An observer and PID-based disturbance control method for wireless power transmission system

Abstract

This invention relates to the field of wireless power transfer (WPT) technology, specifically disclosing a disturbance control method for wireless power transfer systems based on an observer and a PID controller. To address the problem of poor anti-mutual-inductance disturbance control performance in WPT systems using traditional PID control methods, a PID controller based on the internal model is designed based on the system's Hammerstein model. Furthermore, the disturbance observer bandwidth is determined according to the controller bandwidth, and the observer parameters are determined using the pole placement method. This achieves anti-mutual-inductance disturbance control of the WPT system, ensuring that the output voltage U... out Closely track the voltage setpoint u ref This greatly suppresses large voltage fluctuations caused by changes in mutual inductance. Simulation results show that the PID control method for the WPT system based on a disturbance observer proposed in this invention can achieve higher robustness, and the anti-mutual inductance disturbance performance can be adjusted by the observer bandwidth parameter.

Description

Technical Field

[0001] This invention relates to the field of wireless power transfer (WPT) technology, and more particularly to a disturbance control method for wireless power transfer systems based on an observer and PID. Background Technology

[0002] In recent years, wireless power transfer (WPT) technology has become a research hotspot. It can completely isolate the power source and the charging device. Especially in charging systems for drones and underwater UUVs, hovering wireless charging can improve the safety, reliability, and flexibility of the power supply equipment.

[0003] Typically, considering the safety of electrical equipment, a closed-loop control strategy is required for the output voltage. To improve system control performance, researchers have developed various WPT (Wait-and-Put) system control techniques, among which Proportional-Integral-Derivative (PID) control is widely used due to its simplicity and low computational complexity. In actual system operation, drastic changes in system mutual inductance parameters caused by environmental factors can lead to time-varying disturbances, resulting in unstable output voltage. This can be addressed by establishing a time-varying parameter model of the system and designing a controller for the time-varying system to achieve satisfactory control. However, WPT system models are high-order, the parameter changes are stochastic, and the establishment of time-varying parameter models is complex, often resulting in more complex and inaccurate models, leading to unsatisfactory control performance. Typically, WPT systems form a closed loop through a communication link between the primary and secondary sides to control the output voltage, such as... Figure 1 As shown, data sampling, processor calculation, and model simplification in the system will lead to time delays. Furthermore, when using a traditional PID controller, to ensure the closed-loop stability of the time-delay system, it is necessary to reduce the integral effect, which will slow down the closed-loop response and reduce control quality, and cannot effectively eliminate time-varying disturbances. The tuning of traditional PID controller parameters often relies on engineering experience, making the parameter tuning process cumbersome. To better achieve a timely and stable control system using the PID method, it is necessary to adopt a controller design method that is simple to tune, has delay compensation, and is resistant to disturbances in mutual inductance parameters. Summary of the Invention

[0004] This invention provides a disturbance control method for wireless power transfer systems based on observers and PID controllers. The technical problem it solves is that traditional PID control methods for WPT systems have complex parameter tuning and poor control quality against disturbances in mutual inductance parameters.

[0005] To address the above technical problems, this invention provides a disturbance control method for a wireless power transfer system based on an observer and PID controller, comprising the following steps:

[0006] S1. Construct the Hammerstein model of the WPT system;

[0007] S2. Design a closed-loop PID controller based on the linear part and internal model control principle of the Hammerstein model;

[0008] S3. Set the PID parameters of the closed-loop PID controller according to the robustness condition;

[0009] S4. Design a Romberg perturbation observer based on the linear part of the Hammerstein model;

[0010] S5. The WPT system is controlled based on a closed-loop PID controller and a Romberg disturbance observer, so that the output voltage y u Tracking voltage setpoint u ref .

[0011] Furthermore, in step S2, the closed-loop PID controller is designed as follows:

[0012]

[0013] in, λ is the inverse of the minimum phase part of M(s), where M(s) represents the mathematical model of the WPT system object obtained through the data-driven method, s represents the Laplace operator, e represents the delay operator, λ represents the filter coefficient, and L is the lag time.

[0014] Furthermore, for a first-order time-delay WPT system, M(s) is designed as follows:

[0015]

[0016] Where K is the process steady-state gain, K is the process steady-state gain, and ξ is the damping coefficient.

