Design and Implementation of an Adaptive Tuning System Based on... Phase Margin for Digitally Controlled DC–DC Converters

Design and Implementation of an Adaptive Tuning System Based on... Phase Margin for Digitally Controlled DC–DC Converters
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 2, FEBRUARY 2009
559
Design and Implementation of an Adaptive Tuning System Based on Desired
Phase Margin for Digitally Controlled DC–DC Converters
Jeffrey Morroni, Student Member, IEEE, Regan Zane, Senior Member, IEEE,
and Dragan Maksimović, Senior Member, IEEE
Abstract—This letter presents an online adaptive tuning technique for digitally controlled switched-mode power supplies
(SMPS). The approach is based on continuous monitoring of the
system crossover frequency and phase margin, followed by a multiinput–multi-output (MIMO) control loop that continuously and
concurrently tunes the compensator parameters to meet crossover
frequency and phase margin targets. Continuous stability margin
monitoring is achieved by injecting a small digital square-wave
signal between the digital compensator and the digital pulsewidth
modulator. The MIMO loop adaptively adjusts the compensator
parameters to minimize the error between the desired and measured crossover frequency and phase margin. Small-signal models
are derived, and the MIMO control loop is designed to achieve
stability and performance over a wide range of operating conditions. Using modest hardware resources, the proposed approach
enables adaptive tuning during normal SMPS operation. Experimental results demonstrating system functionality are presented
for a synchronous buck SMPS.
Index Terms—Adaptive control, dc–dc power conversion, digital
control.
I. INTRODUCTION
WITCHED-MODE power supply (SMPS) feedback loops
are typically designed conservatively so that closed-loop
regulation and stability margins are maintained over expected
ranges of operating conditions and tolerances in power stage
parameters. Typical designs often lead to degraded closed-loop
performance or loss of stability in the event of significant operating point changes associated with component degradation,
input voltage variations, etc. These types of power stage parameter changes are mitigated by offline controller redesign
to maintain desired dynamic performance requirements. With
the increased feasibility of practical digital control in switching power converters [1], new opportunities exist to incorporate
intelligent control algorithms into the system to improve dynamic responses and reliability over a wider range of possible
operating points.
Recent work in the area of digital control of dc–dc power converters has shown that autotuning algorithms can be completely
integrated into the digital controller with relatively small additional hardware requirements [2]–[8]. In particular, approaches
S
Manuscript received February 16, 2008. Current version published February
6, 2009. Recommended for publication by Associate Editor J. Sun.
The authors are with Colorado Power Electronics Center (CoPEC), Department of Electrical and Computer Engineering, University of Colorado,
Boulder, CO 80309 USA (e-mail: [email protected]; [email protected];
[email protected]).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2008.2007641
to onetime compensator autotuning based on frequency response
data, gathered online, have been proposed in [2]–[8]. However,
the identification of frequency-response information requires
more significant signal processing [2]–[4], or several tuning
steps [5]–[7], and assumes undisturbed, steady-state operation.
The tuning approach proposed in [8], which is capable of operation during load transients, also proceeds in two predefined
steps. For these reasons, it is more difficult to apply [2]–[8]
directly to continuous parameter tuning.
The goal of this letter is to present an approach, based on
a large body of work in adaptive control theory [9]–[12], to
adaptive tuning of digital SMPS controller parameters during
normal closed-loop operation of the converter. The proposed
approach is similar in objectives to the work presented in [13],
but the method in which compensator tuning is performed is
