Distributed Event-triggered Control of Networked Strict-feedback Systems Via Intermittent State Feedback

A directed graph among $N$ agents is denoted by $\mathcal{G}=\{\mathcal{V},\mathcal{E},\mathcal{A}\}$, which consists of a set of nodes $\mathcal{V}=\{v_{1},\cdots,v_{N}\}$, a group of edges $\mathcal{E}$ and a weighted adjacency matrix ${\mathcal{A}}=[a_{ij}]\in\mathcal{R}^{N\times N}$. If there is an edge $(i,j)$ between nodes $i$ and $j$, then $i$ and $j$ are called adjacent, i.e., $\mathcal{E}=\{(i,j)\in\mathcal{V}\times\mathcal{V}:i,j\,\,{\rm{adjacent}}\}$. An edge $(v_{i},v_{j})\in\mathcal{E}$ suggests that node $j$ can receive information from node $i$. For adjacency matrix $\mathcal{A}$, if $(v_{i},v_{j})\in\mathcal{E}$, then $a_{ij}>0$, otherwise, $a_{ij}=0$. $\mathcal{B}={\rm{diag}}\{\mu_{1},\cdots,\mu_{n}\}$ is the leader adjacency matrix, where $\mu_{1}>0$ if there is a directed edge from the leader to the agent i, and $\mu_{1}=0$ otherwise. It is assumed that there are no self-loops or parallel edges in the network. If there is a path from $i$ to $j$, then $i$ and $j$ are called connected. If all pairs of nodes in $\mathcal{G}$ are connected, then $\mathcal{G}$ is called connected. $\mathcal{N}_{i}=\{v_{j}\in\mathcal{V}|{\left(v_{j},v_{i}\right)}\in\mathcal{E},i\neq j\}$ is the indices set of neighbors of node $v_{i}$. The Laplacian matrix $\mathcal{L}=[l_{ij}]\in\mathcal{R}^{N\times N}$ is defined as $l_{ii}=\sum_{j=1,j\neq i}^{N}a_{ij}$ and $l_{ij}=-a_{ij},i\neq j$, satisfying $\mathcal{L}=\mathcal{D}-\mathcal{A}$, where $\mathcal{D}={\rm{diag}}(d_{1},\cdots,d_{N})\in\mathcal{R}^{N\times N}$ is the absolute in-degree matrix with $d_{i}=\sum_{j\in N_{i}}a_{ij}$.

Consider a multiple agent system composed of $N\,(N\geq{2})$ agents under a directed topology $\mathcal{G}$, with the $i$th, $i=1,\cdots,N$ agent modeled as:

	$\displaystyle{{\dot{x}}_{i,k}}=\,$	$\displaystyle{x_{i,k+1}}+{f_{i,k}}\left({{x_{i,1}},\cdots,{x_{i,k}}}\right),\,k=1,2,\cdots,n-1$
	$\displaystyle{{\dot{x}}_{i,n}}=\,$	$\displaystyle{u_{i}}+{f_{i,n}}\left({{x_{i,1}},\cdots,{x_{i,n}}}\right)$
	$\displaystyle{y_{i}}=\,$	$\displaystyle{x_{i,1}}$		(1)

where ${x_{i,k}}:\mathcal{R}_{+}\to\mathcal{R}$, $u_{i}:\mathcal{R}_{+}\to\mathcal{R}$, $y_{i}:\mathcal{R}_{+}\to\mathcal{R}$ are the states, control input and output of the $i$th agent, respectively, ${f_{i,k}}:\mathcal{R}^{k}\to\mathcal{R}$, $k=1,\cdots,n$ are unknown smooth nonlinear functions.

The mismatched nonlinearities ${f_{i,k}}\left({{x_{i,1}},\cdots,{x_{i,k}}}\right)$ are approximated by radial basis function neural network (RBFNN) over a suitable compact set ${\Omega_{\chi}}\in{\mathcal{R}^{\iota}}$, that is

$\displaystyle f_{i,k}(\chi_{i,k})=W_{i,k}^{T}\phi_{i,k}(\chi_{i,k})+\varepsilon_{i,k}{(\chi_{i,k})}$

(2)

where $\chi_{i,k}\in{\mathcal{R}^{\iota}}$ is the NN input vector, $W_{i,k}\in\mathcal{R}^{p}$ is the ideal weight matrix, $\phi_{i,k}(\chi_{i,k})=\left[\phi_{i,k1}(\chi_{i,k}),\cdots,\phi_{i,kp}(\chi_{i,k})\right]^{T}\in\mathcal{R}^{p}$ is the basis function vector, and $\varepsilon_{i,k}{(\chi_{i,k})}\in{\mathcal{R}}$ is the approximate error. The typical choice of $\phi_{i,kh}(\chi_{i,k})$, $h=1,\cdots,p$ is the Gaussian function $\phi_{i,kh}(\chi_{i,k})=\exp{\left[-{{(\chi_{i,k}-C_{h})^{T}(\chi_{i,k}-C_{h})}}/{{b_{h}^{2}}}\right]}$, where $C_{h}=[C_{h1},\cdots,{C_{h\iota}}]^{T}$ is the center of receptive field, $b_{h}$ is the width of the Gaussian function.