[0017] Furthermore, for a second-order time-delay WPT system, M(s) is designed as follows:

[0018]

[0019] Furthermore, the open-loop transfer functions of the first-order time-delayed WPT system and the second-order time-delayed WPT system are designed as follows:

[0020]

[0021] Further, in step S3, based on the maximum sensitivity M s Determine the adjustable parameters of the PID controller, M. s Designed as follows:

[0022]

[0023] Where ω represents the control frequency, and max represents taking the maximum value.

[0024] Furthermore, for a first-order WPT system with time delay, the proportional parameter K of the closed-loop PID controller... p Integral parameter K i Differential parameter K d Set them to:

[0025]

[0026] For a second-order WPT system with time delay, the proportional parameter K of the closed-loop PID controller p Integral parameter K i Differential parameter K d Set them to:

[0027]

[0028] Further, in step S4, the Romberg disturbance observer is connected between the PID controller in the closed-loop PID controller and the Hammerstein model; the Romberg disturbance observer is used to estimate the disturbance of the system;

[0029] The Romberg perturbation observer is represented as follows:

[0030]

[0031] Where L represents the parameter matrix of the Romberg disturbance observer, A, B, and C represent the system state matrix, control matrix, and output matrix, respectively, x1 and z1 represent the system output state and system output state estimate, z represents the system state estimate, and u represents the system input. Denotes the first derivative of z. This indicates the system output estimate.

[0032] Furthermore, for a second-order time-delayed WPT system, H = [β1β2β3] T The three parameters are designed as follows:

[0033]

[0034] Where ω0 is the bandwidth of the Romberg perturbation observer, a0, a1, and b0 are the parameters of the second-order time-delayed WPT system, and the state differential equation of the second-order time-delayed WPT system is expressed as:

[0035]

[0036] y represents the output of the second-order time-delayed WPT system, and d(t) represents the perturbation of the second-order time-delayed WPT system. Let be the first and second derivatives of y, respectively.

[0037] Further, in step S1, the Hammerstein model is expressed as:

[0038] u o =f(d)G p (s)

[0039] f(d) is the nonlinear static characteristic of the WPT system, d is the duty cycle of the primary-side drive signal of the WPT system, and G p (s) is a linear time-invariant model.

[0040] This invention provides a disturbance control method for a wireless power transfer system (WPT) based on an observer and a PID controller. The method includes constructing a Hammerstein model of the WPT system, designing a closed-loop PID controller based on the linear part and internal model control principles of the Hammerstein model, setting the PID parameters of the closed-loop PID controller according to robustness conditions, designing a Romberg disturbance observer based on the linear part of the Hammerstein model, and controlling the WPT system using the closed-loop PID controller and the Romberg disturbance observer to ensure the output voltage y... u Tracking voltage setpoint u ref This invention, based on the Hammerstein model of the system, designs a PID controller based on the internal model. It further determines the disturbance observer bandwidth based on the controller bandwidth and determines the observer parameters using the pole placement method, thereby achieving anti-mutual inductance disturbance control of the WPT system and ensuring the output voltage U... out Closely track the voltage setpoint u ref This greatly suppresses large voltage fluctuations caused by changes in mutual inductance. Simulation results show that the PID control method for the WPT system based on a disturbance observer proposed in this invention can achieve higher robustness, and the anti-mutual inductance disturbance performance can be adjusted by the observer bandwidth parameter. Attached Figure Description

[0041] Figure 1 This is a structural diagram of the closed-loop WPT system provided in an embodiment of the present invention;

[0042] Figure 2 This is an example diagram of the phase shift control principle and the output voltage of the inverter provided in this embodiment of the invention;

[0043] Figure 3 This is an architecture diagram of the Hammerstein model of the WPT system provided in this embodiment of the invention;

[0044] Figure 4 This is an architecture diagram of the closed-loop PID controller provided in an embodiment of the present invention;

[0045] Figure 5 This is an architecture diagram of a closed-loop PID controller based on disturbance observation provided in an embodiment of the present invention;

[0046] Figure 6 This is a structural diagram of the Romberg observer provided in an embodiment of the present invention;

[0047] Figure 7 This is a graph of the identification model provided in the embodiments of the present invention;

[0048] Figure 8 This is a comparison diagram of the transient output response under PID control and PID+observer control provided in the embodiments of the present invention;

[0049] Figure 9 This is a comparison chart of the output responses of closed-loop control, standalone PID control, and PID+observer control provided in the embodiments of the present invention under mutual inductance parameter disturbances. Detailed Implementation

[0050] The embodiments of the present invention are described in detail below with reference to the accompanying drawings. The embodiments are given for illustrative purposes only and should not be construed as limiting the present invention. The accompanying drawings are for reference and illustration only and do not constitute a limitation on the scope of patent protection of the present invention, because many changes can be made to the present invention without departing from the spirit and scope of the present invention.