very different. The proposed approach is based on continuous
monitoring of the system crossover frequency and phase margin
[14], and a multi-input–multi-output (MIMO) control loop that
adaptively tunes the compensator parameters to meet crossover
frequency and phase margin targets. Similar to [8], a digital
signal injection is introduced while the converter operates in
closed loop. However, in contrast to [8], this approach tunes all
parameters of the compensator continuously and concurrently.
Further, the adaptive tuning causes a very small output voltage
perturbation thus allowing tuning without disturbing normal
converter operation. Section II describes the proposed approach
for adaptive tuning. Section III presents a numerical design
procedure for the MIMO control loop. Experimental results are
presented in Section IV using a synchronous buck converter
power stage. Conclusions are presented in Section V.
II. ADAPTIVE TUNING CONTROL SYSTEM
A system block diagram for the proposed tuning approach
is shown in Fig. 1. The digital controller consists of a voltage
A/D converter (ADC), a discrete-time PID compensator, and
a digital pulsewidth modulator (DPWM). There are two main
components to the adaptive tuning system: a stability margin
monitor [14] and a MIMO control loop. The stability margin
monitor is a digital implementation of the analog loop gain
measurement technique using signal injection, as first described
by Middlebrook [15]. The monitor is connected between the
output of the PID compensator and the DPWM input. A digital
square-wave signal Vz , with frequency equal to the frequency
command finj , is injected in the closed-loop system during normal operation. The magnitude of the injection signal Vz is chosen such that only ±1 LSB at the ADC output is triggered, which
is the smallest detectable perturbation. Signals Vx and Vy are
0885-8993/$25.00 © 2009 IEEE
Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on September 1, 2009 at 16:22 from IEEE Xplore. Restrictions apply.
560
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 2, FEBRUARY 2009
Fig. 1. Crossover frequency and phase margin monitor and adaptive tuning control loop block diagram. The outputs of the MIMO control loop are the PID
compensator zero locations and gain. G(z) is the transfer function from compensator inputs K , Z 1 , and Z 2 to stability margin detector outputs.
processed by bandpass filters tuned to finj to remove unwanted
frequency components, and then compared in magnitude using
peak detectors. A simple integral feedback controller adjusts the
injection frequency finj so that ||Vy (finj )|| = ||Vx (finj )||. As a
result, finj represents the system crossover frequency fc
T (finj ) =
Vy (finj )
=1
Vx (finj )
(1)
Simultaneously, a phase detector measures the phase difference ϕ between Vy (finj ) and Vx (finj )
ϕ = Vy (finj ) − Vx (finj )
(2)
to obtain the system phase margin.
The outputs of the stability margin detector, ϕm = ϕ and
fc = finj , are compared to the desired reference values ϕm ref
and fc ref , respectively. These error signals are used to adaptively tune the parameters of the compensator Gc (z) in order to
achieve the target specifications fc = fc ref and ϕm = ϕm ref .
The proposed approach can be applied to adaptive tuning of
various types of compensator structures; however in this paper, a generic PID compensator example is considered with the
following transfer function:
Gc (z) = K
(z − Z1 ) (z − Z2 )
.
z (z − 1)
(3)
As shown in Fig. 1, the adaptive tuning system is driven by
the errors between the desired crossover frequency and phase
margin and the values measured by the stability margin detector.
The error signals, fc error and ϕm error , are inputs to a matrix
of transfer functions used to determine the compensator zero
locations, Z1 and Z2 , and the gain K such that zero error with
respect to the target specifications is achieved. The proposed
adaptive tuning system relies on the assumption that the feedback loop behaves as a linear, time-invariant system. This should
be taken into account when choosing crossover frequency and
phase margin targets for the adaptive tuner.
The matrix S(z) represents the transfer functions from fc error
and ϕm error to K, Z1 , and Z2

