Adaptive Fixed-Time Control of Nonlinear MASs With Actuator Faults

I. Introduction

THE consensus control of multi-agent systems (MASs) has gained much attention in various fields in recent decades [1]–[13]. For consensus control, all followers in MASs should track the trajectory of leader based on messages from their local neighboring agents. The consensus control problem of high-order MASs is challenging. To solve this problem, the backstepping method was used for high-order MASs [14]. In addition, neural networks (NNs) and fuzzy logic systems (FLSs) are usually utilized to deal with nonlinear functions in MASs [15]–[20]. For example, by combining the backstepping technique and FLS, the adaptive finite-time consensus controllers were obtained for high-order nonlinear MASs with unknown nonlinear dynamics in [21]. Although backstepping is an effective method for high-order MASs, it will bring the “explosion of complexity” problem owing to repeated derivation of virtual control signals.

The dynamic surface control (DSC) method can be used to deal with the “explosion of complexity” problem. By constructing the first-order filter and using a virtual signal as the input signal, the output of filter can be used to approximate the derivative of the virtual signal. For example, by combining the DSC approach and minimal learning parameter technique, the adaptive control method was proposed for nonlinear systems [22], in which the “explosion of complexity” problem was avoided. Although the work in [22] can solve the above problem, the initial values of virtual signals were needed in these methods which are hard to obtain in practical systems. To remove the restrictions, the first order sliding mode differentiator was used, which ensured that the filtered tracking errors were bounded, and the initial values were not needed [23] and [24]. Based on the sliding mode differentiator, the tedious analytic computation problem was avoided in [23], and the state observer and disturbance observer were designed simultaneously to estimate unmeasured states and compounded disturbances. In [24], the sliding mode differentiator was utilized in the DSC strategy such that the “explosion of complexity” problem was handled, and the difficulties caused by unknown control directions and state constraints were well addressed.

The above results can only make all error signals converge to steady states when time goes to infinity. With a higher convergence rate, the finite-time control was studied in many practical applications [25]–[29]. Based on the finite-time control theory, the finite-time containment control method of nonlinear MASs was proposed in [30]. A fuzzy finite-time command filtering output feedback control method was presented for a class of nonlinear systems [31]. However, the convergence time of finite-time control depends on the initial condition of the system, which means that the convergence time may be unbounded if the initial condition is too far from the steady states. To overcome such a problem, the stability of fixed-time convergence was first considered in [32]. Different from the finite-time control strategy, the fixed-time control method ensures that the settling time is bounded by a constant, which avoids the influence caused by initial conditions. In [33], the fixed-time consensus of nonlinear MASs was achieved, and the authors proved that the convergence time of fixed-time results is independent of initial conditions. Moreover, a novel fixed-time stability criterion was proposed for high-order MASs in [34], and it was guaranteed that the consensus tracking errors were driven to a small neighborhood of zero in fixed time. Although there are many works on fixed-time control, the adaptive fixed-time control of nonlinear MASs subject to actuator failures has not been fully studied, and the “explosion of complexity” problem has not been well solved.

Motivated by the above results, an adaptive fixed-time control scheme is proposed for a class of nonlinear MASs with actuator failures based on a first order sliding mode differentiator. The main contributions of this paper are summarized as follows.

1) The adaptive fixed-time stability criterion is extended to a more general class of nonlinear MASs with actuator faults. Compared with [35]–[37], this paper proposes a more helpful tool for solving the problem of approximation-based adaptive fixed-time control for nonlinear uncertain systems.

2) A DSC method employing the sliding mode differentiator is used in this paper, it can avoid the tedious analytic computation and “explosion of complexity” problem simultaneously in the conventional backstepping method. Different from the filtering methods in [31] and [38], a filtering error compensation mechanism is proposed in this paper. Compared with the above compensation mechanisms, it not only makes the filtering errors bounded in fixed time, but also easier to design.

3) In addition, an actuator failure model with gain fault, bias fault and unknown control direction is constructed, and the Nussbaum function is used to build the actuator fault compensation mechanism. The proposed control strategy ensures that all error signals in MASs are bounded and the consensus is achieved in fixed time.

The organization of this paper is as follows. In Section II, preliminaries and the dynamic model of MASs with actuator failures are given. In Section III, the process of controller design and stability analysis are given. Simulation results are shown in Section IV. Finally, conclusions are given in Section V.

