A Novel MIMO Control for Interleaved Buck Converters in EV DC Fast Charging Applications

This brief proposes a new multiple input multiple output (MIMO) control for off-board electric vehicle (EV) dc fast chargers. The proposed feedback matrix design avoids multiple tuning of controllers in multiple and interconnected loops while improving the performance of interleaved dc buck converters over classical PI/PID controls. The innovative features of the presented strategy are the reference current monotonic tracking from any initial state of charge with an arbitrarily fast settling time and the fast compensation of both load variations and imbalances among the legs. Numerical results validate the performance improvements of the proposed discrete-time MIMO algorithm for interleaved buck converters over classical PI/PID controls. Full-scale hardware-in-the-loop (HIL) and scaled-down prototype experimental results prove the feasibility and effectiveness of the proposal.


I. INTRODUCTION
A SIGNIFICANT stream of literature has flourished in the past few years aimed at developing compact, lightweight switch power supplies with higher efficiency, fast dynamics and zero steady-state error under parametric component uncertainties and load variations. The interleaved multiphase converter offers several advantages, such as a reduction of switching and conduction losses, a reduction of the size and weight of the converter, an improvement of the total current ripple, and an increase of the switching frequency by interleaving each phase current [1], [2]. Additional advantages include the low cost and the highly standardized parallel modules.
By contrast, the critical aspect of the interleaved technique is its current sharing control, which is strongly affected by the switch on resistance or inductance mismatches [3]. These are due to the unavoidable construction tolerances which are Manuscript  difficult to control by the manufacturer and to temperature variations during the operation of the converter legs. Clearly, when there is unequal current sharing, the losses and heat dissipation will become unbalanced. It follows that the control of this converter requires a different control action for each leg, depending on the variation of multiple sensed signals. Current sharing, then, is a primary requirement when interleaving multiple phases because it improves system performance: it minimizes transient response and thermal problems, [4], [5]. Among all the control methods available in the literature, multiple and interconnected PI/PID controllers are the most widely utilized, see [6], [7], and [8]. Generally, a primary controller computes the average duty cycle on the basis of the transfer function of the converter obtained by statespace averaging. A further group of secondary controllers compensates for the unbalances between one "master" leg and the remaining "slave" legs. The main limitation of the widespread PID-based control strategy is a lack of robustness in the presence of large disturbance and uncertainty [9]. Nonlinear sliding mode control has been proven to overcome this problem. However, more accurate values of the parameters and information on the state vector are required. In [10], the duty ratio generated by the sliding mode control is adjusted for each phase by PI controllers. By contrast, the main advantage of fuzzy controllers is that no prior knowledge of the converter is needed. However, fuzzy logic controllers require substantial computational power due to complex and heuristic decisionmaking processes, namely fuzzification, rule base storage, an inference mechanism, and process of defuzzification [11].
To overcome these limitations, in this brief we propose a new multiple input multiple output (MIMO) control algorithm based on the method introduced first in [12] and further developed in [13]. The key idea behind this approach, which goes under the name of global monotonicity, is to design the feedback matrix that guarantees a monotonic response from any initial condition (and therefore avoids overshoot and undershoot) with an arbitrarily fast settling time, by imposing a closed-loop eigenstructure where each component of the tracking error is driven only by a single real-valued closedloop mode. It is proven in [13] that this is the only way in which the output of the system can be rendered monotonic in each output component regardless of the initial condition of the system. The proposed MIMO control meets the current demand for new control strategies in the automotive sector for a reduction of greenhouse gas (GHG) emissions, as well as an improvement of the quality of urban air [14]. In particular, electric vehicles (EVs) powered by batteries and recharged with electricity from renewable sources allow for a reduction of emissions by up to 30% compared to a car with a thermal engine [15]. Battery technology plays a key role in the diffusion of EVs, due to the significant correlation between battery cost, lifetime, charging time, and the characteristics of the employed battery charger, [16]. Battery chargers can be classified into three main power levels. In particular, level three refers to the fast chargers. This type of chargers is off-board, generally installed in public charging stations, and supplies high dc power directly to the battery. The major challenges for these chargers are the cost reduction and the competitiveness of the charge time against the conventional petroleum refill [17], [18], [19]. Several converter topologies have been proposed in the literature to meet these targets, including the Vienna rectifier [20], and the multilevel neutral point clamped (NPC) converter [21]. The buck converter is widely used in the dc-dc charger stage for its simple structure and well-known design procedures. Generally speaking, the Lithium-ion (Li-ion) battery charging strategy can be divided into three classes as follows: 1) Empirical model-oriented charging controls, which predict the battery states and calculate electrical elements using past experimental data. In these cases, the battery charging can be improved by fuzzy control, linear quadratic control, or model predictive control. 2) Control algorithms based on electrochemistry battery models aimed at minimizing the charging time while performing thermal management of the battery. 3) Model-free methods, including constant-current (CC), CC constant-voltage (CCCV), multistage CCCV, and pulse charging techniques.
The main advantages and drawbacks of these control algorithms are discussed in [22], [23], and [24]. Right now, the most common EV charging method is the CC-CV [25]. The performance of the proposed MIMO control has been experimentally verified and compared against other methods via simulations. The main features of this converter control for EV dc fast charging are as follows.
1) The design of a MIMO control system avoids multiple tuning of PI/PID controllers in multiple and interconnected loops. 2) The system matrix of the interleaved buck converter has two complex conjugate eigenvalues, while the eigenvalues of the controlled system matrix are all real and stable. 3) A real-time procedure controls the steady-state values of the state variables, but the feedback matrix determines the dynamic behavior of the system. 4) A monotonic tracking error decay is guaranteed from any initial condition (and this in turn ensures a nonovershooting and non-undershooting response from any initial state). 5) The procedure has a degree of freedom that can be tuned to impose the reference current step response settling time. 6) The closed-loop system compensates for variations of the set-point current/load and current imbalances among the legs under variations of the system parameters. The brief is organized as follows. Section II presents the discrete-time state-space model of the power converter. The tracking control problem and the description of the proposed n-leg interleaved buck converter control are presented in Sections III and IV, respectively. Section V presents a comparison with other methods while in Section VI numerical, real-time, and experimental results obtained using a Li-ion rechargeable battery pack are provided. In the last section, the conclusion derived from this work is drawn.

