Adaptive control for the nonlinear event-triggered networked control system: a decoupling method

In this paper, the adaptive control for a kind of nonlinear system over the event-driven sensors-controller communication network is addressed. Over this network, the existing strategy has been developed. However, the event-triggered mechanism and adaptive estimators are coupled together, which results in extra information delivery of the adaptive estimators over the network or extra computation equipment on the sensor side. To overcome this limitations, two kinds of adaptive control strategies are proposed in this paper. At first, the control strategy with the new event-triggered adaptive estimator is developed for the system without external disturbances where the event-triggered mechanism does not need the information of the adaptive estimators. It is proved that the closed-loop system is asymptotically stable and the Zeno behaviour is excluded. Furthermore, for the system with external disturbances, a robust control scheme with modified adaptive estimator and event-triggered mechanism is proposed and the practical stability of the closed-loop system is guaranteed. Also, the Zeno behaviour is excluded carefully. Comparing with the existing results, due to the decoupling structure of event-triggered mechanism and adaptive estimators, it is unnecessary to transfer the estimators’ information to the event-triggered mechanism on the sensor side, which saves the communication cost. What’s more, although the system state is sampled and event driven, the asymptotical stability is still guaranteed. Also, the developed strategy can handle with the external disturbances. Finally, two examples are performed to illustrate the effectiveness of the schemes and some comparisons are displayed to explain the advantages of the methods.


