2. Problem formulation and preliminaries

This paper aims to analyze the formation control problem of high-order nonaffine nonlinear MASs. In what follows, some necessary preliminaries are required to describe the model and obtain main results.

2.1. Algebraic graph theory

In this paper, the information interaction between agents in the MASs is represented by graph. The connected relationship between agents is represented by a weighted undirected graph $G=\left ( \nu,\varepsilon,A \right )$ , where $\nu = \left \{ \nu _{1},\nu _{2},\cdots,\nu _{N} \right \}$ represents the set of nodes, $\varepsilon =\left \{ \left ( i,j \right ) \in \nu \times \nu \right \}$ is the ordered set of node pairs, called the edge of the graph, and $A=[a_{ij}]\subset \mathbb{R}^{N\times N}$ is the corresponding adjacency matrix. In the network of agents, $\left ( i,j \right ) \in \varepsilon$ means that the agent $j$ passes state information to the agent $i$ , then $j$ is the neighbor of $i$ and $a_{ij}=1$ , $\left ( i\neq j, i,j=1,2,\cdots N \right )$ . Thus, a neighbor set is denoted as $\mathcal N_{i}\,:\!=\,\left \{ j:\left ( \nu _{i},\nu _{j} \right ) \in \varepsilon \right \}$ . Otherwise, $a_{ij}=0$ represents no interaction. We assume that $a_{ii}= 0$ and the graph is fixed. $D={\rm{diag}} (d_{i})\subset \mathbb{R}^{N\times N}$ is called the in-degree matrix, where $d_{i}= \sum _{j=1}^{N}a_{ij}$ , $i=1,2,\cdots,N$ , is the weighted in-degree of node $\nu _{i}$ , and the Laplacian matrix is defined as $L=D-A$ . In addition, a weighted matrix $C=\rm{diag} (c_{i})$ is defined with $c_{i}=1$ if the $i$ th follower can obtain the information of the leader agent directly, otherwise $c_{i}=0$ .

Figure 1 shows the communication topology of the MASs with one leader and four followers. Let $x_{i}$ represent the location of the $i$ th follower, then the set of vertices is $\nu =\left \{ x_{i}\,:\,i=1,2,3,4 \right \}$ . Therefore, the set of edges of a graph $\varepsilon$ is given as $\varepsilon =\left \{ (1,2),(1,3),(2,3),(3,4)\right \}$ . The communication weight matrix between the leader and followers is defined as $C={\rm{diag}}\left \{ c_{1},0,0,0 \right \}$ and $c_{1}=1$ . The neighbor of $i$ th follower is $ \mathcal N_{1}=\left \{ 2,3 \right \}$ .

Figure 1. Communication topology graph.

Remark The communication graph $G$ reflects the structural relationships between agents in the communication network. Those agents, which have paths to each other, can reach formation. In other words, when the communication topology is connected, every agent in MASs will achieve formation under the action of the controller.

2.2. Dynamical models

Consider a class of $n$ th order nonaffine stochastic nonlinear MASs with one leader agent and $N$ follower agents, the dynamics of follower agent $i$ is described as follows:

(1) \begin{equation} \left \{\begin{array}{l} \dot{x}_{i,1}(t)=f_{i,1}\left ( x_{i,1}(t),x_{i,2}(t) \right ),\\ \quad \quad \quad \quad \quad \quad \quad \vdots \\ \dot{x}_{i,p}(t)=f_{i,p}\left ( \bar{x}_{i,p}(t),x_{i,p+1}(t)\right ),\\ \quad \quad \quad \quad \quad \quad \quad \vdots \\ \dot{x}_{i,n}(t)=f_{i,n}\left ( \bar{x}_{i,n}(t),u_{i}\left ( v_{i}(t) \right )\right ),\\ y_{i}(t)=x_{i,1}(t), \end{array}\right. \end{equation}

where the subscript $i\in \lbrace 1,2,\cdots,N\rbrace$ represents the $i$ th follower, $\bar{x}_{i,p}=\left [ x_{i,1},x_{i,2},\cdots,x_{i,p} \right ]^{T}\in \mathbb{R}^{n}$ are measurable state vector of the $i$ th follower, and $p=1,2,\cdots, n$ represents the order of the system. $y_{i}\in \mathbb{R}^{n}$ is the output vector of the $i$ th follower and $f_{i,p}\left ( \cdot \right )$ denotes unknown continuous nonaffine function of the $i$ th follower. $v_{i}(t)$ is the control input to be designed and $u_{i}\left ( v_{i}(t) \right )$ denotes the input saturation function for the $i$ th agent, which can be computed as:

(2) \begin{equation} u_{i}\left ( v_{i}(t) \right )={\rm{sat}}\left (v_{i}(t)\right )= \begin{cases} -u_{\text{max}} & v_{i}(t)\lt -u_{\text{max}}, \\v_{i}(t), & v_{i}(t) \leqslant u_{\text{max}} \\u_{\text{max}},& v_{i}(t) \gt u_{\text{max}}, \end{cases} \end{equation}

where $u_{\text{max}}$ and $-u_{\text{max}}$ are defined as the upper and lower bounds of the input saturation $u_{i}\left ( v_{i}(t) \right )$ . It is not difficult to find that the saturated nonlinear input of the system is symmetric, which is very commonly used in practical engineering applications. Similarly, the dynamic equation of the leader is expressed as:

(3) \begin{equation} \left \{\begin{array}{l} \dot{x}_{0}(t)=f_{0}\left (x_{0} (t) \right ),\\[3pt] y_{0}(t)=x_{0} (t), \end{array}\right. \end{equation}

where $x_{0} (t)\in \mathbb{R}^{n}$ represents the state of the leader and $y_{0}(t)\in \mathbb{R}^{n}$ is the output of the leader.

To achieve the aforementioned formation control objectives, we need the following definitions and assumptions.

Definition 1. If the control input of a nonlinear MASs is in the form shown in Eq. (4), the nonlinear MASs is said to be nonaffine nonlinear MASs.

(4) \begin{equation} \dot{x}(t)=f (x)+g (x)h (u) \end{equation}

where $x$ is the measurable state, $f (x)$ and $g(x)$ are nonlinear functions, $u$ is the control input to the MASs, $h ( u )$ represents a nonlinear transformation of the controller $u$ , for example, $\sin ( u )$ and $u^{2}$ , and so on.

Definition 2. MASs are said to achieve formation if, for any initial state values, the following conditions hold:

(5) \begin{equation} \left \{\begin{matrix} \vert y_{i}-y_{0}-\chi _{i} \vert \leq \vartheta _{1},\vert \dot{y}_{i}-\dot{y}_{0} \vert \leq \vartheta _{2},\forall t\geq T_{\text{max}},\\[5pt] \lim _{t\rightarrow \infty }\vert y_{i}-y_{0}-\chi _{i} \vert =\lim _{t\rightarrow \infty }\vert \dot{y}_{i}-\dot{y}_{0} \vert =0, \end{matrix}\right. \end{equation}

where $i=1,2\cdots N$ , $\vartheta _{1}\gt 0$ , and $\vartheta _{2}\gt 0$ are positive constants, $T_{\text{max}}=\max \left \{ T_{1},T_{2},\cdots T_{N} \right \}$ is a user-assigned maximum time, which is not related to the system initial states, and $\chi _{i}$ represents the relative position between the $i$ th follower and the leader in the reference formation.

Assumption 1. The graph $G$ contains a spanning tree with the root node representing the leader agent and assumes that at least one follower agent can receive the information from the leader agent, that is, $c_{1}+c_{2}+\cdots +c_{N}\gt 0$ .

Assumption 2. The leader’s output signals $y_{0}(t)$ and $\dot{y}_{0}(t)$ are smooth and bounded and only available to a part of follower agents. It is further assumed that there exist two unknown positive constants $Y_{0}$ and $Y_{1}$ that satisfy $\vert y_{0}(t) \vert \lt Y_{0}$ and $\vert \dot{y}_{0}(t) \vert \lt Y_{1}$ .

