle \frac{\alpha }{2}}{\tilde{w}}^T\tilde{w}+{\displaystyle \frac{2}{\alpha }}{x}_{cl}^T\overline{P}{\overline{B}}_{x\tilde{w}}{\left(x_{cl}^T\overline{P}{\overline{B}}_{x\tilde{w}}\right)}^T\hfill \end{array}$$

(35)

Combining (33)–(35), we obtain that (32) can be guaranteed by the following inequality

$$\left[\begin{array}{ccc}{\mathrm{\Omega }}_{11}& {\mathrm{\Omega }}_{12}& {\mathrm{\Omega }}_{13}\\ \ast & -2{\overline{M}}_m& {\mathrm{\Omega }}_{23}\\ \ast & \ast & {\mathrm{\Omega }}_{33}\end{array}\right]<0$$

(36)

with

$$\begin{array}{cc}\hfill & {\mathrm{\Omega }}_{11}=\overline{P}{\overline{A}}_{cl}+{\overline{A}}_{cl}^T\overline{P}+\alpha \overline{P}+{\displaystyle \frac{2}{\alpha }}\overline{P}{\overline{B}}_{x\tilde{w}}{B}_{x\tilde{w}}^T{\overline{P}}^T,{\mathrm{\Omega }}_{12}=\overline{P}({\overline{B}}_m+{\overline{B}}_{\delta }{\overline{\Delta }}_m)+{\overline{C}}_m^T{\overline{M}}_m-{\overline{H}}_m^T{\overline{M}}_m\hfill \\ \hfill & {\mathrm{\Omega }}_{13}=\overline{P}({\overline{B}}_r+{\overline{B}}_{\delta r}{\overline{\Delta }}_r)+{\overline{C}}_r^T{\overline{M}}_r-{\overline{H}}_r^T{\overline{M}}_r,{\mathrm{\Omega }}_{23}={({\overline{M}}_r\overline{K}[I_N\otimes \left[00I_q\right]{\overline{\Delta }}_m)}^T\hfill \\ \hfill & {\mathrm{\Omega }}_{33}=-2{\overline{M}}_r+{\displaystyle \frac{2}{\alpha }}{\overline{M}}_r{\overline{B}}_{\eta \tilde{w}}{\overline{B}}_{\eta \tilde{w}}^T{\overline{M}}_r+2{\overline{M}}_r{\overline{\Delta }}_r\hfill \end{array}$$

By using Schur’s Complement, the following constraints ensure $\mathcal{E}\left(\overline{P}\right)\subset \mathcal{H}$

$$\left[\begin{array}{cc}\overline{P}& {\overline{H}}_{m\left(p\right)}^T\\ {\overline{H}}_{m\left(p\right)}& m^2/w_{max}^2\end{array}\right]>0,\left[\begin{array}{cc}\overline{P}& {\overline{H}}_{r\left(p\right)}^T\\ {\overline{H}}_{r\left(p\right)}& r^2/w_{max}^2\end{array}\right]>0$$

(37)

Thus, (36) and (37) ensure that $\mathcal{E}\left(\overline{P}\right)$ is invariant, that is, $x_{cl}\left(t\right)$ will go to the origin once $x_{cl}\left(t\right)$ touches the boundary of $\mathcal{E}\left(\overline{P}\right)$, which implies that TVF tracking error $\parallel \zeta \left(t\right)\parallel \le \parallel x_{cl}\left(t\right)\parallel$ is bounded, as long as the initial condition satisfies $x_{cl}\left(0\right)\in \mathcal{E}\left(\overline{P}\right)$.

Next, we simplify (36) and (37) into low-dimensional and solvable LMIs. By orthogonal transformations, after rearranging rows and columns, (36) can be guaranteed by the following inequalities

$$\left[\begin{array}{ccc}{\mathrm{\Pi }}_{11}& {\mathrm{\Pi }}_{12}& {\mathrm{\Pi }}_{13}\\ \ast & -2M_m& {\mathrm{\Pi }}_{23}\\ \ast & \ast & {\mathrm{\Pi }}_{33}\end{array}\right]<0$$