z
A1
K̂
 z−1

Ẑ1  A3 z
 = z − 1


Ẑ2
z
A5
z−1


z 
z − 1 
z 
fˆc error
 fˆc error
A4
= S(z)
.
z − 1
 ϕ̂
ϕ̂m error
z  m error
A6
z−1
(4)
A2
A1 –A6 are designed to achieve stability and desired performance, based on the closed-loop small-signal transfer matrix.
To model the closed-loop system, a transfer matrix from inputs K, Z1 , and Z2 to outputs finj and ϕ must be determined
Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on September 1, 2009 at 16:22 from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 2, FEBRUARY 2009
561
first. In particular, the transfer matrix G(z) of interest can be
written as
 
 
K̂
K̂
Gf i n j −Z 1 Gf i n j −Z 2  
 
Ẑ1  = G(z)Ẑ1 .
Gϕ−Z 1
Gϕ−Z 2
Ẑ2
Ẑ2
(5)
Together, (4) and (5) make up a system of loop gains associated with each loop in the MIMO system (6), as shown at the
bottom of this page.
In (6), the diagonal elements represent the direct-path loop
gains, while the off-diagonal paths represent coupling loop gains
(i.e., from fc → ϕm or ϕm → fc ). Based on (6), each loop gain
entry can be designed, by choosing A1 –A6 , such that the closedloop transfer matrix is stable and well behaved. The closed-loop
transfer matrix can be written, in terms of (6), as
Gf i n j −K
fˆinj
=
Gϕ−K
ϕ̂
fˆc
ϕ̂m
−1
= (I + L(z))
L(z)
fˆc
ϕ̂m
ref
.
(7)
ref
Using (7), the stability of the MIMO control loop can be
directly determined based on the location of the closed-loop
poles [16]. In particular, the closed-loop system described by
(7) is exponentially stable if it is proper and has no poles outside
the unit circle [16].
III. NUMERICAL DESIGN OF THE ADAPTIVE TUNING
CONTROL LOOP
The experimental testbed, from which the numerical design is
performed, is a synchronous buck converter, as shown in Fig. 1,
with a digital feedback loop realized using a Xilinx VirtexIV field-programmable gate array (FPGA). The nominal power
stage parameters in Fig. 1 are L = 4.1 µH, C = 377 µF, R =
2 Ω, Vg = 12 V, Vout = 5 V, and fs = 100 kHz. The ADC is a
TI-THS1030 sampled once per switching cycle with an effective
LSB resolution of 20 mV, or 0.4% of the dc output voltage. The
transistors making up the synchronous rectifier in the power
stage are IRFR024 N power MOSFETs.
In order to design (7) for desired performance and stability, the indices of (5) must be determined first. In this letter,
the modeling is performed based on a design decision to make
the stability margin monitor control loop much faster than the
adaptive tuning control loop. This allows the dynamics associated with the stability margin monitor loop to be neglected with
respect to the adaptive tuning control loop. The waveform illus-
Fig. 2. Effect of a step in K on the theoretical (dashed) and simulated (solid)
models of crossover frequency monitoring.
trating the proposed modeling approach is given in Fig. 2, where
as an example the effect of a perturbation in compensator gain is
considered. The perturbation occurs at time (n−1) causing the
stability margin monitor to update the new monitored crossover
frequency very quickly with respect to the next sample of the
MIMO control loop fc [n]. The small-signal transfer function
from fˆc [n] to k̂[n − 1] is then just a one sample delay td and
a gain scale factor. From a practical point of view, computing
analytical expressions for the gain factors require analytically
solving for the crossover frequency and phase margin as functions of the compensator coefficients. With a second-order plant
model and a second-order compensator (PID), the analytical
solution becomes very complicated yielding little intuitive insight. Therefore, in this letter, a numerical computation of the
desired gains is performed. Also included in Fig. 2 is a simulation of the dynamics of the monitoring control loop after a
perturbation in compensator gain. The simulation, performed in
Simulink, shows that it is possible to design the monitoring control loop dynamics to be much faster than those of the adaptive
tuning loop so that the aforementioned modeling approach is
valid.
In hardware, the sampling rates of both loops are set relative to
the injection frequency (crossover frequency) with the stability
monitoring loop having considerably faster sampling. By doing
so, as the injection frequency changes, the sample rates of each
loop will scale in proportion to each other ensuring that the
stability monitoring loop is much faster than the adaptive tuning
loop despite the injection frequency. In the experimental system,
the sampling rate of the stability margin monitor control loop
is set to 16 times slower than the injection frequency, while the
adaptive tuning loop sampling rate is set 64 times slower than
the injection frequency.
Based on the power stage defined before, a nominal PID
compensator can be designed for the output voltage feedback
loop, using small-signal analysis, to yield slow but guaranteed
stable performance. It is assumed that sufficient information
about the nominal power stage (i.e., at system startup) is known
 z A1 Gf i n j −K + A3 Gf i n j −Z 1 + A5 Gf i n j −Z 2

