A Multiple-Reference Complex-Based Controller for Power Converters

A multiple-reference complex-based controller is proposed for three-phase power converters feeding nonlinear and unbalanced loads. The control scheme incorporates a stable multiple-complex coefficient filter with bandwidths that are set arbitrarily and independently for every harmonic under consideration. The multiple-reference complex-based control scheme is applied to an uninterruptible power supply system with a voltage source converter. Each harmonic is controlled using a standard complex proportional-resonant controller that is designed for stability and robustness using the Nyquist criterion. Similar stability and robustness properties follow for the overall system due to the frequency-domain properties of the filter and of the controllers. The proposed methodology is validated in simulation and experimental tests.


I. INTRODUCTION
T HREE-PHASE power converters often operate in environments with nonlinear elements and under unbalanced conditions. Nonlinearities induce higher harmonic components so that periodic signals are not longer sinusoidal, while unbalanced systems involve positive and negative sequences. Control algorithms for power converters have to deal with the control of multiple (positive and negative) frequencies [1].
The representation of physical systems using complex-valued dynamic models was considered for induction machines in [2] and, in general, for three-phase electrical systems in [3], considering its benefits when used in real applications. The main advantage of this representation is that the analysis is considerably simplified, especially when two-input two-output systems are converted to single-input single-output systems. Extensions of classic control techniques have been proposed in the last years to deal with complex-valued systems. Furthermore, control theory tools for complex-valued systems have been generalized such as the Routh-Hurwitz test for complex polynomials [4]- [6], the root locus technique [7], frequency-domain analysis methods [8], stability conditions for time-varying complex-valued systems [9], or the sliding-mode control technique [10], [11]. Increasingly, the complex domain representation is used not only to simplify the equations of a system, but also to design and analyze control systems in industrial applications. Examples include electrical machines [12]- [14], power converters [15]- [17], and others [8]. Complex controllers also have advantages in applications where harmonics with different sequences need to be controlled, because each sequence component can be treated independently [18], [19].
In applications, the use of an intermediate filter stage to separate the individual components from noisy signals is often advantageous. Complex filters that are commonly used for signal processing [20] have been recently proposed for electrical power systems. One of the most successful scheme is a set of first-order complex-coefficient filters (CCF) known as the multiple-CCF (MCCF) [21]. The main advantage of using an MCCF compared to classical band-pass filters using real coefficients is the polarity-selective property, which makes it possible to extract the positive or negative sequence components in a straightforward manner [22]. This algorithm is used for grid synchronization, but is also valuable for obtaining magnitude and phase information on the respective harmonic components. Applications of the MCCF algorithm include shunt active filters [23], [24] and DSTATCOMs [25].
In this article, the multiple-reference complex-based control (MRCC) scheme is proposed for the control of three-phase power converters. The control algorithm uses the ability of the MCCF of extracting the harmonic components, including positive and negative sequences. For each harmonic component, a complex proportional-resonant (PR) controller is proposed. The overall scheme looks similar to the set of reduced-order generalized integrators (ROGIs) presented in [26], but enables the tuning of the control gains for each harmonic independently. As an example, the MRCC scheme is applied to an uninterruptible power supply (UPS) consisting of a three-phase voltage-source converter (VSC) feeding a nonlinear unbalanced load. This example makes it possible to demonstrate the applicability of the proposed controller. The stability of the closed-loop system, including the filter dynamics that are disregarded in many papers, 0885-8993 © 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
is analysed via the Nyquist criterion applied to transfer functions with complex coefficients [8].
The main contributions are as follows. 1) A new proof of stability of the MCCF. The MCCF is shown to be stable for all values of the cutoff frequencies of the CCFs. This theoretical result is obtained through a relatively simple proof made possible by the complex-domain representation. The result removes the constraint that a single cutoff frequency be used for filtering all harmonics. 2) The main difference with respect to other papers in the literature is the inclusion of the MCCF dynamics as a part of the control loop for design purposes. Then, the design and tuning procedure is made by considering the overall dynamics, including the filter. This is not common and, usually, the filter dynamics are disregarded for stability analysis, see recent examples using the MCCF in [27] and [28]. 3) An MRCC strategy for three-phase power converters.
Thanks to the use of the MCCF that extracts the information for each harmonic, the proposed methodology makes it possible to individually analyse the dynamics of each harmonic and, compared with traditional approaches, facilitates the design by using the complex Nyquist criterion. 4) A practical example is illustrated by a three-phase VSC.
The example includes a detailed design description, as well as simulation and experimental results. This article is organized as follows. In Section II, the complex notation is presented. The MCCF and its stability properties are introduced in Section III. Section IV presents the overall MRCC scheme, and is applied to a VSC in Section V. Then, in Section VI, the proposed control algorithm is validated by simulations and experimental tests. Finally, the conclusion is summarized in Section VII.