(38)

where

$$\begin{array}{cc}\hfill & {\mathrm{\Pi }}_{11}=P{A}_{cl,i}+A_{cl,i}^{T}P+\alpha P+{\displaystyle \frac{2}{\alpha }}P{B}_{x\tilde{w}}{B}_{x\tilde{w}}^T{P}^T,{\mathrm{\Pi }}_{12}=P(B_m+B_{\delta }{\Delta }_m)+C_m^T{M}_m-H_m^T{M}_m\hfill \\ \hfill & {\mathrm{\Pi }}_{13}=P(B_r+B_{\delta r}{\Delta }_r)+C_r^T{M}_r-H_r^T{M}_r,{\mathrm{\Pi }}_{23}={\left(M_{r}K\left[00I_q\right]{\Delta }_m\right)}^T\hfill \\ \hfill & {\mathrm{\Pi }}_{33}=-2M_r+{\displaystyle \frac{2}{\alpha }}{M}_r{B}_{\eta \tilde{w}}{\left(M_r{B}_{\eta \tilde{w}}\right)}^T+2M_r{\Delta }_r\hfill \end{array}$$

Setting $P=Q^{-1}$, $H_{m}Q=Y_m$, $H_{r}Q=Y_r$, $W_m=M_m^{-1}$, $W_r=M_r^{-1}$, $X_1={\Delta }_m{W}_m$, $X_2={\Delta }_r{W}_r$, after congruence transformations, (38) can be guaranteed by (28) due to the fact that Re$\left({\lambda }_1\right)\le$ Re$\left({\lambda }_2\right)\le$...≤ Re$\left({\lambda }_N\right)$ and that $He(A_{cl,1}Q-A_{cl,i}Q)\ge 0$, for any $i=1,\dots ,N$. Thus, the AW gains can be obtained by ${\Delta }_m=X_1{W}_m^{-1}$, ${\Delta }_r=X_2{W}_r^{-1}$.

Then, we demonstrate that (37) is ensured by (29). By permuting rows and columns, the inequalities in (37) are equivalent to

$$\begin{array}{c}\hfill \mathrm{diag}\left\{\left[\begin{array}{cc}P& H_{m\left(k\right)}^T\\ H_{m\left(k\right)}& m^2/w_{max}^2\end{array}\right],\underset{N-1}{\underbrace{P,\dots ,P}}\right\}>0\end{array}$$

(39)

$$\begin{array}{c}\hfill \mathrm{diag}\left\{\left[\begin{array}{cc}P& H_{r\left(k\right)}^T\\ H_{r\left(k\right)}& r^2/w_{max}^2\end{array}\right],\underset{N-1}{\underbrace{P,\dots ,P}}\right\}>0\end{array}$$

(40)

which is equivalent to

$$\left[\begin{array}{cc}P& H_{m\left(k\right)}^T\\ H_{m\left(k\right)}& m^2/w_{max}^2\end{array}\right]>0,\left[\begin{array}{cc}P& H_{r\left(k\right)}^T\\ H_{r\left(k\right)}& r^2/w_{max}^2\end{array}\right]>0$$

(41)

Setting $Q=P^{-1}$, $H_{m}Q=Y_m$ and $H_{r}Q=Y_r$, by congruence transformations, (41) is equivalent to (29).

Finally, using Schur’s complement, the following inequality is guaranteed by $\parallel \zeta \left(t\right)\parallel \le \parallel x_{cl}\left(t\right)\parallel \le ϵw_{max}$

$$\left[\begin{array}{cc}\overline{P}& {\overline{C}}_z^T\\ {\overline{C}}_z& {ϵ}^{2}I\end{array}\right]>0$$

(42)

This is equivalent to (30), thereby completing the proof. □

5. Numerical Example

Two illustrative examples are presented in this section to demonstrate the effectiveness of the proposed methods. First, we examine the advantage of protocol (8) in the first example. Then, we explore the AW compensation for MRS inputs in the second example. The communication topology used in this section is depicted in Figure 2.

5.1. Distributed Extended State Observer-Based Formation Control

An MAS comprising one leader and five followers is considered. The following are the system matrices

$$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -4& -6& -4\end{array}\right],B=\left[\begin{array}{cc}0& 1\\ 0& -1\\ 1& 0\end{array}\right],C=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$$

with $u_0=K_0{x}_0$ and $K_0=\left[\begin{array}{ccc}3.8569& 5.3535& 3.1976\\ -0.0635& 0.0341& 0.7973\end{array}\right]$. This leads to an asymptotically stable closed-loop system ${\dot{x}}_0=(A+B{K}_0)x_0$, where

$$A+B{K}_0=\left[\begin{array}{ccc}-0.0635& 1.0341& 0.7973\\ 0.0635& -0.0341& 0.2027\\ -0.1431& -0.6465& -0.8024\end{array}\right]$$