Assumption 3. The nonaffine function $f_{i,p}\left ( \cdot \right )$ is continuously differentiable $g_{i,p}\left ( \cdot \right )$ with respect to $x_{i,p}$ and $u_{i}$ are bounded. For $x_{i,p}\neq 0$ , $i=1,\cdots N$ , and $p=1,\cdots n-1$ , there exist two unknown positive constants $g_{i,\text{min}}$ and $g_{i,\text{max}}$ such that $0\lt g_{i,\text{min}} \leq \vert g_{i,p}\left ( \cdot \right )\vert \leq g_{i,\text{max}}\lt \infty$ , where

(6) \begin{equation} \begin{aligned} g_{i,p}\left ( \bar{x}_{i,p},x_{i,p+1}\right )&=\frac{\partial f_{i,p}\left ( \bar{x}_{i,p},x_{i,p+1}\right )}{\partial x_{i,p+1}},\\ g_{i,n}\left (\bar{x}_{i,n},u_{i}\right )&=\frac{\partial f_{i,n}\left ( \bar{x}_{i,n},u_{i}\right )}{\partial u_{i}}. \end{aligned} \end{equation}

Remark 1. Due to the fact that $g_{i,p}\left ( \cdot \right )$ is a continuous function, the Assumption 3 implies that $g_{i,p}\left ( \cdot \right )$ is strictly either positive or negative. We mainly consider the scenario that $g_{i,p}\left ( \cdot \right )\gt 0$ in this paper. Mathematically, $g_{i,p}\left ( \cdot \right )\lt 0$ is a matter of a simple mathematical transformation and can be treated in the same way. Therefore, without loss of generality, we only need to consider the case of $g_{i,p}\left ( \cdot \right )\gt 0$ .

2.3. Prescribed performance

In this subsection, we introduce a FTPF, which preserves some desired properties that the controlled MAS should exhibit. The error dynamic system is established based on FTPF, and some useful lemmas are listed at the end of this subsection.

2.3.1. New performance function

In this subsection, we first introduce a new FTPF

(7) \begin{equation} \begin{aligned} \beta _{i}(t)=\left \{\begin{matrix} \gamma _{i}\left ( 1-b_{i} \right )\left ( 1-\frac{t}{T_{i}} \right )^{m} +\gamma _{i}b _{i},& 0\leq t\leq T_{i},\\[4pt] \gamma _{i}b _{i},& t\gt T_{i}, \end{matrix}\right. \end{aligned} \end{equation}

where $0\lt T_{i}\lt \infty$ denotes a user-assigned time. $0\lt b _{i}\lt 1$ and $\gamma _{i}$ are design parameters. Note that the FTPF (7) can reach to $\gamma _ib_i$ in the finite time $T_{i}$ and its trajectory evolves within a prescribed area over time, which are the desired properties that the controlled MAS needs to exhibit in this paper. The aforementioned statements are clearly illustrated in Fig. 2.

Figure 2. Example of prescribed performance function.

The properties associated with the function $\beta _{i}(t)$ stated in the following lemmas are useful for our control design.

Lemma 2.1. [Reference Cao and Song2] If the function $\beta _{i}$ is designed as shown in Eq. (7), then the following properties of the function $\beta _{i}$ are true.

1) $\beta _{i}$ monotonically decreases from $\gamma _{i}$ to $\gamma _{i}b _{i}$ when $0\lt t\lt T_{i}$ and keeps constant $\gamma _{i}b _{i}$ for $t\gt T_{i}$ .

2) $\beta ^{k}_{i}(t) \left ( k= 1,2\cdots m \right )$ are continuously differentiable for $t\in \left [ 0,\infty \right )$ .

3) $0\leq -( \dot{\beta } _{i}(t)/ \beta _{i}(t))\leq m\gamma _{i}\left ( 1-b _{i} \right )/T_{i}b _{i}$ , $t\in [ 0,\infty )$ .

Proof. Based on the expression of function $\beta _{i}$ in Eq. (7), we can see clearly that $\lim _{t\rightarrow T_{i}^{-}}\beta _{i}^{k}(t)=\lim _{t\rightarrow T_{i}^{+}}\beta _{i}^{k}(t)=\gamma _{i}b _{i}$ for $k= 1,2, \cdots, m$ ; thus, $\beta _{i}^{k}(t)$ are continuous functions. In addition, from the definition of $\beta _{i}(t)$ in Eq. (7), it can be seen that

(8) \begin{equation} \dot{\beta }_{i}(t)=\left \{\begin{matrix} -\frac{ m\gamma _{i}\left ( 1-b _{i} \right )}{T_{i}}\left ( 1-\frac{t}{T_{i}} \right )^{m-1},& 0\leq t\leq T_{i},\\[8pt] 0,& t\gt T_{i}. \end{matrix}\right. \end{equation}

Because of $0\lt \left ( 1-t/T_{i} \right )^{m-1}\lt 1$ , we can get $-m\gamma _{i}\left ( 1-b _{i} \right )/T_{i}\leq \dot{\beta }_{i}(t)\leq 0$ for $t\in \left [ 0,\infty \right )$ . Similarly, recalling the fact that $b_{i}\gamma _{i}\leq \beta _{i}(t)\leq \gamma _{i}$ , it can be concluded that $0\leq - \dot{\beta } _{i}(t)/ \beta _{i}(t)\leq m\gamma _{i}$ $\left ( 1-b _{i} \right )/T_{i}b _{i}$ .

Remark 2. In most of the literature on prescribed performance, the performance function is chosen as an exponential form, for example, $\Gamma (t)=\left (\iota _{0}-\iota _{\infty } \right )\text{exp}\left ( -\lambda t \right )+\iota _{\infty }$ , where $\iota _{0}$ , $\iota _{\infty }$ , and $\lambda$ are positive constants. In this setting, the performance function can asymptotically approach some predefined value and thus cannot attain the predefined value in a finite time. Instead, compared with reference [Reference Shi, Zhou and Guo46], the novel prescribed performance function (7) proposed in this paper can satisfy that it can attain the value, which is needed for the prescribed performance, in the time $T_{i}$ .

2.3.2. Transformed formation error dynamic model

Our objective is to design a distributed cooperative control protocol for each agent $i \in N$ such that each agent $i$ preserves the connectivity with its neighbor $j \in \mathcal N_{i}$ while guaranteeing prescribed performance in the presence of unknown dynamics. Define the following neighborhood errors:

(9) \begin{equation} e_{i}=\sum _{j\in \mathcal N_{i}}^{}a_{ij}\left ( y_{i}-\chi _{i} - y_{j}+\chi _{j} \right ) +c_{i}\left ( y_{i}-y_{0}-\chi _{i} \right ). \end{equation}

It should be noted that the prescribed performance bounds are used to ensure transient response performance and steady response performance of MASs. The formation error $e_{i}$ needs to be bounded by the decaying function $\beta _i$ , that is,

(10) \begin{equation} -\beta _{i}(t)\lt e_{i}(t) \lt \beta _{i}(t),\forall t\geq 0. \end{equation}

To solve the formation tracking control problem with constraints, we will transform the formation errors with the constraints (10) into unconstrained forms. First, we need to normalize $e_{i}(t)$ with respect to the prescribed performance function $\beta _{i}(t)$ and define the modulated error $\hat{e}_{i}(t)$ as

(11) \begin{equation} \hat{e}_{i}(t)=\frac{e_{i}(t)}{\beta _{i}(t)}. \end{equation}

The transformation error is given by

(12) \begin{equation} s_{i}=\ln \left ( \frac{1+\hat{e}_{i}}{1-\hat{e}_{i}} \right ), \end{equation}

where $ s_{i}$ represents the error mapping and $\ln \left ( \cdot \right )$ denotes the natural logarithm. Differentiating Eq. (12) with respect to time, we get

(13) \begin{equation} \dot{s}_{i}=\varphi _{i}\left ( \frac{\dot{e}_{i}\beta _{i}-\dot{\beta }_{i} e_{i}}{\beta ^{2}_{i}} \right ), \end{equation}

where $\varphi _{i}=1/\left [\left ( 1+\hat{e}_{i} \right )\left (1-\hat{e}_{i} \right )\beta _{i}\right ]\gt 0$ .