II. COMPLEX REPRESENTATION OF THREE-PHASE SIGNALS
Consider a set of three-phase variables (voltages) v where T ∈ C 3 is defined as for a constant c and an angle θ(t) to be defined. Different values of c are made in the literature, with c = 2 3 preserving the definition of power and c = 2 3 maintaining the signal amplitude. The transformation (1) is not bijective and reconstruction of v abc (t) from v(t) generally requires additional information. In electrical three-phase systems, this information comes from the homopolar component that is the sum of the components of v Often, this component is assumed to be zero (and this is forced to be true for currents in three-wire devices).
The transformation (1) includes the well-known αβ transformation as a special case with θ(t) = 0 and where v α (t), v β (t) are the real and imaginary parts of v(t). The dq-transformation is obtained with θ(t) corresponding to the angle of the d-axis and v d (t), v q (t) being the real and imaginary parts of v(t) [17]. In particular, with dθ dt = ω, v(t) is constant if and only if v a (t), v b (t), and v c (t) are positive-sequence balanced sinusoidal signals of frequency ω. However, v(t) is defined regardless of steady-state or balanced conditions, and generalizes the concept of phasor for v a (t) (as well as v b (t), v c (t) with 2π 3 phase shifts).
A periodic three-phase voltage v abc (t) can be expressed as a sum of multiple frequencies where ω o is the fundamental angular frequency, H r = {h 1r , h 2r , . . . , h pr } ⊂ N p is the set of harmonics of v abc (t), and V xl , φ xl , with x = a, b, c, are the amplitude and phase of the lth harmonic, respectively.
Using (1), with θ = 0 and c = 2 3 , the (scalar) complex variable corresponding to the (vector) three-phase symmetric periodic signal in (4) can be written as, where are the coefficients of the well-known positive and negative sequences. Then, the complex signal in (5) can be written as the sum of v k s complex signals, with positive and negative frequencies, as with a new set of (positive and negative) harmonics defined as where the complex values V k e jφ k correspond to (6) or (7), depending if k = l or k = −l. Remark 1: In a symmetric and balanced case, V al = V bl = V cl = V l and φ al = φ bl = φ cl = φ l . Then, coefficients in (6) and (7) are a l = V l e jφ l , b l = 0 (10) and no negative frequencies appear.

III. MULTIPLE-COMPLEX COEFFICIENT FILTER
The MCCF is composed by a set of CCF that process αβ signals to extract the components for each harmonic. In the complex notation used in Section II, this approach corresponds to estimating some terms of v(t) defined in (8). In [21], all the CCF modules have the same feedback gains, resulting in identical cutoff frequencies. In this article, the MCCF scheme is generalized considering different cutoff frequencies for the CCF modules. Fig. 1 shows the MCCF scheme using the complex notation, where H = {h 1 , h 2 , . . . , h n } ⊂ Z n is the set of considered harmonics. H is usually defined as a subset of H c , including the most representative harmonics in the system, i.e., n ≤ 2p.

A. State-Space Description of the MCCF
Consider an MCCF for n harmonics defined by the set H ⊂ Z n . Then, the CCF for the mth harmonic iŝ wherev m andv m denote the complex input and output signals of the filter for the mth harmonic, respectively, ω o is the fundamental frequency and γ m is the cutoff frequency, which impacts the bandwidth associated with the mth harmonic. The differential form of (11) is and, from Fig. 1, the input for each CCF module is where v is the complex input signal. From (12), the overall model of the MCCF in compact form iṡ v = Av + Bv (14) wherev = (v h 1 , . . . ,v h n ) T , and the matrices A ∈ C n×n and B ∈ C n×1 are given by and respectively. Equation (14) is a complex description and a generalization of the state-space form of the MCCF presented in [21].

