Decentralized Intermittent Feedback Adaptive Control of Non-triangular Nonlinear Time-varying Systems

Large-scale uncertain complex interconnected systems are frequently encountered [1, 2, 3, 4], decentralized adaptive control of such systems is currently facilitated by recent technological advances on computing and communication resources, which serves as an efficient and practical strategy to be employed for many reasons, such as simplicity of controller design and implementation. Communication network is necessary for signal transmission in large-scale nonlinear control systems owing to networked control systems (NCSs) with advantages of lower cost, easier maintenance and higher reliability [5]. However, there is a gap between the decentralized control and the network under such framework, because the sensor data cannot be transmitted/updated in real time on account of limited communication bandwidth and channels, which potentially degrades the control performance of large-scale nonlinear system.

To preserve a trade-off between communication resource usage and control performance, the emergence of event-triggered control is stimulated as an appealing method for saving energy and communication resources, which enables communication only when certain predefined condition is triggered (see e.g., [6, 7] and the references therein). Early available results on event-triggered control mainly focus on linear systems, see [8, 9] for examples. An extension work to nonlinear systems is pioneered in [10], yet the closed-loop dynamic should be input-to-state stable (ISS). Such limitation is removed in [11] by codesigning an event-triggered control algorithm. However, the system models considered in [10, 11] are required to be exactly known. To handle the uncertainties for nonlinear systems, some event-triggered adaptive control schemes are advocated via the backstepping design procedure, see e.g. [12, 13] and the references therein. Nonetheless, those results are dedicated to the case where only the control input is intermittently transmitted over the network while continuous feedback of plant states are required, thus merely saving the communication resources in the controller-to-actuator channel, but not applicable for the senor-to-controller ones.

For the past few years, control design via intermittent state feedback has stirred an increasing amount of attention in the literature due to its more efficient usage of available limited resources. In this direction, two types of strategies should be highlighted. The first one is the state-triggered control using intermittent output only. In this direction, a state-triggered output feedback control scheme is proposed in [14] for delta operator systems with matched uncertainties. In [15], the problem of decentralized adaptive backstepping based output feedback control is addressed for a class of nonlinear interconnected systems. However, in both solutions, the alleviation on communication burden is still limited because only the output is triggered. The second one is the state-triggered control via intermittent full-state feedback. For this scenario, some efforts have been made in [16, 17] by using backstepping based adaptive control, wherein the models are in low-order form [16] or in normal form [17]. More recently, with the aid of dynamic surface control (DSC) technique, such restriction is explicitly relaxed in [18] for a family of nonlinear systems with constant parametric uncertainties. The idea of designing a distributed state-triggered control algorithm for networked nonlinear system with mismatched and nonparametric uncertainties is further introduced in [19]. The result of [20] solves the problem of stabilizing large-scale interconnected systems by distributed state-triggered controllers built on the ISS condition. Nevertheless, the approaches in [18, 19, 20] are tailored for nonlinear systems in triangular form. Meanwhile, the plant parameters involved in the above results are all restricted to constants. In most applications, however, plant parameters may vary with time rapidly [21]. For instance, traffic free speed is considered as a time-varying parameter in freeway traffic systems control, since changes in weather, air pressure, etc., can strongly influences free speed; in automatic train control problems, the mass of the train load that affects the resistance imposed on the train, may not be the same at different runs. Therefore it is vitally important to relax or even remove these strong restrictions, and broaden the applicability of the backstepping-based state-triggered stability theory to cover system models in non-triangular forms with time-varying parameters.