The objective of this paper is to develop a fully distributed neuroadaptive control scheme for system (1) by using intermittent state feedback such that

• $\bullet$

  All subsystem outputs closely track the desired trajectory $y_{0}(t)$, with the output tracking error converging to a residual set around zero.

• $\bullet$

  All signals in the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB).

• $\bullet$

  The Zeno behavior is excluded.

To this end, the following assumptions are needed.

Assumption 1. The directed graph $\mathcal{G}$ is balanced and weakly connected.

Assumption 2. The full knowledge of $y_{0}(t)$ is directly accessible by at least one subsystem, i.e., $\sum\nolimits_{i=1}^{N}{{\mu_{i}}>0}$, and $\dot{y}_{0}(t)$ satisfies $\left|{{{\dot{y}}_{0}}(t)}\right|\leq{Y_{0}}$, where $Y_{0}>0$ is an unknown constant.

Assumption 3. The basis function vector ${\phi_{i,k}(\chi_{i,k})}$ and the approximate error $\varepsilon_{i,k}{(\chi_{i,k})}$ satisfy $\left\|{\phi_{i,k}(\chi_{i,k})}\right\|\leq{\phi_{i,m}}$, $\left|{\varepsilon_{i,k}\left(\chi_{i,k}\right)}\right|\leq{\varepsilon_{i,m}}$, respectively, where ${\phi_{i,m}}$ and ${\varepsilon_{i,m}}$ are unknown positive constants.

The following lemma is introduced, which is crucial in the system stability analysis.

Lemma 1 [19]. Let $Q=\mathcal{L}+\mathcal{B}+(\mathcal{L}+\mathcal{B})^{T}$. Based on Assumption 1, the matrix $Q$ is symmetric and positive definite.

Remark 1. Assumption 1 is quite common in literature [16, 19]. Different from the results in [16, 19], as noted in Assumption 2, only the available information of ${y_{0}}(t)$ and its first derivative is needed, but not the high order derivatives. For Assumption 3, since the basis function vector $\phi_{i,k}$ of the RBNFF chosen in this paper (i.e., Gaussian function) is inherently bounded, and thus it is valid to assume that $\phi_{i,k}$ is bounded.

Denote ${{\bar{x}}_{i,k}}$ and ${{\bar{x}}_{j,k}}$, $i\in\{{0,\mathcal{V}}\}$, $j\in\mathcal{N}_{i}$ as the local states information (or the leader, i.e., $i=0$) and its neighboring states information, respectively, which are broadcast to their connected subsystems according to the designed triggering mechanism. Since $t_{k,l}^{i}$ and $t_{k,l}^{j}$ denote the $l$th event time for agent $i$ and its neighbor $j$ broadcasting their state information, respectively, it holds that the states of the agent $i$ and its neighbor $j$ are kept unchanged as ${{\bar{x}}_{i,k}}\left(t\right)={x_{i,k}}\left({t_{k,l}^{i}}\right)$, $\forall t\in[t_{k,l}^{i},t_{k,l+1}^{i})$, and ${{\bar{x}}_{j,k}}\left(t\right)={x_{j,k}}\left({t_{k,l}^{j}}\right),\forall t\in[t_{k,l}^{j},t_{k,l+1}^{j})$.

The triggering conditions are then designed as:

	$\displaystyle t_{k,l+1}^{i}=\inf\left\{{t>t_{k,l}^{i},\left|{{x_{i,k}}\left(t\right)-{{\bar{x}}_{i,k}}\left(t\right)}\right|>\Delta x_{k}^{i}}\right\}$			(13)
	$\displaystyle t_{k,l+1}^{j}=\inf\left\{{t>t_{k,l}^{j},\left|{{x_{j,k}}\left(t\right)-{{\bar{x}}_{j,k}}\left(t\right)}\right|>\Delta x_{k}^{j}}\right\}$			(14)

where $\Delta x_{k}^{i}$ and $\Delta x_{k}^{j}$ are positive triggering thresholds, $t_{k,0}^{i}$ is the first instant when (13) is fulfilled for subsystem $i$ (or the leader, i.e., $i=0$), and $t_{k,0}^{j}$ is the first triggering time for its neighbor $j$, $l=0,1,2,\cdots$, $k=1,\ldots,n$.