B. Transfer Function of the MCCF
The transfer function of a single CCF has been studied in [21], and references therein. However, the MCCF structure has a more complicated behavior than the CCF. In this section, the transfer function relating the mth outputv m to the filter input v is obtained, and the stability properties of the MCCF are analyzed.
Define the mth output equation for the MCCF aŝ where and Let H m (s) be the transfer function relating the filter input v to the mth outputv m , so that The transfer function is obtained from (14), with (15) and (16) and (18), resulting in where Therefore In other words, the mth harmonic is estimated exactly by the mth component while other harmonics are completely rejected.  The total number of harmonics, as well as the harmonic components, shape the gain plots, but the properties of (24) and (25) are always satisfied. The stability of the MCCF is given by the eigenvalues of (15). Fig. 3 shows the result of calculating the n eigenvalues of A for 5000 different cutoff values. The parameters used for the computation were ω o = 100π rad/s, six harmonics H = {−11, −5, −1, 1, 7, 13} and γ m > 0 for all m ∈ H. One finds that the MCCF remains stable for all values of the parameters γ m . This observation suggests that it may be possible to prove the stability of the algorithm for arbitrary cutoff frequencies.
Indeed, the following result specifies that, if the MCCF (14) does not contain any repeated harmonic, the filter is stable for all choices of cutoff frequencies γ h k > 0, k ∈ H, and the result is true even if the gains are different for distinct harmonics.
Proposition 1: Assume that h k = h l for all k = l and γ h k > 0 for all k. Then, the polynomial D H (s) has all roots in the open left half-plane. 1 The selected set of harmonics is representative of the harmonic content present in the voltage signals due to the nonlinear load used in this work. See the harmonic spectrum in Fig. 16 of the experimental validation. Proof. Using the matrix determinant lemma (see [29,Th. 18.1.1.]), the characteristic polynomial of (15) given by (23) can be written as where The remainder of the proof has two parts: Values of s such that s = jh k ω 0 do not need to be considered. In other words, one needs P (a + jb) = 0 for all a 0, b arbitrary, and a + jb = jh k ω 0 for all k. One has Under the assumptions, which implies that P (a + jb) = 0.

A. Controller Structure
The MRCC is based on Fig. 4. It consists in extracting the harmonic components of a signal,v(t), by means of the MCCF Then, the control action is composed of the sum of individual control actions corresponding to each harmonic The reference signal for the mth harmonic is where V m the desired amplitude of the mth harmonic. The main difference with respect to the set of ROGIs presented in [26] is the inclusion of the CCF modules in Fig. 4 to extract the harmonic components and facilitate the control design. The controller C m (s) for harmonic m consists of a standard complex PR controller with the form [30] where K pm and K im ∈ C are the proportional and resonant gains, respectively. The second term, also known as an ROGI [26], is used to ensure zero error when tracking sinusoidal signals [31] and rejecting disturbances, thanks to the internal model principle. Defining K m = K pm and α m = K im K pm , the PR controller in (33), can be written as where K m , α m ∈ C are the complex gains that will be selected in the design of the controller.

B. Closed-Loop Dynamics
Using (20) in (31), the control input can be written as so that the output signal is where G(s) is the transfer function of the power converter (that may have complex coefficients). Closed-loop stability of (36) is determined by the roots of where L(s) is the loop transfer function

