Digital Autotuning of DC-DC Converters Based on Model Reference ...

Digital Autotuning of dc-dc Converters Based on Model Reference Impulse Response A. Costabeber*, P. Mattavelli**, S. Saggini***, A. Bianco**** *

Dept. of Information Engineering – University of Padova, Italy. E-mail: [email protected] ** DTG– University of Padova, Vicenza, Italy. E-mail: [email protected] *** DIEGM, University of Udine, Udine, Italy. E-mail: [email protected] **** DORA S.p.a., Italy

Abstract—This paper proposes an autotuning technique for digitally controlled voltage-mode dc-dc converters based on the model reference approach. The proposed solution uses the difference between the measured system impulse response and the reference one, determined by the desired dynamic performances, and minimizes the error function acting on the regulator parameters. Two different approaches have been investigated for the evaluation of the impulse response: a deterministic one, based on duty cycle impulse, and a statistical one, based on white noise injection. Compared to existing approaches, this solution has the advantage of simplicity, smallsignal processing and on-line tuning capabilities. Experimental investigation has been performed on a 20 A, 1.5V synchronous buck converter, and both simulation and experimental results confirm the effectiveness of the proposed solution. Keywords – dc-dc converters, autotuning, model reference, impulse response, white noise, on-line tuning

I.

INTRODUCTION

Interest in digital control solutions for Switch Mode Power Supplies (SMPS) is increasing due to availability of digital controller ICs developed for high-frequency switching converters [1-3] and due to some attractive properties not available in analog control as flexibility, immunity to external influences and availability of automated design tools. Moreover, the digital controller is able to include some selftuning capabilities and the automatic adjustments of controller parameters based on the desired dynamic performance. Autotuning techniques [4], developed for electrical drives or high power electronic applications, usually show a degree of complexity which is not compatible with the limits of power consumption and computational efforts in digitally controlled SMPS. This has driven the research toward the study and the development of simpler autotuning algorithms for SMPS. Recently several autotuning solutions [5-10] for high-frequency dc-dc converters have been presented. In [5-6] autotuning is based on the relay feedback technique. In [5] the tuning is performed measuring amplitude and phase of the output voltage oscillations during the start-up, while [6] is based on limit-cycle oscillations. In [7] a solution based on the model-reference approach [4] is presented, by injecting a

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perturbation at the desired crossover frequency into the feedback loop. In [8] a continuous monitoring of the crossover frequency and phase margin is performed and a multi-inputmulti-output control loop continuously tunes the regulator parameters. In [9] a non-parametric method is proposed that requires process identification based on Pseudo-Random Binary Sequence (PRBS) injection. In [10] a tuning technique based on adaptive prediction error filters is addressed, permitting on-line system identification and controller tuning. This paper proposes an autotuning technique, based on the model reference approach [4], that has a low degree of complexity and does not present any specific limitation on the controller structure and converter topology. The proposed algorithm is based on the regulation of loop gain and phase margin at the desired control bandwidth, by injecting a dutycycle impulse, measuring the impulse response, and comparing it with the model reference impulse response. The energy of the difference between the two signals is minimized through a numerical search algorithm. The same approach is then applied for a white-noise injection on the duty-cycle, leading to the possibility of an on-line tuning of the controller, tracking the parametric variations during the normal operation. This second solution guarantees smaller output deviation, but longer convergence time. The paper is organized as follows: in Section II the principle of proposed tuning technique is described, including the numerical search algorithm, the statistical approach and some issues related with measurement noise, while Section III and IV report the simulation and experimental verifications, respectively. II.

PROPOSED TUNING TECHNIQUE

The tuning proposed in this paper is based on the model reference approach [4] where closed-loop system properties are derived from the measurement of the impulse response. The technique is here discussed for the voltage control of a dc-dc buck converter and it can be easily extended to other control configurations. Following Fig.1, a duty-cycle impulse perturbation dP is added to the output of the voltage-mode regulator C(z). The tuning algorithm is based on the comparison between the real

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response hREF[k] and the measured one h[k]:

dP D+d1

D+d2

+

+

Buck Converter

Vin

DPWM

D

L

-

ESR

h[k] HRef(z)

C(z)

- V ref

e[k]

