Robust Maximum Efficiency Tracking Control of Wirelessly Powered Directly Supplied Heart Pumps

Abstract

In recent times, wireless power transfer systems have been identified as a reliable option to supply power to medical implants. Up to now, Wireless Power Transfer Systems (WPTS) have only been used to charge batteries of low-power medical implants. However, for medical implants requiring a relatively higher power, such as a ventricular assist device, which is an implanted blood pump in the patient’s abdominal cavity, an external power supply has been used. When WPTS is used for medical implants, it increases the number of required power converter stages and hardware complexity along with the volume, which tends to reduce the overall efficiency. In addition, the existence of uncertainties in WPTS-based medical implants, such as load and mutual inductance variations, can lead to system instability or poor performance. The focus of this paper is to design a WPTS to supply power to the pump motor directly through its inverter based on the requirements of the motor drive system (MDS) without resorting to an additional DC-to-DC converter stage. To this end, the constraints that the drive system imposes upon WPTS have been identified. In addition, to make a reliable closed-loop operation, a µ-synthesis robust controller is designed to make sure the system maintains its stability and performance with respect to the system’s existing uncertainties. A number of experimental results are provided to verify the effectiveness of the adopted WPTS design approach and the corresponding closed-loop controller for WPTS. Furthermore, the experimental findings for the maximum efficiency tracking (MET) approach (to minimize WPTS coil losses) and constant DC link voltage control approach are shown and compared. According to experimental results and system efficiency analysis, the former appears to perform better. The system dynamic performance analysis, on the other hand, demonstrates the latter’s advantage.

1. Introduction

The number of patients with serious heart failure has been steadily increasing, especially since the recent pandemic. When a patient’s condition has progressed to the point that no other treatment option is available, a heart transplant is considered to be the best option. However, there are a limited number of suitable heart donors. Due to the scarcity of donors, therapies based on Ventricular Assist Device Systems (VADS) have become the standard stop-gap treatment option for patients with severe heart failure [1,2]. In general, VADS can be defined as a motor (heart pump) that is developed to help the heart to compensate for the lack of blood circulation in the body. This therapy could be utilized as a temporary bridge to heart transplantation or as a permanent substitute for heart transplantation if necessary.

The power consumption of conventional VADSs varies between 4 and 12 W [3,4], while for the most recent generations of VADS, this value has reduced to between 2.5 and 5 W [5,6]. Due to the continuous operation of the motor with this relatively high power consumption, a power supply external to the patient’s body is required. Accordingly, a percutaneous driveline has been used to connect the implanted VADS to the external power source. The potential of serious infections associated with the percutaneous driveline is a major issue with the current VADS. Wireless Power Transfer Systems (WPTS) are proposed as a potential solution to eliminate the through-skin driveline [7,8,9].

A typical WPTS consists of an inverter, a rectifier, and a resonant circuit with energy transmission coils and a compensation circuit. The WPTS coils can be represented as a transformer with low magnetizing inductance and high leakage inductance. Compensation networks must be incorporated into the system to cancel the effect of leakage inductances. Various compensation methods have been covered extensively in the past literature, for example, low order compensation circuits such as Series/Series (SS), Series/Parallel (SP), Parallel/Series (PS), Parallel/Parallel (PP) [10,11] and higher order compensation circuits such as LCC, LCL and S/SP compensation topologies, to name a few [12,13,14]. The desirable features of a compensation technique include high efficiency, reliability, load-independent output voltage or current, and mutual independence of output current or voltage [15]. However, each circuit may only offer one or two of these desirable characteristics. Amongst all such circuits, SS compensation has been widely used for medical applications due to its simplicity and desirable features such as load and mutual inductance independent resonance frequency tuning [7,16].

For SS compensation, the load and the mutual inductance, which can be modeled as sources of uncertainty, may vary independently in an arbitrary manner within a particular range. This variation can significantly affect output capability, efficiency, and dynamics. In this regard, closed-loop systems based on linear controllers such as the Proportional and Integral (PI) controller have been widely applied to regulate the output voltage or output current to the desired value. However, as shown in [17,18], conventional linear controllers may not be the best option for a system with variable dynamics and uncertainties. For a motor drive system (MDS) supplied with a WPTS, two main uncertainty sources are the load (the equivalent load value of MDS which can be seen from WPTS perspective) and the mutual inductance of the WPTS. These parameters can vary independently and each of them could highly impact the dynamics of the WPTS. Usually, WPTS controllers are designed for their nominal conditions. Such a design is prone to instability or poor performance when the system operates at a different operating point within the uncertainty range.

Robust controllers are developed to guarantee the Robust Stability (RS) and the Robust Performance (RP) of the system with one or more sources of uncertainty. The former means the closed-loop system remains stable within the range of uncertainties and the latter means the closed-loop system maintains its performance with respect to the system uncertainties. To date, a number of robust controller design methods have been developed. The singular value optimization problem is one of the most powerful tools to design a robust controller which is known as a µ-synthesis robust controller. A controller based on µ-synthesis is designed in [19] for SP compensated WPTS with mutual inductance variation, and, for the same network, a robust controller is designed to deal with load variations [20]. In some references, robust controller design for other types of compensation networks with one or more uncertainty sources have been investigated [21,22]. A common objective for all robust controller design approaches for WPTS is designing a closed-loop system that can offer RS and RP for a system with parameter uncertainties.

