H∞ Control for Oscillator Systems With Event-Triggering Signal Transmission of Internet of Things

This article proposes to design a distributed <inline-formula> <tex-math notation="LaTeX">$H_{\infty} $ </tex-math></inline-formula> optimal control algorithm for Van der Pol oscillators with unknown internal dynamics, input constraints and external disturbances, via event-triggering signal transmission of the Internet of Things (IoT). First, the graph theory for the IoT is introduced. Second, the dynamics of Van der Pol oscillators are transformed into the tracking dynamics which cooperate via the IoT network. Third, unlike the existing online optimal control algorithms using adaptive dynamic programming, we design an <inline-formula> <tex-math notation="LaTeX">$H_{\infty} $ </tex-math></inline-formula> optimal control algorithm employing an event-triggering signal transmission mechanism to reduce the burden of communication resource and computation bandwidth of the IoT network. As the triggering condition and approximation parameter update policies are appropriately designed, the algorithm guarantees that the Zeno phenomenon is free, the consensus errors are uniformly ultimately bounded, and the external disturbance is compensated. Finally, numerical simulation results with comparison to the time-triggering algorithms confirm the effectiveness of the proposed algorithm.


I. INTRODUCTION
Internet of Things (IoT) technology has recently received significant attention from research communities and industrial societies due to the communication ability among devices and intelligent selection ability of perception and execution [1], [2], [3]. The controller for each device in the IoT can interact through the network to exchange data, generate control signals, and send feedback to others [4], [5], [6], [7], thanks to machine learning, distributed control algorithms for devices/plants that have been widely studied for recent years [6], [7], [8].
The non-IoT conventional distributed control algorithms are based on the time-triggering mechanism, where the The associate editor coordinating the review of this manuscript and approving it for publication was Sathish Kumar .
controllers sample the states with the same periods and then exchange information with each other through the communication network. This way of transmitting information is inefficient because a device periodically continuously sends the same information to others or its remote controller [9]. To overcome the burden of communication resource and computation bandwidth, the event-triggering mechanism was first investigated for scheduling stabilizing control tasks [10], where a controller only receives feedback states, updates its parameters and sends control signals to plant only when an event-triggering condition is violated. Inspired by the idea, several works related to event-triggered (ET) control for multiagent have been developed [11], [12], [13], [14], [16]. In [11], the ET decentralized control scheme for interconnected nonlinear systems is proposed. The event-triggering condition is designed suitably to reduce the computational burden on the controllers. Narayanan and Jagannathan [12] proposed a distributed optimal control scheme for interconnected affine nonlinear systems, where observers used the triggered system output to estimate the states of the subsystem. Vamvoudakis et.al. [13] designed the ET optimal tracking control algorithm using actor-critic structure in reinforcement learning theory of machine learning. Tan [14] designed an ET H ∞ distributed control algorithm for large-scale systems with physical interconnection, external disturbances and input constraints, where the subsystems are isolated and exchange states and control signals over the network. Qin et.al. [15] proposed a safe ET control method based on ADP and the zero-sum game theory for nonlinear safety-critical systems with safety constraints and input saturation.
However, the algorithms mentioned above, despite using event-triggering mechanisms, are mainly designed for non-IoT-controlled plants, which have the advantages of stable limit cycles and stable equilibrium points at the origin. For example, distributed control algorithms were devoted to consensus problems of the systems with simple models in a kinematic form or a double integrator (see [17] for more details about the applications). An ET control learning algorithm was designed for an application of voltage source inverters in [16], where the disturbance rejection policy and the optimal control policy are approximated to drive the AC output to the reference while minimizing energy loss. Unfortunately, it only applies to single systems that are not connected to the IoT network.