[0051] The Hammerstein model consists of nonlinear static characteristics and a linear time-invariant autoregressive model (ARX model) with exogenous inputs. The WPT system model can be described using the Hammerstein model. Through data-driven modeling, the model structure and parameters can be obtained simply by identifying the sampled input and output data. Data-driven WPT system modeling can accurately estimate the model parameters after system offset and the system model's time delay. The internal model control (IMC) design method facilitates the design of PID controller parameters. This method considers system delay and is simple to design. The Romberg disturbance observer, built based on the system state equations, can observe system disturbances in real time and perform feedforward compensation, avoiding output disturbances caused by changes in the mutual inductance parameters of the WPT system, and significantly enhancing the robustness and anti-interference performance of the PID control algorithm. Using a combination of PID and disturbance observers in the WPT system can better address the problems of mutual inductance parameter disturbances and communication delays in the control system. Based on these theoretical foundations, this invention provides a disturbance control method for WPT systems based on observers and PID, which solves the problems of complex parameter tuning and poor control quality against mutual inductance parameter disturbances in traditional WPT system PID control methods.

[0052] This invention provides a disturbance control method for a wireless power transfer system based on an observer and a PID controller, comprising the following steps:

[0053] S1. Construct the Hammerstein model of the WPT system;

[0054] S2. Design a closed-loop PID controller based on the linear part and internal model control principle of the Hammerstein model;

[0055] S3. Set the PID parameters of the closed-loop PID controller according to the robustness condition;

[0056] S4. Design a Luenberger perturbation observer based on the linear part of the Hammerstein model;

[0057] S5. The WPT system is controlled based on a closed-loop PID controller and a Romberg disturbance observer, so that the output voltage y u Tracking voltage setpoint u ref .

[0058] (1) Step S1: Construct the Hammerstein model of the WPT system

[0059] As an example, Figure 1This illustrates a typical closed-loop WPT system, consisting of a primary side and a secondary side. The primary side comprises a DC power supply, an inverter, an LCC compensation network, and a transmitting coil L, connected in sequence. p The secondary side includes sequentially connected receiving coils L s Series compensation capacitor C s 1. Rectifier circuit, filter capacitor C d Load resistance R L The LCC compensation network includes a series compensation inductor L f Series compensation capacitor C p and parallel compensation capacitor C f U dc u represents the DC input voltage of the WPT system. ab i represents the first effective value of the inverter output voltage. f u cf L respectively f The current and voltage, i p u cp L respectively p The current and voltage, u cf C represents f The voltage, M represents L p and L s Mutual intuition between them, i s u cs L respectively s The current and voltage, i r u r These represent the current and voltage of the rectifier circuit, respectively. o R represents L The voltage is the system output voltage, and R1, R2, and R3 represent L respectively. f L p and L s The equivalent series resistance.

[0060] In practical applications of WPT systems, the secondary side or load may be removed, leading to system safety and stability issues. The selected LCC-S topology features a constant current characteristic on the primary side. When the receiver or load is removed, the current in the primary coil remains approximately constant, reducing the complexity of system control and the current stress on the inverter circuit. The WPT system employs a primary-side LCC composite compensation method and a secondary-side LC series compensation method. The system's operating frequency is denoted by f0.

[0061] Typically, to meet the requirements of system output voltage stability, the output voltage is adjusted on the primary side via phase shift (PS). The principle of phase shift control and the inverter's output voltage are as follows: Figure 2As shown. d represents the duty cycle of the inverter phase shift angle. α represents the angle of duration for which switches S1 and S4 (or S2 and S3) are simultaneously open. The duty cycle d is calculated using the error between the output voltage and the set voltage. Previous studies have shown that... Figure 1 The system shown is nonlinear.

