IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 5, SEPTEMBER 2005
1093
System Identification of Power Converters
With Digital Control Through
Cross-Correlation Methods
Botao Miao, Regan Zane, Member, IEEE, and Dragan Maksimović, Member, IEEE
Abstract—For digitally controlled switching power converters,
on-line system identification can be used to assess the system
dynamic responses and stability margins. This paper presents a
modified correlation method for system identification of power
converters with digital control. By injecting a multiperiod pseudo
random binary signal (PRBS) to the control input of a power
converter, the system frequency response can be derived by
cross-correlation of the input signal and the sensed output signal.
Compared to the conventional cross-correlation method, averaging
the cross-correlation over multiple periods of the injected PRBS
can significantly improve the identification results in the presence
of PRBS-induced artifacts, switching and quantization noises.
An experimental digitally controlled forward converter with an
FPGA-based controller is used to demonstrate accurate and effective identification of the converter control-to-output response.
Index Terms—Digital control, frequency response, parameter
estimation, system identification.
I. INTRODUCTION
D
IGITAL control of high-frequency switching power
converters offers many potential advantages, including
robustness to noise and parameter variations, reduction of
external components, real-time programmability and simple
integration with advanced features such as adaptive calibration
and health monitoring (diagnostics) [1], [2]. In particular, in
power management and distribution (PMAD) systems, which
commonly include multiple power sources, loads, power buses
and converter modules, uncertainties of system parameters may
compromise static and dynamic performance of the modules,
while interactions among modules may cause system instabilities [3]–[9]. Thus, it is desirable to develop intelligent power
modules capable of individually performing on-line local system
identification, communicating the results to central or distributed
controls, and responding with corrective actions. While complex
PMAD systems with stringent robustness and diagnostics requirements are typical for aerospace applications, it is clear that
successful practical system identification and diagnostics could
also have significant impact in design, testing, and deployment
of switching power supplies in a wide range of applications.
Manuscript received May 25, 2004; revised February 2, 2005. This work is
supported by NASA through the Colorado Power Electronics Center. Recommended by Associate Editor B. Lehman.
The authors are with the Colorado Power Electronics Center (CoPEC), Electrical and Computer Engineering Department, University of Colorado, Boulder,
CO 80309-0425 USA (e-mail: botao.miao@colorado.edu; zane@colorado.edu;
maksimov@colorado.edu).
Digital Object Identifier 10.1109/TPEL.2005.854035
In general, system identification is divided into parametric
and nonparametric methods [10], [11]. In parametric methods,
a system model is assumed, and the identification amounts to an
estimation of the model parameters. In nonparametric methods,
no assumption is made about the system model, and the identification is used to directly compute the system frequency responses. Nonparametric methods include: correlation analysis
[10], [12], [13], transient-response analysis [12], [14], and frequency response, Fourier, or spectrum analysis [10]–[12].
This paper focuses on nonparametric identification, with
the objective of accomplishing on-line assessment of system
dynamic responses and stability margins. For switching power
converters with digital control, the requirements for practical
system identification include the following:
a) signal injection should not disturb normal system operation in terms of static and dynamic voltage regulation;
b) the identification should be immune to switching and
quantization noise;
c) memory and processing requirements should be relatively
low.
With these requirements in mind, we concentrate on the
cross-correlation analysis method [10], [14]–[16]. This method
has been applied to empirical, simulation-based small-signal
modeling of switching converters [13]. In this paper, we present
a modified cross-correlation approach for system identification together with experimental results from an FPGA-based
digital controller realization. Modified cross-correlation is
achieved by first injecting multiperiod pseudo random binary signals (PRBS), meaning that a single period PRBS is
repeated identically a finite number of times, then averaging
the cross-correlation of the input and the output over several
PRBS periods. This approach rejects noise sources and results
in accurate system identification.