II. Preliminaries

In this section, some basic knowledge about graph theory, radial basis function (RBF) NNs, and fixed-time convergence are introduced. Some lemmas are provided for designing the following fixed-time control.

A Graph Theory

Let $G = (N, V, A)$ be a digraph, where $N = \left\{n_{1}, \ldots, n_{M}\right\}$ denotes the node set, $V \subseteq N \times N$ is the edge set, and $A = \left[a_{h m}\right] \in {\mathbb{R}}^{M \times M}$ is the adjacency matrix with $a_{h m}$ being the element of A. $a_{h m} = 1 \; (h\neq m)$ if and only if the information passes from node m to h, otherwise, $a_{h m} = 0$. If there exists an edge from agent h to agent m, then $({n_h}, {n_m}) \in V$ and $M_h = \{ m| {n_h}, {n_m}) \in V\}$ denotes the neighbor set of agent h. Define the in-degree matrix as $B = \text{diag}(b_1, \ldots, b_M)$ with $b_h = \sum_{m\in M_h}a_{h m}$. The Laplacian matrix L is defined by $L = B-A$. Furthermore, the digraph has a direct path between nodes $n_{h}$ and $n_{m}$, if there is a sequence of edge $\left\{\left(n_{h}, n_{r}\right), \left(n_{r}, n_{v}\right), \ldots, \left(n_{t}, n_{m}\right)\right\} .$ Let the leader be considered so the digraph is rewritten by the augmented graph $\bar{G} = (\bar{N}, \bar{V})$, where $\bar{N} = [0, n_1, \ldots, n_M]$ and $0$ means the node of the leader. Let $c_h = 1$ if the message passes from node $0$ to h, otherwise, $c_h = 0$.

B RBFNNs

Based on [37] and [39], RBFNNs are utilized to approximate $\phi_{h, s}\left(\bar{x}_{h, s}\right)$ over a compact set $\Omega_X \subset {\mathbb{R}}^q$ as

where $W^{*}$ is the ideal constant weight vector which is given as $W^* = \text{arg} \min \nolimits_{W\in \bar{{\mathbb{R}}}^l} \left\{ \sup\nolimits_{X\in \Omega_X} |\phi_{h, s}\left(\bar{x}_{h, s}\right)- W^{T}\varphi(X)|\right\}$, and $\varepsilon(X)$ is the approximation error. There exists a positive constant $\varepsilon^{*}\geq |\varepsilon(X)|$ for all $X \in \Omega_X$. The weight vector is $W= \left[W_{1}, \ldots, W_{\ell}\right]^{T} \in {{\mathbb{R}}}^{\ell}$, the RBFNNs node number is $\ell$ and $\ell>1$, the input vector is $X\in \Omega_X \subset {\mathbb{R}}^q$, and the Gaussian basis function vector is $\varphi(X) = \left[\varphi_{1}(X) , \ldots, \varphi_{\ell}(X)\right]^{T}$. Gaussian basis function is defined as

where $\mu_{i} = [\mu_{i 1}, \ldots, \mu_{i q}]^{T}$ and $\eta_{i}$ express the center and the width of the Gaussian function, respectively.

C Fixed-Time Convergence

In fixed-time consensus, if there exists a bounded constant T satisfying $\lim_{t\rightarrow T} x(t) = 0$ for all time $t\geq T$, and T is not associated with initial conditions, the multi-agent systems (1) are fixed-time stable.

Lemma 1 [34]: A system model is considered as follows:

where $\phi(x): {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}$ denotes a smooth vector field, where x is the state vector. Define a positive definite scalar smooth function $V(x)$. If the following inequality holds:

where $c_1, \;c_2,\; \mu,\; \beta, \;\rho$ and κ are positive constants and satisfy $\mu \kappa \geq 1$, $0<\beta \kappa<1$. Let $\Omega_1 = \left\{ x| V(x)\leq 1\right\} , \;\Omega_{a_0} = \{ x| V(x)\leq ( \frac{\rho}{a_0})^{\frac{1}{\mu \kappa}} \}$, and $\Omega_{b_0} = \{ x| V(x)\leq ( \frac{\rho}{b_0})^{\frac{1}{\beta \kappa}}\}$. The following conclusions can be obtained.