z−1
L(z) = G(z)S(z) = 
 z (A G
1 ϕ−K + A3 Gϕ−Z 1 + A5 Gϕ−Z 2 )
z−1

z A2 Gf i n j −K + A4 Gf i n j −Z 1 + A6 Gf i n j −Z 2

z−1
.
z (A2 Gϕ−K + A4 Gϕ−Z 1 + A6 Gϕ−Z 2 ) 
z−1
Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on September 1, 2009 at 16:22 from IEEE Xplore. Restrictions apply.
(6)
562
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 2, FEBRUARY 2009
TABLE I
RANGE OF ALLOWED POWER STAGE VARIATIONS WITHOUT CHANGING ANY
FEEDBACK LOOP SIGNS
Fig. 3. (a) Lines of constant crossover frequency as functions of compensator
gain and zero location. (b) Lines of constant phase margin as functions of
compensator gain and zero locations. In (a), the dashed line indicates where the
feedback loop changes sign.
such that a conservative compensator design can be performed.
Given the experimental system, the following compensator was
used for system initialization:
Gc (z) = 1.0
(z − 0.90) (z − 0.80)
z (z − 1)
(8)
which yields a system crossover frequency fc = 6.2 kHz and
phase margin ϕm = 65◦ . Now, using the defined power stage
and the compensator given in (8) as the adaptive tuning dc
operating point, the small-signal gains given in G(z) have numerically been computed based on the previously described
modeling approach
fˆinj
ϕ̂
 −4577

=
z
470
z
−1138
z
1918
z
2102   K̂ 
z   Ẑ  .
 1 
2239
Ẑ2
z
(9)
Before designing the indices of (4), a large-signal stability
analysis is performed to determine the range over which each
loop gain transfer function is monotonic. In particular, because
the adaptive tuning system is accounting for power stage parameter changes, the dc operating point around which the MIMO
system was designed may significantly change and possibly
cause instability. Contour plots of constant crossover frequency
and phase margin have been plotted as functions of K and
Z1 with Z2 = 0.95. These plots can be used to determine the
range of tuned compensator parameters over which the indices
of (9) do not change sign. Fig. 3(a) and (b) shows that both
crossover frequency and phase margin are monotonic for all
cases presented except the effect of large zero location changes
on crossover frequency. In this case, as the zero location crosses
the dotted line indicated in Fig. 3(a), the feedback loop relating fc to compensator zero location will change sign, and thus
negative feedback cannot be guaranteed about this operating
point. This is dealt with in hardware by choosing S(z) such that
any changes in crossover frequency do not directly affect zero
locations
A3 = A5 = 0.
Fig. 4. Closed-loop frequency response of the MIMO control system from
(a) fc → fc re f , (b) ϕ m → fc re f , (c) fc → ϕ m re f , (d) ϕ m → ϕ m re f . Responses (a) and (d) track well up to a given frequency while (b) and (c) reject
well, as desired.
(10)
Fig. 5. Experimentally observed dynamic performance of the MIMO adaptive
tuning control loop. (a) Monitored phase margin in degrees. (b) PID compensator
z-domain zero locations. (c) Monitored crossover frequency in kilohertz. (d) PID
compensator gain.
Although this constraint does limit the control design, it helps
widen the range of parameter variations over which the control
loops are monotonic.
Similar plots, as given in Fig. 3, can be used to investigate
the range of power stage variations over which each feedback
loop remains stable. A summary of these results are presented
in Table I showing that major variations in all power stage parameters (C, L, and Vg ) do not cause the feedback loop signs to
change. Note that in Table I, each maximum and minimum value
Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on September 1, 2009 at 16:22 from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 2, FEBRUARY 2009
563
Fig. 6. Output voltage (top, Ch. 1) and inductor current (bottom, Ch. 2) under a 2.5 A → 0 A load transient. (a) Conservatively designed controller V g = 12 V.
(b) Same controller design as in (a) with V g = 8 V. (c) Adaptively tuned controller with V g = 12 V. (d) Adaptively tuned controller with V g = 8 V.
assumes that the other power stage parameters are operating at
nominal values.
Beyond the constraint given in (10), the adaptive tuning gains
can be designed to achieve desired performance and stability as
discussed previously. In the experimental system, the adaptive
tuning control loop has been designed to minimize phase margin
error faster than the crossover frequency error. This amounts to
choosing A1 , A2 , A4 , and A6 such that the closed-loop bandwidth of the direct phase margin loop is greater than the direct
crossover frequency path bandwidth. The gains in the experimental system were chosen as follows:
 