Recently, the Van der Pol oscillators, an original model of an electrical circuit with a triode valve [18], have been interconnected in IoT networks due to the requirement of industrial applications [1,Ch. 6], [19]. The model is then extended to dynamics of relaxation oscillations, elementary bifurcations, and chaos [20], [21], [22], [23]. The different models have been used to design various practical IoT applications in radio engineering, power systems, combustion processes, biomedical engineering, and robotics. In [24], [25], and [26], Van der Pol oscillators, including uncertain parameters and unmodeled dynamics, were controlled by adaptive control algorithms using neural networks (NN). In [27], the outputs of the oscillators were forced to track the reference by NN-based feedback linearizing control algorithms. Experimentally, a sliding-mode observer was used to estimate the states of the oscillators [28]. In optimal control, the oscillators were presented by the strict-feedback nonlinear systems [1,Ch. 6], [29], [30] and nonlinear systems with input constraints [1,Ch. 6], [31]. The optimal control laws were derived from adaptive dynamic programming (ADP) principle.
Although Van der Pol oscillators are widely applied to many engineering disciplines, IoT-based control with the event-triggering signal transmission has not yet been concerned about saving communication resources. Furthermore, the oscillators in the IoT network, impacted by unknown internal dynamics, input constraints and external distur-bances, has been not considered. In this paper, the signals of constrained control and disturbance estimation will be exchanged over the IoT network for executing the control policies. The exchange is in the dynamic sampling instants with variable inter-event time rather than fixed sampling periods. These instants are generated by an adaptive event-triggering condition to guarantee closed-loop stability.
Compared with the works mentioned above, the main contributions of this article are three-fold: 1) Unlike the available algorithms of distributed optimal control for nonlinear systems that are not connected to the IoT network [11], [12], [13], [14], [16], we design an algorithm for Van der Pol oscillators in the IoT network dealing with neither controlled stable limit cycles nor stable controllable equilibrium points at the origin. Especially, the system model is in the presence of unknown internal dynamics, input constraint and external disturbance. 2) Unlike the optimal control methods for the Van der Pol oscillators [1, Ch. 6], [29], [31], we further integrate the event-triggering signal transmission of the IoT to ADP and the two-person zero-sum game theory [32], [33] to obtain a new ET control algorithm which can mitigate the communication resource and computational bandwidth in the IoT network. The ET solution of Hamilton-Jacobi-Isaacs (HJI) is approximated online to find the saddle point for control and disturbance compensation policies. In addition, the difference between the ET control algorithm in the paper and one in [30] is that the distributed control via the IoT network is considered instead of decentralized control through non-IoTsubsystem isolation. 3) An adaptive triggering condition is designed and a rigorous proof is made to ensure that the closed dynamics are asymptotically stable while the Zeno phenomenon is excluded. Compared with the sampling period-based control algorithm in terms of communication resource and computation bandwidth in an application, the proposed algorithm is shown to be more effective.
The rest of the paper is organized as follows. Section I introduces the preliminaries including the graph theory for the IoT and the system dynamics, Section III presents the analysis and designs algorithms, Section IV applies the algorithm in numerical simulation studies, and Section V briefly concludes the paper.
Notation 1: Throughout this article, X ∈ R n denotes vector X with the n-dimensional Euclidean space, and Y ∈ R n×m denotes matrix Y with the n × m-dimensional real space. ∥X ∥ and ∥Y ∥ are the Euclidean norm of X and the L 2 -norm of Y , respectively. λ min (.) denotes the minimum eigenvalue of a matrix (·), σ (.) is the minimum singular value of a matrix (.), and diag[X ] transforms vector X into a diagonal matrix.
Definition 1 (Uniformly Ultimately Bounded (UUB) [34]): The equilibrium point x 0 of dynamicsẋ = f (x, u), x ∈ R n is said to be UUB in a compact set ∈ R n if there exists VOLUME 11, 2023 a bound B and a time T (B, x 0 ) for all x 0 ∈ such that

II. PRELIMINARIES AND PROBLEM FORMULATION A. GRAPH THEORY FOR IoT
A graphḠ(V, , A) in the graph theory is employed to construct an IoT topology of devices/plants, where The edges from nodes to the root 0, namely leader, is presented by C = diag [α 1 , α 2 , . . . , α N ]. If no edge from i to 0, α i = 0, otherwise α i = 1. If there exists a directed edge between s i and s j , ∀(s i , s j ) ∈ V, s i ̸ = s j , L and A are irreducible [35].