1) $x(t)\in \Omega_{a_{0}}$ if $\Omega_{a_{0}}$ and $\Omega_{b_{0}}$ belong to $\Omega_{1}$, the settling time is

2) $x(t)\in (\Omega_{a_{0}} \bigcap \Omega_{b_{0}})$ if at least one of $\Omega_{a_{0}}$ and $\Omega_{b_{0}}$ is outside $\Omega_{1}$, and the settling time is

where

Lemma 2 [34]: Let $a_{l}\geq 0$, $F \in Z_{+}$. For $0<\kappa \leq 1$, the following inequality holds:

If $\kappa>1$, one has

Lemma 3 [40]: Define $\tilde{\theta} = \theta-\hat{\theta}$, where θ and $\hat{\theta}$ are two scalar functions. Let q and p be odd integers. If $0 < c =$ $q/p \leq 1$, one has

where $c_{1} = \frac{1}{1+c}\left[1-2^{c-1}+\frac{c}{1+c}+\frac{2^{c}(1-c^{2})}{1+c}\right],\; c_{2} = \frac{2^{c-1}}{1+c}\big[1-2^{c(c-1)}\big].$

Lemma 4 [34]: Let $\theta>0$ be a constant and $\tilde{\theta}<\theta$. For any odd number $p>1,$ the following holds:

Lemma 5 [34]: Consider a differential equation as

where $\hat{\theta}(t)\in {\mathbb{R}},$ ω and σ are positive constants, $\mu =$ $p_{1} / q_{1}, v = q_{2} / p_{2} > 0.5$ with $p_{1}>q_{1}$ and $p_{2}>q_{2}$ being positive odd numbers, $\varrho(t)$ is a positive function. If $\hat{\theta}\left(t_{0}\right) \geq 0$, $\hat{\theta}(t) \geq 0$ holds for $\forall t \geq t_{0}$.

Lemma 6: Define $\tilde{\theta} = \theta-\hat{\theta}$, where θ and $\hat{\theta}$ are two smooth functions. Let q and p be odd integers and $c = q/p$. There exists a positive constant κ such that

Proof: If $\theta = 0$, $\kappa\tilde{\theta}(-\tilde{\theta})^{c}+\kappa\tilde{\theta}^{1+c} = -\kappa\tilde{\theta}^{1+c}+\kappa\tilde{\theta}^{1+c} = 0$ with $c = q/p$, and q and p as odd integers, one has $\theta^{1+c} \geq \kappa(\tilde{\theta}({\theta}- \tilde{\theta})^{c}+\tilde{\theta}^{1+c})$. If $\theta\neq 0$, ${\theta}^{1+c}> 0$ holds, and the inequality $\theta^{1+c} \geq \kappa(\tilde{\theta}({\theta}-\tilde{\theta})^{c}+\tilde{\theta}^{1+c})$ is satisfied while κ is small enough.

Therefore, $\kappa\tilde{\theta}({\theta}-\tilde{\theta})^{c} \leq \theta^{1+c}-\kappa\tilde{\theta}^{1+c}$ always holds while κ is small enough.■

D Nussbaum Function

The Nussbaum technique is utilized to solve the problem of unknown control coefficients [41]. The Nussbaum-type function $N(\varphi)$ is defined as

In this paper, we choose the Nussbaum gain function as $\varphi^2\sin(\varphi)$.

Lemma 7 [39]: Define $V(t, x)$ and $\varphi(t)$ as smooth functions on $[0, t_{f})$ with $V(t, x) \geq 0$, and $N(\varphi)$ as a Nussbaum-type function. If the inequality satisfies

where E and F represent positive constants. $g_h$ stands for a unknown constant, then $V(t, x), \;\varphi(t)$ and $(g_h N'(\varphi)+1)\dot{\varphi}$ are bounded on $[0, t_{f})$.

III. Problem Formulation and Main Results

A Problem Formulation

A leader-follower MAS is considered in this paper, where one of them is the leader (marked with d) and the others are followers (marked with h from 1 to $M\; (M \geq 2)$). The dynamics of the followers are described as:

where $x_{h} = [x_{h, 1}, x_{h, 2}, \ldots, x_{h, n}]^{T} \in {\mathbb{R}}^{n}$, $\bar{x}_{h, s} = [x_{h, 1}, \ldots, x_{h, s}]^{T} \in {\mathbb{R}}^{s}$, $1 \leq s \leq n-1$, denote the states, $u_{h f} \in {\mathbb{R}}$ and $y_{h} \in {\mathbb{R}}$ represent the control output signal and the system output signal, respectively. $\beta_h$ is an unknown constant. ${\phi}_{h, s}$ $\left(1 \leq s \leq n\right)$ are unknown system continuous functions and satisfy ${\phi}_{h, s}(0) = 0$. According to [39], the actuator fault is modeled as follows:

where $u_h(t)$ is a control input signal, $s_h \in (0, 1]$ is the unknown actuation effectiveness and $w_h(t)$ is a time-varying and bounded function.