+

A/D

href[k] KP KD KI

Tuning Algorithm

Fig. 1 – Proposed tuning scheme for impulsive perturbation dP

system impulse response measured in d2 and the reference one. This case is here denoted as deterministic tuning. In Fig. 2 a similar approach is reported where the impulse response is measured via a cross-correlation algorithm, injecting a white noise signal in dP instead of an impulsive excitation and evaluating the impulse response using a correlation algorithm. This case is instead denoted as statistical tuning. In both cases, the transfer function that represents the impulse response from dp to d2 is: H( z ) =

d2( z ) 1 1 = = d P ( z ) 1 + C( z )G( z ) 1 + Tv ( z )

(1)

where G(z) is the sampled-data transfer function between the duty-cycle d2(z) and the output voltage vo(z) and Tv(z) is the voltage loop gain. Comparing the measured impulse response h[k] h [ k ] = Z −1 ( H ( z ))

(2)

with the reference one hREF[k] ⎞ ⎛ 1 ⎟ hREF [ k ] = Z − 1 (H REF ( z )) = Z − 1 ⎜ ⎜ 1 + T* ( z ) ⎟ ⎠ ⎝

(3)

an error ε is computed. The error function ε is defined as the energy of the difference between the reference impulse

+

C( z ) = PID( z ) = K P +

Crosscorrelation

h[k]

- V ref

e[k]

A/D

-

+

- vo[k] +

KP KD KI

C

C(z)

h[k]

G(z)

Vref href[k]

Reference Impulse Response LUT

+

Error Function Minimization

+

Σ|·|2

-

ε(KP, KD , KI) Tuning Algorithm

(5)

d2[k]

C(z)

io(t)

ESR

+ K D ( 1 − z −1 )

+

e[k]

L

Digital Compensator

href[k]

D+d2[k]

Buck Converter

Vin

DPWM

1 − z −1

D dP

D

KI

where KP, KD, KI are proportional, integral and derivative gains, respectively. Fig. 3 summarizes the main features of the proposed solution, considering the deterministic case. The duty cycle impulsive perturbation is injected into the regulator C(z) output, and the closed loop transfer function impulse response h[k] is measured after the injection node. The reference impulse response hREF[k] is stored in a Look-Up-Table (LUT) in the digital controller (DSP or FPGA). The measured response h[k] and the reference one hREF[k] are then used to compute the error (4). The core of the tuning algorithm is the error function minimization block, where a minimum search algorithm is implemented to find the regulator parameters that minimize the distance between h[k] and hREF[k]. For the sake of explanation, an additional simplification on the controller structure is assumed: in the next paragraphs only a two parameters PD (Proportional-Derivative) tuning is addressed, assuming a fixed integral part. Thus, a more intuitive two variables representation is used for the error function analysis and for the minimum search method. Nevertheless, the same approach holds for the more general three parameters tuning, as reported in the experimental results. +

D+d2

(4)

where N is the number of sampling periods that includes the most relevant part of the impulse response. The algorithm is aimed to tune the controller parameters to minimize the error function ε, which is a function of the controller parameters. Tuning targets are given in terms of desired loop gain T*v(z) with specified crossover frequency and phase margin. From the desired loop gain T*v(z), HREF(z) and hREF[k] are derived. Although the proposed approach is not restricted to a specific controller structure, a PID digital regulator is considered, i.e.:

dP White noise D+d1

∑ ( h([ k ] −hREF [ k ])2

k =1

C Digital Compensator

N

ε=

io(t)

KP KD KI

Auto-tuning algorithm

Fig. 2– Proposed tuning scheme for white noise perturbation dP

Fig. 3 – Proposed method architecture

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Having fixed the integral gain, the error function in (4) becomes a function of two variables. In Fig.4 the two parameters error function (4) is reported for the power stage parameters and reference transfer function used in section III for simulations, with KP and KD normalized by their values in the minimum point KPNOM and KDNOM. The numerical search algorithm is aimed to find the minimum, and thus the set of parameters (KPNOM, KDNOM) for which equation (6) holds: ∂ε ∂K P

K PNOM

= 0,

∂ε ∂K D

K DNOM

=0

KP Δ KD

(6)