Remark 3. It can be shown that if the transformation error $s_{i}$ is bounded, then the modulation error $\hat{e}_{i}(t)$ will be constrained within a certain region $\hat{e}_{i}(t)\in \left ( -1,1 \right )$ . This also means that the error $e_{i}(t)$ satisfies the constraints (10). In addition, the parameters $b_{i}$ , $\gamma _{i}$ , and $T_{i}$ $\left ( i=1,2,\cdots,N \right )$ should be chosen appropriately for the formation tracking performance.

2.4. Active disturbance rejection control

ADRC, including extended state observer (ESO) and TD, is employed to estimate system uncertainties in MASs. Consider the following nonaffine nonlinear system:

(14) \begin{equation} \dot{y}(t)=f\left ( y(t),u(t)\right )+b(t)u(t), \end{equation}

where $y(t)$ denotes the measurable state of system, $f\left ( \cdot \right )$ is a nonaffine nonlinear function, $u(t)$ stands for the control input signal of the system, and $0\lt b_{1}\lt b(t)\lt b_{2}$ is the coefficient of uncertain control quantity.

2.4.1. Extended state observer

In the ADRC technique, ESO is used to dynamically estimate system uncertainties and compensate for the unknown total disturbance. According to Assumption 3, the system (14) can be rewritten as

(15) \begin{equation} \begin{aligned} \dot{z}_{1}&=z_{2}+b_{0}u,\\ \dot{z}_{2}&=H(t), \end{aligned} \end{equation}

where $z_{1}=y$ , $z_{2}=f\left ( z_{1},u \right )+ \left (b(t)-b_{0}\right )u$ , $b_{0}\in \left (b_{1},b_{2} \right )$ , and $H(t)$ is unknown. We employ an ESO for the system (15), and the ESO can be designed as

(16) \begin{equation} \begin{aligned} \dot{\phi }_{1}&=\phi _{2}-\varrho _{1}\varepsilon +b_{0}u,\\ \dot{\phi }_{2}&=-\varrho _{2}\vert \varepsilon \vert ^{\delta }{\rm{sgn}}(\varepsilon), \end{aligned} \end{equation}

where $\varepsilon =\phi _{1}-z_{1}$ represents the estimation error of ESO, the parameters $\varrho _{1}$ and $\varrho _{2}$ are positive constant, $\delta \in \left ( 0,1 \right )$ is a designed constant, and $\rm{sgn}\left ( \cdot \right )$ represents the signum function. The ESO states $\phi _{1}$ and $\phi _{2}$ are used to estimate $z_{1}$ and $z_{2}$ , respectively. Let us define the error $\bar{\varepsilon }=\phi _{2}-z_{2}$ , the controller $u(t)$ is designed as

(17) \begin{equation} \begin{aligned} u(t)=\frac{1}{b_{0}}\left ( -\phi _{2}-kz_{1} \right ) \end{aligned} \end{equation}

where $k\gt 0$ . The observation error orbit $\left (\varepsilon,\bar{\varepsilon } \right )$ can converge to neighborhood of origin under appropriate selection of parameters $\varrho _{1}$ , $\varrho _{2}$ , and $\delta$ [Reference Huang and Han47]. In this paper, we will use the state $\phi _{2}$ of ESO to estimate unknown nonaffine nonlinear function.

2.4.2. Tracking differentiator

TD is used to solve the problem of reasonable extraction of continuous and differential signals from measurement signals that are not continuous or with random noise, which can effectively suppress noise and avoid high-frequency tremor. Compared with classic differentiator/filter (noise amplification effect) and sliding mode differentiator (high-frequency tremor), TD is more convenient to implement in real applications. TD will be used to solve the “differential explosion” problem. The TD is expressed as

(18) \begin{equation} \begin{aligned} \dot{\psi } _{1}&=\psi _{2},\\ \dot{\psi } _{2}&=-\kappa ^{2}{\rm{sgn}} \left ( \psi _{1}-\psi _{0} \right )^{\zeta }\vert \psi _{1}-\psi _{0} \vert -\kappa \psi _{2}, \end{aligned} \end{equation}

where $\psi _{1}$ and $\psi _{2}$ are state of the TD, $\zeta \in \left ( 0,1 \right )$ , and $\kappa \gt 0$ is the designed parameter. By selecting the appropriate parameters, we get $\lim _{t\rightarrow \infty }\psi _{1}-\psi _{0}=0$ and $\lim _{t\rightarrow \infty }\psi _{2}-\dot{\psi } _{0}=0$ .

To sum up, the block diagram of ADRC based on ESO and TD for agent $i$ is shown in Fig. 3. It is composed of a first-order TD and a second-order ESO. By identifying system uncertainties and rejecting external disturbances, the ADRC controller can guarantee stability of the controlled system.

Figure 3. Structure diagram of active disturbance rejection control.

3. Control design based on backstepping and ADRC

In this section, a distributed formation control approach with prescribed performance and saturated input will be developed for MASs (1) by using backstepping control method and ADRC technique.

According to Assumption 3, the nonaffine nonlinear MASs (1) can be rewritten as

(19) \begin{equation} \left \{\begin{array}{l} \dot{x}_{i,1}(t)=F_{i,1}\left ( x_{i,1}(t),x_{i,2}(t) \right )+\xi _{i,1}x_{i,2}(t),\\ \quad \quad \quad \quad \quad \quad \quad \vdots \\ \dot{x}_{i,p}(t)=F_{i,p}\left ( \bar{x}_{i,p}(t),x_{i,p+1}(t)\right )+\xi _{i,p}x_{i,p+1}(t),\\ \quad \quad \quad \quad \quad \quad \quad \vdots \\ \dot{x}_{i,n}(t)=F_{i,n}\left ( \bar{x}_{i,n}(t),u_{i}\left ( t\right ) \right )+\xi _{i,n}u_{i}\left ( v_{i}(t) \right ),\\ y_{i}(t)=x_{i,1}(t), \end{array}\right. \end{equation}

where $x_{i,p}(t)$ ( $p=1,2,\cdots,n-1$ and $i=1,2,\cdots,N$ ) represents the measurable state of $i$ th follower. $u_{i}\left ( v_{i}(t) \right )$ denotes the saturation nonlinearity of the control input of $i$ th follower, $\xi _{i,p}\gt 0$ is the parameter to be selected, and $F_{i,p}\left ( \cdot \right )$ is the new uncertainty function, which are defined as

(20) \begin{equation} \begin{aligned} F_{i,p}\left (\bar{x}_{i,p},x_{i,p+1}\right )&=f_{i,p}\left ( \bar{x}_{i,p},x_{i,p+1}\right )-\xi _{i,p}x_{i,p+1},\quad \quad p=1,\cdots,n-1,\\[3pt] F_{i,n}\left ( \bar{x}_{i,n},u_{i}\right )&=f_{i,n}\left ( \bar{x}_{i,n},u_{i}\right )-\xi _{i,n}u_{i}. \end{aligned} \end{equation}

Figure 4 shows the process of designing controller for the $i$ th follower based on backstepping control method in this paper. By introducing ESO and input compensator, the formation tracking controller, in the presence of input saturation and system uncertainties, is capable of achieving the desired formation of the MAS.

Figure 4. Formation tracking controller for the $i$ th follower.