Motivated by the aforementioned discussion, in this work we develop a decentralized event-triggered adaptive backstepping control method for nonlinear interconnected systems with non-triangular structural uncertainties and unknown time-varying parameters. Under such setting it is actually nontrivial to achieve this goal, this is because two major technical difficulties are present in control design and stability analysis. First, the models of subsystems are all in non-trival non-triangular form with coupling interactions and unknown time-varying parameters that directly challenges the traditional back-stepping design procedure, a tailored technique for triangular systems with constant parameters; Second, with intermittent feedback signals arising from event-triggering, the underlying problem becomes even more complicated when carrying out backstepping design because the repetitive differentiation of virtual control signals (with respect to the triggering signals) is no longer feasible (literally undefined due to the nature of the event-triggering). In this work, we propose two globally decentralized adaptive backstepping control design approaches respectively for the cases with and without event-triggering setting. The first one is developed in a continuous fashion that successfully removes the triangular-form-limitation imposed on the system model by properly treating the non-triangular structural uncertainties for the backstepping design, and simultaneously restrains the affects of the parameter-induced perturbation by freezing the time-varying parameters at the centers, thus avoiding the derivative of time-varying parameters and further relaxing the related state-of-the art conditions [22, 23]. It is shown that the partial derivatives of virtual controllers in each subsystem with respect to states are constants. With such property, the second control scheme is then constructed by replacing the states in the first one with the triggering states, thus circumventing the aforementioned non-differentiability in a global manner. Several lemmas are established to facilitate the authentication of the global uniform boundedness of all the closed-loop signals in both strategies with the stabilization performance improvable by appropriately adjusting design parameters. Moreover, a strictly positive lower bound on the inter transmission times is enforced by the proposed event triggering mechanism (ETM), thus the notorious Zeno phenomenon is avoided. To our best knowledge, this is the first adaptive backstepping control solution for interconnected nonlinear systems under event-triggering setting that is able to tolerate non-triangular structural uncertainties and unknown time-varying parameters.

Consider the following nonlinear system consists of $N$ interconnected subsystems, with the $i$th subsystem modeled as:

	$\displaystyle\dot{x}_{i,k}=\,$	$\displaystyle x_{i,k+1}+\sum_{j=1}^{N}f_{ij,k}\left(x_{j},u_{j},t\right),k=1,\cdots,n_{i}-1$
	$\displaystyle\dot{x}_{i,n_{i}}=\,$	$\displaystyle u_{i}+\varphi_{i}^{T}\left(x_{i}\right){\theta_{i}\left(t\right)}+\psi_{i}\left(x_{i}\right)+\sum_{j=1}^{N}f_{ij,n_{i}}\left(x_{j},u_{j},t\right)$
	$\displaystyle y_{i}=\,$	$\displaystyle x_{i,1}$		(1)

for $i=1,\cdots,N$, where ${x_{i,k}}\in\mathcal{R}$, $k=1,\cdots,n_{i}$ is the system state, with $x_{i}=[x_{i,1},\cdots,x_{i,n_{i}}]^{T}$, $u_{i}\in\mathcal{R}$ and $y_{i}\in\mathcal{R}$ are the control input and output, respectively, $\varphi_{i}\left(x_{i}\right)\in\mathcal{R}^{p}$ and $\psi_{i}\left(x_{i}\right)\in\mathcal{R}$ are known functions, with $\varphi_{i}\left(0\right)=0$, ${\theta_{i}\left(t\right)}\in\mathcal{R}^{p}$ is the unknown parameter vector, $f_{ij,k}\left(x_{j},u_{j},t\right)\in\mathcal{R}$ denotes the nonlinear coupling interaction from the $j$th subsystem for $j\neq i$, and the modeling error of the $i$th subsystem for $j=i$ ¹¹1Arguments of some functions will be omitted hereafter if no confusion is likely to occur..

The objective of this paper is to develop the globally decentralized adaptive backstepping control scheme for system (1) using only locally intermittent feedback signals, such that

• $\bullet$

  The global uniform boundedness of the closed-loop signals is ensured, while all the subsystem outputs are steered into an assignable residual set around zero;

• $\bullet$

  The Zeno behavior is precluded.

To move on, we make the following assumptions.

Assumption 1 [24]. The unknown nonlinear function $f_{ij,k}\left(x_{j},u_{j},t\right)$ satisfies the following condition:

$\displaystyle\left|f_{ij,k}\left(x_{j},u_{j},t\right)\right|\leq\hbar_{ij,k}\left\|x_{j}\right\|+\epsilon_{ij,k}$

(2)

for $i,j=1,\cdots,N$, $k=1,\cdots,n_{i}$, where $\hbar_{ij,k}\geq{0}$ is the unknown coupling gain, which denotes the magnitudes or strengths of the modeling errors and coupling interactions, and $\epsilon_{ij,k}\geq{0}$ is an unknown constant.

Assumption 2. The parameter $\theta_{i}(t)$ is piecewise continuous and $\theta_{i}(t)\in\Omega_{i0}$, for all $t\geq 0$, where $\Omega_{i0}$ is an unknown compact set. The “radius” of $\Omega_{i0}$, denoted by $\beta_{\theta_{i}}$, is assumed to be bounded but not necessarily known.

