Proceeding of the IEEE 28th Canadian Conference on Electrical and Computer Engineering Halifax, Canada, May 3-6, 2015
Controller Design for Low-Input Voltage Switching Converters having Non-minimum Phase Characteristics Ghulam Abbas, Umar Farooq, Student Member IEEE, Jason Gu, Senior Member IEEE and Muhammad Usman Asad, Student Member IEEE Regarding the literature review, [2] applied robust digital control to a DC-DC boost converter to improve load and line regulations. A two degree-of-freedom (2 DOF) controller was employed to the switching converters exhibiting nonminimum phase characteristics to overcome the limitations offered by one degree-of-freedom (1 DOF) controllers like linear quadratic regulator (LQR), ȝ-synthesis, H, etc. [3]. The boost converter system robustness has been enhanced at the time of transients through the adaptive fuzzy-neuralnetwork control (AFNNC) scheme [4]. Since fuzzy-neuralnetwork cannot be represented by a transfer function, it becomes quite distressing for the control engineer to get insight into the system dynamics and its stability. In a supercapacitor energy storage system, a combination of feedforward control and the PWM phase-shifted was applied to control the boost converter [5]. Nothing has been said about the RHP-zero and its impact on the control loop bandwidth in the references mentioned above. In [6], a complicated double loop direct digital controller was suggested to eliminate the drastic effects of RHP-zero. Elimination of RHP-zero was carried out using x-shaped diode-capacitor network in [7]. This paper suggests a control technique to nullify the RHP-zero effect to the maximum extent to get the regulated output voltage along with the improvement of line and load regulations.
Abstract— This paper presents the design of feedback loop control for the switching converter having non-minimum phase characteristics. As an example, low-input voltage high frequency boost converter operating in continuous conduction mode having a right-half plane zero is considered as a plant. A frequency response based controller is usually not suggested for the non-minimum phase plants. In order to facilitate the control engineers and designers being well accustomed to the classical control techniques, feedback loop using the classical control theory is introduced to get better dynamic response. Additional phase lag of 90o acquainted by RHP-zero perplexes the loop compensation. Issues of designing the controller are highlighted in a comprehensive way. Simulation results using the MATLAB/Simulink environment are presented to get insight into compensated system.
I.
INTRODUCTION
Modern power electronic equipment needs compact, highly efficient, light, etc., switch mode power supplies. Switch mode power supplies in this regard not only fulfill the above said requirements but also keep the output voltage regulated over a wide input range by dint of feedback control loop. The controller compensates the perturbations in the input voltage or the load current and thus improves the line and load regulation. It also ensure good set-point tracking. In this paper, special attention has been devoted to highlight the difficulties to design an efficient controller for the boost converter showing the non-minimum phase characteristics.
In this paper, a detailed procedure of designing a controller for a non-minimum phase converter is presented. The limitations imposed by RHP-zero for designing a controller to get better performance criteria are highlighted. The crossover frequency is selected in such a way to counterbalance the drastic effect of RHP-zero on performance. MATLAB/Simulink environment is used to authenticate the design criteria.
Designing a controller for buck converters using the classical control theory has not been a problem [1]. But to control switching converters having a right-half plane zero in their control-to-output transfer functions is a tricky one and needs special attention. The presence of a right-half plane zero in the control-to-output transfer function reduces the loop bandwidth approximately to 20-30%. This reduction in loop bandwidth deteriorates the transient response. The complications rendered by the voltage-mode controlled boost converter are highlighted and removed through a feedback controller. A well-accomplished linear control technique is employed to handle the adverse effects of RHP-zero.
The paper is structured in the following way: In order to design a controller, the dynamics of a boost converter are presented in Section II along with parameter values. Section III encapsulates the design methodology of designing a controller considering both the complex and real zeros. The effect of crossover frequency on the compensated system performance is investigated in Section IV by considering both the Bode plot and closed loop response. Some additional MATLAB/Simulink results are provided in section V to validate the design. Finally conclusions are drawn in Section VI.
