A modified cross-correlation method for system identification of power ...
Aaclten, Germany. 2004
2004 35rk Annual IEEE Power Elecrronics Specialisls Conference
A Modified Cross-Correlation Method for System Identification of Power Converters with Digital Control Botao Miao, R.egan Zane, Dragan Maksimovic Colorado Power Electronics Center ECE Department Univeristy 'of Colorado at Boulder, USA Email: {botaomiao, regamzane, maksimov}@,colorado.edu 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 multi-period Pseudo Random Binary Signal (PRBS) to the control input ofa 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. 1.
INTRODUCTION
Digital 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). 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 ofthe modules, while interactions among modules may cause system instabilities. 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. In general, system identification is divided into parametric and nonparametric methods [ I , 41. 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 Gequency responses. Nonparametric methods include:
0-7803-8399-0/04/$20.00 02004 IEEE.
correlation analysis [ 1,4,5], transient-response analysis [4,6], and frequency response, Fourier, or spectrum analysis [1,2,41. 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 [I]. This method has been applied to empirical, simulation-based small-signal modeling of switching converters [SI. 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 multi-period Pseudo Random Binary Signals (PRBS), 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 11, followed by a simulation example to demonstrate performance of the conventional method using a single period PRBS in Section 111. In Section IV a modification is proposed to improve the performance of identification by multi-period PRBS and a form ofaveraging. Experimental results are then shown in Section V for a 90W 5OV to 15V 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.
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
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2004 35th Annual IEEE Power Elecrronics Specialists Conference
Aochen. Germany. 2004
~
y ( n ) = x h ( k ) u ( n - k ) + v(n)
3
(1)
14 2>::
where f i n ) is the sampled output signal; u(k) is the input digital control signal; h(k) is the discrete-time system impulse response; and v(k) represents disturbances, including switching noise, measurement error, quantization noise, etc. The cross-correlation of the input control signal u(k) and the output signal y(n) is:
,-II =,
where R,,,,(rn)is the auto-correlation of the input signal.
ideal delta function and the cross correlation of the white noise input with disturbances v(k) is ideally zero. Under the conditions of (3), the cross-correlation of (2) can be reduced to
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 R,,dm):
R,(m)
OFT
+Wu).
,* ,b
I_
(C)
Now, if the input control signal u(k) is selected to be white noise, then we benefit from the following characteristics:
In other words, the auto-correlation of the input R , , is an
-
I
- - d. I
-(d)
Fig.] White noise (a) and its aulo correlation(b); single period PRES (c) and its auto correlation(d);
process. The PRBS perturbation signal can be easily generated in a digital system using a shift register, as shown in Fig. 2 for a 9-bit PRBS. An n-bit shift register can generate several different sequences, among which the maximum length sequence has the best properties for this application. The maximum length PRBS can be generated by performing an XOR operation between the i-th bit and a specificj-th bit. For a 9-bit shift register, the XOR operation should be performed between the 1" and the 5thbits, as shown in Fig. 2, resulting in a maximum sequence of 5 11. The following section illustrates application ofthe basic correlation method through a simulation example.
#bit9 8 7 6 5 4 3 2 1
(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. The data length for one period of an n-bit maximum length PRBS is given by A4 = 2" - 1 , and the signal itself has only two possible values:+ e . Figure 1 shows a comparison of samples ofwhite noise (a) and a 9-bit single period PRBS (c) in a digital system. Figures I (b) and (d) show the corresponding auto-correlation 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 non-ideal 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 floor in the PRBS autocorrelation will create errors in our identification
PRBS 1 :+e 0 : -e
Fig.2 9-bit PRBS generated by a 9-bit shift register
111.