Introduction
As the advantages in easy maintenance and implementation, networked control system has becoming a powerful and important control solution. Within this framework, event-triggered control as a flexible sampleddata-based control strategy has attracted lots of attention, such as the stabilization control, decentralized control system and distributed control system [1][2][3].
The main idea of the event-triggered control is to construct event detecting mechanism such that the plant information is sampled and released into the transmission network once the mechanism is violated. Comparing with the traditional sampled data-based control method where the sample period is a fixed constant [4,5], the event-triggered-based control strategy can avoid the unnecessary periodical sample of the plant, which results in an aperiodically sampled data control scheme. In this sense, it has the potential advantages in saving computational and communication resources. Inspired by this fact, many important event-triggered mechanisms (ETMs) such as the event-triggered mechanism combining with the time-triggered mechanism [6], the periodical event-triggered control mechanism [7][8][9][10][11][12], dynamical event-triggered control mechanism [13][14][15][16] have been developed to further reduce the communication cost and the stability conservativeness of the closed-loop system by using the input state stable (ISS) property with respect to measurement errors induced by the ETMs. Nowadays, although the eventtriggered control has achieved important progress, the research on the event-triggered control for different systems remains active. For instance, a novel asynchronous event-triggered H ∞ static output feedback tracking control strategy for discrete-time nonlinear networked systems subject to quantization effects was developed in recent paper [17]. In [18], the authors investigated the event-triggered mechanism of the datadriven optimal control for the nonlinear system with input constraints and an elegant stability analysis was performed. Also, the L 2 gain performance guaranteed event-triggered control with packet losses and quantization was considered [19] and a new quantizer and a novel Lyapunov function are constructed, while, for the nonlinear system, how to handle with the parameter uncertainties with the environment of event-triggered networked control system is another important topic.
As well known, adaptive control strategy is a powerful tool to handle with the parameter uncertainties. However, it is not trivial to design the adaptive control algorithm for the nonlinear system with parameter uncertainties under the framework of the eventtriggered networked control system. The main knotty problem is that the system dynamic not only governed by the nonlinear system, but also constrained by the event-triggered communication network. Unfortunately, the traditional adaptive control tools are not enough to deal with the event-triggered communication environment. To overcome this limitation, many adaptive control strategies and ETMs have been developed in three different directions. Adaptive ETMs. To use the communication resources efficiently, the authors in [20][21][22][23][24][25] developed a kind of adaptive ETMs where the coefficients of the ETMs were dynamically adjusted and real-time scheduled dependent on the plant state. Through the adaptively scheduling of the ETMs' coefficients, the ETMs were able to balance the load of the communication network and control performance. Although this method is an adaptive control strategy, it was developed to handle with the uncertainties of communication network instead of the parameter uncertainties of the nonlinear systems. Robust Adaptive Strategy. For the nonlinear system with parameter uncertainties under the event-triggered networked control systems, the controller-actuator and sensor-controller communication links were addressed, respectively. For the controller-actuator cases, the input-dependent eventtriggered adaptive control strategy was first proposed in the literature [26], where the input-dependent ETMs was designed and the adaptive estimators were continuous and updated through the continuous time plant state. The practical stability of the resulted closedloop system was proved and the ISS property was not needed. Motivated by the pioneer result, this method has been further studied and improved in the works [27][28][29][30][31] under the controller-actuator links. However, only the practical stability was guaranteed. For the sensor-controller cases, a robust impulsive adaptive algorithm and the ETMs were presented [32][33][34][35][36], where multiple event-triggering conditions were developed and the practical stability was guaranteed. Also, the fault-tolerant control strategy was presented under the event-triggered impulsive system method [37]. Recently, the impulsive system method has been further improved and a single ETM with minimal-functionapproximation strategy was proposed [38]. But, only the practical stability of the closed-loop system was guaranteed, either. Estimation-based Adaptive Strategy. To derive the adaptive control strategy such that the asymptotical stability of the closed-loop system is guaranteed, the authors in [39,40] developed a different ETM and adaptive algorithm, where the ETM was designed based on the estimation of the feedback and adaptive parameters by using the property of the nonlinear system functions. The semi-global asymptotical stability of the closed-loop system was proved and the ISS property is not needed. However, this strategy was developed only for the controller-actuator networks and the coefficients of the ETMs are not given explicitly (see variable within formula (45) in [40]).
Through the above discussion, it is shown that the adaptive event-triggered control algorithm to achieve the asymptotical stability of the closed-loop system is rare and remains a challenge problem to be further studied. Furthermore, there is no asymptotically stable strategy for the sensor-controller communication links. The reason resulting this fact is not trivial. To handle with the unknown parameters, the adaptive algorithm, an external dynamic, is introduced such that the ISS property of the closed-loop system is not guaranteed within Lyapunov stability theorem. Therefore, a robust adaptive algorithm such as [26][27][28][29][30][31][32][33][34][35][36][37][38] was presented to guarantee the practical stability of the closed-loop system, which decreases the control performance. What's more, all results mentioned above including the robust adaptive strategy [26][27][28][29][30][31][32][33][34][35][36][37][38] and estimation-based adaptive strategy [39,40] have the characteristic such that the ETMs' threshold coefficients are dependent on the adaptive estimators such asθ of the unknown prameters θ . From the implementation viewpoint, to implement the ETM on the sensor side, an external communication channel from controller to the sensors is needed to transmit the information of adaptive estimators (θ) to the ETMs or the adaptive algorithm will be implemented within the ETMs by using the limited computational resources of the sensors, which will increase the communication and computation costs. Recently, the authors of [41] proposed a robust adaptive event-triggered stabilization control for the type-2 fuzzy system in the presence of cyber attacks to avoid the unwanted triggering events, which makes the proposed scheme more reliable and relaxes the conservativeness of stability analysis. It is notable that the "adaptive" of the control scheme is originated from the adaptive event-triggered mechanism. See formulas (24)- (27) of [41], where the parameter σ (t) is adjusted to improve the performance of the event-triggered control. However, for the nonlinear system with unknown parameters, the adaptive estimation for the unknown parameters over the event-triggered control communication network is still unsolved. Also, in [42], the intermittent dynamic event-triggered consensus protocols for the leader-following multiagent systems were developed, where a novel intermittent and dynamic event-triggered mechanism was proposed (see formulas (5)-(6) of [42]). Although the update frequency of consensus protocols was reduced dramatically, the design of the adaptive algorithm for the nonlinear plants with unknown parameters over the event-triggered networks remains a challenge problem.
Motivated by these facts, the event-triggered adaptive control for nonlinear system with the sensorcontroller communication links is addressed. An hybrid adaptive algorithm including continuous and impulsive dynamics is proposed, where the adaptive estimators are updated by using the event-triggered sampled plant state. A simple and decoupled ETM is derived where the ETM does not need the information of adaptive estimators. It is shown that the asymptotical stability of the resulted closed-loop system is guaranteed without the ISS property. Synthesizing with the discussion above and comparing with the existing results, the contributions or advantages of the event-triggered adaptive control strategy are twofold.
(1) The adaptive estimation algorithm for the unknown parameters of the nonlinear systems is developed over the plant-control event-triggered network, where the adaptive algorithm is event-triggered and proposed for the estimation of the unknown parameters instead of the threshold coefficients of ETM (σ (t) in [41] and η(t) in [42]). (2) A novel event-triggered adaptive control strategy including the hybrid adaptive dynamic and ETM is proposed and the asymptotical stability of the closed-loop system is proved without the ISS assumption. While the robust adaptive strategy [26][27][28][29][30][31][32][33][34][35][36][37][38] cannot guarantee the asymptotical stability of the closed-loop system and the estimation-based adaptive strategy [39,40] is invalid for the sensorcontroller links. (3) A decoupled ETM where the threshold coefficients are given explicitly and do not need the information of adaptive estimators is presented, which decreases the communication and computation costs. While, for the robust adaptive strategy [26][27][28][29][30][31][32][33][34][35][36][37][38] and estimation-based adaptive strategy [39,40], the ETMs need the information of adaptive estimators and the ETMs' threshold coefficients are not given explicitly (see variable of formula (45) in [40]).
The rest of the paper is organized as follows: The system formulations and preliminaries are presented in Sect. 2. In Sect. 3, the event-triggered adaptive control strategy and the stability of the closed-loop sys-tem are presented. Besides, the Zeno behaviour of the ETM is excluded. Furthermore, to validate the proposed control strategy, in Sect. 4, several numerical examples and detailed comparisons with the existing results are given. Finally, Sect. 5 concludes this paper.
Notations The symbols R, R m are used to denote the real number set and vector, respectively. Matrix P > 0 means that the matrix P is symmetric positive definitive. Denote λ min (P) and λ max (P) as the minimum and maximum eigenvalues of P. Symbols |x| and y denote the absolute value of a scalar variable and norm of a vector or the induced matrix norm, respectively.

Problem formulation
In this paper, the same nonlinear system with unknown parameters as [39,40] is considered, which is formulated aṡ x n ] ∈ R n , u ∈ R is the system state and control input, respectively. θ ∈ R m is the unknown parameters vector and φ(x) = [φ 1 (x), φ 2 (x), · · · , φ m (x)] ∈ R m denotes the known nonlinear function vector with φ i (x) : R n → R.