Ghulam Abbas is with Department of Electrical Engineering, The University of Lahore, Lahore Pakistan (Email:
[email protected]). Umar Farooq is with Department of Electrical Engineering, University of The Punjab Lahore-54590 Pakistan (Email:
[email protected]). Jason Gu is with Department of Electrical and Computer Engineering, Dalhousie University, Halifax Canada (Email:
[email protected]). Muhammad Usman Asad is with Department of Electrical Engineering, The University of Lahore, Lahore Pakistan (Email:
[email protected]).
978-1-4799-5829-0/15/$31.00 ©2015 IEEE
II.
BOOST CONVERTER DYNAMICS
The circuit diagram of the boost converter along with feedback path including all the parasitic resistances is shown in Fig. 1. For the sake of simplification, H(s) and PWM
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where Ȧ0, ȦESR and ȦRHP represents the double-pole, ESR-zero and RHP-zero frequencies, respectively. Q refers to the quality factor of boost converter power stage.
gains are considered to be one. The boost converter comes under the category of indirect energy transfer converters. When transistor Q1 is switched on, inductor gets energized and stores energy. The capacitor C alone provides power to the load. When transistor Q1 is switched off, the inductor in series with the DC supply supplies the required power to the load.
For the component values mentioned above, the Bode plot of (1) shown in Fig. 2 shows that double-pole frequency exists at 1.1605×105 rad/s. RHP-zero lies at a frequency 8.7041×105 rad/s whereas the zero due to the capacitor ESR lies at 6.4475×106 rad/s. The power stage of the boost converter behaves like a second-order system. At a crossover frequency almost five times below ȦRHP, the boost converter shows only a phase margin of 8.79o. The low phase margin results in a sluggish response. The compensator has to be introduced into the loop to boost phase margin. The positions of ESR and RHP zeros can be changed using the power stage elements.
Figure 1.
Magnitude (dB)
Bode Diagram Gm = -7.93 dB (at 2.16e+005 rad/sec) , Pm = -10.4 deg (at 3.16e+005 rad/sec) 40
Closed-loop voltage-controlled boost converter.
The typical parameter values used are the following: Vin = 1.8 V, Vout = 3.3 V, Vref = 3.3 V, L = 4.7 μH, C = 4.7 μF, RL = 80 mȍ, RC = 33 mȍ, R = 13.75 ȍ (Iout = 240 mA), fs = 1.0 MHz. Using the state space averaging and linearization techniques, the small signal control-to-output transfer function of the boost converter is expressed as [8, 9]:
vˆ ( s) G p ( s ) = out dˆ ( s )
vˆin ( s ) = 0 iˆ ( s ) =0
= Gdo
out
§ s ·§ · s + 1¸ ¨ − + 1¸ ¨ ω ω RHP ¹ © ESR ¹© § s2 · s + 1¸ ¨¨ 2 + ¸ ω Q ω 0 © 0 ¹
20 0 -20
Phase (deg)
-40 360
270
180
90 10
4
5
10
(1)
III.
ω0 ≈
Vin
(1 − D )
1− D
ω ESR = ω RHP =
(3)
LC 1 RC C
(4)
R (1 − D )
2
L
ω0 § RL · 1 + ¨¨ ¸¸ © L C ( R + RC ) ¹
8
10
10
9
CONTROLLER DESIGN
The zeros of the compensator are placed around the double-pole frequency of the converter to nullify the phase reduction effect of the poles. One of the poles of the compensator is placed at ȦESR. The second pole is a low frequency pole (essentially an integrator) which helps in reducing the steady-state error. The third pole is placed in the vicinity of ȦRHP. The compensator quality factor sets almost equal to boost power stage quality factor. Once the zeros and poles of the compensator are placed, the DC gain of the compensator is determined by the well-famous magnitude
(5)
and Q≈
7
10
The controller has to be designed in such a way that it offers enough gain and phase margins at the required crossover frequency in order to improve the dynamic performance. The RHP-zero imposes more constraints to the crossover frequency and complicates the feedback compensation. The RHP-zero behaves like a left-half plane pole in terms of phase and causes a phase reduction of 90o. Consequently the crossover frequency has to be kept well below the RHP-zero by a factor M while shaping the closed loop according to the requirement.