II. INTERLEAVED BUCK CONVERTER CIRCUIT AND DISCRETE-TIME STATE-SPACE MODEL
Standard off-board EV chargers are composed of an ac-dc converter used as a power grid interface followed by a dc-dc converter that controls the battery charging process.
The dc-dc converter considered in this brief is an n-leg interleaved buck converter as shown in Fig. 1. Here, V in represents the input voltage, (dc-link voltage), while i out and V out denote the output current and voltage, respectively.
Each leg includes a power stage composed of two insulatedgate bipolar transistors (IGBTs) switches with on-state resistance R Sw , and an inductor with inductance L and coupling resistance R L . The equivalent series resistances (ESRs) in each j-leg are given by The power switches are toggled by pulse-width modulation (PWM) signals shifted by 360 • /n from each other and with duty cycles The filter capacitance C with parasitic series resistance R C is shared by the phases. Assuming a low capacitor ESR, the effects introduced by R C can be neglected, and the output voltage can be approximated to V C , while the output current i out can be approximated to the total current as The discrete-time state-space model of the interleaved buck converter can be written as : where the vectors represent, respectively, the m-dimensional state (with m = n + 1), the n-dimensional control input (which has duty-cycle signals as components), and the n-dimensional output. Thus, A ∈ R m×m , B ∈ R m×n , C ∈ R n×m , are the system matrix, the input matrix, and the output matrix, respectively. These matrices can be obtained by the continuous-time state-space model of the interleaved buck converter, see [8], using a zeroorder hold discretization method and sampling time T s .