The closed-loop system can be used not only to precisely regulate the output voltage or current, but also to maximize the efficiency of the WPTS coils. The Maximum Efficiency Tracking (MET) method helps to minimize the generated reactive power by tracking the optimum load value seen from the WPTS rectifier perspective. As shown in [23] for an SS compensation network, MET can help to reduce the corresponding losses of WPTS coils and minimize the generated corresponding heat. In addition, the power losses of the primary and the secondary coils are equally distributed leading to uniform heat dissipation. In [23], the authors have designed a battery charger for a VADS with the help of a PI controller to perform MET with an extra DC-to-DC converter to regulate the desired output voltage. However, for lower power ratings, the DC-to-DC converter appears to be the main source of power losses.

To the best of the authors’ knowledge, a motor drive system (MDS) as a direct load for the WPTS has never been explored comprehensively in literature. Mostly, the WPTSs are optimized for a specific output power, and the voltage/current regulation of the load (mostly battery or resistor) is handled with an additional DC-to-DC converter. Direct connection of a MDS to a WPTS imposes extra constraints upon the system, such as maintaining the voltage in a range that is suitable for the MDS. Once the voltage constraints of MDS are identified, it is possible to perform the MET using the WPTS and the inverter of MDS alone without resorting to an external DC-to-DC converter stage. In this work, firstly, the required constraints for the proper operation of MDS are explored. Then a WPTS design recommendation is proposed based on the constraints of the MDS and the requirements for the MET method. Next, a µ-synthesis robust controller is designed to perform MET and regulate the DC link voltage of the MDS. Finally, the dynamic behavior of the closed-loop system and the system efficiency analysis at steady-state with the MET method and conventional constant DC link voltage method are experimentally explored.

2. Heart Pump Operation and Motor Drive

The general configuration of the WPTS as a power supply for the MDS is shown in Figure 1. In this setup, the MDS is directly connected to the output of the WPTS rectifier. Constant speed control of the motor is the most common clinically approved method for a VADS [24]. On the other hand, as discussed in [25], the load pressure on the output of the heart pump is significantly varied in each cardiac cycle leading to significant shaft torque variations. As a result, the motor power consumption varies greatly with the constant speed control of the motor and the fluctuating torque on the shaft. In order to simplify system modeling, it is assumed that the DC-link capacitor is sufficiently large to slow the DC bus dynamics such that the MDS can achieve near perfect DC-bus disturbance rejection. In addition, it is assumed that the MDS has a perfect disturbance rejection. Accordingly, the motor and the drive system (the electrical load after DC-link capacitor) from the WPT perspective can be modeled as an equivalent variable resistor (R_L,eq) [26,27], as shown in Figure 2.

$$R_{L,eq}=\frac{V_{DC}^2}{P_o},$$

(1)

where V_DC is the DC-link voltage and P_o is the consumed power by the motor and the drive system.

Voltage Limitation of Field Oriented Control (FOC)

In order to design a WPTS to supply a motor, it is crucial to design a system in a way that it could fulfil the voltage requirement of the drive system at the required speed and power. As a result, the variation of the motor drive characteristic from the WPTS perspective must be studied first. The governing equations of PMSMs are well-explained in literature [28,29]. A brief review of the required equations for non-salient PMSM (L_d = L_q = L_s) is provided here for the sake of completeness. Stator voltage equations in the dq coordinate system can be written as

$$\begin{array}{l}{v}_q=R_s{i}_q+L_q\frac{d{i}_q}{dt}+{\omega }_e{L}_d{i}_d+{\omega }_e{\lambda }_{af},\\ v_d=R_s{i}_d+L_d\frac{d{i}_d}{dt}-{\omega }_e{L}_q{i}_q,\end{array}$$

(2)

where R_s and L_s are the rotor resistance and rotor inductance and L_d and L_q are d-axis and q-axis inductances, respectively. In addition, λ_af and ω_e are magnet mutual flux linkage and angular electrical velocity of the motor, respectively. The generated torque (T_e) can be calculated as:

$$T_e=\frac{3}{4}P{\lambda }_{af}{i}_q,$$

(3)

where, P is the number of poles. From (3), it is evident that the generated torque only depends on i_q. The relation between the stator current/voltage magnitude and the current/voltage magnitude in d–q frame can be written as:

$$\begin{array}{l}\left[\begin{array}{c}{i}_q\\ i_d\end{array}\right]=i_s\left[\begin{array}{c}\sin \delta \\ \cos \delta \end{array}\right]\Rightarrow i_s=\sqrt{i_q^2+i_d^2},\\ v_s=\sqrt{v_q^2+v_d^2}.\end{array}$$

(4)

In steady-state conditions and by ignoring the resistive voltage drop of the motor windings, (2) can be rewritten as:

$$\begin{array}{l}{v}_q={\omega }_e{L}_d{i}_d+{\omega }_e{\lambda }_{af},\\ v_q=-{\omega }_e{L}_q{i}_q.\end{array}$$