Thus, the leader’s input $u_0$ satisfies Assumption 3. The initial formation error is chosen to satisfy $x_{cl}\left(0\right)\in \mathcal{E}\left(\overline{P}\right)$. The disturbances of the followers are chosen as $w_i\left(t\right)={[0.5,0.5e^{-(0.5\ast i)t}]}^T$, $i=1,\dots ,5$, which also satisfies Assumption 3. The TVF reference is

$$f_i\left(t\right)=\left[\begin{array}{c}10sin(t+2\pi (i-1)/5)\\ 10cos(t+2\pi (i-1)/5)\\ -10sin(t+2\pi (i-1)/5)\end{array}\right],i=1,\dots ,5$$

If the TVF is achieved, the five followers will be positioned at the vertices of a regular pentagon and rotate around the leader as the center. Let $\widehat{B}=\left[\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\end{array}\right],\tilde{B}=\left[\begin{array}{ccc}1& 1& 0\end{array}\right]$ such that $\widehat{B}B=I_2$ and $\tilde{B}B=0$,. Then the feasibility condition (15) for formation is satisfied.

The control gain and observer gain obtained by Theorem 1 are

$$K_1=\left[\begin{array}{ccc}6.8838& -10.2832& 2.4213\\ -4.5545& -4.5962& -8.5859\end{array}\right],L_1=\left[\begin{array}{cc}0.2605& -6.7171\\ 7.1074& 1.0956\\ -5.8660& -3.4955\end{array}\right],L_2=\left[\begin{array}{cc}2.5955& -1.7376\\ -0.1820& -3.0115\end{array}\right]$$

Figure 3 shows the states of the MAS at various time instances. As we can see, the followers successfully form a regular pentagon and track the leader toward the origin. Figure 4, Figure 5 and Figure 6 demonstrate that the TVF tracking errors and the estimation errors of the observers asymptotically converge, which demonstrated that the designed observers are effective.

5.2. Anti-Windup Formation Control with MRS

This section examines the AW compensation for the TVF tracking control of saturated MASs, where each follower is subject to MRS, with $m=2$ and $r=25$. We consider an MAS composed of second-order agents

$$A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right]$$

with $u_0=K_0{x}_0$ and $K_0=\left[\begin{array}{cc}0& 1\end{array}\right]$; therefore, ${\dot{x}}_0=(A+B{K}_0)x_0$, where

$$A+B{K}_0=\left[\begin{array}{cc}0& 1\\ -1& -1\end{array}\right]$$

It is straightforward to have that the leader’s input $u_0$ satisfies Assumption 3. The disturbances are the same as in the first example. The formation reference signal is

$$f_i\left(t\right)=\left[\begin{array}{c}5cos(-t+2\pi (i-1)/5)\\ 5sin(-t+2\pi (i-1)/5)\end{array}\right]$$

The control gain and observer gain obtained through Theorem 1 are

$$K_1=\left[\begin{array}{cc}-0.4040& -1.2120\end{array}\right],L=\left[\begin{array}{c}1.3451\\ 2.0955\\ 0.9355\end{array}\right]$$

Following the obtained controller, the AW gains through Theorem 2 with $\alpha =2$ are

$${\Delta }_m={\left[\begin{array}{cccc}-0.2078& 4.6343& -0.2937& 0.0291\end{array}\right]}^T,{\Delta }_r=-0.5816.$$

Figure 7 presents the states of the leader and followers at various time instants. For generality and without loss of scope, agent 2 is selected to demonstrate the compensation performance. To satisfy the initial constraint in Theorem 2, we set $x_{cl}\left(0\right)\in \mathcal{E}\left(\overline{P}\right)$. The signals after MRS function are depicted in Figure 8, where the solid line represents the signal with AW compensation, and the dotted line is without AW compensation. As is illustrated in Figure 8a, the control input with anti-windup compensation exits the saturation region faster, and its rate is also reduced as shown in Figure 8b. Figure 9 shows that the TVF tracking errors with anti-windup compensation have smaller overshoot. The first component of the tracking error is reduced from 1.25 to 1 by 20%. Thus, this example demonstrates the effectiveness of the anti-windup design in dealing with MRS.

The simulations in Section 5 were executed on a Dell OptiPlex 7080 desktop with an Intel i7-10700 CPU and 16 GB RAM, using MATLAB R2022b.

6. Conclusions

This paper studies the leader-following TVF control problem for general linear MASs under MRS conditions, where the leader is driven by a nonzero control input and the followers are affected by unknown external disturbances. The leader’s input and the unknown disturbances of followers are treated as new states, and estimated by a distributed ESO, which incorporates an output feedback controller to guarantee the asymptotic convergence of TVF errors. To mitigate the MRS impacts, a static AW compensation is designed for followers, and the AW problem is solved by formulating an optimization problem. The future work is to consider the formation control of nonlinear MASs with MRS.