For system (19), a backstepping procedure with $n$ steps is applied to develop the controller $u_{i}$ based on the following coordinate transformation:

(21) \begin{equation} \begin{aligned} e_{i,1}&=e_{i},\\ e_{i,p}&=x_{i,p}-\alpha _{i,p},\quad \quad p=2,\cdots,n-1,\\ e_{i,n}&=x_{i,n}-\alpha _{i,n},\\ \tilde{e}_{i,n}&=e_{i,n}-\eta _{i}, \end{aligned} \end{equation}

where $\alpha _{i,p}$ and $\alpha _{i,n}$ are virtual control inputs designed by backstepping procedure, and $\eta _{i}$ is the compensation parameter, which is adapted to compensate for input saturation.

Step i,1: Based on Eq. (21), taking the derivative of $e_{i,1}$ , we have

(22) \begin{equation} \dot{e}_{i,1}=\sum _{j\in \mathcal N_{i}}^{}a_{ij}\dot{y}_{i}+c_{i}\dot{y}_{i}-\sum _{j\in \mathcal N_{i}}a_{ij}\dot{y}_{j}-c_{i}\dot{y}_{0}, \end{equation}

Note that $\dot{y}_{i}=x_{i,1}=F_{i,1}+\xi _{i,1}x_{i,2}$ , and Eq. (22) can be rewritten as

(23) \begin{equation} \dot{e}_{i,1}=\tilde{F}_{i,1}+\left (d_{i}+c_{i}\right )\xi _{i,1}x_{i,2}-c_{i}\dot{y}_{0}-\sum _{j\in \mathcal N_{i}}a_{ij}\dot{y}_{j}. \end{equation}

where $\tilde{F}_{i,1}=\left (d_{i}+c_{i}\right )F_{i,1}$ . As the iteration of backstepping proceeds, the interactive derivative operations will lead to differential explosion and thus $\dot{y}_{j}$ cannot be obtained directly. To overcome this difficulty, the following TD is designed to approximate the derivative of $y_{j}$ ,

(24) \begin{align} \dot{\psi } _{j,1,1} & = \psi _{j,1,2},\nonumber\\ \dot{\psi } _{j,1,2} & =-\kappa ^{2}_{j,1}{\rm{sgn}}\left ( \psi _{j,1,1}-y_{j}\right )^{\zeta _{j,1}}\vert \psi _{j,1,1}-y_{j} \vert -\kappa _{j,1}\psi _{j,1,2}. \end{align}

where $\kappa _{j,1}$ and $\zeta _{j,1}\in \left ( 0,1 \right )$ are positive constant to design.

For unknown nonaffine nonlinear smooth function $\tilde{F}_{i,1}$ , the ESO is designed as

(25) \begin{equation} \begin{aligned} \varepsilon _{i,1}&=\phi _{i,1,1}-e _{i,1},\\ \dot{\phi } _{i,1,1}&=\phi _{i,1,2}-\varrho _{i,1,1}\varepsilon _{i,1}+\left ( d_{i}+c_{i} \right )\xi _{i,1}x_{i,2}+c_{i}\dot{y}_{0}-\sum _{j=1}^{N}a_{ij}\psi _{j,1,2}, \\ \dot{\phi } _{i,1,2}&=-\varrho _{i,1,2}\vert \varepsilon _{i,1} \vert ^{\delta _{i,1}}{\rm{sgn}}\left ( \varepsilon _{i,1}\right ), \end{aligned} \end{equation}

where $\varepsilon _{i,1}$ represents the estimation error of ESO, the parameters $\varrho _{i,1,1}$ , $\varrho _{i,1,2}$ are positive constant, and $\delta _{i,1}\in \left ( 0,1 \right )$ . From Eq. (25), we know that $\phi _{i,1,2}$ is the estimation value for $\tilde{F}_{i,1}\left ( \cdot \right )$ .

Then, the virtual control law $\alpha _{i,2}$ is designed as

(26) \begin{equation} \begin{aligned} \alpha _{i,2}=-\frac{1}{\xi _{i,1}\left (d_{i}+c_{i}\right )}\left( \rho _{i,1}s_{i}+\phi _{i,1,2}-\frac{\dot{\beta }_{i}}{\beta _{i}}e_{i}-c_{i}\dot{y}_{0}-\sum _{j\in \mathcal N_{i}}a_{ij}\psi _{j,1,2} \right), \end{aligned} \end{equation}

where $\xi _{i,1}$ and $\rho _{i,1}$ are positive parameters.

Consider a Lyapunov candidate function as

(27) \begin{equation} V_{i,1}=\frac{1}{2}s^{2}_{i}. \end{equation}

Differentiating $V_{i,1}$ with respect to time $t$ and using Eqs. (13), (22), and (27), we obtain

(28) \begin{equation} \begin{aligned} \dot{V}_{i,1}&=\varphi _{i}s_{i}\left( \tilde{F}_{i,1}+\xi _{i,1}\left (d_{i}+c_{i}\right )\left ( e_{i,2}+\alpha _{i,2} \right )-\frac{\dot{\beta }_{i}}{\beta _{i}} e_{i}-\sum _{j\in \mathcal N_{i}}a_{ij}\dot{y}_{j}-c_{i}\dot{y}_{0}\right). \end{aligned} \end{equation}

Substituting the virtual control law $\alpha _{i,2}$ into Eq. (28), we have

(29) \begin{equation} \begin{aligned} \dot{V}_{i,1}&=\varphi _{i}s_{i}\left( -\rho _{i,1}s_{i}+ \tilde{F}_{i,1}+\sum _{j\in \mathcal N_{i}}a_{ij}\left (\psi _{j,1,2} -\dot{y}_{j}\right )+\left (d_{i}+c_{i}\right )\xi _{i,1} e_{i,2} -\phi _{i,1,2}\right). \end{aligned} \end{equation}

Step i,2: Note that $e_{i,2}=x_{i,2}-\alpha _{i,2}$ . The derivative of $e_{i,2}$ along the systems (19) and (21) is

(30) \begin{equation} \begin{aligned} \dot{e}_{i,2}=F_{i,2}\left ( \bar{x}_{i,2},x_{i,3}\right )+\xi _{i,2}x_{i,3}-\dot{\alpha }_{i,2}. \end{aligned} \end{equation}

Similarly, $\dot{\alpha }_{i,2}$ estimated by TD is designed as follows:

(31) \begin{align} \dot{\psi } _{i,2,1} & = \psi _{i,2,2},\nonumber\\ \dot{\psi } _{i,2,2}&=-\kappa ^{2}_{i,2}{\rm{sgn}}\left ( \psi _{i,2,1}-\alpha _{i,2} \right )\vert \psi _{i,2,1}-\alpha _{i,2} \vert ^{\zeta _{i,2}}-\kappa _{i,2}\psi _{i,2,2}, \end{align}

where $\kappa _{i,2}\gt 0$ and $\zeta _{i,2}\in \left ( 0,1 \right )$ are the design parameters. The $\psi _{i,2,2}$ of TD’s state is used to track $\alpha _{i,2}$ .

For function $F_{i,2}\left ( \bar{x}_{i,2},x_{i,3}\right )$ , the following ESO is designed:

(32) \begin{equation} \begin{aligned} \varepsilon _{i,2}&=\phi _{i,2,1}-e _{i,2},\\ \dot{\phi } _{i,2,1}&=\phi _{i,2,2}-\varrho _{i,2,1}\varepsilon _{i,2}+\xi _{i,2}x_{i,3}-\psi _{i,2,2}, \\ \dot{\phi } _{i,2,2}&=-\varrho _{i,2,2}\vert \varepsilon _{i,2} \vert ^{\delta _{i,2}}{\rm{sgn}}\left ( \varepsilon _{i,2}\right ), \end{aligned} \end{equation}

where $\varrho _{i,2,1}$ , $\varrho _{i,2,2}$ , and $\delta _{i,2}\in \left ( 0,1 \right )$ are positive constant to design. $\phi _{i,2,2}$ is the estimation value of intermediate variable $F_{i,2}\left ( \bar{x}_{i,2},x_{i,3}\right )$ . Then, the virtual control law $\alpha _{i,3}$ is designed as

(33) \begin{equation} \begin{aligned} \alpha _{i,3}=-\frac{1}{\xi _{i,2}}\left( \phi _{i,2,2}+\rho _{i,2}e_{i,2}+\varphi _{i}s_{i}\left ( d_{i}+c_{i} \right ) \xi _{i,1}-\psi _{i,2,2}\right), \end{aligned} \end{equation}

where $\xi _{i,2}\gt 0$ and $\rho _{i,2}\gt 0$ are parameters to be designed.