SIMULATION EXAMPLE: FORWARD CONVERTER IDENTIFICATION
Figure 3 shows a digitally controlled forward converter with an undamped input filter. The converter parameters are: V,=SOV, Y= I 5 V, C = 3 3 0 WF, L = IO0 pH, and the load current is 6 A. The turns-ratio of the transformer is 1:1:1, The input filter is a simple L-C low-pass filter with L,= 1.9 mH, C,= 66 pF. 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
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Fig.3 Forward converter with input filter and digital controller block diagram
[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 MATLABISimulink envirorunent. A single period maximum length IO-bit PRBS signal (data length is 1023) is injected as a perturbation to the converter digital duty cycle command. 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 e = 0.01. The additional output voltage ripple caused by PRBS perturbation is about k0.6 V, or about ?4% of the DC output voltage. Figure 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 o f 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. It can be observed that the salient features o f 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:MULTI-PERIOD AND AVERAGING APPROACH
Recall from Fig. I(d) that the non-zero noise floor in the auto-correlation of a single-period PRBS was expected to
M
(h) Fig.4 Simulation results of a forward converter with an undamped input filter when input is one period IO-hit PRBS L = I,M = 1023, frequency of PRBS is 100 kHz. (a) cross-Correlation ofthe input and output, and (h) frequency response from correlationmethod (dashed) and ideal averaged model (solid).
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 L times forms an L-period PRBS. If L tends to infinity, it has following properties and frequency spectrum [I]:
=I-$,
else
Equation (6) gives the mean value of an infinite period PRBS, which tends to zero for large M. Interestingly, (7) shows a key result: for an infinite period PRBS, the autocorrelation is given by periodic delta functions with
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(a)
(b)
Fig.6. The auto correlationafa single period IO-bit PRES (a) and a 4-period &bit PRBS (b)
However, there are additional constraints on the selection of M in the context of identification of a digitally controlled switching power converter. The primary consideration in selecting M is based on achieving desired frequency sampling and resolution, as shown in Fig. 5(b). In (loosely defined) comparison to network or spectrum analyzer terms, Fig.5. Auto correlation (a) and frequency spectrum (b) of an the “start” and “stop” frequencies of the effective frequency infinite period PRBS sweep are given byfdM and fd2 (after DFT), respectively, magnitude e* at k equal to zero and multiples ofM, and equal where& is the PRBS frequency. In addition, the equivalent to / M for all the other k‘s, which is also shown in Fig. 5(a). “resolution bandwidth” or spacing between frequency When M is large, e* / M +0,resulting in a periodic sequence samples isfdM. Thus,fo must be sufticiently high to capture of near ideal delta functions in the auto correlation. This is the desired high frequency content, while f d M must be also seen in the frequency domain, as shown in the frequency sufficiently small to capture low frequency content and spectrum of (8). Figure 5(b) shows a plot of (8), where it is achieve the desired frequency resolution. Another way to seen that the frequency content of an infinitely repeating visualize the low frequency requirement is that the sampling PRBS contains delta functions at k f d M for k = l,,..,M-l, window of a single PRES period in the time domain (given where& is the frequency of the PRBS. Thus the infinitely by M )’i must be sufficiently longer than the system settling repeating PRBS can be seen as equivalent to injecting signals time. at M-l discrete frequencies kfdM, resulting in a clear Based on the above constraints, suitablef, and minimum M limitation to the frequency components that can be identified can be selected based on desired frequency sampling, in the power converter. In comparison, injection of white followed by maximum L based on allowed total data length. noise results in a flat line in the frequency domain, or is Also, note that L must be an integer value to maintain the equivalent to signal injection at all frequencies for ideal desired auto-correlation characteristics. The concept of system identification. Thus, for large M, an infinitely trading M for L is demonstrated in Fig. 8, where simulation repeating PRBS injection would result in near ideal results for a 2-period, 9-bit sequence at 5OkHz PRBS can be identification. compared with the single period, IO-bit, IOOkHz PRBS of In practice, due to limitations in memory and computation Fig. 4. The forward converter switching frequency is IOOkHz capability, only finite length data can be used. However, we for both cases. An improved system identification is achieved still see significant improvements in performance through in Fig. 8 (2-period), while Fig.4 (I-period) has a higher use of finite but multi-period PRBS over single-period. This maximum frequency (2x). Both approaches have essentially is partially explained by improvement in the autocorrelation the same total data length. function, as shown in Fig. 6, which compares a single period Up to this point, our discussion has focused on the quality IO-bit PRBS to a 4-period 8-bit PRBS. The total data length is N = L.M. For single period IO-bit PRBS, N = M = 1023. For 4-period 8-bit PRBS, N = L.M = 4x255 = 1020. Thus while the two have essentially the same data length, the 4-period signal has a significantly lower relative noise floor when compared to the single period version. This characteristic is further explored in Fig. 7, which shows the relationship between noise variance and N, M, L. The horizontal axis is the data length N and the vertical axis is the noise variance on a log scale. When N is fixed, smaller M (that is larger L) gives lower noise variance. Thus, if the I lO*l ’ ’ ’ 256612 I024 2048 40% effective “noise floor” in the input auto-correlation were the data hgh N=LM Fig.7 The noise variance vs data length N only consideration, it would be best to use the largest possible withM=(8,9, IO, 11,12)-bit L (multiple periods) for a given allowable data length.
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t -(M-1)
R"y
M-I
0
k
Fig.9 The cross correlation result when input is a multi-period PRBS signal. Note reduced amplitude sidebands due to finite periods in the input PRBS.
HI
Fig.8 Simulation resuk ofa l00kHz fonvard wnverter with an undamped input filter when the input is two period 9-bit PRBS L = 2, M = 511, frequency of PRBS is 50 kHc. Frequency response from wrrelation method (dashed) and ideal averaged model (solid).Compare to Fig.5 for single-period (L=l).
of the PRBS input signal. With the emphasis now on injecting multiple PRBS periods, an additional consideration is how to handle the multi-period data sampled at the switching converter output. As suggested in [I], 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 complexity. This approach achieves a l / L reduction of external noise sources, v(k), from averaging but no additional reduction from the non-ideal behavior of the input PRBS auto-correlation. Alternatively, we propose to perform the cross-correlation operation on the entire set of multi-period 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 ( I ) through (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 reductiori 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 non-ideal components in the PRBS auto-correlation as described below. From (2), we h o w the cross-correlation is equal to the convolution of input sequence auto-correlation and system impulse response. When the input signal is multi-period, the cross-correlation result is a multi-period 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 multi-period correlation result. One is to take the center impulse response directly, because its signal to noise ratio is the best among these impulse responses. The other is to average these impulse responses over 2L. periods. The second option achieves averaging of the correlation results, but due to the reduced amplitude side-bands this would not appear to benefit the result. However, improvement is achieved through the second optiori due to a key property of the PRBS, where the same position points in
each segment of the auto-correlation have following relation [I]: (9) R,,,,(m)+ R,,,(M t m) = -Cler / M , where C is a constant. This shows that by summing each of the resulting impulse responses (rather than taking just the center response) and dividing by L, the noise in the input sequence auto-correlation sums to a small constant noise floor similar to the infinitely repeating case of(7). Based on the above discussions, our proposed procedure for system identification is summarized in the signal flow graph of Fig. IO. First, identification should be performed when the system is operating in steady state. To start, an L-period n-bit PRBS is generated and injected to the control input. At the same time, output of the system is sampled and stored. After the injection and output data collection are finished, the cross-correlation is computed over the entire data sequence. The 2L impulse responses output from the cross-correlation operation are summed, then divided by L. Finally, the DFT is applied to the averaged cross-correlation result to visualize the system frequency response.
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( System operating in steady state )
1 Inject multi-period PRBS
1 Collect output disturbance
(3)
Perform cross correlation to the input and output disturbances
(4)
Average the crms correlation results to derive the impulse reponse
(5)
x System frequency response
Fig. 10 Signal flow graph of the proposed system identification approach in digitally controlled switching power converten.
2004 35th A n n u l lEEE Power Electronics Specialists Conference
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 Ill: the input voltage is V, = 50 V and the output voltage is V = 15 V. The output filter inductor Lis 100 pH, and the output filter capacitor is C = 3 3 0 p F . The converter operates at the nominal load of 6 A. The switching frequency is 5 = 100 kHz. The turns-ratio of the transformer is 1 :I :1. The input filter parameters are L, = 1.9 mH and C, = 66 pF. The digital controller was implemented using a Xilinx Virtex-11 FPGA. The FPGA-based controller includes a IO-bit digital pulse-width modular, a PRBS generator and a data collection unit. The converter output voltage, scaled by a 1O:l resistive voltage divider, is sampled by an A 0 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 ND data collected by the FPGA is transmitted to a PC for off-line processing in the MATLABISimulink 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 3-period 12-bit PRBS was generated by the FPGA and injected to the digital duty cycle command. The total data length is N = 3.(2’’ - 1) = 12285. The PRBS frequency A, equals the switching frequency A, which means that the process of collecting the data takes N/&= 123 ms. A single PRBS sequence lasts M& = 4 I ms, which is sufficiently long to capture the complete impulse response o f the converter. The corresponding frequency resolution is5lM = 24 Hz, which can be compared to the resolution bandwidth setting in a standard analog measurement of converter transfer timctions using a network analyzer. Figure I 1 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.