(5)

With respect to (2), (4) and (5), one can find the relationship between ω_e, i_d, i_q, and stator phase voltage (v_s) as

$$\frac{v_s^2}{{\omega }_e{L}_d}={(i_d+\frac{{\lambda }_{af}}{L_d})}^2+i_q^2$$

(6)

Equations (4) and (6) show the limitations of the stator current, the stator voltage and the achievable speed. It must be noted that v_s limitation is due to V_DC value. With the assumption of employing Space Vector Modulation (SVM) for the MDS, the relation between maximum required stator voltage (v_s,max) and minimum required V_DC can be written as

$$V_{DC,\min }=\sqrt{3}{v}_{s,\max }.$$

(7)

These constraints are shown in Figure 3. It must be noted that (6) describes the limitations of stator current imposed by flux linkage. Consequently, it is represented by a circle with the origin at (0, λ_af/L_d) and the radius of v_s/ω_eL_d. In order to minimize ohmic losses in the machine, the demanded torque from the load must be satisfied with the minimum stator current amplitude. This goal can be achieved with the help of the Maximum Torque Per Ampere (MTPA) strategy [29] (in some references it is known as zero-direct-axis-current control [28]), as shown in Figure 3.

Accordingly, Equation (3) can be rewritten as:

$$T_e=\frac{3}{4}P{\lambda }_{af}{i}_s.$$

(8)

As it is shown in the next section, for a WPTS with a particular set of parameters, there is an optimum load value R_L,opt that can maximize the WPTS efficiency. From (1), since P_o varies significantly, V_DC must be regulated in a way that R_L,opt is always maintained. Accordingly, the lower limit of V_DC to maintain MTPA control strategy (V_DC,min), based on the required speed and maximum torque for steady-state conditions, can be found as follows:

$$V_{dc,\min }=\sqrt{3{\omega }_{e,\max }{L}_d({(\frac{{\lambda }_{af}}{L_d})}^2+i_{q,\max }^2)}.$$

(9)

Below this voltage, with respect to the required speed and torque demand, MTPA control cannot be achieved. Thus, it must be considered before the design of the WPTS.

3. SS Compensated WPTS Design

3.1. Series-Series Compensation

The SS compensation network, as shown in Figure 4, is the most typical compensation topology used in implanted medical devices. Mostly, the SS compensation network is tuned to operate in the load-independent constant output current (COC) mode.

To achieve COC operating mode, the self-inductance of the primary and the secondary coils must be compensated as:

$${\omega }_o=\frac{1}{\sqrt{C_1{L}_1}}=\frac{1}{\sqrt{C_2{L}_2}},$$

(10)

where L₁, L₂, C₁ and C₂ are the primary inductance, secondary inductance, primary compensation capacitor, and secondary compensation capacitor, respectively. In addition, ω_o is the resonance frequency of the compensation network. The corresponding voltage gain equation of COC mode, with respect to Figure 5a, can be found as

$$\frac{v_{CD,1}}{v_{AB,1}}=\frac{R_{L,ac}}{j{\omega }_{o}M},$$

(11)

where M is the mutual inductance between L₁ and L₂ and:

$$R_{L,ac}=\frac{8}{{\pi }^2}{R}_{L,eq},v_{cd}{}_{,1}=\frac{4}{\pi }{V}_{DC},v_{AB,1}=\frac{4}{\pi }{V}_{in}\sin (\frac{\pi }{2}D)$$

(12)

With respect to Figure 4 and Figure 5a, one can see that R_L,ac is the equivalent transferred R_L,eq from the DC side to the AC side of the rectifier. In addition, D is the duty cycle of the phase-shift modulation. Since the system operates at the resonance frequency, in (11) and (12), it is assumed that all voltage harmonics except the fundamental component (v_AB,1 and v_CD,1) are suppressed and all the waveforms are assumed to be purely sinusoidal. In addition, the magnitude of v_AB,1, using phase-shift modulation for the H-bridge converter, can be controlled by changing V_in and D. To simplify the rest of the equations in this paper, it is assumed that L₁ = L₂ (and consequently C₁ = C₂).

3.2. WPTS Maximum Efficiency Tracking

The WPTS resonant tank efficiency can be maximized or, equivalently, the total generated reactive power in the primary and secondary compensation capacitors can be minimized, as explored in detail in [23], if the load with a specific value is acquired:

$$R_{L,ac,opt}=K{\omega }_o{L}_2,$$

(13)

where R_L,ac,opt is the optimal load value that maximizes efficiency and the coupling coefficient (K) can be found as:

$$K=\frac{M}{\sqrt{L_1{L}_2}}.$$

(14)

The condition (13) is valid for the coils with a high-quality factor (Q > 100). Satisfying (13) leads to minimum RMS current value of primary and secondary coils. For applications where the power losses of coils, and the corresponding generated heat, are important, as in the case of medical implants where generated heat can damage the surrounding tissue of the coils, the maximum efficiency tracking is very desirable. Equation (13) indicates that the WPTS can operate with the maximum efficiency at any K and P_o if R_L,ac equals to R_L,ac,opt. Consequently, to set R_L,ac,opt constant for different operating conditions, V_DC must be set as

$$V_{DC}^{*}=\sqrt{\frac{{\pi }^2}{8}{P}_{o}K{\omega }_o{L}_2}.$$

(15)