Firstly, we modify the tracking errors defined in (3)-(5) to the following form:

	$\displaystyle{\bar{z}_{i,1}}$	$\displaystyle={\bar{x}_{i,1}}-{{\hat{y}}_{i,0}}={{\varepsilon}_{i}}+{{\tilde{y}}_{i,0}}$		(15)
	$\displaystyle{\bar{z}_{i,k}}$	$\displaystyle={\bar{x}_{i,k}}-\bar{\alpha}_{i,kf},k=2,\cdots,n-1$		(16)
	$\displaystyle\bar{e}_{i}$	$\displaystyle=\sum\limits_{j=1}^{N}{{a_{ij}}}\left({{\bar{y}_{i}}-{\bar{y}_{j}}}\right)+{\mu_{i}}\left({{\bar{y}_{i}}-{\bar{y}_{0}}}\right)$		(17)

where ${{\varepsilon}_{i}}={\bar{x}_{i,1}}-{\bar{y}_{0}}$, ${\tilde{y}_{i,0}}={\bar{y}_{0}}-{\hat{y}_{i,0}}$, and ${{\bar{\alpha}}_{i,kf}}\left(t\right)={\alpha_{i,kf}}\left({{t_{1k,l}^{i}}}\right)$, $\forall t\in[t_{1k,l}^{i},t_{1k,l+1}^{i})$, $k=2,\cdots,n$, with

$\displaystyle t_{1k,l+1}^{i}{\rm{=}}$

$\displaystyle\inf\left\{{t>t_{1k,l}^{i},\left|{{\alpha_{i,kf}}\left(t\right){\rm{-}}{{\bar{\alpha}}_{i,kf}}\left(t\right)}\right|{\rm{>}}\Delta{\alpha_{kf}^{i}}}\right\}$

(18)

where $\Delta{\alpha_{kf}^{i}}$, $k=2,\cdots,n$, is the positive triggering threshold, and $t_{1k,0}^{i}$ is the first triggering time when (18) is fulfilled.

Based upon the intermittent state feedback, we redesign the fully distributed event-triggered control scheme as follows:

	$\displaystyle{{\bar{\alpha}}_{i,1}}=\,$	$\displaystyle-{\kappa_{1}}{{\bar{e}}_{i}}-\left({c_{i,1}}+1\right){{\bar{z}}_{i,1}}-\hat{W}_{i,1}^{T}{\phi_{i,1}}\left({{{\bar{\chi}}_{i,1}}}\right)+{{\dot{\hat{y}}}_{i,0}}$		(19)
	$\displaystyle{{\bar{\alpha}}_{i,k}}=\,$	$\displaystyle-\left({c_{i,k}}+1\right){{\bar{z}}_{i,k}}-{{\bar{z}}_{i,k-1}}-\hat{W}_{i,k}^{T}{\phi_{i,k}}\left({{{\bar{\chi}}_{i,k}}}\right)$
		$\displaystyle+\frac{{{{\bar{\alpha}}_{i,k-1}}-{{\bar{\alpha}}_{i,kf}}}}{{{\mu_{i,k}}}},\,k=2,\cdots,n$		(20)
	$\displaystyle{u_{i}}=\,$	$\displaystyle{{\bar{\alpha}}_{i,n}}$		(21)

where $\kappa_{1}$ and $c_{i,k}$, $k=1,\cdots,n$ are positive design parameters. The parameter estimator ${\dot{{\hat{W}}}_{i,k}}$ and the distributed estimator ${\dot{\hat{y}}_{i,0}}$ are designed as:

	$\displaystyle{{\dot{\hat{W}}}_{i,k}}=$	$\displaystyle-{\Gamma_{i,k}}{\sigma_{i,k}}{{\hat{W}}_{i,k}}+{\Gamma_{i,k}}{\phi_{i,k}}\left({{{\bar{\chi}}_{i,k}}}\right){{\bar{z}}_{i,k}}$		(22)
	$\displaystyle{{\dot{\hat{y}}}_{i,0}}=$	$\displaystyle-{\gamma_{yi0}}{{\bar{e}}_{i}}-{\gamma_{yi0}}{\sigma_{yi0}}{{\hat{y}}_{i,0}}$		(23)

for $k=1,\cdots,n$, where ${\sigma_{i,k}}$, ${\gamma_{yi0}}$ and ${\sigma_{yi0}}$ are positive design parameters, $\hat{W}_{i,k}$ is the estimate of ${W}_{i,k}$, and $\Gamma_{i,k}$ is a positive definite design matrix.

The proposed distributed control strategy with state-triggering setting is conceptually shown in Fig. 1.

Remark 2. Here, we must emphasize and clarify that, for the system with state-triggering setting, only the triggered states $\bar{x}_{i,k}\,(k=1,\cdots,n)$ are available for the control design. If the standard backstepping design procedure is adopted, the virtual controllers $\bar{\alpha}_{i,k}(k=2,\cdots,n)$ in (20) should be ${{\bar{\alpha}}_{i,k}}=-\left({c_{i,k}}+1\right){{\bar{z}}_{i,k}}-{{\bar{z}}_{i,k-1}}-\hat{W}_{i,k}^{T}{\phi_{i,k}}({{{\bar{\chi}}_{i,k}}})+{\dot{{\bar{\alpha}}}_{i,{k-1}}}$. Note that the intermittent states $\bar{x}_{i,k}$ are used in ${{{\bar{\alpha}}}_{i,{k-1}}}$, it holds that ${{\dot{\bar{\alpha}}}_{i,{k-1}}}$ is discontinuous and unavailable for the backstepping control design. To overcome this obstacle, in this section we utilize the tracking errors $z_{i,k}$ in (3)-(4) and the virtual control functions $\alpha_{i,k-1}$ in (8)-(9) for the Lyapunov stability analysis. Nevertheless, it should be noted that $z_{i,k}$ and $\alpha_{i,k-1}$ are not for the controller design, since the final control $v_{i}$, the parameter updating law ${\dot{\hat{W}}}_{i,k}$ and the distributed estimator ${{\dot{\hat{y}}}_{i,0}}$ designed in (21)-(23) only utilize intermittent states $\bar{x}_{i,k}$. Such a treatment plays a crucial role in handling the adverse effects arising from state-triggering, as detailed in the sequel.