III. GLOBAL MONOTONIC TRACKING
Global monotonic tracking refers to the problem of designing a state-feedback control law for a MIMO linear timeinvariant system such that for all initial conditions the output y tracks a step reference r with zero steady-state error and is monotonic in all components. If y asymptotically tracks the constant reference r and is monotonic, then it is also both non-overshooting and non-undershooting.
If the discrete-time system is right invertible and stabilizable, and has no invariant zeros at 1 (these are the standard assumptions for a tracking problem to be solvable, see, e.g., [26]), given the step reference r to track, we proceed as follows: first, we choose a feedback gain matrix F such that A + B F is asymptotically stable. We then choose two steadystate vectors x ss and u ss that, for the given r , satisfy the linear equation This equation always admits a solution in view of the right-invertibility and the absence of eigenvalues of A at 1 (this latter assumption implies that A − I is invertible). Now, applying the state-feedback control law to (3) and using the change of variable ξ = x − x ss gives the closed-loop autonomous system aut : The dynamics of the closed-loop system are governed by the eigenvalues of A + B F: the uncontrollable eigenvalues are stable from the assumption of stabilizability, while the other eigenvalues can be assigned arbitrarily to be stable and sufficiently fast to improve the speed of the response. Since A + B F is asymptotically stable, x converges to x ss , ξ converges to zero and y converges to r as t goes to infinity.
Our goal is to select F in such a way that global monotonicity is guaranteed from any initial condition, and this is possible if and only if we can ensure that the tracking error ϵ(k) = y(k) − r (k) converges monotonically to 0 in all components, from all initial conditions. This is in turn equivalent to requiring the tracking error to have the structure where λ 1 , . . . , λ n are closed-loop eigenvalues (and they are therefore stable). In other words, if we are able to obtain a tracking error ϵ(t) that consists of a single power per component and |λ k | < 1, the problem is solved since the powers of λ k are monotonic functions of k. The converse is also true: the only way to obtain a feedback matrix ensuring global monotonic tracking is to obtain a tracking error as in (8): hence, in order for the problem of global monotonic tracking to be solvable, we need to distribute (at most) n modes evenly into the tracking error with one mode per error component. To achieve this goal, all the remaining closedloop modes have to be made invisible from the tracking error. The task is now to find F such that for every ξ 0 the error ϵ(k) is given by a single stable real mode per component. Let j = 1, . . . , n. Let λ j ∈ (−1, 1). Consider a solution v j and w j of the linear equation where β j ̸ = 0 and e j is the jth vector of the standard basis of R n : (9) always has a solution in view of the right-invertibility of the system. By choosing F such that F v j = w j , we find Hence, for any ξ 0 ∈ span{v j } the response is where γ j depends on ξ 0 . Considering λ 1 , . . . , λ n ∈ (−1, 1); by applying this argument for all components of the tracking error, we obtain a set of solutions w n of (9). If v 1 , . . . , v n are linearly independent, we can choose F to be such that F v i = w i for all i = 1, . . . , n. Then, for every ξ 0 ∈ span{v 1 , v 2 , . . . , v n }, by superposition we find For this response to be achievable from any initial condition, we also need to render the remaining m −n closed-loop modes invisible at ϵ. This task can be accomplished by exploiting the supremal stabilizability output-nulling subspace V ⋆ g of the system. This subspace can be obtained as the image of for some other matrix [ W 1 W 2 · · · W d ] partitioned conformably, where {µ 1 , . . . , µ t } are the minimum-phase invariant zeros of and {µ t+1 , . . . , µ d } are arbitrary and stable (assume, e.g., that they are real and distinct for simplicity). If the dimension of V ⋆ g + span{v 1 , v 2 , . . . , v n } is equal to m, every initial state ξ 0 can be decomposed as the sum ξ v + ξ r , Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
where ξ v ∈ V ⋆ g and ξ r ∈ span{v 1 , v 2 , . . . , v n }. If dim V ⋆ g = m − n, and we can find a set of linearly independent columns {v n+1 , . . . , v m } from the columns of [ V 1 V 2 · · · V d ] that is linearly independent of {v 1 , . . . , v n }, we can take w n+1 , . . . , w m to be the columns of W g that correspond to v n+1 , . . . , v m , and construct the feedback control ω [ w 1 · · · w n w n+1 · · · w m ], the response associated with ξ v is identically zero, while the one associated with ξ r is still given by (11). Hence, the tracking error can be written as in (8) for any ξ 0 . The closed-loop eigenvalues obtained with F are given by the union of {λ 1 , . . . , λ n }, with the set of µ j associated with the columns {v n+1 , . . . , v m } chosen from