Remark 1
The system in the Brunovsky canonical form (1) is widely used to represent many systems such as the manipulator system [25,32,39,40]. However, to guarantee the asymptotical stability of the controlled system (1) within the framework of networked control system, due to the absence of the ISS property of the adaptive closed-loop system, is not trivial. Although some robust adaptive event-triggered control strategies have been developed such as [26][27][28][29][30][31][32][33][34][35][36][37][38], where the practical stability was proved, it is more preferable to achieve the asymptotical stability which is closely related to the performance of practical system such as the accuracy of the manipulators.
Motivated by the limitations of the existing works , in this paper, the adaptive event-triggered strategy for system (1) in the sensor-controller communication manner will be considered. The control objective is twofold. (1) A sensor-controller control framework including the ETM (decoupled with the adaptive estimators) on the sensors side and the adaptive algorithm (updated by the event-triggered sampled data) on the controller side will be developed. (2) The asymptotical stability of the resulted closed-loop system under the sensor-controller communication network will be proved and the potential Zeno behavior will be excluded.
Before the design of the adaptive event-triggered control algorithm, it is necessary to introduce the sensor-controller control framework proposed in this paper. The sensor-controller control framework is shown in Fig. 1. On the sensors side, the state of system (1) x(t) will be measured by the sensors and examined by the ETM. Once the ETM is violated, the discrete state information x(t i ) will be delivered to the controller side through the sensor-controller communication network. To save the computational resources of the sensors, the adaptive estimators (adaptive algorithm) are embedded into the controller side and the discrete time state x(t i ) with time instants {t i }: formulas (20)- (22) in [40]). However, the threshold coefficient σ 1, j given in formula (45) of [40] is dependent on the adaptive estimatorθ(t j ) (see formula (43) of [40]); therefore, the ETM needs the information of adaptive estimator at controller side, which is actually a control strategy over controller-actuator network. While, for the sensor-controller network, the information on adaptive estimatorθ is unavailable for the ETM. What's more, in [40], the threshold coefficient of ETM is designed as σ 1, j = min{ 1 2 , γ 2(n +n ζ c m +4n ζ ς j φ j ) } (formula (45) in [40]) and the coefficient is designed as = b max +θ(t j )ς c m + 1 + σ 1, j , in which the coefficient σ 1, j may be not existing. Thus, for the sensorcontroller network, the adaptive control for asymptotical stability of the system with ETM is still a challenging problem.
Before the presentation of main results, the following assumptions on system (1) are considered. Assumption 2 (Unknown Parameters Within Compact Sets) There exists a known constant > 0 such that θ ≤ .
3 Adaptive control for networked control system: asymptotical stabilization

Adaptive control without communication networks
In the traditional adaptive control of the system (1), the adaptive controller can be designed as: Then, with the control input (2), the closed-loop system is induced as: such that there exists a symmetric positive-definite matrix P and Q satisfying A T cl P + PA cl = −Q. Then, the stability of the closed-loop system can be proved via the Lyapunov function as: V (x,θ) = x Px +θ −1θ and the LaSalle's invariance principle with derivative Although the adaptive design of system (1) is trivial via the LaSalle's invariance principle, the design of the adaptive algorithm over the communication network is a non-trivial work with the discrete transmission data x(t i ). The main challenges that arise from the adaptive control is the adaptive updating algorithm design.
From Eq. (4), it is known that the exact cancelation is not possible over the limited communication data x(t i ). With this fact, it is not a trivial work to design the adaptive control over the communication network. Fortunately, several important strategies where the adaptive algorithms are designed as the impulsive dynamics have been reported [32][33][34][35]. However, the uniform boundedness of the impulsive adaptive dynamic is guaranteed in a prescribed manner and only the uniformly ultimate boundedness of the closed-loop system is guaranteed. The asymptotical stability of the closed-loop system, to our best of knowledge, remains a challenge problem. To overcome this limitation, in this paper, we will propose a hybrid adaptive enventtriggered control strategy. Although the Brunovsky canonical form (1) is simple, it is widely used to represent many systems such as the manipulator system [25,32]. What's more, the design strategy can be further extended to other more general nonlinear system.

Design of event-triggered adaptive control algorithm
By the definition of measurement error , under the sensorcontroller communication network, being different from the coupling design of ETM and adaptive estimator, the ETM is designed as: where δ is a tuning parameter which will be determined later and independent on the adaptive estimatorsθ(t).
Remark 4 It is shown that the threshold coefficient of ETM (5) is a tunable parameter δ, which is decoupled with the adaptive estimatorθ(t). That is different from the existing event-triggered adaptive control solutions for the sensor-controller network [32][33][34][35][36][37][38][39][40][41], where the threshold coefficient of the ETM is a function of adaptive estimators δ = δ(θ(t)). Since this kind of coupling between ETM and adaptive estimators, it is hard to implement the control solutions over the sensorcontroller communication network and an extra communication link will be added between the ETM and adaptive estimators, while, in formula (5), this problem is avoided which reduces implementation complexity and communication cost.
Different from the robust event-triggered adaptive control strategy and the practical stability of the nonlinear system [26][27][28][29][30][32][33][34][35][36][37][38][39][40], we consider the asymptotical stabilization control solution for the nonlinear system with unknown parameters over the sensorcontroller network. The main difficulty arises from the design of the direct Lyapunov-based adaptive control as shown in Sect. 3.1, where the stability of the closed-loop system is closely related to the dynamic ofθ = θ −θ(t). However, since the limited available event-triggered information x(t i ), the second term of formula (4): 2θ −1 φ (x) B Px −θ is mismatched and not eliminated. Therefore, the stability of x(t) and the adaptive estimatorθ(t) is not guaranteed, yet. Instead of the direct Lyapunov adaptive control based on the estimation errorθ = θ −θ(t), a new estimation error is constructed as: where θ ∈ R m is the unknown parameters vector, φ(·) is the known nonlinear function vector of system (1).
x(t i ) is the sampled system state and x n (t i ) is the n-th element of the vector x(t i ). Therefore, the transformed variable z(t) is unavailable for implementation. However, it is just used for analysis of the stability. Later, it will be shown that the stability of x(t) is not dependent on the stability of the adaptive estimatorθ(t). Under error (6), system (1) is written as: For the dynamic (7), to eliminate the adaptive estimatorθ φ(x(t i )) and the nonlinear term φ(x(t i )) φ(x (t i ))x n (t), the controller is designed as: with t ∈ [t i , t i+1 ), whereθ is the event-triggered adaptive estimator, which will be designed later and K = [k 1 , k 2 , · · · , k n ] denotes the feedback gain vector such that