Define the Lyapunov candidate function as

(34) \begin{equation} \begin{aligned} V_{i,2}=V_{i,1}+\frac{1}{2}e^{2}_{i,2}. \end{aligned} \end{equation}

Taking the time derivatives of $V_{i,2}$ along Eqs. (30) and (33) yields

(35) \begin{equation} \begin{aligned} \dot{V}_{i,2}&=\dot{V}_{i,1}+e_{i,2}\left( F_{i,2}-\rho _{i,2}e_{i,2}-\phi _{i,2,2}+ \xi _{i,2} e_{i,3}-\dot{\alpha }_{i,2} +\psi _{i,2,2}-\left (d_{i}+c_{i}\right )\xi _{i,1} \varphi _{i}s_{i} \right). \end{aligned} \end{equation}

Step i,p, $\left ( 3\leq p\leq n-1 \right )$ : Similarly, the derivative of $e_{i,p}$ along systems (19) and (21) is

(36) \begin{equation} \begin{aligned} \dot{e}_{i,p}=F_{i,p}\left ( \bar{x}_{i,p},x_{i,p+1}\right )+\xi _{i,p}x_{i,p+1}-\dot{\alpha }_{i,p}. \end{aligned} \end{equation}

TD is designed as follows:

(37) \begin{equation} \begin{aligned} \dot{\psi } _{i,p,1}&=\psi _{i,p,2},\\ \dot{\psi } _{i,p,2}&=-\kappa ^{2}_{i,p}{\rm{sgn}}\left ( \psi _{i,p,1}-\alpha _{i,p} \right )\vert \psi _{i,p,1}-\alpha _{i,p} \vert ^{\zeta _{i,p}}-\kappa _{i,p}\psi _{i,p,2}, \end{aligned} \end{equation}

where $\kappa _{i,p}\gt 0$ and $\zeta _{i,p}\in \left ( 0,1 \right )$ . The $\dot{\psi } _{i,p,2}$ is for the estimation of $\dot{\alpha }_{i,p}$ .

The nonaffine function $F_{i,p}\left ( \bar{x}_{i,p},x_{i,p+1}\right )$ can be estimated by ESO as follows:

(38) \begin{equation} \begin{aligned} \varepsilon _{i,p} &=\phi _{i,p,1}-e _{i,p},\\ \dot{\phi } _{i,p,1}&=\phi _{i,p,2}-\varrho _{i,p,1}\varepsilon _{i,p}+\xi _{i,p}x_{i,p+1}-\psi _{i,p,2},\\ \dot{\phi } _{i,p,2}&=-\varrho _{i,p,2}\vert \varepsilon _{i,p} \vert ^{\delta _{i,p}}{\rm{sgn}}\left ( \varepsilon _{i,p}\right ), \end{aligned} \end{equation}

where $\varrho _{i,p,1}$ , $\varrho _{i,p,2}$ , and $\delta _{i,p}\in \left ( 0,1 \right )$ are positive constant. $\phi _{i,p,2}$ is the estimation value for $F_{i,p}\left ( \bar{x}_{i,p},x_{i,p+1}\right )$ . Based on Eqs. (35), (37), and (38), the virtual control law $\alpha _{i,p+1}$ is taken as

(39) \begin{equation} \begin{aligned} \alpha _{i,p+1}&=-\frac{1}{\xi _{i,p}}\left(\phi _{i,p,2}+\rho _{i,p}e_{i,p}+\xi _{i,p-1} e_{i,p-1}-\psi _{i,p,2}\right), \end{aligned} \end{equation}

where $\rho _{i,p}\gt 0$ and $\xi _{i,p}\gt 0$ are design parameters.

Similarly, consider a Lyapunov candidate function

(40) \begin{equation} \begin{aligned} V_{i,p}=V_{i,p-1}+\frac{1}{2}e^{2}_{i,p}. \end{aligned} \end{equation}

Substituting Eqs. (36) and (39) into $\dot{V}_{i,p}$ yields

(41) \begin{equation} \begin{aligned} \dot{V}_{i,p}= \dot{V}_{i,p-1}+e_{i,p}\left( F_{i,p}-\phi _{i,p,2}+ \xi _{i,p} e_{i,p+1}-\dot{\alpha }_{i,p}-\rho _{i,p}e_{i,p} -\xi _{i,p-1} e_{i,p-1}+\psi _{i,p,2}\right). \end{aligned} \end{equation}

Step i,n: In this step, an actual controller will be constructed. Since the control input of the MASs is limited by saturation, there exists a deviation between the designed control input and the actual control input; an input auxiliary system (42) is introduced to compensate for the deviation,

(42) \begin{equation} \begin{aligned} \dot{\eta }_{i}=-\varpi _{i}\tanh \left ( \eta _{i}\right )+\xi _{i,n}\left ( u_{i}-v_{i} \right ), i=1,2,\cdots,N, \end{aligned} \end{equation}

where $\varpi _{i}\gt 0$ and $\xi _{i,n}\gt 0$ are parameters to be designed, $\eta _{i}$ is the compensation parameter, and $\tanh \left ( \cdot \right )\in \left ( -1,1 \right )$ is the hyperbolic tangent function.

We define the state tracking error ${e}_{i,n}=x_{i,n}-\xi _{i,n}{u}_{i}$ and $\tilde{e}_{i,n}=e_{i,n}-\eta _{i}$ , then the derivative of $\tilde{e}_{i,n}$ can be expressed as

(43) \begin{equation} \begin{aligned} \dot{\tilde{e}}_{i,n}=F_{i,n}-\alpha _{i,n}+\varpi _{i}\tanh \left ( \eta _{i}\right )+\xi _{i,n}v_{i}. \end{aligned} \end{equation}

TD is designed as

(44) \begin{equation} \begin{aligned} \dot{\psi } _{i,n,1}&=\psi _{i,n,2},\\ \dot{\psi } _{i,n,2}&=-\kappa ^{2}_{i,n}{\rm{sgn}}\left ( \psi _{i,n,1}-\alpha _{i,n} \right )\vert \psi _{i,n,1}-\alpha _{i,n} \vert ^{\zeta _{i,n}}-\kappa _{i,n}\psi _{i,n,2}, \end{aligned} \end{equation}

where $\kappa _{i,n}$ and $\zeta _{i,n}\in \left ( 0,1 \right )$ are parameters to be determined. $\dot{\psi } _{i,n,2}$ is used to estimate $\dot{\alpha }_{i,n}$ .

Similar to the previous steps, the ESO in this final step is designed as

(45) \begin{equation} \begin{aligned} \varepsilon _{i,n} &=\phi _{i,n,1}-e _{i,n},\\ \dot{\phi }_{i,n,1}&=\phi _{i,n,2}-\varrho _{i,n,1}\varepsilon _{i,n}+\xi _{i,n}u_{i}-\psi _{i,n,2},\\ \dot{\phi } _{i,n,2}&=-\varrho _{i,n,2}\vert \varepsilon _{i,n} \vert ^{\delta _{i,n}}{\rm{sgn}}\left ( \varepsilon _{i,n}\right ), \end{aligned} \end{equation}

where $\varrho _{i,n,1}$ , $\varrho _{i,n,2}$ , and $\delta _{i,n}\in \left ( 0,1 \right )$ are positive constant. $\phi _{i,n,2}$ is the estimation value for $F_{i,n}\left ( \bar{x}_{i,n},u_{i}\right )$ .