The symbol “*” shows the required V_DC for maximum efficiency tracking. From (15) it is apparent that $V_{DC}^{*}$ setpoint depends on P_o and K. The value of P_o is determined by the load demand which can be measured. On the other hand, K is variable in practice, and its value must be approximated based on the known parameters. Since the WPTS operates at the resonance frequency, the reflected impedance to the primary side (Z₁), as shown in Figure 5, is purely resistive and is given by [30]:

$$Z_1=\frac{{({\omega }_{o}M)}^2}{R_{L,ac}}.$$

(16)

With respect to Figure 5b and assuming the lossless operation of the H-bridge, Z_in can be found as

$$P_{in}=P_1\Rightarrow Z_{in}=2Z_1\frac{V_1^2}{V_{in}^2}=2\frac{{({\omega }_{o}M)}^2}{R_{L,ac}}{(\frac{4}{\pi }\sin (\frac{\pi }{2}D))}^2.$$

(17)

By using (14) and (17), K can be estimated based on V_in and I_in as

$$K=\frac{1}{{\omega }_o{L}_{22}}\sqrt{\frac{V_{in}{R}_{L,ac}}{2I_{in}{(\frac{4}{\pi }\sin (\frac{\pi }{2}D))}^2}}.$$

(18)

In (18), R_L,ac can be found by measuring the current and the voltage of the DC-link to calculate the estimated power consumption of the motor. It must be noted that coils relative displacement happens slowly compared with the motor power variation and, consequently from (18), it is possible to use the motor power consumption along with the knowledge of V_in and D to estimate K variations.

3.3. Wireless Power Transfer Network Design

In this work, the WPTS is used to supply the PMSM and the drive system. The MDS lower DC link voltage limit (V_DC,min) is determined by (9). In addition, the upper limit of the DC link voltage V_DC,max is governed by the upper voltage limit of the drive system power devices and the motor drive controller. One important point that should be emphasized is the role of the motor drive inverter in adjustment of R_L,ac,opt. The motor drive system can provide the required v_s by adjusting the inverter modulation index (m) for the SVM as:

$$v_s=\frac{m}{\sqrt{3}}{V}_{DC}.$$

(19)

With the assumption of the lossless operation of the MDS, the motor power equals to P_o. Since the motor drive can adjust the relation of v_s and V_DC, with respect to (1) and (19), it is possible to set V_DC in a way that the motor and the drive system model become equivalent to R_L,ac,opt from the WPTS perspective.

In this regard, the SS compensated WPTS must be capable of providing the required V_DC within the required range (V_DC,min ≤ V_DC ≤ V_DC,max). Consequently, the following WPTS design procedure is suggested and the system is designed based on

Table 1. System parameters.

Parameter	Value
L1, L2	38.2 µH, 38.1 µH
C1, C2	668 pF, 673 pF
DC link capacitor	220 µF
H-bridge switching frequency (Fsw)	1 MHz
Motor	DMB0224c10002
Motor drive switching frequency	20 kHz
K (Coupling factor)	0.4–0.5
Vin	50 V
Vout	18 V–30 V
Po	2.5 W–5 W
RL,ac,d	120 Ω

.

Step 1. Find the maximum equivalent resistance (R_L,eq,max) seen from the WPTS rectifier perspective. This value can be found from the estimated P_o and the corresponding V_DC,min. One can use (4), (8), and the well-known relation between motor torque, mechanical speed ω_m, and P_o to find R_L,eq,max.

$$\begin{array}{l}{P}_o={\omega }_m{T}_e,\\ {\omega }_m={\omega }_e\frac{P}{2}\end{array}$$

(20)

Figure 6 shows the relation between the required V_DC,min at any P_o and motor speed. In addition, the point where R_L,eq,max can be found from the designated point within a circle in Figure 6.

Step 2. Select the design load value (R_L,ac,d) as

$$R_{L,ac,d}=\frac{8}{{\pi }^2}{R}_{L,DC,d}\ge \frac{8}{{\pi }^2}{R}_{L,eq,\max },$$

(21)

where R_L,DC,d is the equivalent value of R_L,ac,d transferred to the DC side of the rectifier. It must be noted that for the final designed system, R_L,DC,d implicitly shows the relation between $V_{DC}^{*}$ and P_o. If R_L,DC,d is selected to be very large, the motor drive modulation index reduces significantly and, consequently, it leads to poor performance of the motor drive system. In practice, we recommend selecting R_L,DC,d two or three times more than R_L,eq,max to provide extra headroom for the motor controller modulation index to avoid saturation at transitions or sudden load changes. It must be noted that by selecting R_L,DC,d as the load value, WPTS must be able to regulate V_DC within the following range:

$$\underset{V_{DC,\min }}{\underbrace{\sqrt{P_{o,\min }{R}_{L,DC,d}}}}\le V_{DC}^{*}\le \underset{V_{DC,\max }}{\underbrace{\sqrt{P_{o,\max }{R}_{L,DC,d}}}}$$