C. Stability Analysis
The Nyquist Criterion is a useful tool to determine the stability of feedback systems. Through the complex transformation, the system has become a single-input single-output system, where the Nyquist diagram is most powerful for stability and robusness analysis. The criterion, based on the argument principle which is valid for any meromorphic function, applies to transfer functions with complex coefficients mostly as in the real case, but with the difference that the complex Nyquist curve is no longer symmetric for negative and positive frequencies. See further details in [32].
Consider first the case where a single harmonic is controlled, and L(s) = L m (s). Assume that the converter transfer function G(s) is stable. Given that H m (s) is stable for all γ m > 0, the loop transfer function L m (s) is stable, except for a pole at s = jmω o . A modified Nyquist contour must be used, where the portion of the imaginary axis close to ω = mω o is replaced by a semicircle of a small radius. Specifically, the segment ω = mω o + δω, where δω varies from −ε to ε and ε is small is replaced by a semicircle where δω = ε e jα and α varies from −π/2 to π/2. The number of encirclements is counted using the image of the modified contour, noting that the loop transfer function does not have any pole within the modified contour. Let where L mo (s) is stable with no poles on the jω-axis. For infinitesimal ε, the image of the semicircle is which is also a semicircle, but with large radius and connecting the frequency response from ω = mω o − ε to ω = mω o + ε in the clockwise direction. A necessary and sufficient condition for closed-loop stability is that the Nyquist curve does not encircle the −1 + j0 coordinate. A sufficient condition is that the Nyquist curve does not cross the real axis for Re(s) < −1. In particular, the large semicircle crosses the real axis in the right-hand side of the plane if the angle of the asymptote for ω → mω − o belongs to (0, π) (or, equivalently, the asymptote for ω → mω + o belongs to (0, −π)). As will be shown, controller gains can always be found to satisfy this condition.
When the controllers for the separate harmonics are used together, stability is determined by the frequency response of the loop transfer function L(s), which is the sum of the individual loop transfer functions L m (s). The modified Nyquist contour includes multiple semicircles. However, due to the bandpass property of H m (s) and to the resonant nature of C m (s), L(jω) is close to one of the L m (jω) whenever its magnitude is large. In fact, the angle of the asymptote of L(jω) for ω converging mω − o (or mω + o ) is the same as the angle of the asymptote of L m (jω), because H l (jmω o ) = 0 for l = m.
As a result, the design of the control system can be performed one harmonic at a time, making sure that the Nyquist curve does not cross the real axis for Re(s) < −1. To check the stability of the overall feedback system, one only needs to verify that the finite part of the Nyquist curve does not intersect the real axis for Re(s) < −1. No adjustment is typically needed, however, because the frequency response is dominated by one of the components, unless its magnitude is small. Interestingly, the MCCF not only serves to extract the harmomic components, but also to separate the frequency bands of the individual controllers. The application of the concepts will be further explained in the practical design considered in the following section.

V. DESIGN OF THE MRCC
In this section, the design of the MRCC is demonstrated for a UPS, which consists in a three-phase VSC feeding a nonlinear unbalanced load. First, the complex-based model of the VSC is presented. Then, parameters are selected for each component of the controller. Afterward, the individual controllers are combined and the stability and robustness properties of the overall controller are evaluated on the basis of the Nyquist curve. Fig. 5 shows the electrical scheme of the three-phase VSC, where L, C, and R represent the inductance, capacitance, and inductor losses of one phase of the LC filter, respectively. When the load is unbalanced and nonlinear, currents are expected to have a high harmonic content.

A. Complex Representation of the VSC
The dynamics associated with the VSC are described in abc coordinates by is the vector of output voltages (i.e., the voltages on the filter capacitors), i T abc (t) = (i a (t), i b (t), i c (t)) is the vector of currents in the inductors, u T abc (t) = (u a (t), u b (t), u c (t)) is the vector of voltages computed by the switching policy (that will be used as control inputs), i T Labc (t) = (i La (t), i Lb (t), i Lc (t)) is the vector of load currents, and the M matrix is Remark 3: The model (43) and (44) assumes that the load currents are independent from the voltages, so that the load currents can be treated as disturbances. Practically, load currents depend on the voltages and constitute an additional feedback path. The effect of this feedback will be considered in Section V-D.
The control objective is to ensure balanced three-phase voltages on the capacitors, that is v d where V d and ω o are is the desired amplitude and frequency, respectively. The three-phase to complex transformation (1) is then used not only to define equivalent complex signals but also a complex equivalent VSC system. In particular, assuming a three-phase connection without neutral (i.e., the homopolar component is zero), the VSC dynamics in (43) and (44) can be transformed into the complex equivalent system (letting θ = 0) where v, i ∈ C are the complex voltage and current, u ∈ C is the complex control input, and i L ∈ C is the complex load current (all are scalar variables). Similarly to (8), the load current is written as where the modulus and phase of I Lk ∈ C are the amplitude and phase of i Lk (t), respectively. The input/output response is determined by the complex transfer function of the system from u to v, or From (46) with (1), the control objective is to track the complex signal where V 1 ∈ C accounts for both the amplitude and the phase of the reference, respectively By applying the Laplace transform, (51) results in The reference signals as defined in (32) are given by

