Improved Online Identification of Switching
Converters Using Digital Network Analyzer
Techniques
Adam Barkley and Enrico Santi
University of South Carolina
Department of Electrical Engineering, Columbia, SC, USA
Abstract— Recent progress in the identification of switching
power converters using an all-digital controller has granted
network analyzer functionality to the control platform. In
particular, the cross-correlation technique provides a
nonparametric identification of a converter’s small-signal
control-to-output frequency response. The literature shows
the viability of this technique as well as a few improvements
to the basic technique. This online network analyzer
functionality allows new flexibility in the areas of online
monitoring and adaptive control. In this paper, several
improvements to the cross-correlation method of system
identification are proposed which aim to further improve
the accuracy of the frequency response identification,
particularly at high frequencies near the desired closed-loop
bandwidth.
Additionally, an extension to the crosscorrelation method is proposed which allows measurement
of the control loop gain without ever opening the feedback
loop. Thus, performance and stability margins may be
evaluated while maintaining tight regulation of the output.
Simulation and experimental results are shown to verify the
proposed improvements.
I.
INTRODUCTION
In recent times, the availability of faster and cheaper
digital control platforms has enabled all-digital control of
switching power converters [1]. The use of digital control
opens new and exciting possibilities to perform online
monitoring, which is crucial in multi-converter power
systems. These systems are prone to interactions that can
reduce stability and performance margins [2].
Additionally, unknown source and load impedances
change the system dynamics and further affect
performance. With some additional software, the digital
controller can also act as an online digital network
analyzer capable of identifying a converter, including
system interactions, in real time [3]. This is accomplished
by injecting a test signal into the control duty cycle as
demonstrated in [4],[5]. A conceptual block diagram is
shown in Fig. 1.
This work has two main focuses. The first is improving
the accuracy of the existing cross-correlation method by
using three different techniques: oversampling the duty
cycle and output voltage signals used for identification,
windowing the measured cross-correlation, and correcting
for the non-ideal spectrum of the injected pseudorandom
binary sequence (PRBS) signal. The second focus is
extending the method to include measurements performed
without opening the feedback loop which maintains tight
output voltage regulation at all times. This technique
allows the measurement of the compensated loop gain and
Fig. 1 Conceptual block diagram showing injection of a test signal into
the control channel
of the closed-loop reference-to-output transfer function.
Simulation and experimental results verify the proposed
improvements and extensions.
II. THEORY
A switching converter operating at steady state can be
considered a linear time-invariant system to small-signal
disturbances [6]. The sampled system can be described
by:
∞
y (n) = ∑ h(k )u (n − k ) + v(n)
(1)
k =1
where y (n ) is the sampled output signal, u (k ) is the
sampled input signal, h(k ) is the discrete-time system
impulse response, and v(n) represents unwanted
disturbances such as switching and quantization noise.
The cross-correlation of the input control signal u (k ) and
the output signal y (k ) is defined in (2) as:
∞
Ruy (m) ≡ ∑ u (n) y (n + m)
n =1
∞
(2)
= ∑ h(n) Ruu (m − n) + Ruv (m)
n =1
where Ruu (m) is the auto-correlation of the input signal,
Ruy (m) is the input-to-output cross-correlation, and
Ruv (m) is the input-to-disturbance cross-correlation [3].
Consider white noise as a choice of input test signal u (k ) .
White noise input exhibits the following properties:
Ruu (m) = δ (m)
Ruv (m) = 0
(3)
These properties allow simplification of equation (2) such
that the input to output cross-correlation becomes the
discrete-time system impulse response [3].
Ruy (m) = h(m)
(4)
Here, a finite-length pseudorandom binary sequence
(PRBS) is chosen as an approximation to white noise.
The discrete-time system impulse response can be
transformed into the system frequency response using a
Discrete Fourier Transform.
Guy ( s) = DFT {h(m)}
Fig. 2 Measured input to output cross-correlation, which is equal to the
input-to-output discrete-time impulse response
(5)
This method has been applied to switching converters in
[4], [5] to measure the control-to-output transfer function.