z −3.05 × 10−5
0

  
z−1
K̂




z 1.11 × 10−4  fˆc error
Ẑ  

 1 = 
0

 ϕ̂
z−1

Ẑ2
 m error

−5 
z 8.38 × 10
0
z−1
(11)
which leads to a system loop gain matrix of

0.14
 (z − 1)
L(z) = 
 −0.0143
(z − 1)
0.05 
(z − 1) 
.
0.4 
and given in Fig. 4. Fig. 4(a) is the closed-loop frequency response from fc to fc ref indicating that crossover frequency reference changes are tracked well up to about 10 Hz, which is the
approximate bandwidth of that tuning loop. Similar results are
presented in Fig. 4(d) showing that the effect of ϕm ref changes
on ϕm is tracked up to about 40 Hz. Conversely, Fig. 4(b) is the
response of ϕm to changes in fc ref showing that any changes
in fc ref do not significantly affect ϕm due to the action of the
feedback loop. Similarly, Fig. 4(c) indicates that fc is not significantly affected by ϕm ref changes. Note that the closed-loop
bandwidth of the ϕm → ϕm ref loop is the largest thus ensuring phase margin errors converge fastest. Finally, based on the
closed-loop transfer functions, the closed-loop poles can be examined to prove both internal and overall system exponential
stability [16]. For the design given by (11), each of the indices
of the closed-loop transfer matrix shares the same closed-loop
poles given by
z1 = 0.8561
z2 = 0.6039
(13)
which lie inside the unit circle.
(12)
(z − 1)
The impact of this design can be discussed based on the
closed-loop frequency responses, computed from (7) and (12),
IV. EXPERIMENTAL VERIFICATION
Experimental verification was performed on the same hardware as described in Section III. The adaptive tuning process
begins by initializing the system to the nominal compensator
Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on September 1, 2009 at 16:22 from IEEE Xplore. Restrictions apply.
564
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 2, FEBRUARY 2009
TABLE II
REQUIRED LOGIC RESOURCES TO IMPLEMENT ADAPTIVE TUNING ALGORITHM
given by (8). For this experiment, the target crossover frequency
is set at fc ref = 14.6 kHz, or about one-seventh of the switching frequency, and the target phase margin is ϕm ref = 40◦ .
Fig. 5 shows how the compensator parameters adjust to meet
target stability specifications from startup, with the compensator
given by (8), to steady state when the stability margin references
are met. As shown in Fig. 5, after a short time, the compensator
parameters have converged and settled to
Gc (z) = 3.63
(z − 0.9492) (z − 0.8203)
.
z (z − 1)
(14)
Based on the discrete-time model of [17], the analytical
crossover frequency associated with the compensator given by
(15) is fc analytical = 14.5 kHz and the analytical phase margin
is ϕm analytical = 39◦ , both of which closely match the target
values and the values measured by the stability margin monitor.
Fig. 6 is a comparison of load transient performance between
the conservatively designed control loop corresponding to the
compensator given by (8) and the aforementioned adaptive loop
with target crossover frequency fc ref = 14.6 kHz and desired
phase margin ϕm ref = 40◦ . The conservatively designed control loop exhibits noticeably worse load transient performance
after a change in input voltage, while the adaptive loop automatically tunes the controller to maintain desired crossover
frequency and phase margin. In Fig. 6(c) and (d), the amplitude of the oscillation due to the signal injection Vz is about
±1 LSB of the voltage sensing ADC, or ±0.4% of the dc output voltage. Such a small output voltage oscillation caused by
the adaptive tuner makes continuous parameter tuning feasible
without disturbing steady-state regulation requirements.
As a final note, Table II lists the required logic resources
to implement the aforementioned adaptive tuning system and
phase margin monitor. Table II indicates that for a reasonable
number of gates and no memory requirements, the adaptive
tuning controller can be added to any digital system to improve
reliability and performance.
V. CONCLUSION
This letter presented a practical method for online adaptive
tuning of digital controllers for SMPS. The compensator tuning