B. Control Design for the mth Harmonic
We begin with a study of the influence of the control parameters on the complex Nyquist plot. From (34), one has with the magnitude and phase equations where φ K m denotes the phase of the complex number K m . Note that the magnitude and the phase of K m have important and independent impacts in (56) and (57), respectively. On the other hand, (57) suggests that α m can be used for a fine-tuning at frequencies away from mω 0 . The discussion is illustrated for m = 1 on Fig. 6, which shows complex Nyquist plots for the loop transfer function L 1 (s) and various choices of control parameters. The VSC and MCCF parameters are the ones used in Section VI-A. The magnitude and phase of K 1 directly affect the magnitude and the phase of the loop transfer function, as shown in the top plots of Fig. 6. The effect of α 1 is shown in the bottom plots. Since changing the imaginary part of α 1 is similar to modifying the phase of K 1 , the real part of α 1 reduces the arc of the unit circumference defined by the crossing points with the curves for ω < mω o and ω > mω o . On the other hand, Re(α 1 ) can be used to adjust the gain margin.
To ensure the stability of the feedback system, one needs to and two different cases appear Conditions (60) and (61) are directly related to the resonance frequency of the LC filter, given by ω r = 1 √ LC . The result shows that φ K m should be used to compensate the negative phase of the third term in (59), depending on whether the harmonic frequency is smaller or larger than the LC resonance frequency. Table I shows the control gains for the numerical example used in Section VI-A. The set of harmonics considered, H = {−11, −5, −1, 1, 7, 13}, is representative for the load used in the experimental validation in Section VI, see Fig. 16(a).  Table I, and the delay margin, T dm , associated to each plot.
Notice that, since ω o = 100π rad/s and the VSC has a resonance frequency at ω r = 3.1427 10 3 rad/s, harmonics with |m| > 10 require φ K m = π rad, which is equivalent to a reversal of feedback polarity. The complex Nyquist plots for each harmonic are shown in Fig. 7.

C. Stability and Robustness of the Overall System
In Fig. 8, the complex Nyquist plot of L(jω) is shown. Given that there are no encirclements of −1 + j0, the closed-loop system is stable. Besides, closed-loop poles on the imaginary axis are avoided because the Nyquist plot does not pass through the (−1,0) point.
In Fig. 9, the complex Nyquist plots of L m (jω) and L(jω) are compared. One finds that the loop frequency response appears as the combination of the loop frequency responses. In particular, L(jω) approaches one of the L m (jω) along the asymptotes and close to the unit circle. This result is obtained each frequency response C m (jω)H m (jω) becomes large in a different frequency band, not only because of the resonant mode of C m (s), but also because of the selectivity of the filter H m (s) (see Fig. 2). The closed-loop poles of L(jω) and L m (jω) are shown in Fig. 10, which suggests that the collective behavior of the individual  Table I.  loops is representative of the overall system in terms of pole locations as well.
The Nyquist curve can also be used to evaluate the robustness margins of the system. Gain and phase margins can be measured directly, but one should remember that they apply to the complex system, or to the real system if the same gain and phase changes are applied to the different components (a, b, and c). The delay margin is defined as the maximum time delay that may be applied while preserving stability, and can be calculated as where ω gc is the gain crossover frequency (where the magnitude of the loop frequency response is equal to 1). Note that both positive and negative frequencies can determine the delay margin. If multiple frequencies are present, the smallest value must be used. In Fig. 7, the result of the computation of the delay margin is given for each individual harmonic. The minimum is T dm = 0.397 ms for m = 13. The value is close to the value obtained with the overall Nyquist plot of L(jω) in Fig. 8, which gives T dm = 0.347 ms. The delay margin is more than sufficient for the sampling rate of 0.05 ms used in the experimental stage in Section VI-B.

D. Robustness to Load Feedback
As pointed out in Section V-A, the model (48) considers the load current i L as a disturbance. In general, this current depends on the voltage v and constitutes a feedback path. It is not possible to consider all possible load dynamics, but the effect of a static (resistive) load is studied more carefully in this section. From the model (47) and (48) Note that, when R L → ∞, the transfer function (50) is obtained. Following the analysis of the asymptotes in the previous section, we have now that shows how different values of the resistive load affect the phase for frequencies close to the resonant frequency. In particular, note that when R L → ∞, condition (59) is recovered. Fig. 11 shows how the asymptotes in (64) evolve with R L . For higher loads (when R L → 0), the last term in (64) converges to the angle of the impedance Z m = R + jmω o L, which is close to π 2 . This effect can also be observed in the complex Nyquist plots, see Fig. 12. In contrast, for high values of R L , the asymptotes of the complex Nyquist plots tend to move in the counterclockwise direction, for harmonics in the range defined by the resonance frequency, ω r , (clockwise for other harmonics), reducing the phase margin. In conclusion, the worst-case scenario is when R L → ∞, which was the one considered for design in Section V-B Fig. 11. Influence on R L on the asymptotes of ∠L m (jω), with φ Km = 0.  Table I  Therefore, from (51), (52), and (54), V d = 100 V, φ = 0 • , and ω o = 2π50 rad/s. The VSC feeds a nonlinear unbalanced load, consisting of three single-phase unbalanced rectifiers connected to each phase and a three-phase rectifier feeding a resistive load. The main harmonic content is H = {−11, −5, −1, 1, 7, 13}, see Fig. 16(a).