(2)
2
6
Figure 2. Bode plot of the open-loop boost converter power stage.
where
Gdo ≈
10
Frequency (rad/sec)
(6)
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condition i.e. G p ( s )
s = sx
observed that at the same crossover frequency and poles and zeros positions, compensators with complex and real zeros show different margins. The compensator with complex zeros shows a phase margin of 49.4° (see Fig. 3) whereas the other one a phase margin of 36.8° (see Fig. 4) at the same crossover frequency of 4.352×105 rad/s. This means that the compensator with complex zeros offers large phase margin and absence of peaking of the amplitude near resonance frequency as compared to the compensator with real zeros. This comes with the conclusion that complex-zero compensation offers better static properties. However, realzero compensation can be fine-tuned to improve the performance.
. Gc ( s) s = s = 1 , where sx = j Ȧx. Ȧx x
signifies the crossover frequency. The compensators have been designed for the maximum possible value of Ȧx i.e. Ȧx = ȦRHP/2 (M = 2). Definitely phase margin can be improved by increasing M. Consequently, the startup overshoots or undershoots can be decreased. The two complex poles of the boost converter can be compensated either by using the complex zeros or the real zeros of the compensator. A. When the Zeros Are Complex in Nature A three-pole two-zero compensator with complex zeros is expressed in terms of a transfer function as
§ s2 · s + 1¸ ¨¨ 2 + ¸ QC ω0 © ω0 ¹ Gc ( s) = Kc § s ·§ s · s¨ + 1¸¨ + 1¸ © ωESR ¹ © k.ωRHP ¹
For change in load, complex-zero compensation may give detrimental performance. Load variation changes the converter dynamics. Converter’s quality factor also gets disturbed. The compensator designed for fixed value of Q cannot track changes in load and shows poor load regulation. From now onwards, unless otherwise specified, simulation results pertaining to the complex-zero compensation will be shown.
(7)
where the parameter k signifies the deviation around ȦRHP. The compensator DC gain is determined by
§ s ·§ s · + 1¸ ¨ + 1¸ s¨ ω ESR ¹ © k .ωRHP ¹ Kc = © § s2 · s + 1¸ ¨¨ 2 + ¸ QC ω z © ωz ¹
.
1 G p ( s)
IV.
(8) s = sx
s = sx
Using the position of poles and zeros and the DC gain, the controller numerically is expressed as Gc ( s ) =
4.924 × 10−6 s 2 + 0.1598s + 6.632 × 104 8.91× 10−14 s 3 + 7.295 × 10−7 s 2 + s
(9)
B. When the Zeros Are Real in Nature Considering the compensator zeros to be real in nature, the transfer function of the compensator is then expressed as
Bode Diagram Gm = 6.73 dB (at 1.23e+006 rad/sec) , Pm = 49.4 deg (at 4.35e+005 rad/sec) 40 20 Magnitude (dB)
§ s ·§ s · + 1¸¨ + 1¸ ¨ © ω z1 ¹© ω z1 ¹ Gc ( s ) = K c § s ·§ s · + 1¸ ¨ + 1¸ s¨ . ω ω k RHP ¹ © ESR ¹©
(10)
1.782 × 10−14 s 3 + 2.7 ×10 −7 s 2 + s
-60 270
Phase (deg)
Gc ( s ) =
0 -20 -40
Placing the real zeros ω z1 and ω z 2 in the vicinity of ω0 and determining the DC gain using the condition mentioned above, the transfer functions of the analog controller is given by 4.183 × 10−6 s 2 + 0.9223s + 4.958 × 104
CROSSOVER FREQUENCY EFFECT ON PERFORMANCE
The crossover frequency Ȧx and RHP-zero frequency ȦRHP should be separated by a factor M which may vary approximately from 2 to 10. Usually a high crossover frequency (bandwidth) results in superior transient response as more of the frequency components pass through the loop. This does not hold good for converters having RHP-zero. There is a practical limit for crossover frequency not to exceed one-third of RHP-zero frequency. Designing compensator beyond this crossover frequency introduces overshoots/undershoots to the output voltage response. Change in crossover frequency solely changes the DC gain of the compensator leaving no effect on compensator poles and zeros positions. Only the magnitude curve of the Bode plot gets disturbed. As the factor M decreases, more overshoots/undershoots are observed at the startup. The situation is depicted in Figs. 5 and 6.