III. IMPROVEMENTS IN THE CROSS-CORRELATION METHOD
This section presents three techniques for improving the
control-to-output transfer function identification accuracy,
particularly at high frequencies near the desired closedloop bandwidth. The first method delays the output
voltage sampling by half of the sequence clock period.
Second, a window function is applied to the input-tooutput cross-correlation data to remove spurious highfrequency noise from the estimated impulse response.
Finally, a correction is made to the control-to-output
transfer function by dividing by the non-ideal spectrum of
the measured perturbation sequence.
A. Delaying Output Voltage Sampling
To improve the identification at high frequencies, it is
possible to delay sampling of the output voltage by half of
the test sequence clock period. During each test sequence
clock cycle, a single value of perturbation is applied to the
duty cycle command. However, the output voltage
changes continuously within this interval in response to
the variation in duty cycle. By sampling the output
voltage at the midpoint of this clock cycle, it was found
that control-to-output dynamics near the injection clock
frequency were better captured.
B. Windowing the Measured Cross-Correlation
The nonzero sidebands of the auto-correlation and the
cross-correlation are due to the finite-length
approximation to white noise used here. This spurious
high-frequency content appears in the estimated impulse
response even after the true impulse response has decayed
to zero, which corrupts the high-frequency estimation. If
the approximate length of the true impulse response is
known (assuming the system is stable), the application of
a window can significantly improve the high-frequency
estimation. Here, a Gaussian window centered at time
zero is used. The width of the window should be adjusted
to pass the majority of the expected impulse response. As
Fig. 3 Zoom of measured input to output cross-correlation before and
after application of a Gaussian window
seen in Fig. 2, the measured impulse response decays to
zero in approximately 4 ms. A zoom of the crosscorrelation and windowed cross-correlation is shown in
Fig. 3. Notice that the window suppresses the spurious
high-frequency content after 4 ms.
C. Correcting for the Non-Ideal Input Spectrum
The choice of a finite-length PRBS perturbation as an
approximation to white noise introduces several unwanted
side effects. First, the autocorrelation of u(k) is not an
ideal delta function as expected from equation (3), but
instead contains nonzero sidebands as shown in Fig. 4.
The authors in [5] show that using a multiple-period test
sequence and averaging the cross-correlations reduces this
effect. Second, this discrete-time approximation is subject
to the effects of zero-order hold (ZOH) sampling, which
introduces a phase shift at high frequencies according to
equation (4), where H ZOH (s ) is the transfer function of a
zero-order hold sampler [7],[8].
H ZOH ( s ) =
1 − e − sT
sT
(6)
At the Nyquist frequency, a phase shift of -180 degrees
is observed. This effect is clearly seen in the spectrum of
the PRBS shown in Fig. 5. Note that the magnitude
G xu ( s ) ≡
xˆ ( s )
1
=
dˆ ( s ) 1 + Tloop ( s)
Tloop ( s)
yˆ ( s )
G yu ( s ) ≡
=−
1 + Tloop ( s )
dˆ ( s )
(5)
With the transfer functions defined in (5), the estimated
control loop gain is constructed according to (6). In this
simple case, Gyu is also the closed-loop reference-tooutput transfer function and Gxu is the sensitivity function.
Fig. 4 Autocorrelation of a single period PRBS showing nonzero
sidebands
Tloop ( s ) ≡ Gvd ( s ) ⋅ Gcontroller ( s )
⎛ Tloop ( s ) ⎞
⎜
⎟
⎜ 1 + T (s) ⎟
loop
⎝
⎠ = − G yu ( s )
=
G xu ( s)
⎛
⎞
1
⎜
⎟
⎜ 1 + T (s) ⎟
loop
⎝
⎠
V.