Control Objective: The control objective of this paper is to design an adaptive neural control scheme with an error compensation mechanism for nonlinear MASs subject to actuator failures, and to guarantee that all signals in the closed-loop system are bounded and the consensus of MASs is achieved in fixed time.

B Adaptive Controller Design

In this paper, at the ${s}{{\rm{th}}}\; (1\leq {s}\leq n)$ step of backstepping design, RBFNNs $\Psi_{h, s}^T\Xi_{h, s}$ are utilized to approximate unknown functions. An unknown constant $\theta_{h, s} = \max \{\epsilon_{h, s}, \|\Psi_{h, s}\| \}$ is approximated by a designed adaptive parameter $\hat{\theta}_{h, s}$, and its update law $\dot{\hat{\theta}}_{h, s}$ will be given later. Symbol $\|\cdot\|$ denotes the Euclidean norm. Let the estimation error be $\tilde{\theta}_{h, s} = {\theta}_{h, s}-\hat{\theta}_{h, s}$. In addition, we set

as the consensus error of $h{{\rm{th}}}$ follower. The output signal of the leader is defined as $y_d\in {\mathbb{R}}$. Define $\xi_{h, i} = x_{h, i}-x_{h, ic}$ for $i = 2, \ldots, n$.

Step 1: For the $h{{\rm{th}}}$ follower, the Lyapunov function is designed as below:

with $\eta_{h, 1}$ being a designed positive constant. Differentiating both sides of $(4)$, we get

From $(1)$ and $(3)$, one obtains

where $\bar{\phi}_{h, 1}(X_{h, 1}) = d_h \phi_{h, 1}(x_{h, 1}) -\sum_{m \in M_{h}} a_{h m} \phi_{m, 1}(x_{m, 1})$ with $d_h = b_h+c_h$ and $X_{h, 1} = \left[x_{h, 1}, x_{m, 1}\right]^{T}$. From the definition of graph theory, $b_{h}>0$ and $c_{h} \in \{0, 1\}$ always hold such that $d_h>0$. Combining $(5)$ with $(6)$, we get

In this part, RBFNNs are utilized to approximate unknown function $\bar{\phi}_{h, 1}$ as

where $\delta_{h, 1}(X_{h, 1})$ stands for the RBFNNs approximation error, and the inequality $|\delta_{h, 1}(X_{h, 1})| \leq \epsilon_{h, 1}$ is satisfied for any given accuracy $\epsilon_{h, 1}>0$. For $\forall \varsigma>0,$ one has

with $\theta_{h, 1} = \max\{\epsilon_{h, 1}, \|\Psi_{h, 1}\|\}$, $\vartheta_{h, 1} = 1+\|\Xi_{h, 1}\|$, $\varepsilon = 0.2785$. Substituting $(9)$ into $(7)$ yields

Therefore, the control signal $x_{h, 2c}$ of the $h{{\rm{th}}}$ subsystem is

where $\mu = p_{1}/q_{1}>1$, $p_{1}, q_{1}$ are odd numbers and $p_{1}>q_{1}$, $\alpha_{h, 1}$ is designed as below to ensure that $\dot{x}_{h, 2c}$ exists in the next step.

where $f\left(\xi_{h, 1}\right) = g_{h, 11} \xi_{h, 1}+g_{h, 12} \xi_{h, 1}^{3},\; k_{h, 11}$ and $k_{h, 12}$ are positive design parameters. Let $\beta = q_{2}/p_{2} \; (0.5<\beta<1),$ where $p_{2}$ and $q_{2}$ are odd numbers and satisfy $p_{2}>q_{2}$. Similar to [34], the curve fitting method is utilized to ensure that $\alpha_{h, 1}$ is derivable when $\xi_{h, 1} = \pm\epsilon_{h, 1}$, such that the following equation holds:

where $g_{h, 11} = -k_{h, 12} \epsilon_{h, 1}^{2 \beta-2}(2-\beta)$ and $g_{h, 12} = -k_{h, 12} \epsilon_{h, 1}^{2 \beta-4}(\beta-1)$. Substituting $x_{h, 2c}$ into $\dot{V}_{h, 1}$, we get

