Optimal event‐triggered control for wireless power transfer system in electric vehicles

This study focuses on stabilizing a bidirectional inductive wireless power transfer (WPT) system using an event‐triggered approach. It is only assumed the inductor currents on both the primary and pickup sides are measurable, and they are sent synchronously to the controller via a digital channel. To estimate the unmeasured states and maintain plant stability, a full‐order state observer and an observer‐based controller have been developed. The control parameters are optimized through a genetic algorithm to achieve the desired output response. An emulation methodology is then applied to create an output‐based event‐triggering condition. This condition ensures the stability of the closed‐loop system even in the presence of communication constraints. To prevent Zeno sampling, a minimal time interval between two transmissions is enforced using time‐regularization techniques. Furthermore, the performance of the event‐triggered controller is enhanced by solving a linear matrix inequality condition, which further reduces the number of transmission instances. The methodology offers a systematic and optimal design for the bidirectional inductive WPT system. It eliminates the need for manual tuning of control parameters, which is particularly beneficial given the system's complex nature. To address both continuous‐time and discrete‐time dynamics, the entire system is represented as a hybrid dynamical system, making it more intuitive for networked control systems. The efficiency of this approach is assessed through numerical simulations of a WPT system, demonstrating its effectiveness. The results show that the average intertransmission interval has been increased from 0.0799 to 0.1782 s, that is, the proposed event‐triggering strategy reduced the number of transmissions to more than 50% compared with conventional periodic sampling.