(22)

where P_o,min and P_o,max are the minimum and maximum required power from the motor and MDS. The modified required DC link voltage (V_DC,mod) to maintain MET can be found as:

$$V_{DC,\mathrm{mod}}=\sqrt{P_o{R}_{L,DC,d}}.$$

(23)

Step 3. Approximate K range variations [K_min, K_max] and select K_max to determine the inductor value from:

$$L_2=\frac{R_{L,ac,opt}}{{\omega }_o{K}_{\max }}.$$

(24)

It must be noted that, at this stage, several iterations might be needed since K variation range is not exactly known before the design of the inductor. Moreover, the selection of K as K_max in (24) helps to decrease the required V_in level.

Step 4. From (11), (12), and (24), one can determine the minimum possible V_in (V_in,min) to provide the required voltage (at different output voltage values and K) as:

$$V_{in,\min }=\frac{{\pi }^2}{8}\frac{\omega K_{\max }{L}_2}{R_{L,DC,d}}{V}_{DC,\max },$$

(25)

where the phase-shift duty cycle is assumed to be maximum (D = 1). It is recommended to set V_in, 20–25% more than V_in,min to ensure the duty cycle does not reach to the saturation level (i.e., D > 1) at transitions. This practical design procedure is summarized in Figure 7.

4. Robust Controller

4.1. System Model and System Uncertainties

As previously stated, WPTSs are prone to system parameter variations such as K and R_Load variations. Accordingly, the system dynamics can be affected significantly. To investigate the system dynamics, firstly, the state-space equations and the small-signal model of the system must be formulated. The derivation of the state-space equations and the small-signal model of the SS compensated WPTS, as it is shown in Figure 4, based on Extended Describing Functions (EDFs), is given in [31].

That model is used in this work for the controller design. The state-space equations and the small-signal equations are given as:

$$\begin{array}{l}{B}_{ss}=A_{ss}{X}_{ss}\\ A_{ss}\in {\Re }^{\left\{8\times 8\right\}},B_{ss}\in {\Re }^{\left\{8\times 1\right\}}\\ Xss={[\begin{array}{cccccccc}{V}_{1s}& V_{2s}& I_{1s}& I_{2s}& V_{1c}& V_{2c}& I_{1c}& I_{2c}\end{array}]}^T\\ \{\begin{array}{c}\frac{d}{dt}\widehat{X}=A.\widehat{X}+B.\widehat{u}\\ \widehat{y}=C.\widehat{X}+D.\widehat{u}\end{array}\\ A\in {\Re }^{\left\{9\times 9\right\}},B\in {\Re }^{\left\{9\times 3\right\}},C\in {\Re }^{\left\{1\times 9\right\}},D\in {\Re }^{\left\{1\times 3\right\}}\\ \widehat{X}={[\begin{array}{ccccccccc}{\widehat{i}}_{1s}& {\widehat{i}}_{1c}& {\widehat{i}}_{2s}& {\widehat{i}}_{2c}& {\widehat{v}}_{1s}& {\widehat{v}}_{1c}& {\widehat{v}}_{2s}& {\widehat{v}}_{2c}& {\widehat{v}}_{Cf}\end{array}]}^T\\ \widehat{u}={[\begin{array}{ccc}{\widehat{\omega }}_o& {\widehat{v}}_{in}& \widehat{d}\end{array}]}^T,\end{array}$$

(26)

where X_ss, A_ss, and B_ss form the steady-state equations and $\widehat{y}$, $\widehat{X}$, A, B, C, and D form the small-signal model. Interested readers are referred to [31] to find the corresponding coefficients of (26) (A_ss, B_ss, A, B, C, and D).

Mostly, WPTS plants are investigated under nominal K and R_L,eq values and, accordingly, the corresponding system transfer function can be found. However, in practice, one or several parameters of the system vary in a particular range.

For example, K and R_L,eq might vary with respect to the position of coils and the equivalent load demand, respectively. Thus, the system transfer function is not unique anymore, and it is changed accordingly. There are several methods to model the plant of such a system. One of the most common methods is the multiplicative lumped uncertainty representation, which describes the system plant with uncertainties as

$$G(s)=G_{nom}(s)(1+w_I(s){\Delta }_I(s)),$$

(27)

where the nominal transfer function, G_nom(s), is the system transfer function with nominal parameters. In addition, Δ_I(s) is any stable norm bounded transfer function with magnitude less than one which represents all the system uncertainties as a lumped block and can be represented mathematically as

$${\Vert {\Delta }_I(s)\Vert }_{\infty }\le 1,$$

(28)

where operator ${\Vert .\Vert }_{\infty }$ gives the infinite norm of the signal [32]. In addition, w_I(s) represents the available information about the weight of uncertainty distribution at each frequency. The multiplicative representation of the plant is shown in Figure 8.