Fig. 5 – Minimum search algorithm

Ideally, the minimum point could be zero, meaning that the measured impulse response exactly matches the reference one. In practice the exact loop gain of the system is assumed to be unknown, and thus the minimum point is where the distance between the reference response and the measured one is minimum. A. Minimum search method The auto-tuning solution is based on the numerical minimization of the error function ε. In literature there are several algorithms to numerically find the minimum of an nvariables function. In this paper a simple algorithm has been selected and described hereafter. It is worth to point out that the tuning principle is independent on the specific minimization technique, and a different search method can be adopted. The basic principle of the proposed minimum search algorithm [12-13] is to test the four possible directions (Fig. 5) of movement from the starting point (KP0, KD0), measuring for each of them the error function value. Δ is defined as the step in the increase or decrease of the parameters, and it is assumed to be the same for KP and KD. The coordinates that guarantee the maximum decrease in the error function are selected as the new point from which the search is restarted. If none of the directions show a decrease in the function, the search step Δ is reduced. This algorithm generally presents slow convergence speed, but, unlike faster search solutions, its convergence is guaranteed under suitable properties of the error function [12]. The numerical search algorithm is now described after the introduction of some definitions.

ε

p0 = ( K P0 , K D0 ) is the initial point, Δ 0 is the initial search is the set of search step, S = s + K P , s − K P , s + K D , s − K D

{

}

directions in the Kp, KD plane, where si are unit vectors. For each iteration n=1,2,.. one of the next two actions is performed. sn is the search direction for the n-th iteration, pn are the starting coordinates for the n-th iteration and Δn the nth iteration search step. •

If s n ∈ S such that ε( p n + Δ n sn ) < ε( pn ) p n +1 = p n + Δ n s n

Δ n+1 = Δ n •

(7) (8)

Else if for each s n ∈ S ε( pn + Δ n sn ) > ε( pn )

p n +1 = p n

(9)

Δn R

(10)

Δ n+1 =

R >1

Thus, two kind of iterations are possible: 1) successful iterations, where a direction of error decrease is found, the step is unchanged and the parameters KP, KD are updated to the decrease direction; 2) unsuccessful iterations, where all the directions lead to an error function increase and the parameters are unchanged while the search step is reduced of a factor R. The convergence of this algorithm is proved for each p0, Δ0 and R, assuming some properties of the error function ε [13]. The convergence result is resumed in (11).

lim ∇ε( pn ) ≤ lim Δ n = 0

n→∞

n →∞

(11)

where ∇ is the gradient of the error function ε. The key of this convergence relies in the unsuccessful iterations, where the reduction of the search step Δn guarantees lim Δ n = 0

n →∞

KP/ KPNOM KD/ KDNOM Fig. 4 – Two-parameters KP, KD error function

(12)

In the proposed tuning algorithm, a slightly modified version of this basic search algorithm is used, where the step reduction is limited to Δmin, and, at each iteration, not only the four directions are tested but also the starting point is

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remeasured and compared with the other directions. These variations address to two important issues. The first is that a noisy measurement of h[k] could lead the algorithm to converge to a non-minimum point. This is because some instantaneous local minima may appear due to sampling noise. Leaving the algorithm running with fix small step and remeasuring at each iteration the initial point, the convergence in the exact minimum point is guaranteed. The second is to add the on-line tuning feature. With a non-zero final step, and remeasuring the iteration initial point, the algorithm sensitivity to the error function modifications caused for example by power stage parameters variations, is maintained and the PID regulator parameters are continuously updated.

B. Statistical tuning algorithm The second solution is based on the same principle of function ε minimization, but injecting in the control loop only a small amplitude uncorrelated white noise added to the duty cycle. A cross-correlation algorithm is then performed to statistically measure the impulsive response. The main advantage of this solution is that the complexity is similar to the deterministic solution one, but the output voltage deviation is smaller, being the additional noise effect similar to a limit cycle oscillation. The major drawback of this proposal is the tuning time which is much longer. The statistical measurement of the impulse response is performed through the cross-correlation approach in [11]. In Fig. 6 an ideal white noise disturbance dP(k) is injected between the regulator C(z) and the process G(z), where the output voltage vo(k) is affected also by current sampled value iout(k). Considering the system output d2(k), and introducing the buck converter open loop output impedance Zout_ol(z), the corresponding transfer functions are: d2( z ) 1 = d P ( z ) 1 + C( z )G( z )