(11)
225 180 135 90 4
The compensators are tuned to ensure enough phase margin which guarantees the optimal performance. The stability margins can be calculated from the Bode diagram of the open-loop compensated boost converter. It has been
10
5
10
6
10
10
7
10
8
Frequency (rad/sec)
Figure 3. Bode plot of the compensated converter using complex zeros.
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V.
Bode Diagram Gm = 7.89 dB (at 2.4e+006 rad/sec) , Pm = 36.8 deg (at 4.35e+005 rad/sec) 60
The performance of the feedback control loop is tested using the MATLAB/Simulink environment [10]. The closed loop step response shows the non-minimum-phase (inverse) response due to the presence of RHP-zero at the startup. The compensator having complex zeros shows maximum overshoot of 15% and settling time of 7.7 μs (see Fig. 7) whereas the compensator bearing real zeros show maximum overshoot of 32% and settling time of 38.2 μs (see Fig. 8). Thus compensator having complex zeros shows better static performance. The compensator provides necessary phase boost at the required crossover frequency to ensure required performance (see Fig. 9). A phase margin of greater than 45o even at a crossover frequency quite close to the RHP-zero (two times below ȦRHP) using the complex zero compensation ensures superior performance.
Magnitude (dB)
40 20 0 -20 -40
Phase (deg)
-60 360
270
180
90 3
10
4
5
10
6
10
7
10
8
10
9
10
SIMULATION RESULTS
10
Frequency (rad/sec)
Step Response
Figure 4. Bode plot of the compensated boost converter using real zeros.
4 3.5
Bode Diagram 40
2.5 Output Voltage (V)
Magnitude (dB)
3 20 0 -20 -40
Phase (deg)
-60 270
ωx = ωRHP/8
225
2 1.5 1 0.5 0
ωx = ωRHP/4
180
ωx = 3*ωRHP/8
135
ωx = ωRHP/2
90 4 10
5
6
10
7
10
-0.5 -1
0
0.2
0.6
0.8
1 -4
Time (seconds)
8
10
0.4
x 10
10
Figure 7. Closed loop response of the complex zero compensation.
Frequency (rad/s)
Figure 5. Bode plot of Gp(s)*Gc(s) for different values of Ȧx.
Step Response 5
Step Response 4
4
3.5 3
Output Voltage (V)
3 Output Voltage (V)
2.5 2
ωx = ωRHP/8
1.5
ωx = ωRHP/4 ωx = 3*ωRHP/8
1
ωx = ωRHP/2
0.5
2 1 0
0
-1
-0.5 -1
0
1
2
3
Time (seconds)
4
5
-2
-5
x 10
0
0.2
0.4
0.6
Time (seconds)
Figure 6. Closed loop response for different values of crossover frequencies.
0.8
1 x 10
Figure 8. Closed loop response of the real zero compensation.
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-4
Rather than considering the transfer function of the boost converter while investigating the static performance, the dynamic performance is investigated at circuit level. The switching frequency is set to 1 MHz. For a load resistance change of 13.75 ȍ to 27.50 ȍ and then from 27.50 ȍ to 13.75 ȍ, the output voltage settles to its steady-state value of 3.3 V. As depicted from Fig. 10, the output volatge shows less spike and recovery time at the time of transients. This clearly depicts that the complex zero compensation offers superior performance.