The paper begins with a review of the basic correlation
method in Section II, followed by a simulation example to
demonstrate performance of the conventional method using a
single period PRBS in Section III. In Section IV a modification is proposed to improve the performance of identification
by multiperiod PRBS and a form of averaging. Experimental
results are then shown in Section V for a 90 W 50 V-to-15 V forward converter with an undamped input filter. An FPGA-based
digital controller is used to demonstrate the performance of the
proposed identification method on the experimental forward
converter.
0885-8993/$20.00 © 2005 IEEE
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 5, SEPTEMBER 2005
II. CROSS-CORRELATION METHOD
Here we review and study application of the cross-correlation
method to digitally sampled and controlled switching power converters. In steady state, for small-signal disturbances, a power
converter can be regarded as a linear time-invariant discrete-time
system, where the sampled system can be described by
(1)
where
is the sampled output signal;
is the input digis the discrete-time system impulse reital control signal;
represents disturbances, including switching
sponse; and
noise, measurement error, quantization noise, etc.
and the
The cross-correlation of the input control signal
is
output signal
(2)
Fig. 1. (a) White noise and (b) its auto correlation. (c) Single period PRBS
and (d) its auto correlation. Sampling frequency is 100 kHz.
is the auto-correlation of the input signal. Now,
where
is selected to be white noise, then
if the input control signal
we benefit from the following characteristics:
(3)
In other words, the auto-correlation of the input
is an ideal
delta function and the cross-correlation of the white noise input
is ideally zero. Under the conditions of
with disturbances
(3), the cross-correlation of (2) can be reduced to
(4)
Thus the cross-correlation of the input and output sampled
signals give the discrete time system impulse response. The control-to-output transfer function of the target power converter in
frequency domain can then be derived by applying the discrete
Fourier transform (DFT) to
(5)
This theoretical result requires the ability to generate white
noise as an input perturbation to the system. A simple compromise in a digitally controlled power converter is to approximate
white noise through use of PRBS perturbations. The PRBS can
be easily generated but is periodic and deterministic [10], [15],
[16]. The data length for one period of an -bit maximum length
PRBS is given by
, and the signal itself has only
.
two possible values:
Fig. 1 shows a comparison of samples of white noise (a) and
a 9-b single period PRBS (c) (only part signal is shown) in a
Fig. 2.
9-b PRBS generated by a 9-b shift register.
digital system. Fig. 1(b) and (d) show the corresponding autocorrelation functions, respectively. We can see that the auto-correlation of a single period PRBS is very close to a delta function,
but now with a nonideal component (or noise) around it. Recall
from (2) that the cross-correlation between the input and output
can be seen as time convolution between the autocorrelation of
the input (ideally a delta function) and the system impulse response. The additional noise in the PRBS autocorrelation will
create errors in our identification process.
The PRBS perturbation signal can be easily generated in a
digital system using a shift register with feedback [10], [13], as
shown in Fig. 2 for a 9-b PRBS. An n-bit feedback shift register
can generate several different sequences, among which the maximum length sequence has the best properties (optimal noiselike characteristic) for this application. The maximum length
PRBS can be generated by performing an XOR operation between the i-th bit and a specific j-th bit. For a 9-b shift register, the XOR operation should be performed between the first
and the fifth bits, as shown in Fig. 2. The output generated by
a one-bit right shift operation produces a maximum length sequence of 511 before repetition occurs. In Section IV, the properties of the maximum length sequence PRBS is described. The
following section illustrates application of the basic correlation
method through a simulation example.
MIAO et al.: SYSTEM IDENTIFICATION OF POWER CONVERTERS
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Fig. 3. Forward converter with input filter and digital controller block diagram.
III. SIMULATION EXAMPLE: FORWARD
CONVERTER IDENTIFICATION
Fig. 3 shows a digitally controlled forward converter with an
50 V,
undamped input filter. The converter parameters are:
15 V,
330 F,
100 H, and the load current is
6 A. The turns-ratio of the transformer is 1:1:1. The input filter is
low-pass filter with
1.9 mH,
66 F.
a simple
The switching frequency, the sampling frequency and the PRBS
frequency are all 100 kHz.