B. DYNAMICS OF VAN DER POL OSCILLATORS
In this section, we present the normal dynamics of Van der Pol oscillators, then by a definition of event-triggering signal transmission, the tracking errors are defined via the IoT network.
Consider Van der Pol oscillator dynamics i, presented as a strict-feedback nonlinear system with unknown internal dynamics, input constraint and external disturbance: where u i ∈ R is the control input constrained by ∥u i ∥ ≤ū i for a positive constantū i . For all l = 1, . . . , n, where . . .
Note that since f i,l (.), l = 1, . . . , n, is unknown, internal dynamicsf i (x i ) is completely unknown. To facilitate the later design, we adopt the following assumption.
Remark 1: Assumption 1 is practical in many industrial applications [28], [29], [31], where internal dynamics of (1) is Lipschitz and the measured output is bounded. The upper bounds in Assumption 1 are only used to prove stability (see Appendix A) and are not used in the control law.
Consider the dynamics of the leader without disturbance: With the event-triggering signal transmission and the topology of IoT, we define the consensus local tracking error among systems i, neighbors j and leader 0, . .}, the control input of (1) is updated when an triggering condition, to be designed later, is violated. The triggered dynamics of (2) is rewritten aṡ where u i , i = 1, 2, . . . , N , is updated at t i k , k = 0, 1, . . . ., and held until t i k+1 by the zero-order hold (ZOH). We define the triggering consensus local tracking error between system i and its neighbors as ∈ R Nn be the overall vectors, the triggering consensus global tracking error vector via parameters of the graphḠ(V, , A) is defined as where ⊗ is the Kronecker product,1 N = [1, . . . , 1] ⊤ ∈ R N , I n is an identity matrix of size n, e = x −1 N ⊗ I n x 0 ∈ R Nn is the event-triggered global tracking error.
The constraint of the triggering consensus local tracking error and triggering consensus global tracking error is followed by Lemma 1.
Lemma 1 ( [35]): The bounded local tracking error leads to bounded global tracking error if the following inequality is satisfied with the minimum singular value of L+C, σ (L+C): Objective: By Lemma 1, the control objective is to design the locally distributed control policy of each system with unknown internal dynamics, input constraints, and external disturbances. The design employs the event-triggering signal transmission of IoT to reduce the burden of communication resources and computation bandwidth.

III. IoT-BASED DISTRIBUTED H∞ ET CONTROL
In this section, an event-triggering signal transmission cost function is defined and the HJI equation is derived. Then, an online algorithm is designed for approximate control policy and disturbance compensation policy. Define Inspired by the work in [37] for H ∞ optimal control with the disturbance compensation, the performance output η i is required to be minimized such that the bounded L 2 -gain holds the following condition: where Q i is a positive definite matrix. By expanding the work in [37], there exists an attenuation level, γ i > 0, for the bounded L 2 -gain condition (9) to be satisfied, ∀i = 1, . . . , N . For constrained input control problems, the nonnegative function U (u i ) is selected by evolving from a single system in [31] and [36] as where R i is the main diagonal elements of a positive definite matrix. Remark 2: The nonnegative function (10) uses the hyperbolic tangent function, which is a one-to-one real-analytic integrable function of class C η , η ≥ 1, used to map R onto the interval (-ū i ,ū i ).
The triggering local consensus performance index function is defined based on [14] as Remark 3: In the cost function (11), different from work in [14], the event-triggering signal transmission is employed for not only the ET control policy but aslo the ET disturbance compensation policy.
Let inputs u i and d i be depended on states. Then, the triggering local consensus value function is written as where By adopting the two-person zero-sum game theory, we introduce the optimal value V ⋆ i δ i [38] as The saddle point (u ⋆ i , d ⋆ i ) to (13) exists if the following Nash condition holds [38] Applying the ET control laws and ET disturbance compensation policies to dynamics (4), the consensus tracking dynamics is rewritten aṡ . Then, we define the Hamiltonian as where Apply the stationary condition to (16), control and disturbance compensation policies are computed as follows: Substituting (18) and (17) to (16) we have the triggered HJI equation: According to [37], there exists a positive definite smooth minimum solution V ⋆ i (δ i ) to the triggered HJI (19). However, asf i (z i ) is unknown and (19) is high-order nonlinear differential, the analytical solution cannot be found. NN combined with event-triggering is our choice to learn the solution. The smooth optimal value function V ⋆ i (δ i ), i = 1, . . . , N , is therefore approximated by the Weierstrass higher-order approximation theorem [36] as where φ i (δ i ) : R n → R h , W i and ε i are the activation functions, the ideal weights and the NN approximation errors, respectively. By the higher-order approximation property we have the following assumption [39].
We obtain the NN-based triggered HJI by substituting (20) to (18)-(19): where Remark 4: Recall Assumption 2 we have the boundedness of ε iH on a compact set, i.e. ∀b iH > 0, ∃N (b iH ) : Since the ideal NN weights are not available, the value function (20) is estimated bŷ From (17), (18) and (23), the disturbance compensation policy and the optimal control policy are approximated bŷ Using (24), (25) for (21), one obtains the approximate Hamilton asĤ jd j . Next, we propose a NN-weight tuning law to forcê W i → W i for all i = 1, . . . , N . In other words, our goal is to obtainĤ i → H ⋆ i ≡ 0. To remove the system identification procedure for internal dynamicsf i (z i ), the integrated reinforcement learning technique [40] is used in the paper, i.e., the residual error function, which we establish to minimize, where T > 0 is a small interval. To ensure the NN-weights converge to global values but avoid using the persistent excitation (PE) condition in adaptive control [41], we follow the concurrent learning technique [42]. The total integral past residual error, E i,P = P i l=1 E i,Ĥ (t l ), is utilized. Then, the NN-weight tuning law is derived from modifying the Levenberg-Marquardt algorithm, such thatẆ where ρ i is an update rate, and It worth noting that { φ i (t l )} P i l=0 must be linearly independent or rank φ 1 (t 0 ), φ 2 (t 1 ), . . . , φ i (t P i ) = P i [42].
8942 VOLUME 11, 2023 Remark 5: As the unknown internal dynamics,f i (z i ), is absent from (29), a system identification procedure for unknown function is unnecessary.
Let the triggering error be defined as e i = δ i − δ i , by the Assumption 2, the following assumption is satisfied: Assumption 3: For a positive constant L i∇φ , ∇φ i (δ i ) is Lipschitz continuous Next, we design the triggering condition based on the triggering errors and Lyapunov theory. The condition guarantees that the closed-system is stable.