Remark 5 In the controller, the third term
is important and it is necessary to guarantee the stability of the closed-loop system. Actually, in the proof of the closed-loop stability via the constructed Lyapunov function (7) is not eliminated by the third term of the controller, then there is a cross-term in the derivative of V (x, z) as 2x PBφ(x(t i )) φ(x(t i ))x n (t) which is indefinite and the stability of the closed-loop system will not be guaranteed.
Under the designed controller (8), the system (7) is reformulated as the following dynamiċ It can be found that the dynamic of x(t) of (9) is related to the measurement error e(t) = x(t) − x(t i ) and the new estimation error z(t) of (6). Therefore, it is time to consider the dynamic of z(t) and the design of event-triggered adaptive estimatorθ(t). From the controlled system (9), the dynamic of the nth element of x(t) (x n (t)) is formulated as: is presented as the form: Considering the flow dynamic (11) Substituting (12) into (11), (11) is of the following forṁ with t ∈ t i , t i+1 ). It is shown from (13) that the flow dynamic of z(t) has a damping term −φ(x(t i ))φ (x(t i ))z(t), which is impossible for the adaptive estimator based on the estimation errorθ = θ −θ(t). Besides, the flow dynamic of z(t) is related to the measurement error e(t) = x(t) − x(t i ). Therefore, it can overcome the difficulty of adaptive control based onθ . Except for the flow dynamic of z(t), in the following, the jump dynamic of z(t) at t = t i is considered.
For the new estimation error z(t), according to the definition (11) where t + i and t − i is the limit on the right and the left of time instant t i , respectively. Due to the state x(t) is continuous with respect to t, to simplify the stability analysis of closed-loop system, the jump dynamic From (16), the jump dynamic of the adaptive estimator is designed such that Synthesizing the design of adaptive estimator, the updating algorithm of the adaptive estimatorθ(t) is organized as: where θ 0 ∈ R m is the initial guess of the adaptive estimator. As the discussion above, the ETM (5), the controller (8) and the adaptive estimator (18) are presented. Furthermore, under the event-triggered adaptive control strategy, the dynamic of x(t) and z(t) can be summarized as follows.
Remark 6 From the ETM (5), the impulsive adaptive dynamic (18) and the event-triggered controller (8), it is shown that the ETM is simple and is decoupled with the adaptive estimatorθ, which avoids the delivery ofθ to the ETM on the sensors side, which is an important advantage comparing with the existing results [26][27][28][29][30][32][33][34][35][36][37][38][39][40] where the ETM is dependent on the adaptive estimatorsθ . Besides, the adaptive estimator (18) is located in the controller side and updated by using the event-triggered state x(t i ) only. Therefore, in this paper, the proposed sensor-controller framework including ETM (5), controller (8) and adaptive estimator (18) has the simple structure and low communication cost. In addition, it is shown that the control (8) is not a ZOH (zero-order holder) manner. However, by using the event-triggered state x(t i ) only, the control is easy in computation and is piecewise linear with respect to time t, which can be implemented easily in the actuators. Furthermore, comparing with the existing results [26][27][28][29][30][32][33][34][35][36][37][38][39][40], the another important advantage is the asymptotical stability of the closedloop networked control system that can be guaranteed under this design, which is shown in later.