Note that the input filter is not properly damped. Therefore,
the converter control-to-output response exhibits a fourth-order
response with a pair of right-half plane zeros [3]. This example
is chosen to represent a situation where a fault in a power
distribution system on the input side of the converter may
cause system instabilities. It is also an example where both
low-frequency and high-frequency dynamics of the converter
are of interest, and the system identification problem is more
challenging.
The converter model and the identification functions are
implemented in the MATLAB/Simulink environment. A single
period maximum length 10-b PRBS signal (data length is 1023)
is injected as a perturbation to the converter digital duty cycle
command. In this paper, denotes the perturbation signal. The
steady-state duty cycle is 0.3. The magnitude of the PRBS
signal should be small enough in order not to disturb normal
system operation. In this simulation, the PRBS magnitude is
a duty cycle perturbation of
0.01. The additional output
voltage ripple caused by PRBS perturbation is about 0.6 V,
or about
of the dc output voltage. Fig. 4 shows the
simulation results: a) the cross-correlation of the input and
output signals and b) the frequency responses obtained by DFT
of the cross-correlation data in a). The solid curves represent
the magnitude and phase responses of the control to output
transfer function obtained for the converter ideal averaged
model (excluding losses) [3], while the dashed curves represent
the responses obtained by the basic cross-correlation method.
Fig. 4. Simulation results of a forward converter with an undamped input filter
when input is one period 10-b PRBS L 1; M 1023, frequency of PRBS is
100 kHz. (a) cross-correlation of the input and output and (b) frequency response
from correlation method (dashed) and ideal averaged model (solid).
=
=
It can be observed that the salient features of the converter
responses are well identified by the method. However, the
high frequency responses obtained by the identification method
are significantly corrupted by noise. In the next section, we
discuss selection of the identification parameters as well as
modifications to the basic method aimed at reducing the effects
of noise.
IV. MODIFIED CORRELATION METHOD: MULTIPERIOD AND
AVERAGING APPROACH
Recall from Fig. 1(d) that the nonzero noise in the auto-correlation of a single-period PRBS was expected to result in errors
in the calculated system impulse response, which can now be
seen in Fig. 4. In this section, we develop options for improving
the identification results through processing multiple periods of
the PRBS sequence.
Consider first the properties of an infinite period PRBS. A
maximum length PRBS repeated times forms an -period
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 5, SEPTEMBER 2005
Fig. 6. Auto correlation of (a) a single period 10-b PRBS and (b) a four-period
8-b PRBS.
Fig. 5. (a) Auto correlation and (b) frequency spectrum of an infinite period
PRBS.
PRBS. If tends to infinity, it has the following properties and
frequency spectrum [10]:
(6)
(7)
(8)
Equation (6) gives the mean value of an infinite period PRBS,
which tends to zero for large . Interestingly, (7) shows a key
result: for an infinite period PRBS, the autocorrelation is given
by periodic delta functions with magnitude at equal to zero
for all the other ’s,
and multiples of , and equal to
is large,
,
which is also shown in Fig. 5(a). When
resulting in a periodic sequence of near ideal delta functions in
the auto correlation. This is also seen in the frequency domain,
as shown in the frequency spectrum of (8). Fig. 5(b) shows a
plot of (8), where it is seen that the frequency content of an infor
finitely repeating PRBS contains delta functions at
, where
is the frequency of the PRBS.
Thus the infinitely repeating PRBS can be seen as equivalent
discrete frequencies
, reto injecting signals at
sulting in a clear limitation to the frequency components that can
be identified in the power converter. In comparison, injection of
white noise results in a flat line in the frequency domain, or is
equivalent to signal injection at all frequencies for ideal system
identification. Thus, for large , an infinitely repeating PRBS
injection would result in near ideal identification.
In practice, due to limitations in memory and computation capability, only finite length data can be used. However, we still
see significant improvements in performance through use of finite but multiperiod PRBS over single-period. This is partially
Fig. 7. Noise variance versus data length
and 12-b.