[0062] In this embodiment, the Hammerstein model of the WPT system is as follows: Figure 3 As shown. Figure 3 In this context, f(d) represents the nonlinear static characteristic of the WPT system, d is the duty cycle (control variable), and G... p (s) is a linear time-invariant model, where s represents the Laplace operator. Voltage output u o The relationship between d and d can be simplified as follows:

[0063] u o =f(d)G p (s) (1)

[0064] The static nonlinearity of the Hammerstein model is described by f(d) = sin(πd / 2). Furthermore, this nonlinear function can also be identified from the static input-output data (measured after all transients have disappeared). The dynamic linear model can be identified from the dynamic input and output data.

[0065] (2) Step S2: Design a closed-loop PID controller based on the linear part of the Hammerstein model and the control principle of the internal model.

[0066] The principle of the PID control method (closed-loop PID controller) based on the Hammerstein model is as follows: Figure 4 As shown. Figure 4 In this context, Q(s) represents the internal model controller, P(s) represents the Hammerstein model of the WPT system, M(s) represents the mathematical model of the WPT system object obtained through a data-driven approach, and e r Indicates the output voltage y u and set voltage u ref The error between them, E r C(s) represents the Laplace transform of the error signal, C(s) represents the feedback controller, i.e., the designed PID controller, and U(s) represents the output of the feedback controller. This represents the inverse nonlinear transformation.

[0067] The control objective of this invention is to force the output voltage y u Tracking set value u ref The classic IMC-based feedback controller is defined as follows:

[0068]

[0069] Where M(s)=M + (s)M _ (s), M + (s) represents the non-minimum phase part of M(s), f(s) represents the inverse of the minimum phase portion of M(s). -1 Let f(s) represent the inverse of the filter transfer function, f(s) = (1 + λs). -1 λ represents the filter coefficient.

[0070] The design of the PID controller uses a first-order plus time delay (FOPTD) system and a second-order plus time delay (SOPTD) system as examples. The system transfer function models are expressed as follows:

[0071]

[0072]

[0073] In the formula, K is the steady-state gain of the process, L is the lag time, τ is the time constant of the process, ξ is the damping coefficient, and e is the delay operator.

[0074] The internal mold controller is:

[0075]

[0076] The feedback controller is:

[0077]

[0078] In the formula, the first-order Taylor expansion e -Ls ≈1-Ls, which gives the PID feedback controller forms of FOPTD and SOPTD.

[0079] (3) Step S3: Set the PID parameters of the closed-loop PID controller according to the robustness condition.

[0080] For first-order time-delay systems (FOPTD) and second-order time-delay systems (SOPTD), the PID parameters (proportional parameter K) of the closed-loop PID controller are... p Integral parameter K i Differential parameter K d The settings are shown in Table 1.

[0081] Table 1. Controller Parameter Settings for FOPTD and SOPTD Models

[0082]

[0083] Maximum sensitivity (M s As a robust performance index for closed-loop systems, it is used to guide the parameter design of PID controllers. The open-loop transfer function of the FOPTD and SOPTD systems is obtained as follows:

[0084]

[0085] That is, the maximum sensitivity can be expressed as:

[0086]

[0087] Where ω represents the control frequency, and max represents taking the maximum value.

[0088] Based on the sensitivity index, the adjustable parameters of the PID controller can be easily determined, and the values ​​of the PID controller parameters can be further determined through Table 1.

[0089] (4) Step S4: Design the Romberg perturbation observer based on the linear part of the Hammerstein model

[0090] Due to environmental factors, the mutual inductance parameters of the system will change significantly during operation. Traditional PID control cannot guarantee the stability of the system under various operating conditions. Therefore, a Romberg disturbance observer is introduced into the traditional PID controller. The Romberg disturbance observer can observe the disturbance of the system in real time, thereby compensating for the system and achieving precise control of the output voltage. A PID controller based on disturbance observation is shown below. Figure 5 Except for the disturbance observer, the rest of the code is consistent with the aforementioned PID controller section. The Romberg disturbance observer is connected between the PID controller in the closed-loop PID controller and the Hammerstein model. The Romberg disturbance observer is used to estimate the disturbances in the system.