For $\left\|\xi_{h, 1}\right\|\geq\epsilon_{h, 1}$, in light of $(12)$, we have

For $\left\|\xi_{h, 1}\right\|<\epsilon_{h, 1}$, under the condition of $\xi_{h, 1}< \epsilon_{h, 1}<1$, we have

From $(14)$ and $(15)$, $\xi_{h, 1}\alpha_{h, 1}\leq -k_{h, 12} \xi^{2\beta}_{h, 1}$ is satisfied. Therefore, $(13)$ can be rewritten as below:

According to $(16)$, the adaptive update law $\dot{\hat{\theta}}_{h, 1}$ is given by

where $\sigma_{h, 11}$ and $\sigma_{h, 12}$ are positive design parameters. Thus, substituting $(17)$ into $(16)$ has

Step $s\; {\emph {(}}{\emph {2}}\leq s \leq n-{\emph {1}}{\emph {)}}$: At this step, the Lyapunov function $V_{h, s}$ is constructed as

where $\eta_{h, s}$ is a positive design parameter. Then, differentiating both sides of $(19)$, we get

where $\ell_{h, s} = d_h$ while $s = 2$, and $\ell_{h, s} = 1$ while $2<s\leq n$.

Considering the problem of tedious calculations, similar to [23], [24] and [42], the first order sliding mode differentiator is used to approximate $\dot{x}_{h, sc}$

where $\varrho_{h, sj}$ and $l_{h, s}$ are system states of the differentiator, and $R_{h, sj}$ are positive design constants $(j = 0, 1)$. Meanwhile, $\varrho_{h, s0}$ and $l_{h, s}$ are the estimations of $x_{h, sc}$ and $\dot{x}_{h, sc}$, respectively. Therefore, we have

where $p_{h, s}$ stands for the estimated error of differentiator.

Remark 1: In addition, to ensure that the estimated error $p_{h, s}$ is fixed-time bounded, a novel filtering error compensation mechanism is designed as follows:

where $\tilde{p}_{h, s}$ is the compensation error and $\hat{p}_{h, s}$ is designed as an adaptive law to approximate filtering error $p_{h, s}$.

Therefore, $(20)$ can be rewritten as

Utilizing RBFNNs to estimate unknown function $\phi_{h, s}(X_{h, s})$, for $\forall \varsigma>0$, we have

where $\theta_{h, s} = \max\{\epsilon_{h, s}, \|\Psi_{h, s}\|\}$ and $\vartheta_{h, s} = 1+\|\Xi_{h, s}\|$. For the error function $\delta_{h, s}(X_{h, s})$, there exists $|\delta_{h, s}(X_{h, s})| \leq \epsilon_{h, s}$, where $\epsilon_{h, s}>0$ is a given accuracy.

Substituting $(25)$ into $(24)$, then, define

where $f\left(\xi_{h, s}\right) = g_{h, s1} \xi_{h, s}+g_{h, s2} \xi_{h, s}^{3},\; k_{h, s1}$ and $k_{h, s2}$ are positive design parameters. Let $g_{h, s1} = -k_{h, s2} \epsilon_{h, s}^{2 \beta-2}(2- \beta)$ and $g_{h, s2} = -k_{h, s2} \epsilon_{h, s}^{2 \beta-4}( \beta-1)$, such that $\xi_{h, s}\alpha_{h, s} \leq -k_{h, s2}\xi^{2\beta}_{h, s}$ holds.

Substituting $(25)$ and $(26)$ into $(24)$, we have

According to $(28)$, the adaptive update law $\dot{\hat{\theta}}_{h, s}$ is given by

where $\sigma_{h, s1}$ and $\sigma_{h, s2}$ are positive design parameters. Therefore, $(28)$ is rewritten as

Step n: For a positive constant $\eta_{h, n}$, the Lyapunov function $V_{h, n}$ is designed as

Utilizing RBFNN to approximate $\phi_{h, n}(X_{h, n})$, $\dot{V}_{h, n}$ is shown as follows:

where $g_h = \beta_h s_h$, $\theta_{h, n} = \max\{ \epsilon_{h, n}, \|\Psi_{h, n}\|\}$ and $\vartheta_{h, n} = 1+\|\Xi_{h, n}\|$, and $\bar{w}_h \geq |s_hw_h|\; (1\leq h\leq M)$.