N with M = 8-, 9-, 10-, 11-,
explained by improvement in the autocorrelation function, as
shown in Fig. 6, which compares a single period 10-b PRBS to
. For
a four-period 8-b PRBS. The total data length is
1023. For four-period 8-b
single period 10-b PRBS,
4 255 1020. Thus while the two have
PRBS,
essentially the same data length, the four-period signal has a significantly lower relative noise magnitude when compared to the
single period version. This characteristic is further explored in
Fig. 7, which shows the relationship between noise variance and
. The horizontal axis is the data length and the vertical axis is the noise variance on a log scale. When is fixed,
(that is larger ) gives lower noise variance. Thus, if
smaller
the effective noise in the input PRBS auto-correlation were the
only consideration, it would be best to use the largest possible
(multiple periods) for a given allowable data length.
However, there are additional constraints on the selection
in the context of identification of a digitally controlled
of
switching power converter. The primary consideration in seis based on achieving desired frequency sampling
lecting
and resolution, as shown in Fig. 5(b). In (loosely defined)
comparison to network or spectrum analyzer terms, the “start”
and “stop” frequencies of the effective frequency sweep are
and
(after DFT), respectively, where
given by
is the PRBS frequency. In addition, the equivalent “resolution
.
bandwidth” or spacing between frequency samples is
must be sufficiently high to capture the desired high
Thus,
frequency content, while
must be sufficiently small to
MIAO et al.: SYSTEM IDENTIFICATION OF POWER CONVERTERS
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Fig. 9. Cross-correlation result when input is a multiperiod PRBS signal. Note
reduced amplitude side-bands due to finite periods in the input PRBS.
Fig. 8. Simulation results of a 100 kHz forward converter with an undamped
input filter when the input is two period 9-b PRBS L 2; M =511, frequency
of PRBS is 50 kHz. Frequency response from correlation method (dashed) and
ideal averaged model (solid). Compare to Fig. 4 for single-period (L = 1).
=
capture low frequency content and achieve the desired frequency resolution. Another way to visualize the low frequency
requirement is that the sampling window of a single PRBS
) must be sufficiently
period in the time domain (given by
longer than the system settling time.
Based on the above constraints, suitable and minimum
can be selected based on desired frequency sampling, followed
by maximum
based on allowed total data length. Also,
note that must be an integer value to maintain the desired
for
auto-correlation characteristics. The concept of trading
is demonstrated in Fig. 8, where simulation results for a twoperiod, nine-b sequence at 50 kHz PRBS can be compared
with the single period, 10-b, 100 kHz PRBS of Fig. 4. The
forward converter switching frequency is 100 kHz for both
cases. An improved system identification is achieved in Fig. 8
(two-period), while Fig. 4 (one-period) has a higher maximum
frequency (2x). Both approaches have essentially the same total
data length.
Up to this point, our discussion has focused on the quality
of the PRBS input signal. With the emphasis now on injecting
multiple PRBS periods, an additional consideration is how to
handle the multiperiod data sampled at the switching converter
output. As suggested in [1], one possibility is to average the sampled output voltages and work only with one period of input and
output data in the cross-correlation for reduced computational
reduction of external
complexity. This approach achieves a
, from averaging but no additional reduction
noise sources,
from the nonideal behavior of the input PRBS auto-correlation.
Alternatively, we propose to perform the cross-correlation
operation on the entire set of multiperiod input and output data,
followed by an effective averaging of the result to estimate the
system impulse response. To estimate the effect on external
noise sources, consider again (1)–(4), where for white noise
input the cross-correlation operation eliminates uncorrelated
noise. Based on the above selection criteria, we achieve a close
approximation to white noise and expect a significant improvement in noise reduction through cross-correlation as compared
to straight output data averaging. Additionally, depending on
how we deal with the output of the cross-correlation data, it is
possible to achieve further cancellation of nonideal components
in the PRBS auto-correlation as described below.