IV. INTERLEAVED BUCK CONVERTER MIMO CONTROL
One of the available and widely adopted Li-ion battery charging strategies is the CCCV technique. This approach reduces the negative effects on the performance in both the converter and the battery caused by voltage and current spikes, by selecting a charging profile composed of two main modes. A constant current control mode leads to fast charging up to high values of SOCs and for this reason it is predominant in EV dc fast charging facilities. Then, the charging current can eventually gradually decrease under constant voltage control up to the final value of the output current [22]. Focusing on the constant current control mode, the proposed method hinges on the implementation of two distinct control algorithms to satisfy the specifications (zero steady-state tracking error under the reference signal i * t , asymptotic stability and nonovershooting/nonundershooting dynamics). The first algorithm calculates the steady-state values of the state and control vectors x ss and u ss during the battery charging process, as functions of the reference current i * t , the open circuit battery voltage V OC , the battery droop coefficient R and the leg resistances (1). It can be easily shown that [22] x ss = where the real-time droop coefficient R of the battery and the steady-state leg resistance R sj are computed as The open-circuit voltage can be estimated, for example, on the basis of the battery state of charge SOC using the lookup table method [27], [28].
The second control algorithm implements the discrete control law (6). In particular, the constant gain matrix F is designed using the global monotonic tracking method described in Section III. The corresponding control system is depicted in Fig. 2.  The interleaved modulation adopted in this brief is described in detail in [4]. Samplings are performed at the peak of each PWM carrier; this leads to a current value equal to the average inductor current (no current ripple detection), see [8]. It follows that an interrupt routine is generated at the sampling frequency f s = n · f sw , where f sw is the PWM switching frequency. Each sampled current is stored and used to update the n duty cycles. Therefore, with three interleaved legs and a 20 kHz switching frequency, the sampling frequency is equal to 60 kHz (3 · 20 kHz).

V. COMPARISON WITH CONVENTIONAL PI AND PIDF
Several tests have been considered to evaluate the performance of the proposed method under parameter variations and model uncertainties in comparison with PI/PIDF average controls described in [6] and [8] with reference to Table Ia) full-scale scenario with a 3.84 resistive load. In these current-balance controls the average duty cycle is computed by a primary controller on the basis of the converter averaging model. Secondary PI controllers calculate the duty ratio to compensate for the unbalances between the legs. In particular, in [8] a direct digital design method for discrete controllers with PID structure + filter, therein referred to as PIDF controllers, is introduced. This method is based on a polezero cancellation technique jointly with an analytical design technique based on "inversion formulae," which explicitly express the parameters of the controller in terms of the design specifications given by the phase/gain margins and the corresponding crossover frequencies, see [29]. For a fair comparison, the PI controller P I = K p +K i (T s /(z − 1)) using the control strategy described in [6] has been designed to get a monotonic response. In particular, the values K p = 0.15e −3 and K i = 18.16 lead to a phase margin of 71 • and a gain crossover frequency of 3000 rad/s. These are the same design specifications given for the PIDF tuning, see [29].
The state-space model of the buck converter is characterized by the following discrete-time system matrices: The step responses of the inductor currents and the total current obtained with constant duty cycles at 79.8% in an open loop and obtained with the three considered methods are shown in Fig. 3. In all cases the requirement of zero tracking error is satisfied. As expected, the advantage of the proposed solution is a faster response, by 58% and 41% compared to the PI and PIDF controller, respectively. Moreover, the settling time can be modified simply by changing the value of λ as a degree of freedom; λ = 0.9 leads to a settling time equal to 1.2 ms. In Fig. 3    velocity of the step response is given by the imposition of the duty cycles to be in the interval [0, 1] in (4).
The state-space trajectories of the averaged inductor currents at steady-state converge as shown in Fig. 4 (right). The instantaneous currents stabilize over a closed path, whose area is related to peak-to-peak ripple of the current, see Fig. 4 (left).
Results for a step-type variation of i * t (t) from the nominal value 125-130 A are shown in Fig. 5. The controls guarantee zero tracking error in all the considered cases. The method presented in this brief leads to the fastest response: 0.3 s compared to 0.6 and 0.5 s of PI and PIDF controls.