Fig. 5 Spectrum of the PRBS perturbation showing effect of ZOH
sampling
spectrum is approximately flat and a phase lag is present
as the Nyquist frequency of 50kHz is approached. Since
this phase shift also appears in the control-to-output
frequency response estimation, the effect can be reduced
by dividing the control-to-output frequency response by
the input perturbation frequency response.
IV.
LOOP GAIN AND CLOSED-LOOP CONTROL-TOOUTPUT IDENTIFICATION
The ability to measure control loop gain without
opening the feedback loop provides valuable information
about performance and stability margins while retaining
output voltage regulation. A PRBS test signal, u, is
injected into the feedback loop via a summing block. The
signals x and y are measured, and the transfer functions
Gxu and Gyu are estimated using the cross-correlation
technique as shown in Fig. 6.
(6)
RESULTS
A. Simulation Results
The literature [4] presents an interesting example
involving the single-transistor forward converter with
undamped LC input filter shown in Fig. 7. This converter
has a control-to-output transfer function with salient
features across a wide frequency range which makes
accurate identification challenging. Using a switchedmode model for this plant, a simulation test bed was
constructed in Matlab/Simulink. No input filter damping
or switch loss was included in these simulations.
1) Base Simulation
A 12-bit, four-period, 1% amplitude PRBS test signal is
injected into the duty cycle command at 100 kHz. The
switching frequency is also 100kHz. The perturbed duty
cycle and resulting output voltage are exported to Matlab,
where the cross-correlation is computed, sliced, and
averaged following the procedure described in [4]. The
resulting estimations of control-to-output transfer function
magnitude and phase are shown in Fig. 8 together with the
analytically calculated transfer function.
Fig. 7 Single transistor forward converter with undamped LC input filter
used for simulation
Fig. 6 Overview of proposed loop gain measurement
Fig. 8 Simulation results showing the control-to-output transfer function
of a single transistor forward converter without any of the proposed
improvements
Fig. 10 Simulation results showing the combined effect of delayed
sampling and applying a Gaussian window to the disturbance-to-output
cross-correlation
Note the good matching at low frequencies, the correct
identification of the input filter interaction at 500 Hz, the
output filter corner frequency at 1 kHz, and the poor
accuracy at high frequencies. The phase information is
erratic approaching the Nyquist frequency. Also visible is
the phase shift at high frequency due to the spectrum of
the input test sequence.
2) Delayed Output Voltage Sampling
Delaying the output voltage sampling by half of the test
sequence clock period yields a more coherent phase
spectrum at high frequencies as seen in Fig. 9. Compared
to the base simulation, the range of usable phase
information has been increased from 20 kHz to 30 kHz.
Additional phase lag due to the spectrum of the input
sequence is still evident.
3) Windowing the Measured Cross-Correlation
The application of the window shown in Fig. 3 to the
measured cross-correlation results in improved
identification precision at high frequencies. Proper choice
of window width improves high-frequency precision
without drastically affecting low frequency results. The
resulting frequency response is shown in Fig. 10. Notice
that the magnitude response fits closer to the analytic
response near the Nyquist frequency.
Also, the phase response is well behaved throughout the
entire identification range. The additional phase lag at
high frequency due to the spectrum of the input sequence
is still visible.
4) Correcting for the Non-Ideal Input Spectrum
Although the low frequency salient features are
captured in Fig. 8, Fig. 9, and Fig. 10, the phase diverges
from expected within a decade of the 50 kHz Nyquist
frequency. This can be explained by examining the
magnitude and phase spectra of the sampled PRBS input
sequence shown in Fig. 5. The magnitude spectrum is flat
within the entire identification range, indicating a good
white noise approximation. Due to the zero-order hold
sampling, however, there is additional phase shift as the
Nyquist frequency is approached [7],[8]. Compensating
the transfer function using the input sequence’s spectrum
results in an improved phase estimation as shown in Fig.
11. The phase estimate is well-behaved throughout the
identification range and it converges to the expected value
of -540 degrees at the Nyquist frequency of 50 kHz.