Condition 1 (Event-triggering condition):
The consensus tracking errors are sampled and the parameters of disturbance compensation policy and the optimal control policy are updated only when the following triggering condition is violated: The IoT-based ET robust optimal control structure for each system is presented in Fig. 1. In the structure, all systems exchange the states, control and disturbance compensation signals over the network. Each system updates the policy parameters only when the triggering gates are enabled. t i k or t j h are governed by the triggering conditions. The parameters of the disturbance compensation policy and the optimal control policy are adjusted via NN outputs while the NN-weights are adjusted by the online weight tuning law. It is worth emphasizing that all consensus tracking errors are sampled non-periodically. Remark 6: Although states δ j , j ∈ N i are continuously transferred though the network for system i to compute the triggering error e i , the control signals and disturbance compensation signals of neighbors j are only transferred when the triggering condition 1 is violated, otherwise they use zero-order hold (ZOH). Compared with the time-triggering mechanism, where the signal is transmitted continuously according to the fixed sampling period, the communication load in the event-triggering mechanism is mitigated.

A. STABILITY ANALYSIS AND ZENO PHENOMENON EXCLUSION
In the following theorem, we analyze the stability and the exclusion of the Zeno phenomenon. The Zeno phenomenon occurs if the minimum inter-event interval is zero, resulting in excessively increasing the cumulative number of events.
Theorem 3.1: Consider the Van der Pol oscillators (1), which are networked with the topology resented by the communication graphḠ (V, , A). Let the consensus tracking errors be defined in (6). Let the ET disturbance compensation policy and the ET distributed optimal control policy be approximated in (25) and (24). Let the triggering threshold be designed in (31). Then, the closed-loop dynamics of each system is stable and the approximation errors are ultimately uniformly bounded (UUB). In addition, the Zeno behavior is excluded since (32) where i , O i are positive upper bounds.
Proof: See Appendix A. Remark 7: In Appendix A, the closed-loop is stable wheṅ