Aachen, German): 2004
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
.mL.> 10‘
,\ 1 Oa
Hz Fig.1 I Experimental frequency response of 100kHz. 90W forward converter based on the proposed system identification method. Dashed result is based on measured data from the digital identification system with a 3-period, 12-bit PRBS, data length N = 12285, and a PRBS frequency of 100kHz. The solid line is measured by a network analyzer.
method is well suited for implementation in digitally controlled switching power converters. As an example of such anapplication, adigitally controlled 50-to-I5 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 successhl control-to-output response identification. The proposed identification approach can be used for off-line system analysis, digital controller design, and even design validation in the presence of non-idealities 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 o f individual power modules and interactions between modules can he monitored and actively compensated locally to achieve global system stability.
VI. CONCLUSION A modified cross-correlation method for 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
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REFERENCES
L. Ljung, Syslem rdentiJkacorion: theory for the mer, 2d Edition, Prentice- Hall, N.J., 1999. G . F. Franklin, J. D. Powell and M. Workman, Digilol Control of DynomicSyslems, 3* Edition, Addison-Wesley, 1997. R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2“ Edition, Kluwer Academic Publishers, 2001. B. Johansson and M. Lenells, “Possibilities of obtaining small-signal models of E€-lo-D€ power converters by means of system identification,” Telecommunicarions EnergV Conference, INTELEC 2000, pp. 65-75. P. Huynh and B. H. Cho, “Empirical small-signal modeling of switching conveners using PSpice,” in Proc. IEEE PESC‘95, 1995, pp. 801-808. D. Maksimavic, “Computer-aided small-signal analysis based on impulse response of DCDC switching Power Converters,” lEEE Trammlions on Power Elecrronics, Vol. 15, No. 6, Nov. 2000, pp. 1183-1191.
2004 35rk Annual IEEE Power Elecrronics Specialisls Conference
A Modified Cross-Correlation Method for System Identification of Power Converters with Digital Control Botao Miao, R.egan Zane, Dragan Maksimovic Colorado Power Electronics Center ECE Department Univeristy 'of Colorado at Boulder, USA Email: {botaomiao, regamzane, maksimov}@,colorado.edu 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 multi-period Pseudo Random Binary Signal (PRBS) to the control input ofa 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. 1.
INTRODUCTION
Digital 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). 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 ofthe modules, while interactions among modules may cause system instabilities. 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. In general, system identification is divided into parametric and nonparametric methods [ I , 41. 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 Gequency responses. Nonparametric methods include:
0-7803-8399-0/04/$20.00 02004 IEEE.
correlation analysis [ 1,4,5], transient-response analysis [4,6], and frequency response, Fourier, or spectrum analysis [1,2,41. 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 [I]. This method has been applied to empirical, simulation-based small-signal modeling of switching converters [SI. 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 multi-period Pseudo Random Binary Signals (PRBS), 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 11, followed by a simulation example to demonstrate performance of the conventional method using a single period PRBS in Section 111. In Section IV a modification is proposed to improve the performance of identification by multi-period PRBS and a form ofaveraging. Experimental results are then shown in Section V for a 90W 5OV to 15V 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.
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
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~
y ( n ) = x h ( k ) u ( n - k ) + v(n)
3
(1)
14 2>::
where f i n ) is the sampled output signal; u(k) is the input digital control signal; h(k) is the discrete-time system impulse response; and v(k) represents disturbances, including switching noise, measurement error, quantization noise, etc. The cross-correlation of the input control signal u(k) and the output signal y(n) is:
,-II =,
where R,,,,(rn)is the auto-correlation of the input signal.
ideal delta function and the cross correlation of the white noise input with disturbances v(k) is ideally zero. Under the conditions of (3), the cross-correlation of (2) can be reduced to
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 R,,dm):
R,(m)
OFT
+Wu).