Stability analysis
Due to the impulsive dynamic of adaptive estimator (18), the stability of the closed-loop networked control system will be performed with the hybrid system theory. At first, through the hybrid system theory tool, it is proved that the closed-loop system is semi-global stable. Then, driven by this fact, it is shown that the Zeno phenomenon is excluded and the interval time of the ETM is bounded below by a positive constant. Finally, with the facts of stability and interval time, the asymptotical stability of the closed-loop system is shown. Firstly, the semi-global stability is summarized as following theorem.
Theorem 1 Assume Assumptions 1 and 2 hold, consider the nonlinear system (1) controlled by the piecewise linear control (8) with the adaptive estimator (18) and the ETM (5), given any initial states x(t 0 ) = x 0 ∈ R n and θ 0 ∈ R m , there exist the feedback gain K and ETM coefficient δ(x 0 , θ 0 ) such that for any ETM coefficient δ ∈ (0, δ(x 0 , θ 0 )), all the signals of the closedloop networked control system including the state x(t), adaptive estimatorsθ(t) and control u(t) are bounded.
Proof As the impulsive dynamic of the adaptive estimators (18), the stability of the resulted closed-loop networked control system is proved through the hybrid system theory. It is composed of three parts: A Flow dynamic of the impulsive dynamic over the interval time [t i , t i+1 ). B Jump dynamic of the impulsive dynamic at the event-triggered time instants t i . C Hybrid system theory.
A. Flow dynamic At first, the following Lyapunov function is considered over where ρ > 0 is a constant and P is a positive-definite matrix satisfying the Lyapunov equation: Synthesizing the flow dynamics of x(t) in (19) and z(t) in (20), the derivative of the Lyapunov function along the closed-loop dynamics over t ∈ [t i , t i+1 ) can be formulated as: To further estimate the derivative of the Lyapunov function, the following approximations will be used.
In formulas (23), (27) and (28), the Young's inequality is used: x y ≤ τ x x + 1 4τ y y where τ > 0 is an arbitrary constant as r > 0, α > 0, β > 0 and in formulas (24)- (27), the property of ETM (5)): e(t) ≤ δ x(t) is used. Besides, for formulas (26) and (28) (18), it is shown that there exists a jump at the event-triggered time instants t i :θ(t − i ) =(θ )(t + i ). Therefore, it is necessary to consider the jump dynamic of x(t) and z(t) at time instant t i . However, it is clear that the state x(t) is continuous with respect to time t and there is no jump dynamic of x(t): Thus, it is the turn to check the jump dynamic of z(t) at t = t i . For the new estimation error z(t), according to the definition x n (t i ) Furthermore, as the jump dynamic of adaptive estimators (18b), the jump dynamic of z(t) at t = t i can be further formulated as follows: Then, synthesizing the impulsive dynamic of z(t) and x(t) at time instants t i , the jump dynamic of Lyapunov function V (x(t), z(t)) is obtained such that at t = t i . C. Hybrid system theory Synthesizing the flow dynamic (29) and jump dynamic (30) of V (x(t), z(t)), the complete dynamic of V (x(t), z(t)) can be summarized as follows.
Remark 7 From the proof of Theorem 1, it is shown that the threshold coefficient δ of ETM (5) exists such that δ ∈ (0, δ(x 0 , θ 0 )). The upper bound of δ: δ(x 0 , θ 0 ) is dependent on the initial guess of adaptive estimatorθ . What's more, δ(x 0 , θ 0 ) is independent on the information ofθ(t) for t > 0. Therefore, the proposed strategy is suitable for sensor-controller communication network where the ETM cannot receive the information of adaptive estimator. Furthermore, when the strategy is implemented, at the beginning of the time, the upper bound δ(x 0 , θ 0 ) can be determined offline with a given initial value θ 0 and initial state x 0 . Proposition 2 Given any initial state x 0 ∈ R n and initial guess of θ 0 ∈ R m , the threshold coefficient δ(x 0 , θ 0 ) can be calculated such that Remark 8 Guideline of determining δ. Given for the compact set of parameters θ , initial guess of the adaptive estimator θ 0 , initial state x 0 , system matrices A and B, and positive-definite matrix Q, Step 1, find the feedback gain K such that A cl P + PA cl = −Q; Step 2, choose 0 < r < λ min (Q) P 2 ; Step 3, choose α > 0, β > 0 such that α + β < 1; Step 4, Choose ρ > 1/(2r (1 − α − β)); Step 5, with the matrix P and ; Step 8, choose the threshold coefficient δ of ETM (5) such that δ ∈ (0, δ(x 0 , θ 0 )). (18) and the ETM (5), given any initial states x(t 0 ) = x 0 ∈ R n , initial estimator θ 0 ∈ R m and ETM coefficient δ ∈ (0, δ(x 0 , θ 0 )), then the ETM (5) is Zeno phenomenon excluded and the interval time of the ETM: t i+1 − t i (i = 1, 2, · · · ) are bounded below by a positive constant τ (x 0 , θ 0 ) which is dependent on the initial states x 0 and θ 0 .

Theorem 3 Assume Assumptions 1 and 2 hold, consider the nonlinear system (1) controlled by the piecewise linear control (8) with the adaptive estimator
Proof Given any states x 0 and initial estimation θ 0 , the compact set M(x 0 ,θ 0 ) is defined as Theorem 1. As proven in Theorem 1, ( With this fact, in the following, it will be proved that there exists a positive constant τ (x 0 , θ 0 ) > 0 such that t i+1 − t i > τ(x 0 , θ 0 ), for all i = 0, 1, 2, · · · . At first, according to (9), the dynamic of x(t) iṡ Based on the dynamic (32), with Assumptions 1 and 2, within the compact set M(x 0 ,θ 0 ) , one has the following estimation: where the inequality: z ≤ x(T ) − x(t i ) > δ x(T ) , the ETM is triggered at t = T . Therefore, t i+1 = T and there is a positive constant τ T such that τ T = T − t i > 0. What's more, once x(T ) = 0, the state x(t) = 0, controller u(t) = 0 and the ETM is not triggered for t > T . Then, the ETM is considered only for finite time interval t ∈ [t 0 , T ) with x(t) = 0.
Case II. x(t) does not reach zero at finite time. Then, for any t, the ETM is considered with x(t) = 0.
Proof Given any x 0 ∈ R n and θ 0 ∈ R m , following the inequality (31), it is shown that Since the impulsive dynamic at the time instant , by induction, according to (38), it is obtained such that for t ∈ [t i , t i+1 ). Furthermore, according to Theorem 3, it is shown that t i+1 − t i ≥ τ (x 0 , θ 0 ) > 0. Therefore, inequality (39) will be further estimated as Furthermore, since the boundedness of all signals of closed-loop system as proven in Theorem 1,ẋ(t) is bounded for t ∈ [0, ∞). Therefore, by the Barbalat's lemma, the asymptotically stability of x(t) is achieved such that lim t→∞ x(t) = 0.