relies on continuous monitoring of phase margin and crossover
frequency, which are outputs of a stability margin monitor. The
monitored phase margin and crossover frequency are input to
a MIMO control loop that minimizes the error between the desired crossover frequency and phase margin and the measured
values. Simple small-signal models are derived and used to design the adaptive tuning control loop to achieve stability over a
wide range of operating points. Experimental results presented
for a synchronous buck SMPS demonstrated load transient performance, indicating that with the adaptive tuning system, more
aggressive and reliable system performance can be achieved
as compared to conventional compensator designs. In addition,
the hardware requirements for the entire adaptive tuning system
are relatively modest making it a practical solution for highperformance power systems.
REFERENCES
[1] D. Maksimovic, R. Zane, and R. Erickson, “Impact of digital control in
power electronics,” in Proc. IEEE Int. Symp. Power Semicond. Devices
ICs, May 2004, pp. 13–22.
[2] J. Morroni, A. Dolgov, M. Shirazi, R. Zane, and D. Maksimovic, “Online
health monitoring in digitally controlled power converters,” in Proc. IEEE
Power Electron. Spec. Conf., Jun. 2007, pp. 112–118.
[3] B. Miao, R. Zane, and D. Maksimovic, “Automated digital controller
design,” in Proc. IEEE Appl. Power Electron. Conf., 2005, pp. 2729–
2735.
[4] M. Shirazi, L. Corradini, R. Zane, P. Mattavelli, and D. Maksimovic,
“Autotuning techniques for digitally controlled point-of-load converters
with wide range of capacitive loads,” in Proc. IEEE Appl. Power Electron.
Conf., Feb. 2007, pp. 14–20.
[5] W. Stefanutti, P. Mattavelli, S. Saggini, and M. Ghioni, “Autotuning of
digitally controlled buck converters based on relay feedback,” IEEE Trans.
Power Electron., vol. 22, no. 1, pp. 199–207, Jan. 2007.
[6] Z. Zhao, Li. Huawei, A. Feizmohammadi, and A. Prodic, “Limit-cycle
based auto-tuning system for digitally controlled low-power SMPS,” in
Proc. IEEE Appl. Power Electron. Conf., Mar. 2006, pp. 1143–1147.
[7] L. Corradini, P. Mattavelli, and D. Maksimovic, “Robust relay-feedback
based autotuning for DC-DC converters,” in Proc. IEEE Power Electron.
Spec. Conf., Jun. 2007, pp. 2196–2202.
[8] L. Corradini, P. Mattavelli, W. Stefanutti, and S. Saggini, “Simplified
model reference-based autotuning for digitally controlled SMPS,” IEEE
Trans. Power Electron., vol. 23, no. 4, pp. 1956–1963, Jul. 2008.
[9] K. J. Astrom, “Adaptive feedback control,” Proc. IEEE, vol. 75, no. 2,
pp. 185–217, Feb. 1987.
[10] K. J. Astrom and B. Wittenmark, “A survey of adaptive control applications,” in Proc. IEEE Conf. Decis. Control, Dec. 1995, vol. 1, pp. 649–
654.
[11] G. H. M. de Arruda and P. R. Barros, “Relay based gain and phase margins
PI controller design,” in Proc. IEEE Instrum. Meas. Technol. Conf., 2001,
pp. 1189–1194.
[12] W. K. Ho, C. C. Hang, and J. H. Zhon, “Performance tuned gain and phase
margins,” IEEE Trans. Control Syst. Technol., vol. 4, no. 4, pp. 473–477,
Jul. 1996.
[13] A. Kelly and K. Rinne, “A self-compensating adaptive digital regulator for switching converters based on linear prediction,” in
Proc. IEEE Appl. Power Electron. Conf., Mar. 2006, pp. 712–
718.
[14] J. Morroni, R. Zane, and D. Maksimovic, “An online phase margin monitor
for digitally controlled switched-mode power supplies,” in Proc. IEEE
Power Electron. Spec. Conf., Jun. 2008, pp. 859–865.
[15] R. D. Middlebrook, “Measurement of loop gain in feedback systems,”
Int. J. Electron., vol. 38, no. 1, pp. 485–512, Apr. 1975.
[16] J. M. Maciejowski, Multivariable Feedback Design. Reading, MA:
Addison-Wesley, 1989.
[17] D. Maksimovic and R. Zane, “Small-signal discrete-time modeling of
digitally controlled DC-DC converters,” IEEE Trans. Power Electron.
Lett., vol. 22, no. 6, pp. 2552–2556, Nov. 2007.
Authorized licensed use limited to: UNIVERSITY OF COLORADO. Downloaded on September 1, 2009 at 16:22 from IEEE Xplore. Restrictions apply.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement