Improved Online Identification of Switching Converters Using Digital ...
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.
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]
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