From (2), we know the cross-correlation is equal to the
convolution of input sequence auto-correlation and system
impulse response. When the input signal is multiperiod, the
cross-correlation result is a multiperiod impulse response, as
shown in Fig. 9. The reduced amplitude side-bands are due to
the finite periods in the input sequence. There are two options to
deal with the multiperiod correlation result. One is to truncate
impulse
the sidebands and take only the center of the
responses as our result based on the observation that its signal
to noise ratio is the best among the impulse responses. The
other is to average all of the resulting impulse responses over
periods. Intuitively, it appears that the would reduce signal
integrity due to the reduced sideband amplitudes. However, we
have observed improvement in the identification result using
the averaging approach over use of the center impulse only,
and we use the averaging approach in our experimental results
presented in Section V. The improvement through averaging
can be understood qualitatively by recognizing a key property
of the PRBS, where the same position points in each segment
of the auto-correlation have following relation [1]:
(9)
where
is a constant and the right hand term in (9) is very
small when compared to the noise levels seen in Fig. 6 for use
of finite PRBS periods. Now recall that the cross-correlation between the input and output is equivalent to convolution of the
auto-correlation and the system impulse response. If the system
had an ideal delta function as its impulse response, the output of
the cross-correlation would be the auto-correlation of the PRBS.
In this case, averaging of the output responses would result in
a cancellation of the noise created by the PRBS according to
(9) and we would have near perfect identification similar to use
of ideal white noise. With a real system impulse response, the
convolution operation adds in additional noise points that are not
cancelled according to (9) when averaging the output responses.
However, by design, the system impulse response has a settling
time significantly less than the sample time of the PRBS, and a
portion of the PRBS noise is cancelled according to (9). The extent of noise cancellation is dependant on the system response,
PRBS, and sampling time.
Based on the above discussions, our proposed procedure for
system identification is summarized in the signal flow graph
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 5, SEPTEMBER 2005
Fig. 10. Signal flow graph of the proposed system identification approach in
digitally controlled switching power converters.
of Fig. 10. First, identification should be performed when the
system is operating in steady state. To start, an -period n-bit
PRBS is generated and injected to the control input. At the
same time, the output of the system is sampled and stored. After
the injection and output data collection are finished, the crosscorrelation is computed over the entire data sequence. The
impulse responses output from the cross-correlation operation
are summed, and then divided by . Finally, the DFT is applied
to the averaged cross-correlation result to visualize the system
frequency response.
V. EXPERIMENTAL VERIFICATION
The digitally controlled forward converter of Fig. 3 was constructed and used to experimentally verify the proposed system
identification method. The converter parameters are the same
as in the simulation example of Section III: the input voltage
50 V and the output voltage is
15 V. The output
is
100 H, and the output filter capacitor is
filter inductor is
330 F. The converter operates at the nominal load of 6 A.
100 kHz. The turns-ratio of
The switching frequency is
the transformer is 1:1:1. The input filter parameters are
1.9 mH and
66 F.
The digital controller was implemented using a Xilinx
Virtex-II FPGA. The FPGA-based controller includes a 10-b
digital pulse-width modular, a PRBS generator and a data
collection unit. The converter output voltage, scaled by a
10:1 resistive voltage divider, is sampled by an A/D converter
(TI-THS1230). The sampling rate equals the switching frequency. Although the FPGA also includes a discrete-time
compensator to implement closed-loop output voltage regulation, in the experiments reported in this section the converter
is operated open loop. The PRBS and A/D data collected by
the FPGA is transmitted to a PC for off-line processing in the
MATLAB/Simulink environment.
The modified cross-correlation method described in Section IV is used to identify the control to output transfer
function of the experimental forward converter. A three-period 12-b PRBS was generated by the FPGA and injected
to the digital duty cycle command. The total data length is
Fig. 11. Experimental frequency response of 100 kHz, 90 W forward converter
based on the proposed system identification method. Dashed result is based on
measured data from the digital identification system with a three-period, 12-b
12 285, and a PRBS frequency of 100 kHz. The solid
PRBS, data length
line is measured by a network analyzer.
N=
. The PRBS frequency
equals
the switching frequency , which means that the process
of collecting the data takes
123 ms. A single PRBS
sequence lasts
41 ms, which is sufficiently long to
capture the complete impulse response of the converter. The
corresponding frequency resolution is
24 Hz, which
can be compared to the resolution bandwidth setting in a standard analog measurement of converter transfer functions using
a network analyzer.
Fig. 11 compares the magnitude and phase responses obtained by the modified correlation method (dotted line) and by
the network analyzer measurement (solid line) under the same
operating conditions. It can be observed that the matching between the responses is quite good in a wide range of frequencies.
The results obtained by the identification method show a relatively low level of noise even at high frequencies.
VI. CONCLUSION
A modified cross-correlation method for experimental system
identification is presented for switching power converters with
digital control. Multiple periods of a pseudo random binary
signal (PRBS) are injected to a control input (such as the duty
cycle) of a power converter, and the output is sampled over
multiple PRBS periods. The computed cross-correlation is averaged over multiple periods to get the system impulse response,
which is then used to compute the system frequency response.
Simulations and experimental results show that the proposed
method can give reliable identification results in the presence
of PRBS artifacts, switching and quantization noise (in digital
systems). The method is well suited for implementation in
digitally controlled switching power converters. As an example
of such an application, a digitally controlled 50-to-15 V forward converter operating at 100 kHz is constructed and the
identification method is demonstrated using an FPGA-based
digital controller. The experimental results show successful
control-to-output response identification.
MIAO et al.: SYSTEM IDENTIFICATION OF POWER CONVERTERS
The proposed identification approach can be used for off-line
system analysis, digital controller design, and even design validation in the presence of nonidealities such as losses, delays
and switching and quantization noise. In addition, the concepts
can be extended for use in on-line applications, such as PMAD
systems where static and dynamic performance of individual
power modules and interactions between modules can be monitored and actively compensated locally to achieve global system
stability.
1099
Botao Miao received the B.S., M.S., and Ph.D.
degrees in electrical engineering from Tsinghua
University, Beijing, China, in 1998, 2001, 2003,
respectively.
He is now doing his Post-doctoral work at the Colorado Power Electronics Center (CoPEC), University
of Colorado at Boulder. His current research work is
the digital control of power converters.
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Regan Zane (M’99) received the B.S., M.S., and
Ph.D. degrees in electrical engineering from the
University of Colorado at Boulder in 1996, 1998,
and 1999, respectively.
From 1999 to 2001, he was a Research Engineer
with General Electric’s Global Research Center,
Niskayuna, NY, where he developed mixed-signal
IC controllers for electronic ballasts. Since 2001,
he has been with the Department of Electrical and
Computer Engineering, University of Colorado,
where he is currently an Assistant Professor and
faculty member of the Colorado Power Electronics Center (CoPEC). His
research interests include mixed-signal IC design, modeling, analysis and
control of switching power converters, and power generation and management
in low-power wireless systems.
Dr. Zane received the 2004 National Science Foundation (NSF) CAREER
Award.
Dragan Maksimović (M’89) received the B.S. and
M.S. degrees in electrical engineering from the University of Belgrade, Belgrade, Yugoslavia, in 1984
and 1986, respectively, and the Ph.D. degree from
the California Institute of Technology, Pasadena, in
1989.
From 1989 to 1992, he was with the University of
Belgrade. Since 1992 he has been with the Department of Electrical and Computer Engineering, University of Colorado at Boulder, where he is currently
an Associate Professor and Co-Director of the Colorado Power Electronics Center (CoPEC). His current research interests include
power electronics for low-power portable systems, digital control techniques,
and mixed-signal IC design for power-management applications.
Dr. Maksimović received the 1997 National Science Foundation (NSF) CAREER Award, and a Power Electronics Society Transactions Prize Paper Award.