Finally, the controller of $i$ th follower be designed as

(46) \begin{equation} \begin{aligned} v_{i}&=-\frac{1}{\xi _{i,n}}\left(\rho _{i,n}\tilde{e}_{i,n}+\xi _{i,n-1}e_{i,n-1}+\phi _{i,n,2}-\psi _{i,n,2}+\varpi _{i}\tanh \left ( \eta _{i}\right )\right). \end{aligned} \end{equation}

where $\rho _{i,n}$ is the parameter to be designed. In particular, $v_{i}$ is a controller designed for agent $i$ based on backstepping method.

Consider the Lyapunov candidate function as

(47) \begin{equation} \begin{aligned} V_{i,n}=V_{i,n-1}+\frac{1}{2} \tilde{e}^{2}_{i,n}. \end{aligned} \end{equation}

Substituting Eqs. (46) into (2), the saturated control input $u_{i}(v_i(t)\!)$ of the MASs can be written as

(48) \begin{equation} u_{i}\left ( v_{i} \right )=\left \{\begin{array}{l} -u_{\text{max}}, \quad \quad \quad \quad \quad \quad \quad v_{i}\lt -u_{\text{max}},\\[3pt] -\dfrac{1}{\xi _{i,n}}\Big (\rho _{i,n}\tilde{e}_{i,n}+\xi _{i,n-1}e_{i,n-1}+\phi _{i,n,2}-\psi _{i,n,2}+\varpi _{i}\tanh \left ( \eta _{i}\right )\Big ),\vert v_{i}\vert \leqslant u_{\text{max}}, \\[3pt] u_{\text{max}},\quad \quad \quad \quad \quad \quad \quad \quad v_{i}\gt u_{\text{max}}. \end{array}\right. \end{equation}

By Eqs. (43) and (48), the derivative of the Lyapunov candidate function (47) is written as

(49) \begin{equation} \begin{aligned} \dot{V}_{i,n}&=\dot{V}_{i,n-1}+\tilde{e}_{i,n}\left ( F_{i,n}+\xi _{i,n}u_{i}-\dot{\alpha }_{i,n}- \dot{\eta _{i}} \right )\\ &=\dot{V}_{i,n-1}+\tilde{e}_{i,n}\left({-}\rho _{i,n}\tilde{e}_{i,n}+F_{i,n}-\xi _{i,n-1}e _{i,n-1}+\psi _{i,n,2}-\phi _{i,n,2}-\dot{\alpha }_{i,n}\right). \end{aligned} \end{equation}

The pseudo code of a finite-time ADRC formation controller with prescribed performance and saturated input designed based on backstepping is described in Algorithm 1.

4. Stability analysis of the MAS

The following theorem presents the stability result of the ADRC tracking controller for nonaffine nonlinearity MASs with prescribed performance.

Proof. We will use Lyapunov stability theorem to prove the stability of nonaffine nonlinear MASs (1). For the analysis of stability, choose the Lyapunov candidate function $V$ as

(50) \begin{equation} \begin{aligned} V=\sum _{i=1}^{N} V_{i,n}, \end{aligned} \end{equation}

From Eqs. (29), (35), (41), and (49) and with the help of Young’s inequality, one has

(51) \begin{equation} \begin{aligned} \dot{V}_{i,1}\leq & -\left ( \varphi _{i}\rho _{i,1}-\frac{1+d_{i}}{2} \right )s_{i}^{2}-\sum _{j=1}^{N}\frac{a_{ij}}{2}\rho ^{2}_{i,1}\left ( \psi _{j,1,2} -\dot{y}_{j}\right )^{2}+\frac{\rho ^{2}_{i,1}}{2}\left (\tilde{F}_{i,1}-\phi _{i,1,2} \right )^{2}\\ &+\varphi _{i}\left (c_{i}+ d_{i} \right )\xi _{i,1}s_{i}e_{i,2}, \end{aligned} \end{equation}

(52) \begin{equation} \begin{aligned} \dot{V}_{i,2}\leq & -\left ( \varphi _{i}\rho _{i,1}-\frac{1+d_{i}}{2} \right )s_{i}^{2}+\frac{1}{2}\left ( F_{i,2}-\phi _{i,2,2} \right )^{2}-\sum _{j=1}^{N}\frac{a_{ij}}{2}\rho ^{2}_{i,1}\left (\psi _{j,1,2} -\dot{y}_{j}\right )^{2}\\ &+\rho _{i,2}e_{i,2}e_{i,3}+\frac{\rho ^{2}_{i,1}}{2}\left (\tilde{F}_{i,1}-\phi _{i,1,2} \right )^{2}+\frac{1}{2}\left ( \psi _{i,2,2}-\dot{\alpha } _{i,2} \right )^{2}\\ &-\left ( \xi _{i,2}-1\right )e^{2}_{i,2}, \end{aligned} \end{equation}

(53) \begin{equation} \begin{aligned} \dot{V}_{i,p}\leq &-\left ( \varphi _{i}\rho _{i,1}-\frac{1+d_{i}}{2} \right )s_{i}^{2}+\frac{\rho ^{2}_{i,1}}{2}\left (\tilde{F}_{i,1}-\phi _{i,1,2} \right )^{2}-\sum _{j=1}^{N}\frac{a_{ij}}{2}\rho ^{2}_{i,1}\left ( \psi _{j,1,2} -\dot{y}_{j}\right )^{2}\\ &-\sum _{j=2}^{p}\left[\left ( \rho _{i,j}-1\right )e^{2}_{i,j}-\frac{\rho ^{2}_{i,j}}{2}\left ( \psi _{i,j,2}-\dot{\alpha } _{i,j} \right )^{2}-\frac{1}{2}\left ( F_{i,j}-\phi _{i,j,2} \right )^{2}\right]\\ &+\xi _{i,p}e_{i,p}e_{i,p+1},\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad p=3,\cdots,n-1, \end{aligned} \end{equation}

(54) \begin{equation} \begin{aligned} \dot{V}_{i,n}\leq &-\left ( \varphi _{i}\rho _{i,1}-\frac{1+d_{i}}{2} \right )s_{i}^{2}+\frac{1}{2}\left (\tilde{F}_{i,1}-\phi _{i,1,2} \right )^{2}+\sum _{j=2}^{N}\frac{1}{2}\left ( \psi _{i,j,2}-\dot{\alpha }_{i,j} \right )^{2}\\ &-\left (\rho _{i,n-1}-\frac{3}{2}\right )e^{2}_{i,n-1}+\frac{1}{2}\xi _{i,n-1}^{2}\eta _{i}^{2}-\sum _{j=1}^{N}\frac{a_{ij}}{2}\rho ^{2}_{i,1}\left ( \psi _{j,1,2} -\dot{y}_{j}\right )^{2}\\ &-\left ( \rho _{i,n}-1 \right )\tilde{e}^{2}_{i,n}-\sum _{j=2}^{n-2}\left (\rho _{i,j}-1\right )e^{2}_{i,j}+\sum _{j=2}^{n}\frac{1}{2}\left ( F_{i,j}-\phi _{i,j,2} \right )^{2}. \end{aligned} \end{equation}

To prove the boundedness of auxiliary system parameters $\eta _{i}$ , we consider a Lyapunov candidate function

(55) \begin{equation} \begin{aligned} V_{\eta }=\sum _{i=1}^{N}\frac{1}{2}\eta ^{2}_{i}. \end{aligned} \end{equation}

Taking the derivative of $V_{\eta }$ , we have

(56) \begin{equation} \begin{aligned} \dot{V}_{\eta }&=\sum _{i=1}^{N} \eta _{i}\dot{\eta }_{i}\\ &=-\sum _{i=1}^{N} \eta _{i}\left ( \varpi _{i}tanh\left ( \eta _{i}\right )-\xi _{i,n}\left ( u_{i}-v_{i} \right )\right )\\ \leq &-\sum _{i=1}^{N}\vert \eta _{i}\vert \left ( \varpi _{i}tanh\left (\eta _{i}\right )-\xi _{i,n}\vert u_{i}-v_{i} \vert \right ), \end{aligned} \end{equation}

If $\varpi _{i}\geq \varpi _{i}\tanh \left (\eta _{i} \right )\geq \xi _{i,n}\vert u_{i}-v_{i} \vert$ , then $\dot{V}_{\eta }\leq 0$ . Therefore, $\eta _{i}$ is bounded.