| INTRODUCTION
The increasing popularity and global demand for electric vehicles (EVs) as an alternative to fossil fuel vehicles has led to great research efforts in this direction.The EVs provide attractive advantages in terms of reduced pollution, contribution to handling the global warming and climate change phenomena, and independence from scarce fossil fuels. 1,2In this regard, the charging capability of EVs is considered one of the main technical challenges of the deployment of such EVs due to the long charging time and the need for installing sufficient charging stations to plug in the battery.To overcome these problems, wireless power transfer (WPT) technology has been developed to make the charging process easier and more convenient for consumers.In this case, the user only needs to park and leave the EV at any stationary WPT station to charge the vehicle battery.Although the concept is tempting, implementing WPT presents considerable difficulties due to the sluggish charging speed and complicated control mechanisms, see References [3-10].Note that the charging of EVs can have a notable impact on the stability of transfer charging systems.The addition of multiple EVs to a charging network can strain the power grid and create fluctuations in power demand, potentially leading to voltage drops or instability.To mitigate these issues, advanced management and control systems are required to balance and coordinate the charging loads across different vehicles and charging stations, ensuring a stable and reliable power transfer process.Moreover, implementing smart grid technologies, such as demand response and load shedding, can further enhance the stability of transfer charging for EVs, enabling the efficient and widespread adoption of electric mobility without compromising the integrity of the power supply infrastructure.Moreover, the control design of WPT is usually implemented over a communication channel between the sensors and the controller, which requires taking into account the sampling-induced errors due to the network. 11Hence, the research question of interest here is how to stabilize the WPT system while reducing the amount of communication between sensors, controllers, and actuators.
An efficient design approach to deal with networked control design is the event-triggered control (ETC), in which the sampling times are produced by a statedependent rule rather than in a periodic manner.The approach has attracted great research attention in the last few decades due to its superior performance in reducing the utilization of limited bandwidth of digital channels, see, for example, References [12-20], and the reference therein.Hence, ETC is well adapted for many control architectures, such as networked control systems, [21][22][23] cyber-physical systems, 24 multiagent systems, [25][26][27] robotics applications, 28,29 sensors networks, 30,31 and distributed control systems. 23,32,33his paper is dedicated to the problem output feedback stabilization for bidirectional inductive WPT systems under ETC.Specifically, it is assumed that only the inductor currents on the primary and pickup sides are available for measurement and transmitted to the controller via a digital channel.The approach involves an emulation procedure to design the ETC.Initially, the focus is on stabilizing the continuous-time plant while disregarding network effects.Then, considering the sampling impact, an event-triggering condition is formulated to uphold the closed-loop system's stability.To achieve this, a full-order observer is developed for state estimation, and an observer-based controller is designed for plant stabilization.The control gains are optimized via a genetic algorithm to automatically compute the desired parameters for the linear quadratic regulator (LQR) controller, eliminating the need for manual tuning, which is particularly beneficial for large-scale plant models.The methodology also accounts for communication constraints and optimizes an outputbased ETC to reduce transmission instants.Moreover, the triggering mechanism is optimized to minimize transmissions further.The entire closed-loop system is depicted as a hybrid dynamical system, encompassing both continuous-time and discrete-time dynamics due to sampling.The conditions necessary for applying this technique are expressed as linear matrix inequalities (LMIs).The effectiveness of the proposed method is assessed through numerical simulations within a WPT system.
The contribution of this study is outlined below.
• A systemic design procedure of optimal output feedback ETCs for bidirectional inductive WPT systems is developed.• The feedback law is optimized using a genetic algorithm.• The performance of the ETC is enhanced by solving an LMI optimization condition.• The closed-loop system is developed as a hybrid dynamical system to intuitively describe the mixed continuous-time and discrete-time dynamics of the system.• The proposed approach achieves the desired closedloop performance while reducing the amount of transmissions.
5][36][37][38][39] The result of Ahmad et al. 34 is concerned with the ETC of the active suspension system based on a T-S fuzzy active suspension model.The proposed approach in Ahmad et al. 34 combines inwheel motor technology, active suspension control, decentralized dynamic event-triggered communication, and dynamic damping.By integrating these components, the study aims to optimize vehicle dynamics, improve ride comfort, and achieve energy savings.The approach of Ahmad et al. 34 assumes that the full-state measurement is available for feedback, which can be difficult to satisfy in practice.The approach of Pham et al. 35 explored a novel event-triggered mechanism for EVs to provide multiple frequency services in smart grids.The study focuses on the integration of EVs into the grid to enhance grid reliability and stability, especially in scenarios where multiple grid frequencies are present.The proposed mechanism employs ETC strategies to enable EVs to adapt to varying grid conditions and contribute to grid services, thus supporting the broader integration of renewable energy sources and improving grid efficiency.The developed feedback law in Pham et al. 35 is a static output feedback law, which requires more restrictive conditions to apply compared with observer-based controllers.Han et al. 36 investigated an ETC approach for managing hybrid energy storage systems in EVs by employing active disturbance rejection control with event-triggered mechanisms.By implementing this control strategy, the research aims to minimize energy losses and optimize the use of various energy sources, ultimately contributing to the improvement of EV power management and the extension of the EV range.The approach of Han et al. 36 is adapted to the case of state feedback control, which cannot be feasible for practical systems.The technique of Saxena and Fridman 37 developed a load frequency control in power systems using event-triggered techniques.The study explored a switching-based control strategy to enhance the stability and performance of load frequency control.By incorporating event-triggered mechanisms, the research aims to reduce the communication and control overhead while maintaining the desired system stability and frequency regulation.The result of Saxena and Fridman 37 is dedicated to static output feedback control, which is a conservative approach compared with dynamic controllers and observer-based feedback laws.The technique of Incremona and Ferrara 38 investigated the problem of ecodriving in urban traffic networks for EVs.The study employs ETC and sliding mode control techniques to optimize driving strategies for EVs, with a focus on enhancing energy efficiency and reducing environmental impact.The feedback law in Incremona and Ferrara 38 is designed based on the full-state measurement, which may not be available in practice.Wu et al. 39 proposed an event-triggered model predictive control approach for efficiently managing the dynamic energy needs of EVs within microgrids.The study focused on optimizing energy usage and distribution in microgrid systems where EVs are integrated.By employing ETC strategies, the research aims to enhance the flexibility and responsiveness of energy management, which is crucial for maximizing the utilization of renewable energy sources, minimizing grid stress, and promoting sustainable transportation solutions.The technique of Wu et al. 39 relies on the availability of full-state measurements, which can be difficult to satisfy in practice.
The novelty of the proposed approach with respect to existing techniques is summarized in Table 1.
As shown in Table 1, the addressed control problem in this study is different from those considered in References [34-39].Moreover, it is not required in this study the knowledge of the full-state measurement as in References [34, 36, 38, 39], and the plant is stabilized using dynamic (observer-based) controllers, which is more general and less restrictive than static output feedback control as in References [35, 37].Furthermore, the overall system is modeled as a hybrid dynamical system to account for both continuous-time and discretetime dynamics, which is not the case in the previously mentioned results.