From Table 1, K_nom is chosen to be 0.45. However, in practice, it is assumed that the coupling coefficient may change from 0.4 to 0.5 for medical implants. Another important point is that for the maximum efficiency tracking of the WPTS, the value of R_L,eq should remain constant (=R_L,DC,d) to get to the maximum efficiency of the WPTS, as discussed in the WPT design section. However, this assumption is correct only for steady-state conditions. For transitions, when the torque in the motor changes, R_L,eq may deviate from the desired setpoint and this variation depends on the motor dynamics and the motor drive controller performance. Consequently, R_L,eq can be assumed as a source of uncertainty in the system during transitions. Therefore, to make sure R_L,eq variation at transitions are included in the parameter uncertainties, it is assumed that R_L,eq, might deviate up to 50% from R_L,DC,d within the following range:

$$[0.5R_{L,DC,d}\le R_{L,eq}\le 1.5R_{L,DC,d}].$$

(29)

It must be noted that the assumption of ±50% R_L,eq variation (as it is found in practice) provides a reliable margin to make sure all load dynamic variations are taken into consideration for the controller design process. To investigate the effect of parameter variation, it is possible to define the relative error (e_R) as

$$e_R=\frac{\left|G(j\omega )-G_{nom}(j\omega )\right|}{\left|G_{nom}(j\omega )\right|},$$

(30)

which shows the relative frequency response deviation of the systems with uncertainty from the frequency response of the system with nominal parameters. The e_R variations for several plants for a range of parameter variations that are randomly selected are shown in Figure 9.

K and R_L,eq variations lead to the considerable frequency response deviation from the nominal plant. To be able to describe a system with a single transfer function G(s) as it is shown in (27), w_I must be calculated. By using (30) and considering (27) and (28), one can select w_I as:

$$e_R=\frac{\left|G(j\omega )-G_{nom}(j\omega )\right|}{\left|G_{nom}(j\omega )\right|}\le w_I(j\omega ).$$

(31)

Thus, it can be concluded that w_I must be selected in a way to cover the e_R region. In addition, to simplify the uncertainty representation and avoid the unnecessary expansion of the uncertainty region, it is preferable to use a low-order high-pass transfer function. As such, in Figure 9, w_I is selected as a first-order high-pass transfer function which fully covers the e_R region.

4.2. Controller Design

Robust controller methods are developed to deal with system uncertainties similar to those that are presented in the WPTS in hand. The principles underpinning robust controller design are well-investigated in [32,33], and only an overview is provided here. Robust controllers can offer RS and RP for a range of parameter variations. It is worth mentioning that RS means the closed-loop system stability is guaranteed under the whole range of parameter variations, and RP means the system performance is guaranteed for all possible plants in the range of parameter variations. To find such a controller, one can assume Figure 10 as the general configuration of the closed-loop WPTS. The system plant is shown with the multiplicative representation, and the robust controller is shown with C_µ. In addition, w_P and w_U are two weighting functions that determine the closed-loop system performance and the controller effort performance, respectively [32], which are discussed in the following section. To formulate the robust controller design problem, Figure 8 can be rearranged to the well-known M-Δ configuration or P–Δ–C_µ configuration, as shown in Figure 10. For the former, Δ (which is equivalent to Δ_I in Figure 8 and contains the system uncertainties) can be pulled out from the rest of the system, and, for the latter, Δ and C_µ are pulled out from the rest of the system. It is worth mentioning that P consists of w_I, w_p, w_U, and G_nom, and M contains P and C_µ. With respect to Figure 10, one can express the robust controller design problem as finding C_µ that can guarantee RS and RP for the nominal system and the system in the presence of Δ.

One of the most powerful tools to find C_µ is the structured singular value µ which is expressed mathematically as:

$$\mu (M)\triangleq \frac{1}{{\min }_{\Delta }\{\overline{\sigma }(\Delta )|\det (I-M\Delta )=0\}},$$

(32)

where $\overline{\mathsf{\sigma }}$(.) is the maximum singular value and M is the transfer function from the output (u_Δ) to the input (y_Δ) of Δ. As it is proved in [32,34], a controller that can provide RS and RP conditions must satisfy the following criterion:

$$\underset{\omega \in \Re }{\sup }\mu [M(P,C_{\mu })(j\omega )]<1.$$

(33)

In (33), M(P, C_µ) is a lower linear fractional transformation and explicitly emphasizes that M is formed by P and C_µ. Alternatively, to define the robust controller design as an optimization problem, (33) can be expressed as:

$$\underset{C_{\mu }(s)}{\inf }\underset{\omega \in \Re }{\sup }\mu [M(P,C_{\mu })(j\omega )]<1.$$

(34)

This optimization problem determines C_µ that can minimize the peak value of µ of the closed-loop transfer function, M. The designed controller based on this approach is called a µ-synthesis robust controller, and the index of C_µ emphasizes the acquired approach for the controller design. One of the methods to solve (34) is the DK-iteration algorithm, which is used in this work to design the required controller for the WPTS. This algorithm is based on a numerical solution method, and it is readily available in MATLAB software. It is important to note that the DK-iteration algorithm does not lead to a globally optimized solution. Thus, several iterations should be performed to obtain the desirable controller performance. Further explanations about this algorithm is outside of the scope of this paper, and interested readers are referred to [32,33] for further information.