VI. NUMERICAL AND EXPERIMENTAL VALIDATION
The proposed MIMO control structure has been tested numerically, in real-time hardware-in-the-loop (HIL), and experimentally employing the converter parameters in Table I  and the battery models in Table II. In particular, the values in Tables Ia) and IIa) represent a full-scale scenario chosen for comparative analysis against other methods and for the realtime HIL validation, while the values in Tables Ib) and IIb) refer to the scaled-down experimental prototype.

A. Numerical Results Based on Shepherd's Li-Ion Battery Model
Numerical simulation results have been acquired using the MATLAB-Simulink (MathWorks) battery block, in which  the battery is modeled based on the Shepherd's model with parameters given in Table II [30]. The control is implemented using the same feedback gain matrix F discussed in Section V.
The results shown in Fig. 6 refer to a step variation of i * t from 0 to 125 A, λ = 0.85 in the case of SOC equal to 20%. The chosen reference current corresponds to a charging rate of 2.5 C likely in low-SOC battery packs connected to fast charging facilities. The currents monotonically track the reference value i * t /3 with zero steady state error in 0.5 ms. Moreover, to test if this result is obtained from any initial condition, simulations have been carried out from SOC 20% and 70%. In all the cases, the tracking error vanishes monotonically.
The control of the output current and inductor currents under variations of R s1 are shown in Fig. 7. At t = 0 ms, the resistance of the first leg series has been modified from 0.32 to 0.62 . The current balancing among the legs has been reached in 0.7 ms, confirming that the control compensates for resistance variations among the legs in contrast with the openloop scenario.
Finally, simulations have been carried out to evaluate the robustness of the proposed control method under uncertainties on the value of the inductor-inductors are among the components with highest nominal value uncertainty. In particular, Fig. 8 shows the total and inductor currents under variations of the reference step when L 2 = 0.9 L. Despite the dissymmetry in the peak-to-peak ripples of the inductor current, the total current mean value tracks the reference signal without steadystate error in about 0.8 ms.

B. HIL Real-Time Validation
To validate the real-time feasibility, HIL results have been acquired employing the experimental setup in Fig. 9. The fullscale converter plant [ Table Ia)] was emulated in real-time inside the RT-Box 1 (Plexim) while the TMS320F28379D (Texas Instruments) digital signal processor (DSP) run the controller on a separated control card. Both RT-Box 1 and DSP execution codes have been deployed in a PLECS (Plexim) environment and the results have been collected using the external mode functionality. This test allows one to verify the feasibility of a commercial DSP with actual execution times, PWMs generation synchronization, and sensing/conversion delays. The full-scale converter has been emulated with a 2.38 µs discretization step and using the same battery parameters of Section VI-A [see Table IIa)] with 50% initial SOC, which accounts for all the main phenomena tackled by a real full-scale prototype. The control is implemented using the same feedback gain matrix F as in Section V.
Similar to what was done for our simulation results, the step response, the current balancing, and the robustness under parameter uncertainty have been verified. The same R s1 step shown in Section VI-A is visible in Fig. 10; the real DSP restores the HIL current balancing in about 0.6 ms instead of 0.7 ms as visible in Fig. 7. Fig. 11 shows the HIL step response under inductor uncertainty (L 2 = 0.9L). Other than a slightly different current ripple peak-to-peak range, no average current drift/mismatch is observed. The proposed MIMO control guarantees a monotonic tracking response in about 0.6 ms.