| METHODOLOGY
In this section, the used notation is first given below, and then the problem formulation, the hybrid model, and the control design approach are presented.

| Notation and preliminaries
In this section, we present the mathematical notation that we adopt throughout the analysis in this study.We also introduce the form of the hybrid dynamical systems that we use to model the overall system.
We represent the set of real numbers by the symbol and the set of integers by .For a real symmetric matrix .
In this paper, we consider hybrid systems of the following form 40 : where ∈ ξ n ξ is the state variable,  is the flow set, F is the flow map,  is the jump set, and G is the jump map.

| Setup description
The equivalent circuit of the inductive WPT system is shown in Figure 1.The pick-up output linked to the load is depicted as a direct current source capable of either absorbing or providing power.A steady track current i t ( ) flows from the primary supply through L T , magnetically linked to the pick-up coil.Both primary and pick-up circuits utilize nearly identical electronics, featuring a reversible rectifier and a tuned (resonant) inductor-capacitor-inductor setup, enabling two-way power transfer between the track and the pick-up.
A common setup for a bidirectional WPT system typically comprises bidirectional high-frequency converters, compensation circuits, and coupling coils on both the primary side (power source) and the secondary side (EV).These coils are inductively coupled to enable WPT and are fine-tuned to resonate at the system operating frequency using the compensation circuits, which can be categorized based on their configurations.Among these, the series-series topology compensation circuits have demonstrated superior power factor, increased power transfer efficiency, and a consistent output-current profile.These attributes play a pivotal role in enhancing the performance of a bidirectional WPT system while streamlining its power regulation.Through inductive couplings, the energy is moved from the primary side to the secondary side over the air gap.For further information on the schematic circuit of the WPT system, we direct the reader to Swain et al. 41 and Rana et al. 42 The WPT dynamic model is given below Primary controller

Secondary controller
F I G U R E 1 Schematic equivalent circuit of the wireless power transfer system.
are available for measurement and can be sent to the controller.Moreover, the control input of the system (2) consists of the voltages υ pi and υ so at the primary and secondary sides, respectively.As a result, system (2) can be modeled in the following state space form: where , , , ,  , , ) pi cpi pt T so cso st si denotes the state vector, pi so is the control input and T so is the output.The matrices A B C , , are given by The transmitting and receiving converters typically function at a predetermined, constant frequency to maintain resonance within the system and achieve efficient power transmission between the two endpoints.Consequently, such fixed frequencies can be conservative and lead to unnecessary computations and control updates.Our objective is to design an event-triggered controller that can reduce the converters' operating frequency while maintaining the desired system response.
We assume that the sensors transmit output data to the controller over a digital channel at discrete-time instants.Then, our objective is to synthesize an ETC to stabilize the WPT system in the presence of sampling.To that end, the problem is solved by emulation as we first stabilize the WPT system in continuous time, then the effect of sampling is considered as explained in the sequel.
Due to the fact that only a portion of the state can be measured, we use a Luenberger full-order observer to stabilize the system as follows: where x ˆdenotes the observer state and K L , are gain matrices with appropriate dimensions.The estimation error is defined as follows: Consequently, in view of (3) and ( 5), we have that On the basis of ( 3) and ( 7), we have The stability of the closed-loop system can, therefore, be ensured if K L , are designed such that the eigenvalues of A BK − and A LC − are strictly negative (assuming A B ( , ) are controllable and A C ( , ) is observable).

| Hybrid dynamical model
Since the output y t ( ) is broadcast to the controller at discrete-time instants ∈ t k , k , then the digital implementation of (3) and ( 5) leads to We define the sampling-induced error as ).
Since y t ( ) k is updated to y t ( ) at each transmission instant, it should be noted that the sampling error s t ( ) is reset to zero at each t k .Accordingly, ( ) ( ) From ( 10) and ( 9), we get and where We define the following variable ∈  τ 0 to keep track of the intervals between transmissions: We also define the following function 0 that we will use later in the design of the ETC: where for some ς > 0 and  β 0, γ > 0 will be defined later.
Hence, in view of ( 9)- (15), we obtain the following impulsive model: The closed-loop system clearly involves both continuous-time and discrete-time dynamics.Therefore, representing the closed-loop system as a hybrid dynamical system is more intuitive.Define , then we have where  and  denote the flow set and the jump set, respectively, and (Ξ), (Ξ)   denote the flow map and the jump map, respectively and are given by The flow set  is used to define the state region at which system (18) evolves in continuous time while the jump set  determines the state region of discrete transitions.Intuitively, these sets will be designed based on the ETC, as we will explain later.