The state of ESO ( $\phi _{i,1,2}$ and $\phi _{i,j,2}$ ) is used to estimate the unknown nonlinear functions $\bar{F}_{i,1}\left ( \cdot \right )$ and $F_{i,j}\left ( \cdot \right )$ , $i=1,2,\cdots N$ and $j=2,3,\cdots n$ . According to Eq. (16), the estimate errors of the ESO can be sufficiently small by choosing the right parameters $\varrho _{i,j,1}$ , $\varrho _{i,j,2}$ , and $\delta _{i,j}$ . Similarly, the estimate errors of the TD, $\psi _{j,1,2}-\dot{y}$ and $\psi _{i,j,2}-\dot{\alpha }_{i,j}$ , can be enough small by selecting appropriate parameters $\kappa _{i}$ and $\zeta _{i,j}$ , $i=1,2,\cdots,N$ , $j=1,2,\cdots,n$ . Denote the entire errors from the ESO and TD as a positive parameter $\theta$

(57) \begin{equation} \begin{aligned} \theta &=\underset{t\in (0,+\infty )}{\sup } \Bigg \{ \sum _{i=1}^{N}\rho _{i,1}\vert \tilde{F}_{i,1}-\phi _{i,1,2} \vert +\sum _{i=1}^{N}\sum _{j=2}^{n}\Big ( \vert F_{i,j}-\phi _{i,j,2}\vert + a_{ij}\vert \psi _{i,j,2} -\dot{\alpha }_{i,j} \vert \Big ) \\ &+\sum _{i=1}^{N}\frac{1}{2}\xi _{i,n-1}^{2}\eta _{i}^{2}+\sum _{i=1}^{N}\sum _{j=1}^{N}a_{ij}\rho _{i,1}\vert \psi _{j,1,2} -\dot{y}_{j} \vert \Bigg \}. \end{aligned} \end{equation}

It follows from inequalities (51) to (54) that

(58) \begin{equation} \begin{aligned} \dot{V}\leq &-\sum _{i=1}^{N}\sum _{j=2}^{n-2}\left ( \rho _{i,j} -1\right )e^{2}_{i,j}-\sum _{i=1}^{N}\left ( \rho _{i,n}-1 \right )\tilde{e}^{2}_{i,n} \sum _{i=1}^{N}\left ( \rho _{i,n-1} -\frac{3}{2}\right )e^{2}_{i,n-1}\\ &-\sum _{i=1}^{N} \left ( \varphi _{i}\rho _{i,1}-\frac{1+d_{i}}{2} \right )s_{i}^{2}+\frac{\theta ^{2}}{2}\\ \leq &-\lambda V+\frac{\theta ^{2}}{2}, \end{aligned} \end{equation}

where $\lambda =\min \left [\varphi _{i}\rho _{i,1}-\frac{1+d_{i}}{2},\rho _{i,j} -1, \rho _{i,n-1}-\frac{3}{2}\right ]$ , $i=1,2,\cdots,N$ , and $j=2,3,\cdots,n-2,n$ . Integrating both sides of the inequality (50) yields

(59) \begin{equation} \begin{aligned} V\leq \frac{\theta ^{2}}{2\lambda }+\left ( V\left ( 0 \right )-\frac{\theta ^{2}}{2\lambda }\right )e^{-\lambda t}. \end{aligned} \end{equation}

From Eqs. (50) and (59), it can be seen that $V(0)$ is bounded and $V$ can asymptotically converge to a neighborhood of radius $\theta ^{2}/2\lambda$ as $t\rightarrow \infty$ . The inequality in Eq. (48) implies that transformation error $ s_{i}$ and coordinate transformation $e_{i,p}$ , $\tilde{e}_{i,n}$ can satisfy

(60) \begin{equation} \begin{aligned} \vert s_{i}\vert &\leq \sqrt{2\theta ^{2}/\lambda +\left ( 2V\left ( 0 \right )-2\theta ^{2}/\lambda \right )\exp \left ( -\lambda t \right )}\\ \vert e_{i,p}\vert &\leq \sqrt{2\theta ^{2}/\lambda +\left ( 2V\left ( 0 \right )-2\theta ^{2}/\lambda \right )\exp \left ( -\lambda t \right )},\\ \vert \tilde{e}_{i,n}\vert & \leq \sqrt{2\theta ^{2}/\lambda +\left ( 2V\left ( 0 \right )-2\theta ^{2}/\lambda \right )\exp \left ( -\lambda t \right )},\\ \end{aligned} \end{equation}

which implies that $s_{i}$ , $e_{i,p}$ $ \left ( p=2,\cdots,n \right )$ and $\tilde{e}_{i,n}$ converge exponentially to a small neighborhood around zero in a finite time $T_{i}$ by choosing small enough $\lambda$ .

Due to the candidate transformation function defined in Eqs. (11) and (12), the neighborhood error $e_{i}$ satisfies

(61) \begin{equation} \begin{aligned} \vert e_{i}\vert & \leq \vert \beta _{i}\vert \frac{\exp \left ( 2\sqrt{\frac{\theta ^{2}}{2\lambda }+\left ( V\left ( 0 \right )-\frac{\theta ^{2}}{2\lambda }\right )e^{-\lambda t}}\right )-1}{\exp \left ( 2\sqrt{\frac{\theta ^{2}}{2\lambda }+\left ( V\left ( 0 \right )-\frac{\theta ^{2}}{2\lambda }\right )e^{-\lambda t}}\right )+1}\\ &\leq \vert \beta _{i}\vert, i=1,2,\cdots,n, \end{aligned} \end{equation}

where $\beta _{i}$ are defined by Eq. (7).

The transient and steady states of the formation tracking error $e_{i}(t)$ evolve strictly within the bounds generated by the FTPF $\beta _{i}(t)$ . In addition, in order for the formation tracking errors to converge to a small neighborhood around the origin, the design parameters of ESO, TD, and controllers should be selected appropriately. It is shown that the proposed controllers (48) with input saturation can ensure that, under prescribed performance, all formation tracking errors can asymptotically converge to predefined regions in a prescribed time and therefore all followers can converge to a small neighborhood of the trajectory of the leaders within a finite time $T_{i}.$

5. Simulation

To verify the effectiveness of the proposed control method, numerical simulation results are provided. Consider a group of multi-agent networks composed of five followers and one leader. Figure 5 shows the network communication topology among the leader and the followers. According to the network communication topology, it is obvious that Assumption 1 is satisfied.

Figure 5. The communication topology of the MASs.

These robotic agents are assumed to be linked via an undirected communication graph in Fig. 5. The Laplacian matrix for the graph is computed as

(62) \begin{equation} L=\begin{bmatrix} 2& \quad 0 & \quad -1 & \quad 0 & \quad -1 \\[2pt] 0& \quad 1 & \quad 0 & \quad -1 & \quad 0 \\[2pt] -1& \quad 0 & \quad 3& \quad -1& \quad -1\\[2pt] 0 & \quad -1 & \quad -1 & \quad 2 & \quad 0 \\[2pt] -1& \quad 0 & \quad -1 & \quad 0 & \quad 2 \end{bmatrix} \end{equation}

Based on the network communication topology (Fig. 5), we know that follower agents 1 and 2 can communicate with leaders, so matrix $C$ can be written as $C=\rm{diag}\left \{1,1,0,0,0 \right \}$ .