,* ,b
I_
(C)
Now, if the input control signal u(k) is selected to be white noise, then we benefit from the following characteristics:
In other words, the auto-correlation of the input R , , is an
-
I
- - d. I
-(d)
Fig.] White noise (a) and its aulo correlation(b); single period PRES (c) and its auto correlation(d);
process. The PRBS perturbation signal can be easily generated in a digital system using a shift register, as shown in Fig. 2 for a 9-bit PRBS. An n-bit shift register can generate several different sequences, among which the maximum length sequence has the best properties for this application. The maximum length PRBS can be generated by performing an XOR operation between the i-th bit and a specificj-th bit. For a 9-bit shift register, the XOR operation should be performed between the 1" and the 5thbits, as shown in Fig. 2, resulting in a maximum sequence of 5 11. The following section illustrates application ofthe basic correlation method through a simulation example.
#bit9 8 7 6 5 4 3 2 1
(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. The data length for one period of an n-bit maximum length PRBS is given by A4 = 2" - 1 , and the signal itself has only two possible values:+ e . Figure 1 shows a comparison of samples ofwhite noise (a) and a 9-bit single period PRBS (c) in a digital system. Figures I (b) and (d) show the corresponding auto-correlation 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 non-ideal 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 floor in the PRBS autocorrelation will create errors in our identification
PRBS 1 :+e 0 : -e
Fig.2 9-bit PRBS generated by a 9-bit shift register
111.
SIMULATION EXAMPLE: FORWARD CONVERTER IDENTIFICATION
Figure 3 shows a digitally controlled forward converter with an undamped input filter. The converter parameters are: V,=SOV, Y= I 5 V, C = 3 3 0 WF, L = IO0 pH, and the load current is 6 A. The turns-ratio of the transformer is 1:1:1, The input filter is a simple L-C low-pass filter with L,= 1.9 mH, C,= 66 pF. 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
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(a)
Fig.3 Forward converter with input filter and digital controller block diagram
[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 MATLABISimulink envirorunent. A single period maximum length IO-bit PRBS signal (data length is 1023) is injected as a perturbation to the converter digital duty cycle command. 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 e = 0.01. The additional output voltage ripple caused by PRBS perturbation is about k0.6 V, or about ?4% of the DC output voltage. Figure 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 o f 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. It can be observed that the salient features o f 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:MULTI-PERIOD AND AVERAGING APPROACH
Recall from Fig. I(d) that the non-zero noise floor in the auto-correlation of a single-period PRBS was expected to
M
(h) Fig.4 Simulation results of a forward converter with an undamped input filter when input is one period IO-hit PRBS L = I,M = 1023, frequency of PRBS is 100 kHz. (a) cross-Correlation ofthe input and output, and (h) frequency response from correlationmethod (dashed) and ideal averaged model (solid).
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 L times forms an L-period PRBS. If L tends to infinity, it has following properties and frequency spectrum [I]:
=I-$,
else
Equation (6) gives the mean value of an infinite period PRBS, which tends to zero for large M. Interestingly, (7) shows a key result: for an infinite period PRBS, the autocorrelation is given by periodic delta functions with
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(a)
(b)
Fig.6. The auto correlationafa single period IO-bit PRES (a) and a 4-period &bit PRBS (b)
However, there are additional constraints on the selection of M in the context of identification of a digitally controlled switching power converter. The primary consideration in selecting M is based on achieving desired frequency sampling and resolution, as shown in Fig. 5(b). In (loosely defined) comparison to network or spectrum analyzer terms, Fig.5. Auto correlation (a) and frequency spectrum (b) of an the “start” and “stop” frequencies of the effective frequency infinite period PRBS sweep are given byfdM and fd2 (after DFT), respectively, magnitude e* at k equal to zero and multiples ofM, and equal where& is the PRBS frequency. In addition, the equivalent to / M for all the other k‘s, which is also shown in Fig. 5(a). “resolution bandwidth” or spacing between frequency When M is large, e* / M +0,resulting in a periodic sequence samples isfdM. Thus,fo must be sufticiently high to capture of near ideal delta functions in the auto correlation. This is the desired high frequency content, while f d M must be also seen in the frequency domain, as shown in the frequency sufficiently small to capture low frequency content and spectrum of (8). Figure 5(b) shows a plot of (8), where it is achieve the desired frequency resolution. Another way to seen that the frequency content of an infinitely repeating visualize the low frequency requirement is that the sampling PRBS contains delta functions at k f d M for k = l,,..,M-l, window of a single PRES period in the time domain (given where& is the frequency of the PRBS. Thus the infinitely by M )’i must be sufficiently longer than the system settling repeating PRBS can be seen as equivalent to injecting signals time. at M-l discrete frequencies kfdM, resulting in a clear Based on the above constraints, suitablef, and minimum M limitation to the frequency components that can be identified can be selected based on desired frequency sampling, in the power converter. In comparison, injection of white followed by maximum L based on allowed total data length. noise results in a flat line in the frequency domain, or is Also, note that L must be an integer value to maintain the equivalent to signal injection at all frequencies for ideal desired auto-correlation characteristics. The concept of system identification. Thus, for large M, an infinitely trading M for L is demonstrated in Fig. 8, where simulation repeating PRBS injection would result in near ideal results for a 2-period, 9-bit sequence at 5OkHz PRBS can be identification. compared with the single period, IO-bit, IOOkHz PRBS of In practice, due to limitations in memory and computation Fig. 4. The forward converter switching frequency is IOOkHz capability, only finite length data can be used. However, we for both cases. An improved system identification is achieved still see significant improvements in performance through in Fig. 8 (2-period), while Fig.4 (I-period) has a higher use of finite but multi-period PRBS over single-period. This maximum frequency (2x). Both approaches have essentially is partially explained by improvement in the autocorrelation the same total data length. function, as shown in Fig. 6, which compares a single period Up to this point, our discussion has focused on the quality IO-bit PRBS to a 4-period 8-bit PRBS. The total data length is N = L.M. For single period IO-bit PRBS, N = M = 1023. For 4-period 8-bit PRBS, N = L.M = 4x255 = 1020. Thus while the two have essentially the same data length, the 4-period signal has a significantly lower relative noise floor when compared to the single period version. This characteristic is further explored in Fig. 7, which shows the relationship between noise variance and N, M, L. The horizontal axis is the data length N and the vertical axis is the noise variance on a log scale. When N is fixed, smaller M (that is larger L) gives lower noise variance. Thus, if the I lO*l ’ ’ ’ 256612 I024 2048 40% effective “noise floor” in the input auto-correlation were the data hgh N=LM Fig.7 The noise variance vs data length N only consideration, it would be best to use the largest possible withM=(8,9, IO, 11,12)-bit L (multiple periods) for a given allowable data length.
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2004 35fh A n n u l IEEE Power Elecrronics Speciolisrs Conference
t -(M-1)
R"y
M-I
0
k
Fig.9 The cross correlation result when input is a multi-period PRBS signal. Note reduced amplitude sidebands due to finite periods in the input PRBS.
HI
Fig.8 Simulation resuk ofa l00kHz fonvard wnverter with an undamped input filter when the input is two period 9-bit PRBS L = 2, M = 511, frequency of PRBS is 50 kHc. Frequency response from wrrelation method (dashed) and ideal averaged model (solid).Compare to Fig.5 for single-period (L=l).
of the PRBS input signal. With the emphasis now on injecting multiple PRBS periods, an additional consideration is how to handle the multi-period data sampled at the switching converter output. As suggested in [I], 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 complexity. This approach achieves a l / L reduction of external noise sources, v(k), from averaging but no additional reduction from the non-ideal behavior of the input PRBS auto-correlation. Alternatively, we propose to perform the cross-correlation operation on the entire set of multi-period 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 ( I ) through (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 reductiori 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 non-ideal components in the PRBS auto-correlation as described below. From (2), we h o w the cross-correlation is equal to the convolution of input sequence auto-correlation and system impulse response. When the input signal is multi-period, the cross-correlation result is a multi-period 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 multi-period correlation result. One is to take the center impulse response directly, because its signal to noise ratio is the best among these impulse responses. The other is to average these impulse responses over 2L. periods. The second option achieves averaging of the correlation results, but due to the reduced amplitude side-bands this would not appear to benefit the result. However, improvement is achieved through the second optiori due to a key property of the PRBS, where the same position points in
each segment of the auto-correlation have following relation [I]: (9) R,,,,(m)+ R,,,(M t m) = -Cler / M , where C is a constant. This shows that by summing each of the resulting impulse responses (rather than taking just the center response) and dividing by L, the noise in the input sequence auto-correlation sums to a small constant noise floor similar to the infinitely repeating case of(7). Based on the above discussions, our proposed procedure for system identification is summarized in the signal flow graph of Fig. IO. First, identification should be performed when the system is operating in steady state. To start, an L-period n-bit PRBS is generated and injected to the control input. At the same time, output of the system is sampled and stored. After the injection and output data collection are finished, the cross-correlation is computed over the entire data sequence. The 2L impulse responses output from the cross-correlation operation are summed, then divided by L. Finally, the DFT is applied to the averaged cross-correlation result to visualize the system frequency response.
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( System operating in steady state )
1 Inject multi-period PRBS
1 Collect output disturbance
(3)
Perform cross correlation to the input and output disturbances
(4)
Average the crms correlation results to derive the impulse reponse
(5)
x System frequency response
Fig. 10 Signal flow graph of the proposed system identification approach in digitally controlled switching power converten.
2004 35th A n n u l lEEE Power Electronics Specialists Conference
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 Ill: the input voltage is V, = 50 V and the output voltage is V = 15 V. The output filter inductor Lis 100 pH, and the output filter capacitor is C = 3 3 0 p F . The converter operates at the nominal load of 6 A. The switching frequency is 5 = 100 kHz. The turns-ratio of the transformer is 1 :I :1. The input filter parameters are L, = 1.9 mH and C, = 66 pF. The digital controller was implemented using a Xilinx Virtex-11 FPGA. The FPGA-based controller includes a IO-bit digital pulse-width modular, a PRBS generator and a data collection unit. The converter output voltage, scaled by a 1O:l resistive voltage divider, is sampled by an A 0 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 ND data collected by the FPGA is transmitted to a PC for off-line processing in the MATLABISimulink 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 3-period 12-bit PRBS was generated by the FPGA and injected to the digital duty cycle command. The total data length is N = 3.(2’’ - 1) = 12285. The PRBS frequency A, equals the switching frequency A, which means that the process of collecting the data takes N/&= 123 ms. A single PRBS sequence lasts M& = 4 I ms, which is sufficiently long to capture the complete impulse response o f the converter. The corresponding frequency resolution is5lM = 24 Hz, which can be compared to the resolution bandwidth setting in a standard analog measurement of converter transfer timctions using a network analyzer. Figure I 1 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.
Aachen, German): 2004
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
.mL.> 10‘
,\ 1 Oa
Hz Fig.1 I Experimental frequency response of 100kHz. 90W forward converter based on the proposed system identification method. Dashed result is based on measured data from the digital identification system with a 3-period, 12-bit PRBS, data length N = 12285, and a PRBS frequency of 100kHz. The solid line is measured by a network analyzer.
method is well suited for implementation in digitally controlled switching power converters. As an example of such anapplication, adigitally controlled 50-to-I5 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 successhl control-to-output response identification. The proposed identification approach can be used for off-line system analysis, digital controller design, and even design validation in the presence of non-idealities 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 o f individual power modules and interactions between modules can he monitored and actively compensated locally to achieve global system stability.
VI. CONCLUSION A modified cross-correlation method for 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
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REFERENCES
L. Ljung, Syslem rdentiJkacorion: theory for the mer, 2d Edition, Prentice- Hall, N.J., 1999. G . F. Franklin, J. D. Powell and M. Workman, Digilol Control of DynomicSyslems, 3* Edition, Addison-Wesley, 1997. R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2“ Edition, Kluwer Academic Publishers, 2001. B. Johansson and M. Lenells, “Possibilities of obtaining small-signal models of E€-lo-D€ power converters by means of system identification,” Telecommunicarions EnergV Conference, INTELEC 2000, pp. 65-75. P. Huynh and B. H. Cho, “Empirical small-signal modeling of switching conveners using PSpice,” in Proc. IEEE PESC‘95, 1995, pp. 801-808. D. Maksimavic, “Computer-aided small-signal analysis based on impulse response of DCDC switching Power Converters,” lEEE Trammlions on Power Elecrronics, Vol. 15, No. 6, Nov. 2000, pp. 1183-1191.