According to $(32)$, the control input signal $u_h$ and adaptive update law $\dot{\hat{\theta}}_{h, n}$ are given as

where $k_{h, nj}$ and $\sigma_{h, nj}$ are positive design constants $(j = 1, 2)$, and the Nussbaum function $N(\varphi_h)$ is chosen as $\varphi_h^2 \sin(\varphi_h)$.

Substituting $(33)-(35)$ into $(32)$, we have

Define $D = \left(g_hN(\varphi_h)+1\right)\dot{\varphi}_h$, such that $(36)$ can be rewritten as

Based on Lemma 3, we have

where $c_{1} = \frac{1}{2\beta}\left(1-2^{2\beta-2}+\frac{2\beta-1}{2\beta}+\frac{2^{2\beta-1}\left(1-(2\beta-1)^{2}\right)}{2\beta}\right)$, $c_{2} = \frac{2^{2\beta-2}}{2\beta}\big(1- 2^{(2\beta-1)(2\beta-2)}\big)$.

Before utilizing Lemma 4, $\hat{\theta}_{h, s} = {\theta}_{h, s}-\tilde{\theta}_{h, s}>0$ should be guaranteed. For $\hat{\theta}_{h, s}\left(0\right) \geq 0$ and $\forall t>0$, Lemma 5 shows that $\hat{\theta}_{h, s}(t)>0$ holds with adaptive law $\dot{\hat{\theta}}_{h, s}$ $(17)$, $(29)$ and $(35)$.

Therefore, from Lemma 4, we can get

Substituting $(38)$ and $(39)$ into $(37)$ yields

where ${r}_{h} = \sum_{s = 1}^{n} \frac{\sigma_{h, s1}}{\eta_{h, s}} \theta_{h, s}^{2 \mu}+\sum_{s = 1}^{n} \frac{\sigma_{h, s2}}{\eta_{h, s}} c_{1} \theta_{h, s}^{2 \beta} +\sum^{n}_{m = 1} \varepsilon \varsigma \theta_{h, m}+\frac{\bar{w}_{h}^2}{2}$.

Theorem 1: For the MASs $(1)$ with actuator faults $(2)$, under the first order sliding mode differentiator $(21)$, the error compensation mechanism $(23)$, the control input signal $(34)$, adaptive laws $\dot{\hat{\theta}}_{h, s}$, and the adaptive update laws

where $\gamma_{h, s}$ and $\tau_{h, sj} (j = 1, 2)$ are positive design parameters, it can guarantee that the tracking errors $\xi_{h, 1} (h = 1, \ldots, M)$ converge to a small neighborhood of origin and all signals are fixed-time bounded in the closed-loop system.

Proof: For the compensating mechanism $(23)$, choose the Lyapunov function $\bar{V}_n$ as

Differentiating $\bar{V}_h$, one gets

Substituting $(41)$ into $(43)$, we have

Define $\kappa_{h, s1}$ and $\kappa_{h, s2}$ as positive design parameters. By utilizing Lemma 6, we get

Substituting $(45)$ into $(44)$, $\dot{\bar{V}}_{h}$ is rewritten as

Choose $V_{h} = V_{h, n}+{\bar{V}}_{h}$ as the Lyapunov function candidate of $h{{\rm{th}}}$ MASs $(1)$. Differentiating $V_{h}$, $\dot{V}_{h}$ can be calculated as

where $\varpi_{h, 1} = \min \{2^{\mu} k_{h, s1}, 2^{\mu} \eta_{h, s}^{\mu-1} \sigma_{h, s1}, 2^\mu\gamma_{h, s}^{\mu-1}\tau_{h, s1} \}$, $\varpi_{h, 2} =$ $\min \left\{2^{\beta} k_{h, s2},\;\;\; 2^{\beta}c_2\eta_{h, s}^{\beta-1} \sigma_{h, s2},\;\; \;2^\beta\gamma_{h, s}^{\beta-1}\tau_{h, s2} \right\}$ and $\bar{r}_{h} =\sum^n_{s = 2}\times$ $\frac{\tau_{h, s1}}{\kappa_{h, s1}\gamma_{h, s}}p^{2\mu}_{h, s} + \sum^n_{s = 2}\frac{\tau_{h, s2}}{\kappa_{h, s2}\gamma_{h, s}}p^{2\beta}_{h, s} + r_{h}$.