Adaptive control for networked control system with external disturbances
In this section, system (1) with time-varying disturbances is considered which is of the following form: where is the timevarying disturbance where we have the following assumption.

Assumption 3 The external disturbance d(t) is bounded such that there exists an unknown constant D > 0 and d(t) ≤ D.
In the presence of the no-vanishing external disturbances, the asymptotical stability of x(t) will not be guaranteed. Instead, the practical stability of the networked control system is established with the modified ETM and adaptive estimators such that ETM: with > 0. Adaptive estimator: where γ > 0 is the coefficient of σ modification.

Remark 9
Comparing with the adaptive estimator (18), the robust adaptive algorithm (43) has the extra term −γ (θ + φ(x(t i ))x n (t i )) in the flow dynamic. Actually, according to the new estimation error z = θ −θ − φ(x(t i ))x n (t), the flow dynamic in (43) is equivalent ). The term γ z in the adaptive estimator will be a damping term in the flow dynamic of z which will be shown in formula (44). With this type of damping term, the practical stability of the controlled system with external disturbances will be guaranteed.

Assumption 4
The nonlinear function φ(x) is upper bounded such that φ(x) ≤ φ M for x ∈ R n with φ M being a known constant.
Remark 10 Assumption 4 widely holds in the mechanical system with bounded nonlinear mechanism such as the manipulator system [25,32,39,40].
Theorem 5 Assume Assumptions 1-4 hold, consider the nonlinear system (41) controlled by the feedback strategy (8) with the adaptive estimator (43) and the ETM (42). Given any initial state x 0 ∈ R n and initial estimation θ 0 ∈ R m ,

(1) All signals of the closed-loop system including x(t), θ(t) and u(t) are bounded and the state x(t) is uniformly ultimately bounded.
(2) The interval time t i+1 − t i with i = 0, 1, 2, · · · is bounded below by a constant τ > 0.
Proof (1) Flow dynamic With the same transform (6) as Theorem 1, by substituting the control (8), robust adaptive estimator (43) and the modified ETM (42), the dynamic of x(t) and z(t) are formulated aṡ Then, according to the Lyapunov function (21) defined as V (z(t), x(t)) = x Px +ρ/2z z, the derivatives along the dynamics (44) and (45) iṡ where with the modified ETM ((42), Assumptions 1-3, we have the following inequalities where r i (i = 1, 2 · · · , 9) are positive constants. With the inequalities above in mind, furthermore, by using Assumption 4, the dynamic of V (x(t), z(t)) along the closed-loop system can be further formulated aṡ where In addition, it can be concluded from the above inequality that, if choosing the coefficients ρ, r i (i = 1, 2, · · · , 9) suitably such that c 1 > 0, c 2 > 0, the flow dynamic of V (x(t), z(t)) can be further estimated aṡ with c = min{2c 1 , γ }. Jump dynamic Being similar to formula (16) induced by the jump property of the adaptive law (43), the jump dynamic of z(t) is continuous such that Given a compact set ) at time instants t = t i . While, for the state on the boundary such that ( Therefore, if initial states x 0 and θ 0 are given such that (x 0 , z(t 0 )) ∈ , then (x(t), z(t)) ∈ , for t ∈ [0, +∞), while, if initial states (x 0 , θ 0 ) are given such that (x(t), z(t)) / ∈ , the Lyapunov function V (x(t), z(t)) will decrease and (x(t), z(t)) will converge to in finite time, which implies that all the signal including x(t), z(t),θ(t) and u(t) is bounded and the state x(t) is robustly convergent to the compact set in finite time. In addition, x(t) ≤ √ 2 (φ M , , D, )/c as t → ∞.
(2) Consider the robust ETM (42) with constant > 0, the dynamic of measurement error e(t) = x(t) − with initial state e(t + i ) = 0. As the proof above, for any initial state (x 0 , θ 0 ), the boundedness of x(t), z(t), θ(t) is guaranteed and x(t) ≤ x M , z(t) ≤ z M with x M > 0 and z M > 0. Therefore, according to Assumptions 1, 2 and 3, the dynamic of e(t) has the following approximation: with M = A cl x M + BK +φ M z M +φ 2 M + L + D. As the initial state e(t + i ) = 0, we have the approximation as which implies that there exists a constant τ = /M such that t i+1 − t i ≥ τ > 0 for i = 0, 1, 2, · · · . The proof is completed.

Remark 12
Comparing with the adaptive control for the event-triggered network without external disturbance where the asymptotical stability is achieved, the practical stability of the closed-loop networked control system is achieved by the robust adaptive control strategy including ETM (42), adaptive estimators (43) and controller (8) for the system with external disturbance. Furthermore, it is shown that the state is convergent into a small residual set with upper bound √ 2 (φ M , , D, )/c which implies a small disturbance upper bound D, ETM threshold coefficient and a large coefficient c > 0 will decrease the robust convergent upper bound with more frequently ETM-driven communication.