[0091] If the system has a disturbance d(t), the state equation of the linear part of the system can be expressed as:

[0092]

[0093] In the formula, x is the state variable of the system. Let x represent the first derivative of x, u be the system input, y be the system output, and A, B, and C represent the system state matrix, control matrix, and output matrix, respectively. Then, using the feedback control principle, a Romberg state observer for the system is constructed:

[0094]

[0095] Where H = [β1β2β3] TThis represents the parameter matrix of the Romberg state observer, where x1 and z1 represent the output state and the system output state estimate, respectively, and z represents the system state estimate. Denotes the first derivative of z. This indicates the system output estimate.

[0096] The Romberg observer can observe model mismatch errors, eliminate high-frequency noise, and reduce the impact of parameter perturbations. The structure of the Romberg observer is as follows: Figure 6 As shown.

[0097] Depend on Figure 6 The system state estimation error can be obtained as follows:

[0098]

[0099] To ensure the system state error is zero, the A-HC eigenvalues ​​must be less than zero. This can be achieved by configuring the eigenvalues, such that H = [β1β2β3]. T Solve for it.

[0100] Based on the foregoing, it can be known that the linear part G(s) of the system can be a first-order or second-order system. The method for the first-order system is similar to that for the second-order system. Therefore, this embodiment of the invention only provides the parameter configuration method for the second-order system. The first-order system can refer to the configuration method for the second-order system.

[0101] For a second-order system with perturbations, its state differential equation can be written in the following form:

[0102]

[0103] Where a0, a1, and b0 are system parameters. Let be the first and second derivatives of y, respectively. Then the state variables are expressed as:

[0104]

[0105] Where z1 tracks the output y, and z2 tracks the first-order state of the output. The disturbance d(t) of the z3 tracking system.

[0106] According to equation (11), using the pole placement method, the observer parameters can be configured as follows:

[0107]

[0108] ω0 is the bandwidth of the Romberg disturbance observer, which is generally taken as 3 to 5 times the control bandwidth ω.

[0109] In summary, this invention provides a disturbance control method for a wireless power transfer system based on an observer and a PID controller. The method includes constructing a Hammerstein model of the WPT system, designing a closed-loop PID controller based on the linear part and internal model control principles of the Hammerstein model, setting the PID parameters of the closed-loop PID controller according to robustness conditions, designing a Romberg disturbance observer based on the linear part of the Hammerstein model, and controlling the WPT system using the closed-loop PID controller and the Romberg disturbance observer to ensure the output voltage y... u Tracking voltage setpoint u ref This invention, based on the Hammerstein model of the system, designs a PID controller based on the internal model. It further determines the disturbance observer bandwidth based on the controller bandwidth and determines the observer parameters using the pole placement method, thereby achieving anti-mutual inductance disturbance control of the WPT system and ensuring the output voltage U... out Closely track the voltage setpoint u ref This greatly suppresses large voltage fluctuations caused by changes in mutual inductance. Simulation results show that the PID control method for the WPT system based on a disturbance observer proposed in this invention can achieve higher robustness, and the anti-mutual inductance disturbance performance can be adjusted by the observer bandwidth parameter.

[0110] The simulation verification will be performed below.

[0111] This embodiment provides simulation results to verify the effectiveness of the proposed control method. Phase-shift control is used to achieve stable output voltage control. In the actual simulation, a controllable voltage source in the form of a fundamental frequency is used instead of the switching transistor, and a delay element is added to the system. The main parameters of the system are shown in Table 2, and the model parameter estimates are shown in Table 3.

[0112] Table 2 Main parameters of the simulation

[0113]

[0114] Table 3 Estimated parameters for steady-state operating point

[0115]

[0116] Figure 7 The curves of the identification model are shown, with a fitness of 95.37%, which accurately describes the WPT system. To verify the good performance of the proposed control method, transient response tests and parameter disturbance variation tests were performed in this example. All tests have the same trajectory: they consist of two setpoint variations, i.e., u ref =140→160V and u ref=160→180V, and the sinusoidal change of mutual inductance after steady state. The period of mutual inductance change is 10Hz, the amplitude is 7uH, and the control period is 60us. After the system output reaches a steady state, the setpoint and mutual inductance change.