| Problem statement
The objective of this study is to synthesize an optimal output feedback controller (5) and an optimal eventtriggering mechanism such that the stability of the closed-loop system is ensured while the communication load between the sensors and the controller is reduced.

| Optimal design of the eventtriggered controller
The design procedure follows the emulation approach in the sense that we first stabilize the closed-loop system in continuous time, that is, by ignoring the effect of the network, and then we take into account the effect of sampling and we synthesize the event-triggering mechanism.

| First step: Design of the feedback law
Let us start by assuming that y x = , that is, the whole state measurement is accessible.Using the subsequent quadratic cost function, an LQR controller can be developed to optimize the state response and the control effort.
where Q R , 1 1 are symmetric positive definite diagonal matrices.Then, by solving the algebraic Riccati equation the control gain matrix K is given by We now take into account that only y is monitored, not the entire state.The state is then estimated using the observer (5).The Kalman filter approach is utilized to design the observer gain L. Hence, the following algebraic Riccati formula needs to be solved: where the ⩾ Q R > 0, 2 2 are symmetric real matrices.The gain L is therefore given by The structures of the control matrices depend mainly on the desired output performance and the produced control effort and are usually manually tuned.However, due to the large dimension of the WPT system, this manual tuning of 2 matrices will be hard until a satisfactory response can be obtained.An efficient methodology to automatically compute such matrices according to the required performance is the genetic algorithm, see, for example. 43Figure 2 provides an outline of such a technique.
The process starts with generating a random initial population.At each iteration, individuals with superior fitness values are chosen to form the subsequent population.Crossover operations are then performed by merging pairs of parents from the current population.Following this, mutations are introduced by randomly altering the genes of individual parents.Parents for the next generation are selected using a scaling function based on their fitness values.This process continues until optimal values are attained.
We follow this approach to optimize the control matrices as in Abdelrahim and Almakhles. 44

| Second step: Design of the eventtriggering rule
After the controller gain parameters have been optimized, we now proceed to the design of the ETC such that the closed-loop stability is preserved in the presence of sampling.An important property of the ETC is to ensure that Zeno sampling does not occur, that is, to prevent infinite transmission times in a finite time.For this purpose, we use the timeregularization technique, and we design ETC conditions of the following form: where σ T , > 0 are design parameters to be specified later.According to (25)  45 and Nešić et al. 46 To properly design this time T , we need to verify certain sufficient conditions to ensure closed-loop stability as stated below, see Carnevale et al. 45 and Nešić et al. 46 Assumption 1.Consider the closed-loop systems ( 13) and ( 14).There exists ε γ > 0 and a positive definite symmetric real matrix P such that the following condition holds: If we define , then it is straightforward to show that the feasibility of ( 26) implies We also note that Accordingly, the time constant T in ( 25) with γ from Assumption 1.Note that the time γ ( )  corresponds to the time it takes for ζ t ( ) in ( 16) to decrease from λ −1 to λ, see Carnevale et al. 45 and Nešić et al. 46 for more detail.
It is obvious from ( 27) that if we restrict the sampling error to satisfy Now we can design the flow and jump sets in (18) based on the ETC as follows: where ≔ σ ε γ .The construction of the ETC (31) ensures that the Zeno phenomenon is automatically ruled out thanks to the enforced dwell-time T between two samples.

| RESEARCH RESULT
In this section, we present the derived stability result and the simulation results.

| Stability result
We present here the main result of this paper.
Theorem 1.Consider systems (18) and (31).Suppose that Assumption 1 holds.Take σ = ε γ 31).There exist Ω > 0 and ∈ K Γ  such that any solution Φ = (Φ , Φ , Φ ) and The first property means that the origin of systems (18) and ( 31) is globally asymptotically stable.The second property implies that all triggering times are separated by at least T amount of time.