Remark 13
The event-triggered adaptive control strategy proposed for the system (1) can be directly extended to the block cascaded multi-input multioutput system such thaṫ where the state is x n ] and the input vector is u = [u 1 , u 2 , · · · , u m ] ∈ R m . Besides, the unknown parameters matrix θ = [θ 1 , θ 2 , · · · , θ n ] ∈ R l×n with θ i ∈ R l which is consistent with the basis function vector φ(x) ∈ R l . This system can be used to model the inverted double pendulums as shown in [35], which will be illustrated to validate the effectiveness of our approach in the following section.

Numerical examples
In this section, the adaptive strategy including ETM (5), adaptive estimator (18) and control (8) for the asymptotical stability (Theorems 1-4) without disturbance and the robust adaptive strategy including ETM (42), adaptive estimator (43) and control (8) for the practical stability (5) with disturbance are illustrated. Besides, comparisons with the existing backsteppingbased adaptive strategy [34,35,38] are made. Finally, as shown in Remark 13, the proposed adaptive control scheme is extended to stabilize the inverted double pendulums system.

Networked manipulator: tracking a set point
The following single-link manipulator system will be performed to validate our approach.
where variables ω,ω andω denote the angle position, the angle velocity and the angle acceleration of the link, respectively. Parameter M = 2.0kg · m 2 is the inertia, g = 9.8m/s 2 is the acceleration of gravity, m = 1.0kg and l = 1.5m are the mass and length of the link, respectively. ς is the control torque. To validate the effectiveness of the proposed adaptive control algorithm over the sensor-controller communication network, the control objective for the manipulator is to enforce the angle position ω to track a given set point y * = π/2. Therefore, by introducing the new variables x 1 = ω − y * and x 2 =ω, the system is reformulated as the forṁ with θ = 0.5mgl/M and φ(x 1 + y * ) = − sin(x 1 + y * ).
Through the numerical validation of the algorithms, the simulation time is given as T = 30s and the discrete step is t = 0.001s. In addition, the control algorithms for the networked manipulator system are given as follows: (a) Without disturbances(d(t) = 0). For this system, the ETM (5), adaptive algorithm (18) and control input (8) for the asymptotically stable result (Theorems 1-4) are validated, where the feedback gain is given as K = [3,2] . Besides, the nonlinear function φ is known with a Lipschitz constant L = 1. Given the matrix Q = diag{20, 20}, therefore, following the procedure in Proposition 2, the threshold coefficient of the ETM (5) Fig. 8. (c) Backstepping-based adaptive strategy [34,35,38].
For this system, following [34,35,38], the adaptive control strategy is designed as The event-triggered-based adaptive algorithm is designed as: The threshold coefficients are given as κ 1 = 1 (k 1 −5/4)/(k 1 ) and κ 2 = 2 (k 2 − 5/4)/(k 2 + θ L) with Lipschitz constant L = 1. Symbol ∧ denotes the two inequalities will be satisfied simultaneously. Based on the adaptive law and virtual control, finally the actual control input is formulated as: The initial states and guess of the unknown parameter are same with the approach in this paper. Besides, the feedback gains are given as k 1 = 10.0, k 2 = 5.0; within the threshold coefficients of the ETM, the parameters are given as 1 = 0.1 and 2 = 0.1. In addition, the adaptive coefficients are given as = 1.0,γ = 0.005 and μ = 1.0. For the system without the disturbance d(t) = 0, the results are shown in Figs. 2, 3 and 4. For the system with disturbance d(t) = 0.05 sin(t), the control performance of the networked manipulator is shown in Figs. 5, 6 and 7. Finally, total number of the events generated by the ETM is compared within Fig. 9. For the networked system without external disturbances, the event-triggered adaptive control strategy developed in Theorems 1 is implemented. Comparing with the backstepping-based method [34,35,38], in Fig. 2, it is shown that the tracking performance of our approach is better than the backstepping-based one, where the asymptotically tracking is achieved by our approach and the practical stability is guaranteed by the backstepping-based one. Besides, the transient performance is better than the backstepping-based one with less fluctuation in states. Besides, in Fig. 3, it is shown that the event-triggered adaptive estimators are both bounded. What's more, in Fig. 4, the control torque of our approach is much less than the backsteppingbased one. Synthesizing the displayed in Figs. 2, 3 and Fig. 2 The tracking performance of the networked manipulator of the event-triggered control(ETC) in this paper and the backstepping-based method [34,35,38] with d(t) = 0 Fig. 3 The variation of networked adaptive estimators (18) and backstepping-based method with d(t) = 0 4, comparing with the backstepping-based method, the adaptive control strategy developed for the networked system without disturbance is better in tracking accuracy (asymptotical stability) and transient performance with less control torque.
For the system with external disturbances, the robust event-triggered adaptive control strategy is implemented. It is shown from Fig. 5 that the tracking performance is better than the backstepping-based method with a smaller tracking error. The adaptive estimators are bounded as shown in Fig. 6. Especially, the backstepping-based method needs a larger control Tracking performance of the robust strategy in this paper and the backstepping-based method with disturbance torque than our approach, which is shown in Fig. 7.
Besides, Fig. 8 shows that the auxiliary variable z(t) of the asymptotically stable scheme and the robust strategy are continuous, which is consistent with the Theorem 1 and Theorem 5. Finally, the total number of events generated by ETMs are displayed in Fig. 9. It is shown that the asymptotically stable method presented in (8), adaptive estimator (18) and ETM (5) has a better tracking performance while a large amount of communication is needed than the backstepping-based method. For the backstepping-based method, the disturbance has a little impact on the total number of events as shown in Fig. 9. For the robust event-triggered adap-  Control torque of the robust strategy in this paper and backstepping-based method with disturbance tive control algorithm presented in this paper, the total number of events is dramatically decreasing comparing with the backstepping-based method. However, a better tracking performance is achieved with less control torque.