(13)

d 2 ( z ) Z out _ ol ( z )C( z ) = iout ( z ) 1 + C( z )G( z )

(14)

H( z ) =

Q( z ) =

In the time-domain, the signal d2(k) is the sum between the convolution of dp(k) with the impulsive response h(k) of H(z) and the convolution of dp(k) with the impulsive response q(k) of Q(z). The convolution is identified with the symbol *.

d 2 ( k ) = ( d P * h )( k ) + ( iout * q )( k ) = =

+∞

+∞

n = −∞

n = −∞

∑ d P ( n )h( k − n ) + ∑ iout ( n )q( k − n )

(15)

The input-output cross-correlation for k greater or equal than zero is: φ d P d 2 ( k ) = ( d P ⊕ d 2 )( k ) = 1 N d P ( n − k )d 2 ( n ) n → +∞ N n =0

= lim

∑

White noise +

dP[k]

+

d2[k]

d1 [k]

C(z)

e[k]

-

Correlation

vo[k]

+

Vref

Cross-

+

dP[k]

G(z)

-

Zout_ol(z)

iout [k]

h[k]

Fig. 6 – Impulse response measurement through cross-correlation

where ⊕ is the symbol for the correlation. Substituting (15) into (16), the cross-correlation is:

φ d P d 2 ( k ) = ( d P ⊕ ( d P * h + iout * q ))( k ) = = ( φ d P d P * h )( k ) + ( φ d P iout * q )( k )

(17)

The second convolution in (17) is constant if dP(k) and iout are independent and, in particular, it is zero if at least one of the two signals has zero mean. In (17) dP(k) is assumed to be a white noise signal, with zero mean. In the hypothesis of an ideal white noise with power spectral density P, the autocorrelation of dP(k) is an impulse: 1 δ( k ) P

(18)

1 1 δ * h )( k ) = h( k ) P P

(19)

φ dPdP ( k ) =

Then (17) becomes:

φ d Pd2 ( k ) = (

where h(k) is exactly the impulse response. It is worth to note that in (17) iout can be considered as a disturbance in the impulse response measurement, but the correlation guarantees immunity to any uncorrelated noise. The drawback of the correlation is that the measured impulse response is exact only if the observation time N tends to infinite. A finite observation window N leads to measurements errors, which are limited by using a long observation period.

C. Measurement Noise Effects Both in deterministic and statistical case the previous analysis assumes a noiseless measurement of the impulse response. In the real system a noisy measurement is due to different causes. For any case, the possible presence of a limit cycle oscillation is translated in an error in the response measurement. In the statistical case, another noise source is the limited observation window in cross-correlation algorithm. Sampling noise coming from the prototype adds to these terms. Neglecting the exact form of the noise sources, it is important to observe that the measured response hmeas[k] results in an aleatory variable given by:

(16)

hmeas [ k ] = h[ k ] + H N [ k ]

(20)

where h[k] is the impulse response without noise and HN[k] a zero mean stochastic function accounting for the measurement

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SIMULATION RESULTS

The proposed solution has been simulated in Simulink/Matlab® for the voltage mode control of a dc-dc buck converter with power stage parameters: Vin=5 V, Vo=1.3 V, L=0.8 uH, C=2000 uF, fsw=500 kHz. Due to the absence of ESR and assuming one step control delay in the control implementation, a PID regulator is required. This regulator is designed to have a control bandwidth equal to fsw/15=33.3 kHz with a phase margin of 50°. The integral part KI NOM is supposed to be fixed for simplicity. Under these design constraints, nominal PD parameters KPNOM and KDNOM have been designed, assuming the knowledge of all the converter parameters, and the reference impulse response (3) has been obtained. To validate the proposed tuning algorithm, the PD controller gains are shown to converge to their final nominal values KPNOM and KDNOM starting from an arbitrary initial condition and applying the deterministic tuning. Regulator gains are initialized to KP=0.01KPNOM and KD=0.05KDNOM and in spite of their mismatch, the proposed algorithm succeeds in leading them to their normalized nominal values in about 15 ms, as shown in Fig. 7. In Fig. 8 the pre-tuning and post-tuning loop gains are reported, showing a good match with the tuning targets: before the tuning the crossover frequency is fC=1.42kHz and the phase margin mФ=80°, while after the PD tuning fC=35.5kHz and mФ=52°, with a 6% error in the target frequency and a 4% error in the target margin. The same simulations have been performed for the