Applying Lemma 2 to $(47)$, one shows

where $\bar{\varpi}_{h, 1} = \varpi_{h, 1}n^{1-\mu}3^{1-\mu}$, and

Further, substituting $(48)$ and $(49)$ into $(47)$, we have

From $(50)$, on one hand, if $V_h \geq 1$, one has

On the other hand, if $0\leq V_h<1$, we get $\dot{V}_{h} \leq - \varpi_{h, 2} V_{h} + \bar{r}_{h}+D$. By using Lemma 7, $|D|$ is bounded. Define $\bar{\lambda}_h$ as a positive constant and let $|D|<\bar{\lambda}_h$. For $\bar{D} = \bar{r}_{h}+\bar{\lambda}_h$, we get

Let $\kappa = 1$, $\Omega_{\xi_h} = \left\{ \xi_h | V_h(\xi_{h, s}) \leq \left( \frac{{r}_h}{\bar{\varpi}_{h, 1}}\right)^{\frac{1}{\mu}} \right\}$, $\Omega_{\varepsilon_h} = \left\{ \xi_h| \xi_{h, s} \leq \varepsilon_{h, s} \right\}$ and $\Omega_h = \max\{\Omega_{\xi_h}, \Omega_{\varepsilon_h} \}$. From Lemma 1, $(52)$ shows that the tracking error $\xi_{h, 1}$ and all error signals $\xi_{h, s}$ will converge to $\Omega_h$ in fixed time.

Remark 2: Compared with Lemmas 3 and 4, Lemma 6 can be used in the both conditions $0<c\leq 1$ and $p>1$. Moreover, the limitations where $\theta>0$ and $\tilde{\theta}<\theta$ are removed. However, utilizing Lemma 6 will create a lager $\bar{D}$ in $(52)$, which makes the stability analysis of closed-loop system more conservative. ■

IV. Simulation Examples

In this section, two examples are provided to verify the effectiveness of fixed-time command filtering adaptive neural controller. In simulation examples, the communication graph G of MASs is shown in Fig. 1. Consider the MASs with a leader and 4 followers.

Figure  1.  Directed graph G.

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Example 1: Consider the model of $h{{\rm{th}}}(h = 1, \ldots, 4)$ agent

where $u_{if} = u_i\; (i = 1, 2, 3)$, and

The trajectory of leader is defined as $y_d = \sin(0.4t)+\sin(0.8t)$, the filter parameters are $R_{h, 10} = 20, R_{h, 11} = 5$ and other parameters are shown in Table I.

Table  I.  Selection of Parameters $(h = 1, \ldots, 4)$

Initial conditions
$x_{h, 1}(0) = 0$, $x_{h, 2}(0) = 1$, $\hat{\theta}_{h, 1}(0)=\hat{\theta}_{h, 2}(0)=0.1$
$\varrho_{h, 10}(0)=\varrho_{h, 11}(0)=0$, $\hat{p}_{h, 2}(0)=-0.5$, $\phi_{h}(0)=4$
Controller parameters $(s=1, 2)$
$k_{h, s1}=k_{h, s2}=2$, $\gamma_{h, 2}=50$, $\tau_{h, 21}=\tau_{h, 22}=0.01$, $\varepsilon_{h, 1}=0.01$
$\eta_{h, s}=5$, $\sigma_{h, s1}=\sigma_{h, s2}=0.05$, $\varsigma=0.01$, $\mu=\frac{9}{7}$, $\beta=\frac{9}{11}$

 | Show Table

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Simulation results are shown in Figs. 2-7. The trajectories of output signal $y_{h}$ and reference signal $y_d$ are shown in Fig. 2. Fig. 3 shows the tracking error $e_{h}$. From Fig. 3 it can be seen that, after 20 s, tracking error $e_4$ rises temporarily when actuator failure occurs and returns to a small neighbourhood of the origin again, which shows that the actuator fault-tolerant control works in the systems. Compared with Fig. 2, Fig. 4 shows that the systems will be out of control without the actuator failure compensation. In addition, the filtering error compensation adaptive law $\hat{p}_{h, 2}$ is shown in Fig. 5, the adaptive laws $\hat{\theta}_{h, j}\; (j = 1, 2)$ are shown in Fig. 6, and the control input signal $u_h$ is shown in Fig. 7. With $\hat{p}_{h, 2}$, $\hat{\theta}_{h, j}$ and $u_h$, the fixed-time consensus of MASs $(1)$ is achieved.