As the proposed distributed state triggering control is constructed by replacing the states with their triggered ones, it is important to show that such replacement not only saves communication and energy resources but also preserves consensus stability. To this end, we establish the following lemmas.

Lemma 2. For all $i=1,\cdots,N,k=1,\cdots n$, there exist bounds $\Delta{\phi_{i,k}}>0$, such that

$\displaystyle\left\|{{\phi_{i,k}}\left({{\chi_{i,k}}}\right)-{\phi_{i,k}}\left({{{\bar{\chi}}_{i,k}}}\right)}\right\|\leq\Delta{\phi_{i,k}}$

(24)

where $\Delta{\phi_{i,k}}>0$ is a constant that depends on triggering thresholds $\Delta y_{0}$, $\Delta x_{k}^{i}$, $\Delta x_{k}^{j}$ and design parameters $b_{h}$, $j\in\mathcal{N}_{i}$, $h=1,\cdots,p$.

Proof. See Appendix B.

Lemma 3. The effects of state-triggering are bounded as follows:

	$\displaystyle\left|{{z_{i,1}}-{{\bar{z}}_{i,1}}}\right|\leq\,$	$\displaystyle\Delta{x_{1}^{i}}\buildrel\Delta\over{=}\Delta{z_{i,1}}$		(25)
	$\displaystyle\left|{{z_{i,r}}-{{\bar{z}}_{i,r}}}\right|\leq\,$	$\displaystyle\Delta{x_{r}^{i}}+\Delta{\alpha_{rf}^{i}}\buildrel\Delta\over{=}\Delta{z_{i,r}},\,r=2,\cdots,n$		(26)
	$\displaystyle\left|{{\alpha_{i,k}}{\rm{-}}{{\bar{\alpha}}_{i,k}}}\right|\leq\,$	$\displaystyle\sum\limits_{q=1}^{k}{{\rho_{i,kq}}\left\|{{{\tilde{W}}_{i,q}}}\right\|}+{\tau_{i,k}}\buildrel\Delta\over{=}\Delta{\alpha_{i,k}}$		(27)

for $k=1,\cdots,n$, where $\Delta{z_{i,k}}$, ${\rho_{i,kq}}$ and ${\tau_{i,k}}$ are positive constants that depend on the triggering thresholds $\Delta y_{0}$, $\Delta x_{k}^{i}$, $\Delta x_{k}^{j}$, $\Delta\alpha_{rf}^{i}$, topology parameters $d_{i}$, $\mu_{i}$, design parameters $\kappa_{1}$, $\mu_{i,r}$, $c_{i,k}$ and $b_{h}$, $r=2,\cdots,n$, $h=1,\cdots,p$.

Proof. See Appendix C.

With those lemmas, we establish the following result.

Theorem 2. Consider a strict-feedback nonlinear multiple agent system of $N$ agents (1) with a desired trajectory $y_{0}(t)$ under the directed topology $\mathcal{G}$, satisfying Assumptions 1-2, if using distributed neuroadaptive controller (21), with adaptive law (22) and distributed estimator (23) under triggering conditions (13), (14) and (18), it then holds that: i) all signals in the closed-loop system are SGUUB; ii) the output tracking error ${\varepsilon}=[\varepsilon_{1},\cdots,\varepsilon_{N}]^{T}$ converges to a residual set around zero, and the upper bound for ${\left\|\varepsilon\right\|_{[0,T]}}$ can be decreased with some proper choices of the design parameters; and iii) the Zeno phenomenon is precluded.
Proof. The proof involves three parts: stability analysis, performance analysis and exclusion of Zeno behavior.
1) Stability analysis. This part is comprised of the following $n$ steps.

Step 1: Consider a Lyapunov function ${V_{1}}=\sum\nolimits_{i=1}^{N}\frac{1}{2}{z_{i,1}^{2}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{\tilde{W}_{i,1}^{T}\Gamma_{i,1}^{-1}{{\tilde{W}}_{i,1}}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{y_{i,2}^{2}}+\sum\nolimits_{i=1}^{N}{\frac{{{\kappa_{1}}}}{{{2\gamma_{yi0}}}}\tilde{y}_{i,0}^{2}}$. From (1), (3), (4), (7), (8) and using Young’s inequality, the derivative of ${V_{1}}$ is computed as

	$\displaystyle{{\dot{V}}_{1}}\leq$	$\displaystyle-\sum\limits_{i=1}^{N}{{\kappa_{1}}{e_{i}}{z_{i,1}}}-\sum\limits_{i=1}^{N}{{c_{i,1}}z_{i,1}^{2}}+\sum\limits_{i=1}^{N}{\frac{{{\kappa_{1}}}}{{{\gamma_{yi0}}}}{{\tilde{y}}_{i,0}}{{\dot{\tilde{y}}}_{i,0}}}$
		$\displaystyle+\sum\limits_{i=1}^{N}\left({\tilde{W}_{i,1}^{T}\left({{\phi_{i,1}}\left({{\chi_{i,1}}}\right){z_{i,1}}-\Gamma_{i,1}^{-1}{{\dot{\hat{W}}}_{i,1}}}\right)}+{l_{i,1}^{2}}\right)$
		$\displaystyle-\sum\limits_{i=1}^{N}{\mu_{i,2}^{*}y_{i,2}^{2}}+\sum\limits_{i=1}^{N}{{z_{i,1}}{z_{i,2}}}+\sum\limits_{i=1}^{N}\frac{1}{2}\varepsilon_{i,m}^{2}$		(28)

where ${l_{i,1}}=-{\dot{\alpha}_{i,1}}$, $\mu_{i,2}^{*}$ is a positive design parameter, with $1/{\mu_{i,2}}\geq\mu_{i,2}^{*}+3/4$.

Step k $(k=2,\cdots,n-1)$: Consider a Lyapunov function ${V_{k}}={V_{k-1}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{z_{i,k}^{2}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{\tilde{W}_{i,k}^{T}\Gamma_{i,k}^{-1}{{\tilde{W}}_{i,k}}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{y_{i,k+1}^{2}}$. From (1), (4), (7), (9), (28) and using Young’s inequality, ${\dot{V}_{k}}$ is expressed as

	$\displaystyle{{\dot{V}}_{k}}\leq$	$\displaystyle-\sum\limits_{i=1}^{N}{{\kappa_{1}}{e_{i}}{z_{i,1}}}-\sum\limits_{i=1}^{N}\sum\limits_{r=1}^{k}{{c_{i,r}}z_{i,r}^{2}}+\sum\limits_{i=1}^{N}{\frac{{{\kappa_{1}}}}{\gamma_{yi0}}{{\tilde{y}}_{i,0}}{{\dot{\tilde{y}}}_{i,0}}}$
		$\displaystyle+\sum\limits_{i=1}^{N}\sum\limits_{r=1}^{k}\left({\tilde{W}_{i,r}^{T}\left({{\phi_{i,r}}\left({{\chi_{i,r}}}\right){z_{i,r}}{\rm{-}}\Gamma_{i,r}^{-1}{{\dot{\hat{W}}}_{i,r}}}\right)}+{l_{i,r}^{2}}\right)$
		$\displaystyle{\rm{-}}\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{k}{\mu_{i,r+1}^{*}y_{i,r+1}^{2}}}{\rm{+}}\sum\limits_{i=1}^{N}{{z_{i,k}}{z_{i,k+1}}}+\sum\limits_{i=1}^{N}\frac{k}{2}\varepsilon_{i,m}^{2}$		(29)

where ${l_{i,k}}=-{\dot{\alpha}_{i,k}}$, $\mu_{i,k+1}^{*}$ is a positive design parameter, with $1/{\mu_{i,k+1}}\geq\mu_{i,k+1}^{*}+3/4$.

Step n: Consider a Lyapunov function ${V_{n}}={V_{n-1}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{z_{i,n}^{2}}+\sum\nolimits_{i=1}^{N}\frac{1}{2}{\tilde{W}_{i,n}^{T}\Gamma_{i,n}^{-1}{{\tilde{W}}_{i,n}}}$. From (1), (4) and (29), it holds that

	$\displaystyle{{\dot{V}}_{n}}\leq$	$\displaystyle-\sum\limits_{i=1}^{N}{\kappa_{1}}{e_{i}}{z_{i,1}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{{c_{i,r}}z_{i,r}^{2}}}+\sum\limits_{i=1}^{N}{\frac{{{\kappa_{1}}}}{{{\gamma_{yi0}}}}{{\tilde{y}}_{i,0}}{{\dot{\tilde{y}}}_{i,0}}}$
		$\displaystyle+\sum\limits_{i=1}^{N}{{z_{i,n}}\left({{u_{i}}+W_{i,n}^{T}{\phi_{i,n}}\left({{\chi_{i,n}}}\right)-{{\dot{\alpha}}_{i,nf}}+{z_{i,n}}}\right)}$
		$\displaystyle+\sum\limits_{i=1}^{N}\sum\limits_{r=1}^{n-1}{\tilde{W}_{i,r}^{T}\left({{\phi_{i,r}}\left({{\chi_{i,r}}}\right){z_{i,r}}-\Gamma_{i,r}^{-1}{{\dot{\hat{W}}}_{i,r}}}\right)}$
		$\displaystyle-\sum\limits_{i=1}^{N}{\tilde{W}_{i,n}^{T}\Gamma_{i,n}^{-1}{{\dot{\hat{W}}}_{i,n}}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{\mu_{i,r+1}^{*}y_{i,r+1}^{2}}}$
		$\displaystyle+\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{l_{i,r}^{2}}}+\sum\limits_{i=1}^{N}z_{i,{n-1}}z_{i,n}+\sum\limits_{i=1}^{N}\frac{{2n{\rm{-}}1}}{4}\varepsilon_{i,m}^{2}.$		(30)

From (23), it is seen that $\sum\nolimits_{i=1}^{N}\frac{{{\kappa_{1}}}}{{{\gamma_{yi0}}}}{{\tilde{y}}_{i,0}}{{\dot{\tilde{y}}}_{i,0}}-\sum\nolimits_{i=1}^{N}{\kappa_{1}}{e_{i}}{z_{i,1}}\leq\sum\nolimits_{i=1}^{N}{\kappa_{1}}{\sigma_{yi0}}{{\tilde{y}}_{i,0}}{{\hat{y}}_{i,0}}+\sum\nolimits_{i=1}^{N}\frac{{{\kappa_{1}}}}{{{\gamma_{yi0}}}}\left|{{{\tilde{y}}_{i,0}}}\right|\left|{{{\dot{\bar{y}}}_{0}}}\right|+\sum\nolimits_{i=1}^{N}{\kappa_{1}}\left|{{\tilde{y}}_{i,0}}\right|\Delta{e_{i}}-\sum\nolimits_{i=1}^{N}{\kappa_{1}}{e_{i}}{\varepsilon_{i}}$, where $\Delta{e_{i}}$ is defined in (47)). By using Young’s inequality, one has

	$\displaystyle\sum\limits_{i=1}^{N}{\kappa_{1}}{\sigma_{yi0}}{{\tilde{y}}_{i,0}}{{\hat{y}}_{i,0}}\leq$	$\displaystyle-\sum\limits_{i=1}^{N}\frac{{3{\kappa_{1}}{\sigma_{yi0}}}}{4}\tilde{y}_{i,0}^{2}+\sum\limits_{i=1}^{N}{\kappa_{1}}{\sigma_{yi0}}y_{0}^{2}$		(31)
	$\displaystyle\sum\limits_{i=1}^{N}\frac{{{\kappa_{1}}}}{{{\gamma_{yi0}}}}\left|{{{\tilde{y}}_{i,0}}}\right|\left|{{{\dot{\bar{y}}}_{0}}}\right|\leq$	$\displaystyle\sum\limits_{i=1}^{N}\frac{{{\kappa_{1}}{\sigma_{yi0}}}}{4}\tilde{y}_{i,0}^{2}+\sum\limits_{i=1}^{N}\frac{{{\kappa_{1}}}}{{\gamma_{yi0}^{2}{\sigma_{yi0}}}}{{Y}_{0}^{2}}$		(32)
	$\displaystyle\sum\limits_{i=1}^{N}{\kappa_{1}}\left|{{\tilde{y}}_{i,0}}\right|\Delta{e_{i}}\leq$	$\displaystyle\sum\limits_{i=1}^{N}\frac{{{\kappa_{1}}{\sigma_{yi0}}}}{4}\tilde{y}_{i,0}^{2}+\sum\limits_{i=1}^{N}\frac{{{\kappa_{1}}}}{{{\sigma_{yi0}}}}\Delta e_{i}^{2}.$		(33)

In view of (30), (31), (32) and (33), it holds that

	$\displaystyle{{\dot{V}}_{n}}\leq$	$\displaystyle-{\frac{{{\kappa_{1}}{\lambda_{\min}}\left(Q\right)}}{2}{{\left\|\varepsilon\right\|}^{2}}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{{c_{i,r}}z_{i,r}^{2}}}{\rm{-}}\sum\limits_{i=1}^{N}{\frac{{{\kappa_{1}}{\sigma_{yi0}}}}{4}\tilde{y}_{i,0}^{2}}$
		$\displaystyle+\sum\limits_{i=1}^{N}\sum\limits_{r=1}^{n-1}{\tilde{W}_{i,r}^{T}\left({{\phi_{i,r}}\left({{\chi_{i,r}}}\right){z_{i,r}}{\rm{-}}\Gamma_{i,r}^{-1}{{\dot{\hat{W}}}_{i,r}}}\right)}+\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{l_{i,r}^{2}}}$
		$\displaystyle+\sum\limits_{i=1}^{N}{{z_{i,n}}\left({{u_{i}}+W_{i,n}^{T}{\phi_{i,n}}\left({{\chi_{i,n}}}\right){\rm{-}}{{\dot{\alpha}}_{i,nf}}+{z_{i,n}}}{\rm{+}}z_{i,{n-1}}\right)}$
		$\displaystyle-\sum\limits_{i=1}^{N}{\tilde{W}_{i,n}^{T}\Gamma_{i,n}^{-1}{{\dot{\hat{W}}}_{i,n}}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{\mu_{i,r+1}^{*}y_{i,r+1}^{2}}}+M$		(34)

where $M=\sum\nolimits_{i=1}^{N}{{{\kappa_{1}}{\sigma_{yi0}}y_{0}^{2}}}+\sum\nolimits_{i=1}^{N}{{\frac{{{\kappa_{1}}}}{{\gamma_{yi0}^{2}{\sigma_{yi0}}}}{Y_{0}}^{2}}}+\sum\nolimits_{i=1}^{N}{\frac{{{\kappa_{1}}}}{{{\sigma_{yi0}}}}\Delta e_{i}^{2}}+\sum\nolimits_{i=1}^{N}\frac{{2n-1}}{4}\varepsilon_{i,m}^{2}$. Notice that the actual control law $u_{i}$ in (21) can be rewritten as

	$\displaystyle{u_{i}}=$	$\displaystyle-\left({{c_{i,n}}+1}\right){z_{i,n}}-{z_{i,n-1}}-\hat{W}_{i,n}^{T}{\phi_{i,n}}\left({{\chi_{i,n}}}\right)$
		$\displaystyle+\frac{{{\alpha_{i,n-1}}-{\alpha_{i,nf}}}}{{{\mu_{i,n}}}}+\left({{c_{i,n}}+1}\right)\left({{z_{i,n}}-{{\bar{z}}_{i,n}}}\right)$
		$\displaystyle+\hat{W}_{i,n}^{T}\left({{\phi_{i,n}}\left({{\chi_{i,n}}}\right)-{\phi_{i,n}}\left({{{\bar{\chi}}_{i,n}}}\right)}\right)+{z_{i,n-1}}$
		$\displaystyle-{{\bar{z}}_{i,n-1}}+\frac{{{{\bar{\alpha}}_{i,n-1}}-{\alpha_{i,n-1}}}}{{{\mu_{i,n}}}}+\frac{{{\alpha_{i,nf}}-{{\bar{\alpha}}_{i,nf}}}}{{{\mu_{i,n}}}}.$		(35)

Using (22) and (35), then ${{\dot{V}}_{n}}$ becomes

	$\displaystyle{{\dot{V}}_{n}}\leq$	$\displaystyle-{\frac{{{\kappa_{1}}{\lambda_{\min}}\left(Q\right)}}{2}{{\left\|\varepsilon\right\|}^{2}}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n}{{c_{i,r}}z_{i,r}^{2}}}{\rm{-}}\sum\limits_{i=1}^{N}{\frac{{{\kappa_{1}}{\sigma_{yi0}}}}{4}\tilde{y}_{i,0}^{2}}$
		$\displaystyle-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n}{\frac{{{\sigma_{i,r}}}}{2}\tilde{W}_{i,r}^{T}{{\tilde{W}}_{i,r}}}}+\sum\limits_{i=1}^{N}{\Delta{W_{i,\phi}}+}\sum\limits_{i=1}^{N}\left|z_{i,n}\right|\Delta{\alpha_{i,n}}$
		$\displaystyle-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{\mu_{i,r+1}^{*}}y_{i,r+1}^{2}}+\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{l_{i,r}^{2}}}+M_{1}$		(36)

where $\Delta{\alpha_{i,n}}=({c_{i,n}+1})\Delta{z_{i,n}}+\Delta{\alpha_{i,n-1}/{{\mu_{i,n}}}+{\Delta{\alpha_{i,nf}}}}/{{\mu_{i,n}}}+\Delta{z_{i,n-1}}+\left|{\hat{W}_{i,n}^{T}\left({{\phi_{i,n}}\left({{\chi_{i,n}}}\right)-{\phi_{i,n}}\left({{{\bar{\chi}}_{i,n}}}\right)}\right)}\right|$, ${\Delta{W_{i,\phi}}}=\sum\nolimits_{r=1}^{n}\tilde{W}_{i,r}^{T}\left({{\phi_{i,r}}\left({{\chi_{i,r}}}\right){z_{i,r}}-{\phi_{i,r}}\left({{{\bar{\chi}}_{i,r}}}\right){{\bar{z}}_{i,r}}}\right)$, $\Delta{\alpha_{i,{nf}}}={\left|{{\alpha_{i,{nf}}}-{\bar{\alpha}_{i,{nf}}}}\right|}$ and $M_{1}=M+\sum\nolimits_{i=1}^{N}{\sum\nolimits_{r=1}^{n}}{\frac{{{\sigma_{i,r}}}}{2}{{\left\|{{W_{i,r}}}\right\|}^{2}}}$. Invoking Lemmas 2-3 and using Young’s inequality, it holds that

	$\displaystyle\left|z_{i,n}\right|\Delta{\alpha_{i,n}}\leq$	$\displaystyle\frac{{{c_{i,n}}}}{2}z_{i,n}^{2}+\sum\limits_{r=1}^{n}{\frac{{\rho_{i,nr}^{2}}}{{{c_{i,n}}}}\tilde{W}_{i,r}^{T}{{\tilde{W}}_{i,r}}}+\frac{\tau_{i,n}^{2}}{{{c_{i,n}}}}$		(37)
	$\displaystyle\Delta{W_{i,\phi}}\leq$	$\displaystyle\sum\limits_{r=1}^{n}{\left({\Delta{\phi_{i,r}}\left|{{z_{i,r}}}\right|+\phi_{i,m}}\Delta{z_{i,r}}\right)}\left\|{{{\tilde{W}}_{i,r}}}\right\|$
	$\displaystyle\leq$	$\displaystyle\sum\limits_{r=1}^{n}{\frac{{{c_{i,r}}}}{4}z_{i,r}^{2}}+\sum\limits_{r=1}^{n}\frac{{\Delta\phi_{i,r}^{2}}}{{{c_{i,r}}}}\tilde{W}_{i,r}^{T}{{\tilde{W}}_{i,r}}$
		$\displaystyle+\frac{1}{4}\tilde{W}_{i,r}^{T}{{\tilde{W}}_{i,r}}+\sum\limits_{r=1}^{n}{{\phi_{i,m}^{2}}\Delta z_{i,r}^{2}}.$		(38)

From (37) and (38), it can be derived from (36) that

	$\displaystyle{{\dot{V}}_{n}}\leq$	$\displaystyle-{\frac{{{\kappa_{1}}{\lambda_{\min}}\left(Q\right)}}{2}{{\left\|\varepsilon\right\|}^{2}}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n}{c_{i,r}^{*}z_{i,r}^{2}}}$
		$\displaystyle-\sum\limits_{i=1}^{N}{\frac{{{\kappa_{1}}{\sigma_{yi0}}}}{4}\tilde{y}_{i,0}^{2}}-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n}{\frac{{{\sigma_{i,r}^{*}}}}{2}\tilde{W}_{i,r}^{T}{{\tilde{W}}_{i,r}}}}$
		$\displaystyle-\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{\mu_{i,r+1}^{*}y_{i,r+1}^{2}}}+\sum\limits_{i=1}^{N}{\sum\limits_{r=1}^{n-1}{l_{i,r}^{2}}}+{\Gamma_{n}}$		(39)

where ${\Gamma_{n}}=M_{1}+\sum\nolimits_{i=1}^{N}{\frac{{\tau_{i,n}^{2}}}{{{c_{i,n}}}}}+\sum\nolimits_{i=1}^{N}{\sum\nolimits_{r=1}^{n}}{\phi_{i,m}^{2}}\Delta z_{i,r}^{2}$. Since for any $\rho>0$, the set defined by ${\Omega_{v}}:=\{\sum\nolimits_{i=1}^{N}{\sum\nolimits_{r=1}^{n}}{z_{i,r}^{2}}+\sum\nolimits_{i=1}^{N}{\sum\nolimits_{r=1}^{n}}{\tilde{W}_{i,r}^{T}\Gamma_{i,r}^{-1}{{\tilde{W}}_{i,r}}}{\rm{+}}\sum\nolimits_{i=1}^{N}{\sum\nolimits_{r=1}^{n-1}}{y_{i,r+1}^{2}}{\rm{+}}\sum\nolimits_{i=1}^{N}\frac{{{\kappa_{1}}}}{{\gamma_{yi0}}}\tilde{y}_{i,0}^{2}$ $\leq 2\rho\}$ is a compact one. Thus $\left|{{l_{i,r}}}\right|\leq\varpi_{i,r}$ on $\Omega_{v}$, $r=1,\cdots,n-1$, it follows that

$\displaystyle{{\dot{V}}_{n}}\leq-{\frac{{{\kappa_{1}}{\lambda_{\min}}\left(Q\right)}}{2}{{\left\|\varepsilon\right\|}^{2}}}-\lambda{V_{n}}+{\Delta_{n}}$

(40)

where $\lambda=\min\left\{\right.2{c_{i,1}^{*}},\cdots,2{c_{i,n}^{*}},\frac{{\sigma_{i,1}^{*}}}{{\lambda_{\max}}\{{\Gamma_{i,1}^{-1}}\}},\cdots,\frac{{\sigma_{i,n}^{*}}}{{\lambda_{\max}}\{{\Gamma_{i,n}^{-1}}\}}$, $2\mu_{i,2}^{*},\cdots,2\mu_{i,n}^{*},\frac{1}{2}{{\sigma_{yi0}}{\gamma_{yi0}}}\left.\right\}$, $c_{i,r}^{*}=\frac{3}{4}{c_{i,r}}$, $r=1,\cdots,n-1$, $c_{i,n}^{*}=\frac{1}{4}{c_{i,n}}$, $\sigma_{i,k}^{*}\geq\frac{{{\sigma_{i,k}}}}{2}-\frac{{\Delta\phi_{i,k}^{2}}}{{{c_{i,k}}}}-\frac{{\rho_{i,nk}^{2}}}{{{c_{i,n}}}}-\frac{1}{4}$, $k=1,\cdots,n$, and ${\Delta_{n}}={\Gamma_{n}}+\sum\nolimits_{i=1}^{N}{\sum\nolimits_{r=1}^{n-1}}{\varpi_{i,r}^{2}}$.