Fig. 11 Simulation results after delayed output voltage sampling,
correcting for the spectrum of the input sequence, and applying a
Gaussian window
Fig. 9 Simulation results showing the effects of delayed output voltage
sampling
TABLE I.
PARAMETERS FOR THE EXPERIMENTAL 300W BUCK
CONVERTER
Parameter Name
Vg
L
C
Lfilt
Cfilt
Rload
Fswitch
Fig. 12 Simulation results showing estimated loop gain, Tloop, and the
transfer functions Gxu and Gyu
5) Loop Gain Measurement
The simulation demonstrating loop gain measurement is
performed on the forward converter without the LC input
filter. A 12-bit, four-period, 1% amplitude PRBS test
signal is injected into the feedback loop via a summing
block at 100 kHz. The signals u, x, and y are exported to
Matlab, where the improved correlation techniques
described above are used to estimate the transfer functions
Gxu and Gyu. As seen in Fig. 12, good matching with
analytic results is obtained. It is important to note that the
estimates are very well matched near the crossover
frequency where stability margins are typically calculated.
B. Experimental Results
To further validate the proposed improvements,
experimental verification was performed on the 300 W
buck converter with undamped L-C input filter shown in
Fig. 13. Parameter values for the converter are given in
Table 1. A Xilinx VirtexII-Pro based Field Programmable
Gate Array (FPGA) was used to control the output voltage
and generate the PRBS test sequence. A LeCroy
WaveRunner 6100A 1GHz DSO was used for data
logging. Data was then exported for post processing in
Matlab. To establish a reference frequency response, an
Agilent 4395A 10Hz-500MHz Network Analyzer was
used. Network analyzer data was then exported to Matlab
for comparison with the improved identification method
described above.
Value [units]
60 [V]
70 [μH]
69 [μF]
200 [μH]
69 [μF]
10 [Ω]
110 [kHz]
1) Base Experimental Results
A 14-bit, four-period, PRBS test signal was injected
into the duty cycle command at 110 kHz, which was
chosen to avoid the intermediate frequency of the network
analyzer at 100kHz. The switching frequency was also
110kHz. The logged data was exported from the DSO
into Matlab, where the proposed improvements were
tested. As a baseline for comparison, the methods
described in [5] were used to estimate the control-tooutput transfer function of the buck converter.
Fig. 14 shows the estimated control-to-output frequency
response and the reference frequency response from the
network analyzer without any of the proposed
improvements. The low-frequency salient features are
well captured in both the magnitude and phase response:
the input filter interaction at 1.3 kHz and the output filter
at 2.3 kHz. The network analyzer measurement also
shows a high frequency zero due to the equivalent series
resistance (ESR) of the output capacitor. Within a decade
of the Nyquist frequency of 55 kHz, both the magnitude
and phase estimations are corrupted. The phase shift due
to the non-ideal input spectrum is also visible from 3 to10
kHz, after which the phase information becomes unusable.
Since the desired closed-loop bandwidth is likely to be in
this frequency range, improved accuracy within this range
would significantly benefit the control designer or control
adaptation mechanism.
2) Windowing and Delayed Sampling
Fig. 15 shows the estimated control-to-output frequency
response and the reference frequency response from the
network analyzer with delayed sampling and the
application of a Gaussian window to the input-to-output
cross correlation. The magnitude and phase responses are
more precise throughout the identification range and very
little information is lost at low frequencies. The phase
response is still inaccurate due to the phase spectrum of
the input perturbation, and it diverges at 30 kHz.
Fig. 13 Buck Converter with undamped LC input filter showing
injection of a test signal into the control duty cycle
Fig. 14 Experimental results showing the converter’s control-to-output
transfer function without any of the proposed improvements
Fig. 17 Adaptive control structure using the nonparametric crosscorrelation technique for system identification
Fig. 15 Experimental results showing the combined effect of delayed
sampling and applying a Gaussian window to the input-to-output crosscorrelation
3) Correcting for the Non-Ideal Input Spectrum
A significant improvement in the phase estimation was
obtained by dividing the control-to-output transfer
function by the spectrum of the input perturbation. The
improvements are shown in Fig. 16. Good phase
matching was obtained up to 30 kHz, including the effect
of the high-frequency zero associated with the ESR of the
output capacitance.