Figure  2.  The trajectories of followers $y_{h}\; (1\leq h\leq 4)$ and the leader $y_{d}$.

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Figure  3.  The tracking errors of followers $e_h\; (1\leq h\leq 4)$.

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Figure  4.  The tracking errors of followers $e_h\; (1\leq h\leq 4$) without actuator failure compensation.

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Figure  5.  The trajectories of filtering error compensation adaptive parameters $\hat{p}_{h, 2}\; (1\leq h\leq 4)$.

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Figure  6.  The trajectories of adaptive parameters $\hat{\theta}_{h, j}\; (1\leq h\leq 4, j = 1, 2)$.

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Figure  7.  The trajectories of control inputs $u_{h}\; (1\leq h\leq 4)$.

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Example 2: A practical example is shown to further illustrated the proposed control method. Consider the motion of i-th marine surface vehicle, it has the following nonlinear Norrbin dynamical model with stable line-movement:

where $\dot{W}_{i}$ is the course angle, $\nu_{i}$ and $r_{i}$ show the course angle rate and command rudder angle with $i = 1, 2, 3, 4$. N represents the Norrbin coefficient. $T_{i}$ and $\tau_{i}$ are defined as time and gain constants, respectively. Then, by defining and choosing parameters, one has

where $u_{1f} = u_1,\; u_{2f} = u_2+0.2\cos(t),\; u_{3f} = 0.9u_3$, and

Define the trajectory of leader as $y_d = \sin(t)$, the filter parameters are $R_{h, 10} = 20$, $R_{h, 11} = 5$, and other parameters are shown in Table II.

Table  II.  Selection of Norrbin Model Parameters

Initial conditions
$x_{h, 1}(0) = x_{h, 2}(0) = 0$, $\hat{\theta}_{h, 1}(0)=\hat{\theta}_{h, 2}(0)=0.01$
$\varrho_{h, 10}(0)=\varrho_{h, 11}(0)=0$, $\hat{p}_{h, 2}(0)=0$, $\phi_{h}(0)=8$
Controller parameters
$k_{h, s1}=2, k_{h, s2}=5$, $\gamma_{h, 2}=50$, $\tau_{h, 21}=\tau_{h, 22}=0.01$,
$\varepsilon_{h, 1}=0.01$, $\eta_{h, s}=5$, $\sigma_{h, s1}=\sigma_{h, s2}=0.05$, $\varsigma=0.01$,
$\mu=\frac{9}{7}$, $\beta=\frac{9}{11}$

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The algorithm results based on this model are shown in Figs. 8 and 9. From Fig. 8, with the filtering error compensation mechanism and actuator fault-tolerant control mechanism, the output signals can track the reference signal even if actuator failures occur, and all signals are bounded in fixed time. In addition, when considering the communication constraints and time delay, the consensus problem of MASs will become more challenging [43], [44].

Figure  8.  The trajectories of followers $y_{h}\; (1\leq h\leq 4)$ and the leader $y_{d}$.

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Figure  9.  The trajectories of control inputs $u_{h}\; (1\leq h\leq 4)$.

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V. Conclusions

The fixed-time consensus problem of nonlinear MASs with actuator failures has been studied in this paper. RBFNNs have been used to approximate the unknown functions. Command filtering has been utilized to solve the computation complexity problem. Furthermore, a novel fitering error compensation mechanism has been proposed to make filtering errors converge in fixed time. To avoid the effects of actuator failures, an actuator fault-tolerant control method based on Nussbaum gain theory has been utilized. Finally, the designed adaptive RBFNNs fixed-time command filtering controller guarantees that all signals are fixed-time bounded, and the effectiveness of the proposed control method has been verified by a numerical example and a practical example. In future work, the optimization algorithm for MASs will be considered to further improve control performance [45], [46]. Furthermore, human-in-the-loop control is also an important method to improve the control scheme of MASs. For example, a novel human-in-the-loop control framework was developed, which was a major breakthrough and had important reference value [47]–[49].