4.3. Selection of Weighting Factors

As mentioned before, w_P and w_U are weighting factors associated with the performance specifications and the controller effort specifications of the system which must be determined along with w_I. In the process of designing a µ-synthesis robust controller, both weighting factors are absorbed into P block in Figure 10 (and consequently M). Thus, it is essential to define them. References [32,34,35] provide a guideline for the selection of these weighting factors. For the case of w_P, it is recommended to select it as a low pass filter.

$$w_P(s)=\frac{\frac{s}{\alpha }+{\omega }_B}{S+{\omega }_B\beta }.$$

(35)

Here ω_B, α and β are related to the required bandwidth, overshoot of the time response (mostly selected between 1.5 and 2), and the maximum permissible steady-state error of the system. Ideally, the steady-state error should be zero, however, in practice, it is selected as a small value to make the process of controller optimization easier for the DK-iteration algorithm. The reason for selecting w_P as a low pass filter can be explained with respect to the outcomes of (33). By satisfying (33), as it is proved in [32], the following equations hold true:

$${\Vert w_{P}S\Vert }_{\infty }<1\&{\Vert w_u{C}_{\mu }S\Vert }_{\infty }<1,$$

(36)

where the sensitivity function S, with respect to Figure 8, is the closed-loop transfer function from r to e, or from d to e, and it is defined as:

$$S(s)=I-G(s)C_{\mu }(s).$$

(37)

The closed-loop system exhibits good reference tracking and disturbance rejection characteristics if S ≈ 0. On the other hand, the system shows a good noise rejection ability if S ≈ I. It might seem there is a conflict between good reference tracking and good noise rejection criteria. However, it must be noted that the noise is prominent in higher frequency ranges compared to the reference tracking requirement, which is handled in low frequency ranges. This feature requires shaping of S as a transfer function with low gain at low frequencies and high gain at high frequencies. Consequently, by selecting w_P as a low pass filter shaped function, with respect to (35), the controller is designed in a way to provide the desired S shape as ${\Vert S\Vert }_{\infty }<1/{\Vert w_P\Vert }_{\infty }$.

The other weighting function, w_U, affects the controller effort. Mostly, as the first choice, it is preferable to select w_U as a constant value. However, if the controller effort shows a significant overshoot or rapid changes, it is possible to smooth it by selecting w_U as a differentiator [34]. The selected weighting factors with respect to the parameters in Table 1 are given in

Table 2. µ-Synthesis Controller Parameters.

Title 1	Title 2
wI	$\frac{3(s+2.5\times {10}^5)}{s+2.5\times {10}^6}$
wP	$\frac{0.067(S+3.75\times {10}^2)}{(S+2.5\times {10}^{-1})}$
wU	5
Controller order	15
Maximum µ value	0.96

.

5. Experimental Results

In order to show the efficacy of the proposed WPTS design method and verify the effectiveness of µ-synthesis controller to perform maximum efficiency tracking, the experimental system is established with the design parameters of Table 1. Based on the rate of load change in a VADS and the DC- link capacitor value of the MDS, the settling time of 100 ms and 10% overshoot in step response are chosen as the closed-loop step response of WPTS. Accordingly, C_µ is designed with the weighting factors provided in Table 2 and the step responses of the closed-loop system for a number of plants with random parameters that lie within uncertainty ranges of WPTS are shown in Figure 11.

In addition, the schematic and the picture of the experimental setup are shown in Figure 12 and Figure 13, respectively. To investigate the performance of the proposed system, it is evaluated at the maximum coupling coefficient (K_max = 0.5) and the minimum coupling coefficient (K_min = 0.4) values, and for each K value, the motor power demand with a step change varies from minimum to maximum values (2.5 W to 5 W). It must be noted that K_max and K_min are selected based on discussion with our medical team and simulation of body tissue in ANSYS MAXWELL software for a 10–15 mm coil distance. However, the methodology presented is applicable for other coupling factor ranges. The motor drive controller and C_µ are implemented on CPU 1 and CPU 2 of the F28379D TI microcontroller, respectively, and they share data via inter-processor communication (IPC).

For example, V_DC and I_DC are measured with CPU 1 and their values are transferred to CPU 2 for $V_{DC}^{*}$ calculations. It is worth mentioning that 20 kHz discretization is used for C_µ and it is digitally implemented with the approach proposed in [21]. In addition, L₁ and L₂ are composed of 23 turn litz wire (330 strands and 46 AWG) to decrease skin effect. The inner diameter of 35 mm and outer diameter of 87 mm are chosen for coils that are deemed suitable for VADS applications [23]. The corresponding resistance values are measured with a network analyzer (R&S ZVL13) at 1 MHz and give 891 mΩ and 853 mΩ, respectively.

In order to investigate the proposed controller design and the WPTS design, various steady-state and dynamic analyses are performed for both the maximum efficiency tracking and constant voltage control of the DC-link. For the latter, V_DC is kept constant at 24 V. The steady-state waveforms of the maximum efficiency tracking corresponding to K_min and K_max are shown in Figure 14a,b and Figure 15a,b.