The dynamics of the follower agent is described in Eq. (63)

(63) \begin{equation} \begin{aligned} \left \{\begin{array}{l} \dot{x}_{i,1}=f_{i,1}(x_{i,1},x_{i,2}),\\ \dot{x}_{i,2}=f_{i,2}\left ( \bar{x}_{i,2},u_{i}\left ( v_{i} \right ) \right ),\\ y_{i}=x_{i,1}, \end{array}\right. \end{aligned} \end{equation}

where $i=1,\cdots, 5$ , and the nonaffine nonlinear functions $f_{i,1}\left ( x_{i,1},x_{i,2} \right )$ and $f_{i,2}\left ( \bar{x}_{i,2},u_{i}\left ( v_{i} \right )\right )$ are represented as

(64) \begin{equation} \begin{aligned} f_{i,1}\left (x_{i,1},x_{i,2}\right )&=3x_{i,2}+\cos \left (x_{i,1}x_{i,2}\right ),\\ f_{i,2}\left ( \bar{x}_{i,2},u_{i}\left ( v_{i} \right )\right )&=1.6u_{i}\left ( v_{i} \right ) +\sin \left ( x_{i,1} x_{i,2} \right )+\cos \left ( u_{i}\left ( v_{i} \right ) \right ), \end{aligned} \end{equation}

and the controller with saturated input is designed as

(65) \begin{equation} u_{i}\left ( v_{i}(t) \right )={\rm{sat}}(v_{i}(t))=\left \{\begin{matrix} -30, & v_{i}(t)\lt -30,\\ v_{i}(t),& \vert v_{i}(t)\vert \leqslant 30, \\ 30,& v_{i}(t) \gt 30, \end{matrix}\right. \end{equation}

The leader’s dynamic is described by

(66) \begin{equation} \begin{aligned} \left \{\begin{array}{l} \dot{x}_{0,1}(t)=0.7,\\ \dot{x}_{0,2}(t)=0.15x_{0,1}(t). \end{array}\right. \end{aligned} \end{equation}

In the nonaffine nonlinear MASs, the initial positions of the agents are given by $x_{0}\left ( 0 \right )=\left [ 0;\, 0\right ]$ , $x_{1,1}\left ( 0 \right )=\left [ 2;\, 1\right ]$ , $x_{2,1}\left ( 0 \right )=\left [1 ;\,2 \right ]$ , $x_{3, 1}\left ( 0 \right )=\left [ 0;\,1.5 \right ]$ , $x_{4,1}\left ( 0 \right )=\left [ -0.3 ;\,0.4\right ]$ , $x_{5,1}\left ( 0 \right )=\left [ 1 ;\,0\right ]$ , and $x_{i,2}\left ( 0 \right )=\left [0;\,0\right ]$ , where $i=1,\cdots, 5$ . The prescribed performance function is given by

(67) \begin{equation} \begin{aligned} \beta _{i}(t)=\left \{\begin{matrix} 4\left ( 1-0.02 \right )\left ( 1-\dfrac{t}{5} \right )^{5} +0.08,& 0\leq t\leq 5,\\[5pt] 0.08,& t\gt 5, \end{matrix}\right. \end{aligned} \end{equation}

The parameters of the prescribed performance function are set to $T_{i}=5$ , $b_{i}=0.02$ , $m=5$ , and $\gamma _{i}=4$ .

Based on the PPC and ADRC techniques, the proposed control law can allow the MASs to realize a circle formation, in which the leader is located at the center of a circle and all the followers are evenly distributed on the circle. The relative position between the $i$ th follower and the leader in the desired formation is designed as

(68) \begin{equation} \begin{aligned} \chi _{i}=\left [ \cos \left (\frac{2\left ( i-1 \right ) \pi }{5} \right );\,\sin \left (\frac{2\left ( i-1 \right ) \pi }{5} \right ) \right ]. \end{aligned} \end{equation}

The design parameters in our controllers are chosen as: $\varrho _{i,1,1}=5$ , $\varrho _{i,1,2}=50$ , $\kappa _{j,1}=2$ , $\zeta _{i,1}=0.8$ , $\delta _{i,1}=0.9$ , $\kappa _{i,2}=2$ , $\zeta _{i,2}=0.8$ , $\varrho _{i,2,1}=20$ , $\varrho _{i,2,2}=150$ , $\delta _{i,2}=0.9$ , $\varpi _{i}=30$ , $\xi _{i,1}=1$ , $\xi _{i,2}=4$ , $\rho _{i,1}=30$ , and $\rho _{i,2}=40$ , where $i,j=1,\cdots, 5$ . The initial values of ESOs and TDs are all zero.

Simulation results for the distributed ADRC controller (48) are presented in Figs. 6–8. Figure 6 plots the trajectory and formation process of the MASs and the initial and final positions of all agents. It can be clearly seen that all agents can follow the trajectory of the leader and eventually form the predetermined formation and maintain the formation at the velocity of the leader.

Figure 6. Trajectory of the leader and followers.

Figure 7. Control input of the followers in X-direction.

Figure 8. Control input of the followers in Y-direction.

The saturated input signals for each agent in the X and Y directions are shown in Figs. 7 and 8. During the tracking control process, the control signals are always maintained between $u_{\text{max}}$ and $-u_{\text{max}}$ . It is interesting to see that no larger control effort is required from our control approach compared with that from ref. [Reference Tee, Ge and Tay48]. Figures 9 and 10 show the observation effects of the ESO designed for Node 1. It can be seen that, in a very short time, the output of the ESO $\phi _{1,1,2}$ and $\phi _{1,2,2}$ can accurately estimate the nonaffine nonlinearity function $\tilde{F}_{1,1,1}$ and $F_{1,2,1}$ and provide internal compensation for the unknown nonlinear term of the closed-loop system. The curves show that the ESO designed in this paper can deal with external disturbance and nonaffine nonlinearity well.

Figure 9. The performance of the ESO in the first step.

Figure 10. The performance of the ESO in the second step.

The curves of the error and the predefine performance function are plotted in Figs. 11 and 12. In the figures, the red and pink dashed lines represent the trajectories of the prescribed performance function, and the blue dashed lines represent the steady state of Eq. (7). It can be seen that with the controller (48) proposed in this paper, the formation tracking errors $e_{i}$ of agents can quickly converge in a finite time and are bounded by the decaying function, which is predefined before the control design. Therefore, the prescribed performance can be guaranteed by the proposed control scheme. It is worth noting that the formation errors are preserved within the prescribed performance bounds for all cases and that our proposed asymptotically formation tracking control scheme has higher tracking precision than the uniformly ultimately bounded formation tracking control scheme in ref. [Reference Tee, Ge and Tay48].

Figure 11. Formation tracking error in X-direction.

Figure 12. Formation tracking error in Y-direction.

To evaluate the transient response to the sensitivity to disturbance and performance of controllers, we introduce integral squared errors (ISE) defined by Eq. (69), which is used to verify the formation tracking accuracy of the system under PPC. Table I shows the settling time, overshoot, control input, ISE, and tracking effect of the different control algorithms under the condition of uncertainty interference.

(69) \begin{equation} \begin{aligned} \text{ISE}=\int _{t_{1}}^{t_{2}} e_{x}^{2} ( t )+e_{y}^{2} ( t )dt \end{aligned} \end{equation}

where $e_x,e_y$ are the tracking errors for MASs in $x$ and $y$ coordinates, respectively.

Table I. Performance comparisons of various formation control algorithms.

By analyzing the results of each control method in formation tracking performance in Table I, it can be seen that the ADRC designed in this paper with predefined performance and saturated input can not only make the nonaffine nonlinear MASs form the reference formation successfully but also track the lead agent trajectory in any initial states within a finite time. Moreover, it performs better than other control algorithms in adjusting time, overshoot, control input, ISE, and tracking uncertain interference. At the same time, compared with ref. [Reference Yang, Si, Yue and Tian3], the controller designed in this paper sacrifices the estimation time of uncertain interference and greatly reduces the control input and cost of the system. In general, the controller designed in this paper shows very good performance in terms of response speed, convergence speed, and control input for MASs formation tracking when the interference is determined.