C. Experimental Validation on a Scaled-Down Prototype
The performance of the proposed MIMO control for the dc charger has been evaluated using the set-up shown in Fig. 12.
In particular, the three-phase interleaved buck converter is Fig. 11. Proposed method HIL step response with an L 2 10% inductor reduction. characterized by the parameters shown in Tables Ib) and IIb). The currents of the inductors have been detected by three LES 6-NP (LEM) Hall effect current sensors which guarantee galvanic isolation between the converter power board and the DSP side. The output signals of the current sensors are proportional to the sensed current except for a 2.5 V offset. A dedicated conditioning circuit has been designed to interface with the 3.3 V 12-bit analog-to-digital converter (ADC) of the DSP control card. The same DSP used for HIL results generates the PWM signals to drive the bipolar junction transistors (BJTs) of the converter. The battery is a Li-ion rechargeable pack consisting of four 18650 cells having nominal voltage 3.7 V each [see Table IIb)]. This kind of battery cell is widely employed in high-performance battery packs from first-tier EV manufactures (e.g., Tesla Inc.). The proposed MIMO control has been implemented in MAT-LAB/Simulink, then the executable C code has been deployed to the DSP target employing Code Composer Studio (Texas Instruments) IDE. Experimental results have been acquired using the oscilloscope DS1054Z (Rigol) at a sampling rate ranging from 5 to 250 MSa/s. The model of the experimental converter with zero initial condition is characterized by the matrices   Step responses using Li-ion battery pack with i * t = 3 A, λ = 0.985 (top) and λ = 0.980 (bottom). closed-loop system has been selected to be equal to the invariant zero of the system to be controlled, while the values of the other three eigenvalues have been chosen in the stability unit circle and equal to each other in order to keep a balanced current share among the converter legs. Setting these values equal to λ ∈ {0.972, 0.985, 0.985, 0.985}, the feedback gain matrix is   Similar to what can be observed in Fig. 3, a smaller value of λ leads to a faster response. In this particular case, λ = 0.985 leads to a settling time of 1.1 ms, while λ = 0.98 leads to a value of 1 ms. Again, the steady state is reached without any overshoot.
The interleaved currents, under a variation of the set point i * t from 1.5 to 2 A, are shown in Fig. 15. The +33% current increment that can be observed is likely to occur in large EV charging facilities when power reallocation after the EV disconnection takes place [31]. Notice that the proposed MIMO control reaches the steady-state value without oscillations in less than 1 ms. Finally, the proposed method has been also tested under variation of R s1 , by reducing its initial value by 1.5 . The imbalanced current shared among the legs has been compensated in 1.2 ms, see Fig. 16. During the evolution, i 1 rises by about 74%, reaching a value of 1.15 A while i 2 and i 3 almost identically decrease by about 38%, dipping to 410 mA.
Overall, the effectiveness of our MIMO control has been validated numerically on a full-scale scenario. The same scenario has been tested with a real-time HIL setup. Finally, a scaled-down prototype has confirmed that the proposed control architecture effectively meets the expected performance.

VII. CONCLUSION
A new MIMO control for dc-dc interleaved buck converters in EV dc fast charging applications was presented in this brief.
The proposed control avoids multiple tuning of controllers in multiple and interconnected loops, improving, at the same time, the dynamical behavior of the converter with respect to standard PI/PID regulators. The results of our simulations have confirmed that the tracking error vanishes monotonically from any initial state of the battery charge, and under reference current step and load variations. Moreover, the proposed control is shown to compensate for imbalances among the legs due to either different parasitic resistances or dissimilar aging of the components. The proposed control framework has been experimentally tested to validate the theoretical and simulated analysis utilizing full-scale HIL and scaled-down setups.
Future developments are aimed at achieving a current balancing action without the use of a dedicated transducer if a sensor-less approach through estimators is carried out [32].