2
, it holds that As a result, in view of ( 35) and ( 39), it holds that, for all ∈ ξ  which implies global asymptotic stability of the closedloop system.□

| Optimization of the ETC
We note from (31) that to optimize the performance of the ETC, that is, reduce the number of transmissions, we need to enlarge the threshold σ.Consequently, the sampling error s t ( ) will take a longer time until the condition is violated, which leads to enlarging the intertransmission times.To that purpose, since σ = ε γ , we need to reduce γ and enlarge ε provided that the LMI (26) is feasible.This leads to solving the following optimization problem: Remark.The implementation of the eventtriggering mechanism in (31) requires continuous verification of the triggering rule ( ) after the elapse of the enforced dwell-time T .On the one hand, this continuous check of the triggering condition provides an instantaneous response against any change in the system state due to external disturbances, which is a desirable robustness property in practice.On the other hand, the continuous verification of the triggering rule entails the employment of feedback sensors that are equipped with some computational capabilities to implement this mechanism to determine the next transmission instantly.Alternatively, periodic event-triggering control (PETC) schemes can be investigated which the triggering rule is only verified at periodic time instants.However, this requires a complete set of assumptions and different analysis tools compared with those used in this study.Moreover, the utilization of PETC techniques is often less robust than continuous event-triggering methods as the developed one in (31).

| Simulation results
To demonstrate the efficiency of the proposed approach, the result is applied to the WPT with the parameters in Rana et al., 42 as shown in Table 2.
Then, we apply the developed ETC approaches by following the procedure in Algorithm 1.
Algorithm 1. Guidelines on how to apply the ETC approaches.

Start
1: Define the wind turbine parameters and compute the matrices 2: Check controllability and observability If the system is controllable and observable, do: , and compute the observer gain L 3: Construct the matrices , , , The GA-based is executed with 100 individuals, 10 chromosomes, 100 generations, and the following fitness function f t : where t t M , , > 0 r s p denote the rise time, the settling time, and the maximum peak, respectively.Consequently, we get Then, we take the eigenvalues of the observer to be λ A LC λ A BK ( − ) = 5 ( − ) and compute the corresponding value of the gain matrix L.
Figure 4 shows that the capacitor voltage υ t ( ) cpi and its estimate.It can be noted that the estimated state captures the actual state in a short time and both the states converge asymptotically to the origin.
Figure 5 shows that the capacitor voltage υ t ( ) pt and its estimate.We note that the estimation error rapidly vanishes, and both states converge asymptotically to the origin.
Figure 6 shows that the inductor current t ( ) T and its estimate.It can be noted that the estimated state captures the actual state in a short time, and both the states converge asymptotically to the origin.
Figure 7 shows that the inductor current i t ( ) so and its estimate.We note that the estimation error rapidly vanishes and both the states converge asymptotically to the origin.
Figure 8 shows that the capacitor voltage υ t ( ) cso and its estimate.It can be noted that the estimated state captures the actual state in a short time, and both the states converge asymptotically to the origin.
Figure 9 shows that the capacitor voltage υ t ( ) st and its estimate.It can be noted that the estimated state captures the actual state in a short time, and both the states converge asymptotically to the origin.
Figure 10 shows that the inductor current i t ( ) si and its estimate.We note that the estimation error rapidly vanishes, and both the states converge asymptotically to the origin.
The control input signals υ t ( ) pi and υ t ( ) so are shown in Figure 11 where it can be noted that the control signal converges to zero once the plant state is stabilized at the origin.
The intertransmission times in Figure 12 show how the transmission instants are generated by the ETC according to the status of the output response.It is worth mentioning that the genetic algorithm used to compute the LQR matrices Q and R is greatly beneficial due to the large dimension of the plant, which allowed us to compute their parameters automatically in an optimized manner.Otherwise, the manual tuning of such matrices would require too many iterations until a satisfactory output response can be obtained.To better illustrate the benefit of the ETC compared with traditional periodic time-triggering, we present here the simulation result of the latter case.Hence, we assume that the sequence of transmission instants is generated periodically based on the elapse of a fixed time constant T .In this way, the triggering condition (25) becomes Consequently, in view of (18), the flow and jump sets become We run the simulation for the new hybrid system (18)-( 46) with the same parameters and initial conditions as in Section 3.3.The obtained state and control trajectories are almost the same as in Figures 3-11, however, the intertransmission times are shown in the below.
As expected, the transmission instants are periodically generated, which results in many more transmissions than with event-triggered implementation, as shown in Figure 13.The advantage of the proposed event-triggering scheme is highlighted in Figure 14.We note that the average intertransmission times generated by the ETC τ = 0.1782 avg,ETC is more than twice the average value of periodic sampling τ = 0.0799 avg,periodic .In other words, the amount of transmissions under ETC has been reduced to more than 50% than the traditional periodic sampling.