A. Simulation Results
Simulations have been performed using SimPowerSystems of MATLAB/Simulink with a realistic model of the power converter including losses, switching effects, and noise in the measured signals. This model does not emulate the microcontroller (and its problems related to execution times or quantification), nor thermal or electromagnetic interferences.
The proposed test consists in three scenarios. First, only the fundamental component is controlled (controllers C m (s) for m = 1 are switched OFF). Then, at t = 0.2 s, the complete MRCC is activated (with zero reference for the harmonic components). Finally, at t = 0.4 s, a reference change is applied to the fundamental component from 100 to 80 V amplitude. Figs. 13-16 show the results. For simplicity, the initial transient is omitted and only reference changes are discussed.
During the first part of the test, the fundamental component tracks its reference. See in Fig. 13(a) that e 1 (t) remains close to zero, but other harmonic components are still present because they are not controlled. See also errors in Fig. 13 and voltage/current waveforms in Fig. 14. At t = 0.2 s, all the harmonic components are controlled and their errors converge to zero (see Fig. 13). In Fig. 14, the voltage waveforms are improved. Finally, when the reference changes at t = 0.4 s, the fundamental component achieves the desired value in approximately 30 ms (one cycle and a half of the output voltage), and the errors of other harmonics also come down to zero after a short transient (see Fig. 13).
Frequency spectrums for the three-phase output voltages are shown for the first and second scenarios in Fig. 16. The total harmonic distortion when the full MRCC is used drops from 14.12% to 3.01%, which is a low value compared with to common standards for UPS.

B. Experimental Results
The experimental platform consists in the LC three-phase VSC described earlier. The converter was assembled from a SEMiX-101GD12E4s IGBT module of Semikron and the controller algorithm was implemented in a TMS320F28335 floating point DSP of Texas Instrument. The experimental testbed is shown in Fig. 17.
The testing procedure includes the same scenarios described with the numerical simulations. Fig. 18(a) shows the voltage signals during the first scenario (when only the control of the fundamental component is running with V d = 100 V). The amplitude and frequency of the three-phase voltages (CH1 to CH3) are controlled, but the waveforms are distorted because there is no harmonic compensation of the components of the load current (see phase b in CH4). In contrast, the voltage is improved  when all the significant harmonic components are controlled [see Fig. 18(b)].
Finally, Fig. 18(c) and (d) shows the transient behavior when the full MRCC is engaged, and under an amplitude reference change, respectively. In Fig. 18(c), the waveform of the voltage is improved in approximately one cycle and a half, and the   reference of the amplitude and frequency are maintained during the change. In Fig. 18(d), the amplitude of the fundamental component changes from 100 to 80 V, and the new reference value is reached in less than one period and a half.

VII. CONCLUSION
In this article, a new scheme is presented for controlling threephase power converters. The methodology is based on complexparameter models, which transform the systems to single-input single-output systems and enable powerful analysis tools. The proposed controller is based on an MCCF combined with multiple reference controllers. An intermediate contribution is a new proof of stability of the MCCF for arbitrary values of the gains. Filtering of separate harmonic components can be tuned independently. As opposed to traditional schemes, the design of the controllers is simplified by considering one harmonic at a time, a procedure that is made possible by the frequency selectivity of the combined MCCF and reference controllers. Original contributions are the inclusion of the dynamics of the MCCF during the control design, and a guarantee of stability of the overall feedback system. The analysis using the Nyquist diagram also makes it possible to evaluate the robustness margins of the system. Due to the frequency separation, the margins are close to the ones predicted from the designs carried out for separate harmonics. Simulation and experimental results validated the proposed control methodology in a practical converter design.