VI. FUTURE WORK
Closed loop converter loop gain has not yet been
verified experimentally due to limitations in the memory
storage space within the FPGA chosen. A larger FPGA
with onboard RAM will be used to log the u, x, and y
signals.
A high-accuracy frequency response monitoring tool
could potentially be used for adaptive control design using
the nonparametric estimation [9],[10]. Future work will
include the design of an adaptation law to dynamically
adjust the digital controller parameters based on changes
in the control-to-output transfer function or the control
loop gain. One possible adaptive control structure is
shown in Fig. 17. This adaptive controller could reduce
the sensitivity of the bandwidth and stability margins to
changes in the source and/or load systems connected to
the converter.
Fig. 16 Experimental results showing the control-to-output transfer
function after delayed output voltage sampling, windowing the crosscorrelation, and correcting for the spectrum of the input sequence.
VII. CONCLUSIONS
Three improvements to the nonparametric crosscorrelation method of system identification were
presented. First, delayed sampling of the output voltage
with respect to the input perturbation was shown to
improve high-frequency accuracy. Second, windowing
the cross-correlation drastically enhances high-frequency
precision. Third, a correction for the input perturbation
spectrum was shown to improve the phase measurement
within a decade of the Nyquist frequency. Additionally, a
technique was introduced which allows measurements of
the feedback loop gain and the closed-loop reference-tooutput transfer function without opening the feedback
loop. These improvements and extensions improve highfrequency system identification accuracy near the desired
closed-loop bandwidth, potentially enabling more robust
control designs using adaptive control techniques.
ACKNOWLEDGMENT
This work was supported by the National Science
Foundation under grant ECS-0348433.
REFERENCES
[1]
Maksimovic, D.; Zane, R.; Erickson, R., "Impact of digital control
in power electronics," Power Semiconductor Devices and ICs,
2004. Proceedings. ISPSD '04. The 16th International Symposium
on , vol., no., pp. 13-22, 24-27 May 2004
[2] R. D. Middlebrook, “Input filter considerations in design and
application of switching regulators,” in Proc. IEEE Industry
Applications Soc. Annu. Meeting, 1976, pp. 366-382.
[3] L. Ljung, System Identification: Theory for the User, 2nd ed.
Englewood Cliffs, NJ: Prentice-Hall, 1999.
[4] Miao, B.; Zane, R.; Maksimovic, D., "System identification of
power converters with digital control through cross-correlation
methods," Power Electronics, IEEE Transactions on , vol.20, no.5,
pp. 1093-1099, Sept. 2005
[5] Miao, B.; Zane, R.; Maksimovic, D., "Practical on-line
identification of power converter dynamic responses," Applied
Power Electronics Conference and Exposition, 2005. APEC 2005.
Twentieth Annual IEEE , vol.1, no., pp. 57-62 Vol. 1, 6-10 March
2005
[6] R. W. Erickson and D. Maksimovic, Fundamentals of Power
Electronics, 2nd ed. Boston, MA: Kluwer, 2001.
[7] S. Mitra, Digital Signal Processing: A Computer-Based Approach,
2nd ed. New York, NY: McGraw-Hill Irwin, 2001.
[8] C. Phillips, H. Nagle, Digital Control System Analysis and
Design, 3rd ed. Upper Sallde River, NJ: Prentice-Hall, 1995.
[9] K.J. Astrom and B. Wittenmark. Adaptive Control. Addison
Wesley, second edition,1995.
[10] G.A. Dumont, M. Huzmezan "Concepts, methods and techniques
in adaptive control" American Control Conference. Proceedings of
the 2002, vol. 2, pp.1137-1150. 2002.