| CONCLUSION
An optimal control design for a bidirectional inductive WPT system has been explored.Firstly, a genetic algorithm has been employed to construct an optimal LQR observer-based controller for the continuous-time plant to optimize the power transfer efficiency and ensure overall system stability.Subsequently, an output feedback event-triggering mechanism has been constructed to enhance the closed-loop system performance.Moreover, the proposed ETC mechanism has been  optimized by solving an LMI condition to minimize the communication load.Simulation results indicate that the optimized ETC has reduced transmissions by over 50% compared with traditional periodic sampling schemes.The overall system has been analyzed as a hybrid dynamical system, and the stability under variable intertransmissions has been investigated using appropriate Lyapunov functions.Numerical simulations demonstrate the effectiveness of our approach, showing improved output response while significantly reducing power transmissions, thereby enhancing energy efficiency and reducing network usage.
The charging of EVs can introduce challenges related to harmonic distortion in the power distribution system.When EVs are charged, the power converters and chargers often generate harmonic currents that can distort the quality of electrical voltage within the grid.These harmonics can interfere with other sensitive electronic equipment connected to the same power network and potentially compromise the overall power quality.To address this issue, it is crucial to employ harmonic filtering and advanced power quality control mechanisms in charging infrastructure, which can help minimize the introduction of harmonics and maintain the integrity of the electrical grid, ensuring that EV charging remains efficient and harmonically clean, without adversely affecting other connected systems.
This work can be extended in different directions: • In the current study, we assumed that the WPT system is not affected by external disturbances or measurement noises.Hence, it would be practically important to consider these issues and provide a more robust control design.F I G U R E 14 Comparison between periodic sampling and event-triggering.
• While we have considered in this work the sampling effect of the communication network, it would also be important to study other phenomena induced by the network, such as quantization and delays.• The control design approach has been carried out based on the linearized model of the WPT system, which does not take into account the coupling effect and/the mutual induction element.The extension of the methodology to the nonlinear model is challenging yet important.• The practical application of the technique on an experimental setup would further support the obtained result in this study.

4 : 2 :
Check the feasibility of LMI(27)  If LMI is feasible, do:1: Find γ L, and compute σ T , Set the initial condition and start the simulation End

T A B L E 2
Parameters of the wireless power transfer system.

F I G U R E 3
State trajectories of x 1 and x ˆ1 for the first 0.5 s.F I G U R E 4 State trajectories of x 2 and x ˆ2 for the first 0.5 s.F I G U R E 5 State trajectories of x 3 and x ˆ3 for the first 0.5 s.F I G U R E 6 State trajectories of x 4 and x ˆ4 for the first 0.5 s. control

7 8
State trajectories of x 5 and x ˆ5 for the first 0.5 s.State trajectories of x 6 and x ˆ6 for the first 0.5 s.

9
State trajectories of x 7 and x ˆ7 for the first 0.5 s.U R E 10 State trajectories of x 8 and x ˆ8 for the first 0.5 s.

F I G U R E 11 1 F
Trajectories of the control input for the first 0.5 s F I G U R E 12 Intertransmission times for 10 s.T A B L E 3 Summary of the obtained results.I G U R E 13 Intertransmission times with time-triggered implementation.
The transpose of a matrix A is written as A T while n denotes the identity matrix with dimension n.For two vectors max denote the minimum and maximum eigenvalues of Q, respectively.T T T .The Euclidean norm on n is denoted by  • and for a matrix ∈ Q , a new transmission instant t + 1 k is only permitted after the passage of time T since the previous triggering instant t k and such that The time T is designed as the Maximally Allowable Transmission Interval bound developed in Carnevale et al.