The problems regarding the design of a controller using the classical control theory for a switching converter having nonminimum phase characteristics like boost converter have clearly been mentioned. It has been observed that the crossover frequency and RHP-zero frequency should be separated by a factor. If both the frequencies fall close to each other, large overshoots or undershoots are observed. Existence of RHP-zero limits the performance. Effects of changing various parameters like crossover frequency, load, etc. on the performance are also investigated. Unlike buck converter, boost converter shows non-minimum phase response at the startup due to the presence of RHP-zero. Special attention has been devoted to get superior response even if the constraints on bandwidth. MATLAB/Simulink results are provided to validate the design. Buck-boost (inverting), Cuk, Sepic, and indirect energy transfer converters may be expected to exhibit the same response using the analog controller. Derivation of analog controller’s digital counterpart and its implementation are the objectives of future work.
Bode Diagram 40 Magnitude (dB)
Gp(s) 20
Gc (s) Gp(s)*Gc (s)
0 -20 -40 -60 360
REFERENCES
Phase (deg)
270
[1]
Abbas, Ghulam, Umar Farooq, Jason Gu, and Muhammad Usman Asad. "Graphical user interface based controller design for switching converters." In Information and Automation (ICIA), 2014 IEEE International Conference on, pp. 1149-1153. IEEE, 2014. [2] Ohta, Y.; Higuchi, K.; Kajikawa, T., "Robust digital control for boost DC-DC converter," 8th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), pp. 525-528, 17-19 May 2011. [3] Dey, J.; Saha, T. K., "Design and performance analysis of two degree-of-freedom (2 DOF) control of DC-DC boost converter," 2013 IEEE International Conference on Industrial Technology (ICIT), pp. 493-498, 25-28 Feb. 2013. [4] Rong-Jong Wai; You-Wei Lin; Li-Chung Shih, "Design of adaptive fuzzy-neural-network control for DC-DC boost converter," the 2012 International Joint Conference on Neural Networks (IJCNN), pp.1-6, 10-15 June 2012. [5] Wei Jiang; Renjie Hu; Jinlong Zhang, "The PWM phase-shifted plus feed-forward control of the boost converter applied in supercapacitor energy storage system," 2011 International Conference on Electrical and Control Engineering (ICECE), pp.455-458, 16-18 Sept. 2011. [6] Ellabban, O.; Hegazy, Omar; Van Mierlo, Joeri; Lataire, Philippe, "A DSP digital controller design and implementation of a high power boost converter in hybrid electric vehicles," 2010 IEEE Vehicle Power and Propulsion Conference (VPPC), pp. 1-6, 1-3 Sept. 2010. [7] Yan Zhang; Jinjun Liu; Xiaolong Ma, "Using RC type damping to eliminate right-half-plane zeros in high step-up DC-DC converter with diode-capacitor network," 2013 IEEE ECCE Asia Downunder (ECCE Asia), 2013 IEEE , pp. 59-65, 3-6 June 2013. [8] Huliehel, F. A.; Lee, F. C.; Cho, B. H., "Small-signal modeling of the single-phase boost high power factor converter with constant frequency control," 23rd Annual IEEE Power Electronics Specialists Conference, 1992. PESC '92 Record, pp. 475-482, vol.1, 29 Jun-3 Jul 1992. [9] B. Bryant, M. K. Kazimierczuk, “Small-signal duty cycle to inductor current transfer function for boost PWM DC-DC converter in continuous conduction mode,” in Proc. ISCAS, Vol. 5, pp. 856-859, 2004. [10] MATHWORKS, Matlab, The Language of Technical Computing, Simulink for model-based and system level design”, 2010.
180 90 0 -90 4 10
5
6
10
7
10
8
10
9
10
10
Frequency (rad/s)
Figure 9. Bode plot of the various systems. 3.6 3.5
V out(V)
3.4 3.3 3.2 3.1 3
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5 -4
x 10 0.25
iL(A)
0.2
0.15
0.1
4
4.5
5
5.5
6 Time (s)
6.5
7
7.5
8
8.5 -4
x 10
Figure 10. Load regulation offered by the compensated boost converter.
VI.
CONCLUSION
In this paper, an efficient controller has been designed for boost converter working in continuous conduction mode.
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