[0117] Based on the identified system model, the PID parameters of the closed-loop control system can be designed as k p =3.678·10 -4 k i =1.015·10 -4 k d =8.051·10 -3 The observer bandwidth ω0 = 9.666·10 3 . Figure 8 These are the transient output response waveforms under standalone PID control and PID+observer control. These waveforms show that the control system can track these points well at different steady-state operating points. Both the standalone PID and PID+observer schemes achieve good and similar control performance, that is, they achieve fast dynamic response characteristics at different operating points, with a settling time of only 5ms.

[0118] Figure 9 The output responses of no-closed-loop control, standalone PID control, and PID+observer control under mutual inductance parameter disturbances are demonstrated respectively. Without closed-loop control, the output voltage fluctuation amplitude is 22.5V; with standalone PID control, it is 6V; and with the PID+observer scheme, it is 1.2V.

[0119] In summary, the simulation results show that the control simulation using PID + observer in the WPT system can achieve higher robustness, and the design process can be simplified by designing the PID controller using the internal model concept.

[0120] The above embodiments are preferred embodiments of the present invention, but the embodiments of the present invention are not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A disturbance control method for a wireless power transfer system based on an observer and PID controller, characterized in that, Including the following steps: S1. Construct the Hammerstein model of the WPT system; S2. Design a closed-loop PID controller based on the linear part and internal model control principle of the Hammerstein model; the closed-loop PID controller is designed as follows: ， in, Let be the inverse of the minimum phase part of M(s), where M(s) represents the mathematical model of the WPT system object obtained through the data-driven method, s represents the Laplace operator, and e represents the delay operator. The filter coefficients are represented by L, where L is the lag time. For a first-order time-delay WPT system, M(s) is designed as follows: ， Where K is the steady-state gain of the process, and τ is the time constant of the process; For a second-order time-delay WPT system, M(s) is designed as follows: ， Where ξ is the damping coefficient; The open-loop transfer functions of the first-order time-delayed WPT system and the second-order time-delayed WPT system are designed as follows: ； S3. Set the PID parameters of the closed-loop PID controller according to the robustness condition; S4. Design a Romberg disturbance observer based on the linear part of the Hammerstein model; the Romberg disturbance observer is connected between the PID controller in the closed-loop PID controller and the Hammerstein model; the Romberg disturbance observer is used to estimate the disturbance of the WPT system; The Romberg perturbation observer is represented as follows: ， Where H represents the parameter matrix of the Romberg perturbation observer. , , These represent the state matrix, control matrix, and output matrix of the closed-loop system, respectively. , This indicates the system output state and the system output state estimate. This represents the system state estimate. Indicates the system input, express The first derivative, This represents the system output estimate; S5. The WPT system is controlled based on a closed-loop PID controller and a Romberg disturbance observer, so that the output voltage y u Tracking voltage setpoint u ref .

2. The disturbance control method for a wireless power transfer system based on an observer and PID according to claim 1, characterized in that: In step S3, based on the maximum sensitivity M s Determine the adjustable parameters of the PID controller, M. s Designed as follows: ， in, This indicates the control frequency, and max indicates taking the maximum value.

3. The disturbance control method for a wireless power transfer system based on an observer and PID according to claim 2, characterized in that, For a first-order WPT system with time delay, the proportional parameter K of the closed-loop PID controller p Integral parameter K i Differential parameter K d Set them to: K p = ,K i =τ, K d =0; For a second-order WPT system with time delay, the proportional parameter K of the closed-loop PID controller p Integral parameter K i Differential parameter K d Set them to: K p = ,K i = ,K d = 。 4. The disturbance control method for a wireless power transfer system based on an observer and PID according to claim 3, characterized in that, For a second-order time-delay WPT system The three parameters are designed as follows: ， Where ω0 is the bandwidth of the Romberg perturbation observer, a0, a1, and b0 are the parameters of the second-order time-delayed WPT system, and the state differential equation of the second-order time-delayed WPT system is expressed as: ， Let d(t) represent the output of the second-order time-delayed WPT system, and d(t) represent the perturbation of the second-order time-delayed WPT system. , They are respectively The first and second derivatives.

5. The disturbance control method for a wireless power transfer system based on an observer and PID according to claim 4, characterized in that, In step S1, the Hammerstein model is represented as: ， f(d) is the nonlinear static characteristic of the WPT system, d is the duty cycle of the primary-side drive signal of the WPT system, and G p (s) is a linear time-invariant model.

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