Assumption 3. The functions $\varphi_{i}\left(x_{i}\right)$ and $\psi_{i}\left(x_{i}\right),i=1,\cdots,N$ satisfy the global Lipschitz continuity condition such that

	$\displaystyle\left\|\varphi_{i}\left(x_{i}\right){\rm{-}}\varphi_{i}\left(\bar{x}_{i}\right)\right\|\leq\,$	$\displaystyle L_{\varphi_{i}}\left\|x_{i}-\bar{x}_{i}\right\|$		(3)
	$\displaystyle\left|\psi_{i}\left(x_{i}\right)-\psi_{i}\left(\bar{x}_{i}\right)\right|\leq\,$	$\displaystyle L_{\psi_{i}}\left\|x_{i}-\bar{x}_{i}\right\|$		(4)

where $L_{\varphi_{i}}$ and $L_{\psi_{i}}$ are unknown bounded constants.

Remark 1. Notice from (2) that $f_{ij,k}\left(x_{j},u_{j},t\right)$ is bounded by a function that allows dependence on all subsystems states, in other words, the uncertainties under consideration fails to satisfy the triangular structure. In addition, it holds that the larger the value of $\hbar_{ij,k}$ is, the stronger the influence degree would be. Such interactions are rather general in numerous real-world systems, such as power systems, water systems, traffic systems and flexible space structures [25, 26], which tend to degrade the system performance and thus challenge the reliability and safety of the system.

Remark 2. Assumption 1 can be commonly found in literature, for example, [24, 27]. As noted in Assumption 2, only the “radius” of $\Omega_{i0}$, i.e., $\beta_{\theta_{i}}$ is assumed to be bounded, which is more general than the existing results [28, 29]. Specifically, it is assumed that $\dot{\theta}\in\mathcal{L}_{\infty}$ with $\|\dot{\theta}\|_{\infty}\leq h(\cdot)\leq\theta_{0}$ for all $t\geq 0$ in [28], where $h$ is a known continuous function and $\theta_{0}$ is a known positive constant. Besides, $\beta_{\theta_{i}}$ considered in [29] is required to be available for the control design. Assumption 3 is quite common, see [15, 18] for examples.

In this section, a decentralized adaptive backstepping control scheme is developed using locally continuous state signals, which can also be regarded as the basis of the control scheme via intermittent state feedback in next section. To this end, we first carry out the following change of coordinates:

	$\displaystyle{z_{i,1}}=\,$	$\displaystyle{x_{i,1}}$		(5)
	$\displaystyle{z_{i,k}}=\,$	$\displaystyle{x_{i,k}}-{\alpha_{i,k-1}},k=2,\cdots,n_{i}$		(6)

The decentralized adaptive backstepping control scheme under continuous state feedback is designed as:

	$\displaystyle\alpha_{i,1}=\,$	$\displaystyle-c_{i,1}z_{i,1}-\frac{1}{4\varpi_{ii,1,1}}z_{i,1}-\frac{1}{4\varpi_{ii,1,2}}z_{i,1}$
		$\displaystyle-\sum_{j\neq i}\frac{1}{4\varpi_{ij,1,1}}z_{i,1}-\sum_{j\neq i}\frac{1}{4\varpi_{ij,1,2}}z_{i,1}$		(7)
	$\displaystyle\alpha_{i,k}=\,$	$\displaystyle-c_{i,k}z_{i,k}-\sum_{j=1}^{N}\left(\frac{1}{4\varpi_{ij,k,1}}+\frac{1}{4\varpi_{ij,k,2}}\right)z_{i,k}$
		$\displaystyle-\sum_{j=1}^{N}\sum_{l=1}^{k-1}\left(\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ij,k,1}}+\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ij,k,2}}\right)z_{i,k}$
		$\displaystyle-z_{i,k-1}+\sum_{l=1}^{k-1}\xi_{k-1,l}^{i}x_{i,l+1},\,k=2,\cdots,n_{i}$		(8)
	$\displaystyle u_{i}=\,$	$\displaystyle\alpha_{i,n_{i}}-\varphi_{i}^{T}\left(x_{i}\right)\hat{\theta}_{i}-\psi_{i}\left(x_{i}\right)$		(9)

where $c_{i,k}$, $\varpi_{ij,k,1}$ and $\varpi_{ij,k,2}$, $k=1,\cdots,n_{i}$ are positive design parameters, $\xi_{k-1,l}^{i}(k=2,\cdots,n_{i},l=1,\cdots,k)$ is the partial derivative of $\alpha_{i,k-1}$ to $x_{i,l}$, which is a constant that relies on $c_{i,k}$, $\varpi_{ij,l,1}$ and $\varpi_{ij,l,2}$. The updating law of ${{\hat{\theta}}_{i}}$ is designed as:

$\displaystyle\dot{\hat{\theta}}_{i}=\Gamma_{i}\left[-\sigma_{i}\hat{\theta}_{i}+\varphi_{i}\left({x}_{i}\right){z}_{i,n_{i}}\right]$

(10)

where $\sigma_{i}$ is some positive design parameter, $\hat{\theta}_{i}$ is the estimate of ${\theta}_{i}$, with ${\tilde{\theta}}_{i}={\theta}_{i}-\hat{\theta}_{i}$, and $\Gamma_{i}$ is a positive definite design matrix.

At this stage, the following lemma is introduced.

Lemma 1. [24] The state vector $x_{i}$ and its transformation vector $z_{i}$ obey the following relationship, with $z_{i}=[z_{i,1},\cdots,z_{i,n_{i}}]^{T}$:

$\displaystyle\left\|x_{i}\right\|\leq\left\|A_{i}^{-1}B_{i}\right\|_{F}\left\|z_{i}\right\|$

(11)

where $A_{i}$ and $B_{i}$ are constant matrices defined in (63) and (68).

Proof. See Appendix A.

Now we are ready to state the following theorem.

Theorem 1. Consider the interconnected nonlinear non-triangular system (1) under Assumptions 1-3, if using the decentralized adaptive controller (9), with the adaptive law (10), then it holds that: i) the global uniform boundedness of the closed-loop signals is ensured; and ii) all the subsystem outputs are steered into a residual set around zero, and the stabilization performance can be improved with some proper choices of the design parameters.

Proof. The proof is composed of the following $n_{i}$ steps.

Step 1: Consider the Lyapunov function ${V_{i,1}}=\frac{1}{2}z_{i,1}^{2}$. From (1), (5) and (6), the derivative of ${V_{i,1}}$ is computed as

$\displaystyle{\dot{V}_{i,1}}=z_{i,1}\left({z_{i,2}}+{\alpha_{i,1}}\right)+z_{i,1}f_{ii,1}+z_{i,1}\sum_{j\neq i}f_{ij,1}.$

(12)

According to Assumption 1, it is derived that

	$\displaystyle\left|z_{i,1}f_{ii,1}\right|\leq\,$	$\displaystyle\frac{1}{4\varpi_{ii,1,1}}z_{i,1}^{2}+\varpi_{ii,1,1}\hbar_{ii,1}^{2}\left\|x_{i}\right\|^{2}$
		$\displaystyle+\frac{1}{4\varpi_{ii,1,2}}z_{i,1}^{2}+\varpi_{ii,1,2}\epsilon_{ii,1}^{2}$		(13)
	$\displaystyle\left|z_{i,1}\sum_{j\neq i}f_{ij,1}\right|\leq\,$	$\displaystyle\sum_{j\neq i}\left(\frac{1}{4\varpi_{ij,1,1}}z_{i,1}^{2}+\,\varpi_{ij,1,1}\hbar_{ij,1}^{2}\left\|x_{j}\right\|^{2}\right.$
		$\displaystyle\left.+\frac{1}{4\varpi_{ij,1,2}}z_{i,1}^{2}+\varpi_{ij,1,2}\epsilon_{ij,1}^{2}\right).$		(14)

By utilizing (7), (13) and (14), $\dot{V}_{i,1}$ is expressed as

	$\displaystyle\dot{V}_{i,1}\leq$	$\displaystyle-c_{i,1}z_{i,1}^{2}+z_{i,1}z_{i,2}+\varpi_{ii,1,1}\hbar_{ii,1}^{2}\left\|x_{i}\right\|^{2}+\varpi_{ii,1,2}\epsilon_{ii,1}^{2}$
		$\displaystyle+\sum_{j\neq i}\varpi_{ij,1,1}\hbar_{ij,1}^{2}\left\|x_{j}\right\|^{2}+\sum_{j\neq i}\varpi_{ij,1,2}\epsilon_{ij,1}^{2}.$		(15)

Step $k$ $(k=2,\cdots,n_{i}-1)$: Consider the Lyapunov function ${V_{i,k}}={V_{i,k-1}}+\frac{1}{2}z_{i,k}^{2}$. Using (1) and (6), $\dot{V}_{i,k}$ is derived as

	$\displaystyle\dot{V}_{i,k}=\,$	$\displaystyle\dot{V}_{i,{k-1}}+z_{i,k}(z_{i,k+1}+\alpha_{i,k})+z_{i,k}f_{ii,k}+z_{i,k}\sum_{j\neq i}f_{ij,k}$
		$\displaystyle-z_{i,k}\sum_{l=1}^{k-1}\xi_{k-1,l}^{i}\left(x_{i,l+1}+f_{ii,l}+\sum_{j\neq i}f_{ij,l}\right)$		(16)

Based on Assumption 1 and using Young’s inequality, it is seen that

	$\displaystyle\left|z_{i,k}f_{ii,k}\right|\leq\,$	$\displaystyle\frac{1}{4\varpi_{ii,k,1}}z_{i,k}^{2}+\varpi_{ii,k,1}\hbar_{ii,k}^{2}\left\|x_{i}\right\|^{2}$
		$\displaystyle+\frac{1}{4\varpi_{ii,k,2}}z_{i,k}^{2}+\varpi_{ii,k,2}\epsilon_{ii,k}^{2}$		(17)
	$\displaystyle\left|z_{i,k}\sum_{j\neq i}f_{ij,k}\right|\leq\,$	$\displaystyle\sum_{j\neq i}\left(\frac{1}{4\varpi_{ij,k,1}}z_{i,k}^{2}+\varpi_{ij,k,1}\hbar_{ij,k}^{2}\left\|x_{j}\right\|^{2}\right.$
		$\displaystyle\left.+\,\frac{1}{4\varpi_{ij,k,2}}z_{i,k}^{2}+\varpi_{ij,k,2}\epsilon_{ij,k}^{2}\right)$		(18)

	$\displaystyle\left|z_{i,k}\sum_{l=1}^{k-1}\xi_{k-1,1}^{i}f_{ii,l}\right|\leq\,\sum_{l=1}^{k-1}\left(\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ii,k,1}}z_{i,k}^{2}\right.$
	$\displaystyle\left.+\,\varpi_{ii,k,1}\hbar_{ii,l}^{2}\left\|x_{i}\right\|^{2}+\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ii,k,2}}z_{i,k}^{2}+\varpi_{ii,k,2}\epsilon_{ii,l}^{2}\right)$			(19)
	$\displaystyle\left|z_{i,k}\sum_{l=1}^{k-1}\xi_{k-1,l}^{i}\sum_{j\neq i}f_{ij,l}\right|\leq\,\sum_{j\neq i}\sum_{l=1}^{k-1}\left(\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ij,k,1}}z_{i,k}^{2}\right.$
	$\displaystyle\left.+\,\varpi_{ij,k,1}\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}{\rm{+}}\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ij,k,2}}z_{i,k}^{2}+\varpi_{ij,k,2}\epsilon_{ij,l}^{2}\right).$			(20)

By using (8), (15), (17)-(20), it can be derived from (16) that

	$\displaystyle\dot{V}_{i,k}\leq$	$\displaystyle-\sum_{\tau=1}^{k}c_{i,\tau}z_{i,\tau}^{2}+z_{i,k}z_{i,k+1}+\sum_{\tau=1}^{k}\sum_{l=1}^{\tau}\sum_{j=1}^{N}\left(\varpi_{ij,\tau,1}\right.$
		$\displaystyle\left.\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}+\varpi_{ij,\tau,2}\epsilon_{ij,l}^{2}\right).$		(21)

Step $n_{i}$: Consider the following Lyapunov function:

$\displaystyle V_{i,n_{i}}=V_{i,n_{i}-1}+\frac{1}{2}z_{i,n_{i}}^{2}+\frac{1}{2}({k_{\theta,i}-\hat{\theta}_{i}})^{T}\Gamma^{-1}_{i}({k_{\theta,i}-\hat{\theta}_{i}})$

(22)

where $k_{\theta,i}$ is an unknown bounded constant vector. Different from the related control designs [12, 16, 17], here we construct the adaptive parameter term $\frac{1}{2}({k_{\theta,i}-\hat{\theta}_{i}})^{T}\Gamma^{-1}_{i}({k_{\theta,i}-\hat{\theta}_{i}})$, instead of $\frac{1}{2}({{\theta_{i}}-\hat{\theta}_{i}})^{T}\Gamma^{-1}_{i}({{\theta_{i}}-\hat{\theta}_{i}})$, which is one of the key steps to avoid the appearance of $\dot{\theta}_{i}$, while ensuring system stability simultaneously, as detailed in the sequel. From (1), (6), (9), (22) and using Young’s inequality, $\dot{V}_{i,n_{i}}$ is evaluated as

	$\displaystyle\dot{V}_{i,n_{i}}{\rm{\leq}}$	$\displaystyle-\sum_{\tau=1}^{n_{i}}c_{i,\tau}z_{i,\tau}^{2}+\sum_{\tau=1}^{n_{i}-1}\sum_{l=1}^{\tau}\sum_{j=1}^{N}\left(\varpi_{ij,\tau,1}\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}\right.$
		$\displaystyle\left.+\,\varpi_{ij,\tau,2}\epsilon_{ij,l}^{2}\right){\rm{-}}({k_{\theta,i}{\rm{-}}\hat{\theta}_{i}})^{T}\Gamma^{-1}_{i}\dot{\hat{\theta}}_{i}+\varphi_{i}^{T}(x_{i})\tilde{\theta}_{i}z_{i,n_{i}}$
		$\displaystyle+\sum_{j=1}^{N}\left(\varpi_{ij,n_{i},1}\hbar_{ij,n_{i}}^{2}\left\|x_{j}\right\|^{2}+\varpi_{ij,n_{i},2}\epsilon_{ij,n_{i}}^{2}\right)$
		$\displaystyle+\sum_{j=1}^{N}\sum_{l=1}^{n_{i}-1}\left(\varpi_{ij,n_{i},1}\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}+\varpi_{ij,n_{i},2}\epsilon_{ij,l}^{2}\right)$		(23)

Substituting (10) into (23) yields

	$\displaystyle\dot{V}_{i,n_{i}}\leq$	$\displaystyle-\sum_{\tau=1}^{n_{i}}c_{i,\tau}z_{i,\tau}^{2}+\sum_{\tau=1}^{n_{i}}\sum_{l=1}^{\tau}\sum_{j=1}^{N}\left(\varpi_{ij,\tau,1}\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}\right.$
		$\displaystyle\left.+\,\varpi_{ij,\tau,2}\epsilon_{ij,l}^{2}\right)-\frac{\sigma_{i}}{2}\left({k_{\theta,i}-\hat{\theta}_{i}}\right)^{T}\left({k_{\theta,i}-\hat{\theta}_{i}}\right)$
		$\displaystyle+\varphi_{i}^{T}\left({x}_{i}\right)\Delta_{\theta_{i}}{z}_{i,n_{i}}+\frac{\sigma_{i}}{2}\left\|k_{\theta,i}\right\|^{2}$		(24)

where $\Delta_{\theta_{i}}={\theta_{i}}-{k_{\theta,i}}$. In accordance with Lemma 1 and Assumptions 1-2, it can be obtained that

	$\displaystyle\left|\varphi_{i}^{T}({x}_{i})\Delta_{\theta_{i}}{z}_{i,n_{i}}\right|\leq\,$	$\displaystyle\frac{1}{2}{\beta_{{\theta}_{i}}}L_{\varphi_{i}}^{2}\left\|{x}_{i}\right\|^{2}+\frac{1}{2}{\beta_{{\theta}_{i}}}\left\|{z}_{i}\right\|^{2}$
	$\displaystyle\leq\,$	$\displaystyle\frac{1}{2}{\beta_{{\theta}_{i}}}\left(L_{\varphi_{i}}^{2}\left\|A_{i}^{-1}B_{i}\right\|_{F}^{2}+1\right)\left\|z_{i}\right\|^{2}$		(25)

Here, we pause to stress that, the effect of the parameter-induced perturbation term $\left|\varphi_{i}^{T}({x}_{i})\Delta_{\theta_{i}}{z}_{i,n_{i}}\right|$ is handled in a non-compensatory manner due to the involvement of ETM, rather than making compensation for it as proposed in [29], see Remark 7 for more details. By using (25), then $\dot{V}_{i,n_{i}}$ can be further bounded as

	$\displaystyle\dot{V}_{i,n_{i}}\leq$	$\displaystyle-\sum_{\tau=1}^{n_{i}}c_{i,\tau}z_{i,\tau}^{2}+\frac{1}{2}{\beta_{{\theta}_{i}}}\left(L_{\varphi_{i}}^{2}\left\|A_{i}^{-1}B_{i}\right\|_{F}^{2}+1\right)\left\|z_{i}\right\|^{2}$
		$\displaystyle+\sum_{\tau=1}^{n_{i}}\sum_{l=1}^{\tau}\sum_{j=1}^{N}\left(\varpi_{ij,\tau,1}\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}+\,\varpi_{ij,\tau,2}\epsilon_{ij,l}^{2}\right)$
		$\displaystyle-\frac{\sigma_{i}}{2}\left({k_{\theta,i}-\hat{\theta}_{i}}\right)^{T}\left({k_{\theta,i}-\hat{\theta}_{i}}\right)+\frac{\sigma_{i}}{2}\left\|k_{\theta,i}\right\|^{2}.$		(26)

Consider the Lyapunov function $V=\sum_{i=1}^{N}V_{i,n_{i}}$, we can obtain

	$\displaystyle\dot{V}\leq$	$\displaystyle-\sum_{i=1}^{N}\underline{c}_{i}\left\|z_{i}\right\|^{2}+\sum_{i=1}^{N}\frac{1}{2}{\beta_{{\theta}_{i}}}\left(L_{\varphi_{i}}^{2}\left\|A_{i}^{-1}B_{i}\right\|_{F}^{2}+1\right)\left\|z_{i}\right\|^{2}$
		$\displaystyle+\sum_{i=1}^{N}\sum_{\tau=1}^{n_{i}}\sum_{l=1}^{\tau}\sum_{j=1}^{N}\left(\varpi_{ij,\tau,1}\hbar_{ij,l}^{2}\left\|x_{j}\right\|^{2}+\,\varpi_{ij,\tau,2}\epsilon_{ij,l}^{2}\right)$
		$\displaystyle-\sum_{i=1}^{N}\frac{\sigma_{i}}{2}\left({k_{\theta,i}{\rm{-}}\hat{\theta}_{i}}\right)^{T}\left({k_{\theta,i}{\rm{-}}\hat{\theta}_{i}}\right)+\sum_{i=1}^{N}\frac{\sigma_{i}}{2}\left\|k_{\theta,i}\right\|^{2}.$		(27)

where $\underline{c}_{i}=\min\{c_{i,1},\cdots,c_{i,n_{i}}\}$. By applying Lemma 1, it can be derived from (27) that

	$\displaystyle\dot{V}\leq$	$\displaystyle-\sum_{j=1}^{N}\underline{c}_{j}\left\|z_{j}\right\|^{2}+\sum_{j=1}^{N}\frac{1}{2}{\beta_{{\theta}_{j}}}\left(L_{\varphi_{j}}^{2}\left\|A_{j}^{-1}B_{j}\right\|_{F}^{2}+1\right)\left\|z_{j}\right\|^{2}$
		$\displaystyle+\sum_{j=1}^{N}\left(\sum_{\tau=1}^{n_{i}}\sum_{l=1}^{\tau}\sum_{i=1}^{N}\varpi_{ij,\tau,1}\hbar_{ij,l}^{2}\left\|A_{j}^{-1}B_{j}\right\|_{F}^{2}\right)\left\|z_{j}\right\|^{2}$
		$\displaystyle-\sum_{i=1}^{N}\frac{\sigma_{i}}{2}\left({k_{\theta,i}-\hat{\theta}_{i}}\right)^{T}\left({k_{\theta,i}-\hat{\theta}_{i}}\right)+\Delta$
	$\displaystyle\leq$	$\displaystyle-\sum_{j=1}^{N}{c}_{j}^{*}\left\|z_{j}\right\|^{2}{\rm{-}}\sum_{i=1}^{N}\sigma_{i}^{*}\left({k_{\theta,i}{\rm{-}}\hat{\theta}_{i}}\right)^{T}\left({k_{\theta,i}{\rm{-}}\hat{\theta}_{i}}\right)+\Delta$		(28)

where ${c}_{j}^{*}=\underline{c}_{j}-\sum_{\tau=1}^{n_{i}}\sum_{l=1}^{\tau}\sum_{i=1}^{N}\varpi_{ij,\tau,1}\hbar_{ij,l}^{2}\left\|A_{j}^{-1}B_{j}\right\|_{F}^{2}-\frac{1}{2}{\beta_{{\theta}_{j}}}L_{\varphi_{j}}^{2}\left\|A_{j}^{-1}B_{j}\right\|_{F}^{2}-\frac{1}{2}{\beta_{{\theta}_{j}}}>0$ by choosing $\underline{c}_{j}$, i.e., $\min\{c_{j,1},\cdots,c_{j,n_{i}}\}$ larger enough, $\sigma_{i}^{*}=\frac{\sigma_{i}}{2}>0$, and $\Delta=\sum_{i=1}^{N}\sum_{\tau=1}^{n_{i}}\sum_{l=1}^{\tau}\sum_{j=1}^{N}\varpi_{ij,\tau,2}\epsilon_{i,j,l}^{2}+\sum_{i=1}^{N}\frac{\sigma_{i}}{2}\left\|k_{\theta,i}\right\|^{2}$. Then it holds that $\dot{V}\leq-lV+\Delta$, with $l=\min\left\{2{c}_{1}^{*},\cdots,2{c}_{N}^{*},\frac{2{\sigma}_{1}^{*}}{\lambda_{\max}\{\Gamma_{1}^{-1}\}},\cdots,\frac{2{\sigma}_{N}^{*}}{\lambda_{\max}\{\Gamma_{N}^{-1}\}}\right\}$.

Now we are ready to prove in detail that the results i) and ii) in Theorem 1 are ensured.

$\bullet$ Stability Analysis. From the above analysis, we have ${V}\left(t\right)\leq{e^{-lt}}{V}\left(0\right)+\frac{{\Delta}}{l}\left({1-{e^{-lt}}}\right)\in{\mathcal{L}_{\infty}}$, it follows that ${z_{i,k}}$ and ${\tilde{\theta}_{i}}$ are bounded, $k=1,\cdots,n_{i}$. From (5), (6), (7) and (8), it is established that ${x_{i,k}}$ is bounded, $k=1,\cdots,n_{i}$. Then it can be derived from (9) that $u_{i}$ is bounded. Therefore, all signals in the closed-loop system are globally uniformly bounded.

$\bullet$ Performance Analysis. As $\dot{V}\leq-lV+\Delta$, we have $\left|{z_{i,1}}\right|\leq\sqrt{2V}=\sqrt{2\left({V\left(0\right)-\frac{\Delta}{l}}\right){e^{-lt}}+2\frac{\Delta}{l}}$, which implies that $z_{i,1}$ is ensured to attenuate to a residual set around zero. In addition, from the definition of $V$, $l$ and $\Delta$, it holds that the upper bound of $\left|{z_{i,1}}\right|$ can be decreased by increasing design parameters $c_{i,k}$ and $\Gamma_{i}$, or decreasing design parameter $\varpi_{ij,k,1}$ and $\varpi_{ij,k,2}$, $k=1,\cdots,n_{i}$. $\hfill{\blacksquare}$

Remark 3. The non-triangular structural uncertainties involve both modeling errors and nonlinear coupling interactions, thus are difficult to tackle. For the first part, by constructing a nonlinear compensation term $-\sum_{j\neq i}^{N}\left(\frac{1}{4\varpi_{ij,k,1}}+\frac{1}{4\varpi_{ij,k,2}}\right)\bar{z}_{i,k}-\sum_{j\neq i}^{N}\sum_{l=1}^{k-1}\left(\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ij,k,1}}+\frac{\left(\xi_{k-1,l}^{i}\right)^{2}}{4\varpi_{ij,k,2}}\right)\bar{z}_{i,k}$ in the virtual controllers of each subsystem $\alpha_{i,k-1}(k=2,\cdots,n_{i})$, as seen in (8), we naturally offset the terms related to $z_{i,k}$ in (18) and (20). Whereas the uncertainties in the second part contain nonlinear coupling interactions among subsystems, it is even more challenging to handle. Here, inspired by the ideas in [24], we first keep all terms associated with $\left\|x_{j}\right\|^{2}(j=1,\cdots,N)$ in each recursive step, and then cope with them in the final step with the aid of Lemma 1. It is worth noting that what is considered here is an entirely different and more difficult implementation scenario than the one in [24], where the traditional backstepping method can be directly used in [24] as state/input-triggering is not considered. Meanwhile the work in [24] involves only constant parametric uncertainties which therefore can be easily handled by using adaptive parameter estimates methods.

In this section, a decentralized adaptive backstepping control scheme under event-triggering setting is constructed upon the previous scheme, with feasibility and stability analysis provided. Such strategy not only inherits the ability of handling non-triangular structural uncertainties and time-varying parameters in the continuous scheme, but also evades the non-differentiability issue.