Inverted double pendulums
In the following, the inverted double pendulums [35] is illustrated to show the extension of the presented approach in this paper to the block cascaded system as Total number of events generated by the ETM of this paper and the backstepping-based method (52). The system can be modelled as follows: where ω 1 and ω 2 ,ω 1 andω 2 ,ω 1 andω 2 are the angular positions, angular rates, angular angular acceleration, respectively; m 1 and m 2 denote the pendulum masses; J 1 and J 2 are the inertia; g = 9.8m/s 2 is the gravity acceleration, the spring coefficient is k, pendu-lum height is r and the distance between the pendulum hinges is denoted as b. By defining the variables X 1 = [x 1,1 , x 1,2 ] = [ω 1 , ω 2 ] and X 2 =[x 2,1 , x 2,2 ] = [ω 1 ,ω 2 ] , then the system (54) can be further reformulated as the block cascaded form: where X = [X 1 , X 2 ] is the system state, θ = [θ 1 , θ 2 ] is the rearranged unknown parameters vec- ; Nonlinear function matrix (X(t)) = diag{φ 1 (X(t)), φ 2 (X(t))} with φ 1 (X) = [sin(x 1,1 ), 1, sin(x 2,2 )] and φ 2 (X) = [sin(x 1,2 ), 1, sin(x 2,1 )] . The control gain matrix is B J = diag{1/J 1 , 1/J 2 } and the control input is formulated as U = [u 1 , u 2 ] . With this block cascaded form in mind, following the controller (8), the control is designed as: with the linear feedback gain matrix K ∈ R 4×2 . For the disturbance-free case, the ETM and adaptive estimators are designed as: θ(t + i+1 ) = −( (X(t i+1 )) − (X(t i )))X 2 (t i+1 ) +θ(t − i+1 ) Besides, for system with external disturbances, the robust ETM and adaptive estimators are of the following form: +θ(t − i+1 ) In this simulation, the parameters of the inverted double pendulums are given as m 1 = 1.0kg, m 2 = 1.5kg, J 1 = 2.0kg·m 2 , J 2 = 5.0kg·m 2 , k = 0.1N /m, l = 0.1m, r = 0.5m, b = 0.01m. The initial states are given such that x(0) = [−0.5, 0.1, 0.5, −0.1] and the initial guess of the estimators areθ(0) = 0, whereθ = [θ 1,1 ,θ 1,2 ,θ 1,3 ,θ 2,1 ,θ 2,2 ,θ 2,3 ] . The feedback gain is chosen as K = 10 0 10 0 0 10 0 10 and the simulation time step is chosen as 0.001s. With the ETM (5, adaptive estimator (18, following the proposition 2, the threshold coefficients can be calculated as δ = 0.0198. While for the robust ETM(42) and adaptive estimators (43), the coefficient is γ = 0.01 and the threshold is given as = 0.01. The simulation results are displayed in Figs. 10, 11, 12, 13, 14 and 15. It is shown that in Fig. 10, the asymptotical stability of the closed-loop system is achieved under the adaptive controller (8), ETM (5) and adaptive estimator (18), which is consistent with Theorem 4. Also, in Fig. 11, the variation of the adaptive estimatorsθ is shown, from which it is shown that the adaptive estimators are bounded and convergent to a constant. Furthermore, as shown in Fig. 12, the time intervals t i+1 − t i is bounded below by a constant which means the Zeno behaviour of the ETM is excluded, which is consistent with Theorem 3, while, for the robust control scheme presented in this paper, Fig. 13 displays the state of the inverted double pendulums. The asymptotical stability is not achieved, instead the practical stability of the closed-loop system is obtained which is the result of Theorem 5. Besides, Fig. 14 shows the dynamic of the adaptive estimators. It is shown that the adaptive parameters are bounded. Finally, the time intervals t i+1 − t i are shown in Fig. 15. It is displayed that the time intervals are bounded below by a constant and the Zeno behaviour is excluded. Furthermore, comparing with Fig. 12, it is shown that the proposed asymptotical stability guaranteed method needs more communication data than the robust method and there must be a tradeoff between the communication cost and the control performance.

Conclusions
In this paper, the adaptive estimation and control for the networked control system with ETM-driven sensorcontroller communication channel have been investigated. Two kinds of adaptive control strategy have    been developed, and the event-triggered mechanism has been designed decoupling with the adaptive estimators. With this decoupling structure, the external information of adaptive estimators has not been delivered to the ETM, which the communication cost has been saved. At first, the asymptotical stability guaranteed adaptive algorithm has been developed and the Zeno behaviour of the ETM has been excluded. Furthermore, for the external disturbance, a robust adaptive strategy and ETM have been presented, the practical stability of the closed-loop system has been proved and the Zeno behaviour of the robust ETM has been excluded. Throughout this paper, it has shown that the adaptive control strategy over the sensor-controller com-  (42) munication network with the ETM decoupling with the estimators is possible. The adaptive control strategy including the controller, ETM and adaptive estimators over the sensor-controller network with attack and network-induced disturbances such as delays and package dropout will be an interesting topic within this framework.