As can be seen, the output voltage has been controlled with the help of H-bridge duty-cycle adjustment (phase-shift modulation) at 1 MHz switching frequency. In addition, the output voltage is successfully adjusted with respect to P_o and K. Since the calculation of $V_{DC}^{*}$ depends on K, and K calculation depends on the output power, it is recommended to take the average over a few samples of K to find its value. The average operation does not pose any considerable restriction over the controller since K variations are mostly due to mechanical movement, and its time constant is much larger than the electrical time constant of the system.

The waveforms of V_DC and i₂ to evaluate the dynamic performance of C_µ for a sudden step change of the motor load for maximum efficiency tracking and constant DC-link voltage control are shown in Figure 14c, Figure 15c, Figure 16 and Figure 17. Since a conventional sensorless FOC algorithm is used for the drive system, only the speed of the motor as an indicator of motor dynamics is presented here. When applying the maximum load over the motor, the speed of the motor drops; however, the power demand from the motor increases leading to an increase in $V_{DC}^{*}$. For both maximum efficiency tracking and constant DC-link voltage control, C_µ successfully regulates the output voltage within the defined settling time. However, the settling time of constant output voltage control shows better performance (50 ms for K = 0.5) compared with the maximum efficiency tracking strategy (98 ms for K = 0.5).

The main reason for this difference is the effect of the motor dynamic as a load on the DC-link. By changing V_DC, the motor controller requires more time to return the speed to the reference value. One of the potential solutions to improve the response time of the motor controller is to use the speed drives based on variable DC-link [36]. However, these methods are beyond the scope of this paper, and they are not discussed. It is worth mentioning that the motor driver remains unchanged for all cases. The power loss analysis of the system is provided in Figure 18. The equations given in [1] for coil and rectifier power losses and the equations given in [37] for H-bridge power losses are used here. The total power loss of the rectifier stage P_Loss,rec can be calculated as:

$$P_{Loss,rec}=2(\frac{P_o}{V_{DC}}{V}_{DR,0}+i_{2,rms}^2{R}_{DR,0})$$

(38)

where R_DR,0 is the differential resistance value of the diode and V_DR,0 is the voltage drop across the diode. The power losses of the primary coil (P_L1) and secondary coil (P_L2) of the WPTS are obtained from:

$$P_{L1}=R_1{i}_{1,rms}^2,P_{L2}=R_2{i}_{2,rms}^2$$

(39)

The total power loss of GaN based H-bridge inverter (P_Loss,HB) can be calculated as:

$$P_{Loss,HB}=\underset{\mathrm{Conduction}\mathrm{Losses}}{\underbrace{4RDs{I}_{1,rms}^2}}+\underset{\mathrm{Switching}\mathrm{Losses}}{\underbrace{2V_{in}{I}_{1,rms}{f}_s{T}_{TR}}}$$

(40)

where R_DS is the GaN on-resistance and T_TR is the switch transition time from on mode to off mode and vice versa.

As it can be seen from (38)–(40), the global optimization of the total power loss is a complicated problem since it depends on many factors, and, for the application under-study, the system dynamic varies significantly. On the other hand, due to the location of coils and the sensitivity of body tissue to the generated heat from them, it is desirable to use a maximum efficiency tracking algorithm.

The analysis of the power loss distribution in Figure 18 shows that, for all scenarios, the majority of power losses occur in the motor and drive system. The total power loss of the MDS depends on many factors, such as the switching frequency of the drive system, type of modulation, DC link voltage, type of the motor, etc. Thus, further investigation on the effect of maximum efficiency tracking of the WPTS on the MDS remains as future work. However, in this study, we only focus on the WPTS coil losses. The target of the maximum efficiency tracking is to minimize and share the power losses equally between two coils. As it can be seen, with optimum efficiency tracking, the power losses of coils (R_L1 and R_L2) are equal, and their total power loss is less than that of the non-optimized control system. For low power operation of the system, the share of coil power loss in the total power loss distribution is lower compared to that in high-power operation. The reason is simply that the lower current in the coils in low power operation results in comparatively lower losses. Figure 18b shows almost equal power losses since the desired $V_{DC}^{*}$ is almost the same as V_DC reference voltage for the non-optimized scenario. Thus, the two systems show almost identical power loss characteristics.

6. Conclusions

In this paper, a WPTS is used to directly supply power to a MDS in a VADS without resorting to an intermediate DC-to-DC converter. A major hurdle with the implementation of such a system is the parameter variations in the WPTS, such as motor power demand and mutual inductance variations, which can lead to a significant dynamic stability issue. In addition, the required constraints to ensure the MDS remains stable with the defined power and speed are identified. Thereafter, these constraints are extended to the MET method to be able to achieve minimum coil power losses. Accordingly, a design methodology for a WPTS is suggested which takes the limitations into consideration. In addition, a robust controller based on a µ-synthesis controller is designed which can guarantee RS and RP of the WPTS within the parameter uncertainty ranges. A number of experiments are performed and evaluated to verify and compare the MET and constant output voltage control methods. As it is shown in this paper, MET control leads to a better efficiency performance. In contrast, the constant DC-link voltage control offers the better dynamic performance in comparison with MET.