Kd/Kd

NOM

2 1.5 1

NOM

0 -50

Pre tuning

-100 0 10

10

1

2

3

10 10 Frequency [Hz]

10

4

10

5

10

5

Post tuning

-90 Pre tuning -180 10

1

2

3

10 10 Frequency [Hz]

10

4

Fig. 8– Loop gain Bode plots before and after the PD tuning

complete three parameters PID tuning, starting from a low integral part KI=0.001KINOM and confirming the convergence to the precalculated parameters. The same results were obtained with the statistical tuning. In Fig. 9 the effect of a uniformly distributed noise ±1% superposed to the measured duty cycle response is presented, showing that the original algorithm fails to converge for both the parameters (lines a) and b) are the convergence values with the original algorithm), while the modified version converges successfully, accepting an oscillation around the minimum point, caused by the non-zero final step size. Times t1 and t2 are the instants at where the search step saturates at its minimum value. IV.

EXPERIMENTAL RESULTS

The proposed algorithm has been tested on a voltage-mode control of a synchronous buck. The parameters of the experimental prototype are: Vin=5 V, Vo=1.5 V, L=0.8 μH, C= 660 uF, fsw=200 kHz, IoNOM=20 A. For the sake of rapid prototyping, the control has been implemented in a digital signal processor (TMS320F2808), which is suitable for highfrequency dc-dc converters having an embedded Digital Pulse Width Modulator (DPWM) with resolution of 150 ps. The control loop targets of bandwidth and phase margin are respectively fC=17 kHz and mФ=60°. Starting from these 1.4 1.2

0.005

0.01 t [s]

0.015

0.02

a)

KP

1

1.5

Kp/Kp

Post tuning

50

-270 0 10

0.5 0 0

100

0 Phase [deg]

III.

Magnitude [dB]

noise. Substituting (20) in the error function (4), results in a noisy error function. The search algorithm is modified as described in section II.A, to account for the noise. In particular, a limit in the search step decrement is imposed to ensure a continuous cycling of the search that is required to average the noise effect, at the cost of a precision loss in the minimum detection. The result is a small steady-state oscillatiton of the regulator parameters near the minimum point.

b)

0.8

KD

0.6

1

0.4 0.5

0 0

0.2 0.005

0.01 t [s]

0.015

0 0

0.02

Fig. 7– Normalized derivative gain Kd/ KdNOM tuning, normalized proportional gain Kp/ KpNOM tuning

t1 0.005

t2 0.01 t [s]

0.015

0.02

Fig. 9– Parameters KP, KD normalized to their final values during deterministic tuning with noisy measurement

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specifications an approximate reference loop gain transfer function Tv*(z) has been derived:

z−A

2

iL

Fig. 11– Output voltage during deterministic tuning, Vin=5V(vo 20mV/div, iL 0.5A/div, time 100μs/div)

Module [dB]

where the gain G and the zero A are derived from bandwidth and phase margin constraints. Once Tv*(z) is known, the reference impulse response follows from equation (3). Fig.10 shows the parameters KP,KD during the PD tuning with fixed integral part, normalized to their final values, the error function, normalized to the impulse perturbation amplitude. Fig.11 reports the output voltage and inductor current during the deterministic tuning. The parameters KP and KD are both initialized to zero. The voltage deviation due to the duty cycle impulse is Δvo=60 mV, and the distance between two consecutive duty impulses is the tuning period TTUNING=250 μs. From Fig.10, after 60 tuning periods, corresponding to 15ms, the tuning reaches the step saturation, where the error for KP is 20% and for KD is 10%. This is caused by the noisy measurement, and leaving the minimum search algorithm cycling with fixed step this error is reduced, leading to the post tuning loop gain in Fig.12, measured for Vin=5 V, that is very closed to the target of fC=17 kHz and mФ=60°. The final measured parameters errors are 4% for KP and 6% for KD. In Fig. 13-14-15-16 the experimental results with the statistical method are reported. Fig.13 addresses the most important limitation of the statistical method, which is the long tuning time. In Fig. 13 KP and KD oscillate around their nominal value and the error approaches zero in about 60 periods. Being the duration of each period TTUNING=25 ms the total tuning time to compute a good impulse response through cross-correlation is 0.5 s, making this approach unsuitable for a fast start-up tuning. In Fig.14 the small output voltage deviation ±10 mV during the tuning can be appreciated, showing the feasibility of an on-line tuning. Fig.15 reports the tuning result for Vin=5 V, with the same targets of the deterministic case, while in Fig. 16 the input voltage was changed from 5 to 8 V. Despite its intrinsic slowness, this solution can be active continuously to monitor the error minimization and thus tracking the process variations. 2.5