IV. NUMERICAL SIMULATION
In this section, a numerical simulation study of the proposed distributed robust optimal control algorithm for Van der Pol oscillator agents is conducted. The comparison results between the ET control algorithm and the time-triggered (TT) control algorithm [1,Ch. 6] are performed. The consensus network topology of IoT is presented in Fig. 2, where the leader (a 0 ), sends its states x 0 and control signalû 0 to agents 1 and 2 (a 1 , a 2 ). Agent 1 sends its information, including x 1 , control signalû 1 , and disturbance compensation signald 1 to agent 2. Then, agent 1 and agent 2 send their triggered information, including x 1 and x 2 , control signalsû 1 andû 2 , and disturbance compensation signalsd 1 andd 2 , to agent 3 (a 3 ) at the triggering moments.
The states of the leader is generated by applying the following control law to dynamics (3) withū 0 = 0.1:  whereŴ 0 is updated by the law (28) withK 0 = U (û 0 ). The triggering condition of the leader is The dynamics of all agents are presented in the form of (1), , the update rates ρ h = 25, the sampling period T c = T = 0.1(s). The dimension of past data P h = 20. In the first 2 seconds, a small probing noise sin 2 (t) + 0.5 cos(t) − 0.1sin 2 (t) cos(t) + sin 5 (t) is added to the control inputs to excite the system and collect the past data fully. Figures 3 and 4 show that after 20s, when NN weights converge, the state trajectory of agents 1 and 2 synchronize with the states of the leader while the state trajectory of agent 3 synchronizes with the states of agents 1 and 2. From  Fig. 5, it can be seen that the state trajectory of the leader and all agents approach to the origin in finite time. The costs of the leader and agents are presented in Fig. 6, where all signals converge to the near-optimal values.
The control inputs of both ET and TT algorithms, an example of agents 0 and 2, are compared in Figs. 7 and 8. Although the control signals are saturated at the maximum and minimum values, the closed systems are always stable. The control inputs of the time-triggering algorithm are smoother         which are the same values as previous. Within the inter-event times, the systems are controlled by last triggered control signals.  In Figs. 9-12, the thresholds ∥e hT ∥, h = 0, 1, 2, 3 are reduced accordingly the consensus achievement. The triggering errors are less than the thresholds all the time. The inter-event intervals generated by the ET control algorithm is shown in Fig. 13, where the minimum inter-event time is 0.3s. Observing Figs. 9-13 we see that the Zeno phenomenon is excluded.
The effectiveness of reducing burden of communication cost is described in Table 1. The total number of communication times for the event-triggering algorithm is 236 while the total number for the time-triggering algorithm is 1200.

V. CONCLUSION
This paper proposed a method based on the event-triggering signal transmission of the IoT. Such a method is required for the design of the algorithm of distributed H ∞ optimal control for Van der Pol oscillators with unknown internal dynamics, input constraint, and external disturbance. The system dynamics have been transformed into the triggering consensus tracking dynamics, for which the distributed robust optimal control algorithms have been constructed. The algorithm have employed the event-triggering signal transmission of the IoT to reduce the burden of the communication resource and computation bandwidth. As a result, the optimal control policy and disturbance compensation policy have been derived based on the adaptive dynamic programming and two-player zero-sum game theory. The triggering VOLUME 11, 2023 condition has been established such that the Zeno phenomenon is excluded and the stability of the overall closed systems is guaranteed. The numerical simulation results with comparison to the time-triggering algorithms have confirmed the effectiveness of the proposed algorithm. In the future work, we shall concentrate on switching IoT topologies with time-delay.

APPENDIX A PROOF OF THE THEOREM 1
Proof: Followed by Lemma 1, one only needs to prove the stability of each agent. Note that by settingk 0 = 0, x 0 = δ i , and x 0 = δ i , it is easy to infer the proof for the leader based on the proof for the agent. First, we propose a Lyapunov function candidate for agent i as for the optimal value functions V ⋆ i (δ i ) and V ⋆ i (δ i ). We divide the proof into two situations: within the triggering intervals and at the triggering time.
Situation 2: ∀t = t i k , ∀k ∈ N, taking the difference of (33) one obtains From (53), asL i < 0 and the trajectories of (28) and (15) are continuous, we have Then, we rewrite L i as where κ i is in a class-κ function [41]. Recalling (54), it can be seen that the Lyapunov function (33) it is continuously reducing at any triggering time, t = t i k , k ∈ N. From (53) and (57), it can be included that the closed dynamics has been asymptotically stable.
To prove the Zeno behavior is excluded, we observe (24), (25) and Lipschitz property of f i (z i ). The dynamics (15), ∀t ∈ [t i k , t i k+1 ), satisfies For a small positive real number a i , we have where i = i0 + b if . Formally, we can follow from [43] to prove the rest of the proof. This completes the proof.