vo

(21)

( z − 1 )2

0.1

mF =76.6 °

-180

before

10

2

3

10 10 Frequency [Hz]

4

Fig. 12– Bode diagram before and after the PD tuning for the deterministic case with Vin=5V 3

ε

2.5

0.08

0.5

KP

0.4

ε

2

1.5

0.3 0.06

1.5

0.04

1

0.02

0.5

1

0.2

0.5

KD 0 0

mF =57 °

after

-90

-270

0.12

KP

30 fC=17.7 kHz after 20 10 0 -10 fC=1.2 kHz -20 -30 before -40 -50 2 3 4 10 10 10 Frequency [Hz] 0

Phase [°]

Tv* ( z ) = G

PWM

20 40 N TUNING

60

0 0

20 40 N TUNING

0 0

60

Fig. 10– Parameters KP, KD normalized to their final values and tuning error normalized to the duty cycle impulse amplitude during deterministic tuning. NTUNING is the number of tuning periods TTUNING

0.1

KD 20 40 N TUNING

60

0 0

20 40 N TUNING

60

Fig. 13– Parameters KP, KD normalized to their final value and tuning error normalized to the duty cycle impulse amplitude during statistical tuning NTUNING is the number of tuning periods TTUNING

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Module [dB]

PWM

vo

iL

30 fC=17.3 kHz 20 after 10 0 -10 -20 fC=2.2 kHz before -30 -40 -50 2 3 4 10 10 10 Frequency [Hz]

Phase [°]

0

Fig. 14– Output voltage during statistical tuning, Vin=5V(vo 10mV/div, iL 0.5A/div, time 100μs/div)

mF =59°

after

-90 mF =64 °

-180

before

30 20 10 0 -10 -20 -30 -40 -50 2 10

10

fC=17.4 kHz

3

10 10 Frequency [Hz]

4

after

mΦ=58 °

after

-180 mΦ=77 ° -270

before 3

10 10 Frequency [Hz]

after

Fig. 15– Bode diagram before and after the PD tuning for the statistical case with Vin=5V

In Fig. 17 a complete three parameters PID tuning is presented, computed through the statistical method and with the same target bandwidth and phase margin. Similar results were obtained with the deterministic tuning of 3-parameters PID. V.

30 fC=17.2 kHz 20 10 0 -10 -20 fC=1.2 kHz -30 before -40 -50 2 3 4 10 10 10 Frequency [Hz]

0

4

Phase [°]

Phase [°]

4

before

-90

10

3

10 10 Frequency [Hz]

Fig. 16– Bode diagram before and after the PD tuning for the statistical case with Vin=8V

fC=1.24 kHz

0

2

2

after

Module [dB]

Module [dB]

-270

mΦ=58 °

-90 mΦ=76.6 °

-180

before

-270 10

2

3

10 10 Frequency [Hz]

4

Fig. 17– Bode diagram before and after the PID tuning for the statistical case with Vin=5V

CONCLUSIONS

In this paper a closed loop autotuning technique is proposed. The tuning is performed injecting a disturbance in the duty cycle suitable to measure in either a deterministic or a statistical way the impulse response of the system. The impulse response is compared with the reference one, obtained from the desired loop frequency response and the energy of the error signal is minimized through a numerical search method to meet the tuning targets. The experimental results for the two proposed solutions show good agreement with the theoretical expectations, confirming that the deterministic

solution guarantees faster tuning. Instead, the statistical solution presents smaller output voltage deviation and on-line tuning capabilities, but much longer tuning time.

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