High-performance cascade-type repetitive controller for CVCF PWM inverter: analysis and design Y. Ye, B. Zhang, K. Zhou, D. Wang and Y. Wang Abstract: A novel cascade-type repetitive controller with phase-cancellation filter is proposed for a constant-voltage constant-frequency pulse-width-modulated DC–AC converter to achieve very low total harmonic distortion and fast transient response. The paper reveals the principle of phase cancellation for the performance improvement of RC. In the form of real-time window filtering, a phase-cancellation filter is developed for the repetitive controller. Experimental results show that a high-quality AC source can be achieved by the proposed scheme even under nonlinear rectifier load and parameter uncertainties.
1
Introduction
Constant-voltage constant-frequency (CVCF) pulse-width modulated (PWM) converters are widely employed in various AC power-conditioning systems, such as automatic voltage regulators and uninterruptible-power-supply (UPS) systems. Output-voltage total harmonics distortion (THD) is one important index to evaluate the performance of converters, as it leads to communication interference, excessive heating in capacitors and transformers etc. Nonlinear loads, causing periodic distortion, are major sources of THD in AC power systems. To minimise THD, several feedback-control schemes have been proposed for CVCF PWM DC–AC converters, such as the deadbeat or one-sampling-ahead-preview (OSAP) controller [1, 2], sliding-mode controller (SMC) [3] and hysteresis controller (HC) [4]. Unfortunately, conventional feedback control alone cannot eliminate the periodic distortion caused by nonlinear loads and parameter uncertainties. Repetitive control (RC) [5–7] is a scheme that has the ability to track exactly or reject periodic signals. By adjusting the control input period by period, based on the errors of previous period(s), periodic errors are forced asymptotically toward zero The periodicity of output makes CVCF PWM converters an ideal application field of RC [8– 11] and THD can be suppressed significantly by RC. However, in practice, if RC is not well designed, it will bring poor control performance, i.e. low tracking accuracy and a slow convergence rate. Hence it is an interesting issue to investigate how to improve the design of RC. In this paper, a ‘cascade-type’ RC is applied to a statefeedback-controlled CVCF PWM converter for convenience of implementation, periodically adjusting the commands that are sent to the closed-loop system. The principle r The Institution of Engineering and Technology 2007 doi:10.1049/iet-epa:20050518 Paper first received 30th June 2005 and in final revised form 18th May 2006 Y. Ye is with the School of Information, Zheliang University of finance & economics, Hangzhou 310012, China B. Zhang, K. Zhou, D. Wang and Y. Wang are with the school of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 E-mail:
[email protected]
112
of phase cancellation for the performance improvement of RC is revealed. To allow online implementation, the error is filtered by a moving window filter. The design of the window filter utilises the information of frequency response of the closed-loop system and inverse discrete fourier transform (IDFT) to implement the idea of phase cancellation for improvement of control performanceF tracking accuracy and transient response. Experiments show that a high-quality AC source is obtained regardless of nonlinear load and parameter uncertainties by using the phase-cancellation RC. 2
2.1
Window-filtering-based RC
‘Cascade-type’ configuration
Typically, RC is employed to complement the action of a conventional feedback controller. A conventional feedback control stabilises and makes the system robust, and RC ensures high-accuracy steady-state tracking. There are two popular configurations of the combination of RC and conventional feedback control, as shown in Figs. 1 and 2, where Yd ðzÞ is the periodic reference signal of period N , Y ðzÞ is the output, DðzÞ is the periodic disturbance of period N , EðzÞ is the tracking error, Gs ðzÞ is the plant, Gc ðzÞ is the feedback controller, and Gr ðzÞ is the repetitive controller. Figure 1 shows the widely used ‘plug-in’ scheme [7, 12]. The repetitive controller connects in parallel with a closed loop. The output of RC is inserted into the feedback loop and produces a modified error signal. Figure 2 shows the ‘cascade-type’ configuration. The repetitive controller connects in cascade with the feedback closed loop. The output of the RC adjusts the input of the feedback loop. In [13], the ‘cascade-type’ configuration is used and referred to as a ‘two-layer control structure’. When RC applies to a commercial product whose feedback loop is encapsulated, the ‘cascade-type’ scheme is relatively easier to implement. Figure 3 shows the repetitive controller Gr ðzÞ. If QðzÞ ¼ 1, the RC law can be written in z-transform as Ur ðzÞ ¼ zN Ur ðzÞ þ kr zN Gf ðzÞEðzÞ
ð1Þ
where N ¼ fc =f is the period with fc being the sampling IET Electr. Power Appl., Vol. 1, No. 1, January 2007
Fig. 1
‘Plug-in’ configuration
Fig. 2
‘Cascade-type’ configuration
Fig. 3
Repetitive controller
frequency and f being the reference-signal frequency, kr is the repetitive control gain and Gf ðzÞ is a filter. We can superimpose yd ðkÞ on the repetitive control output ur ðkÞ [10]: ) Uc ðzÞ ¼ Yd ðzÞ þ Ur ðzÞ ð2Þ Ur ðzÞ ¼ zN Ur ðzÞ þ kr zN Gf ðzÞEðzÞ with Uc ðzÞ being the command signal to the closed-loop system as shown in Fig. 2.
2.2
Stability and design
Rearrange the terms in (2), and we can obtain kr zN Gf ðzÞ EðzÞ þ Yd ðzÞ 1 zN The output in Fig. 2 is Uc ðzÞ ¼
Y ðzÞ ¼ Uc ðzÞH ðzÞ þ ¼
EðzÞ þ Yd ðzÞH ðzÞ þ
¼
DðzÞ 1 þ Gc ðzÞGs ðzÞ ð4Þ
EðzÞ Yd ðzÞ Yd ðzÞH ðzÞ DðzÞ=½1 þ Gc ðzÞGs ðzÞ zN 1 zN 1 þ kr Gf ðzÞH ðzÞ
ð5Þ
8 0ooop
ð6Þ
the RC system is stable (o is a normalised frequency with p representing the Nyquist frequency). Note that the ‘plug-in’ configuration RC [12] has the same stability condition as (6) [14]. If the angular frequency o of the reference input IET Electr. Power Appl., Vol. 1, No. 1, January 2007
ð7Þ
In practice, it would be difficult to satisfy condition (6) in the whole frequency band, especially in the high-frequency band. Therefore a low-pass zero-phase filter jQðzÞj 1 can be introduced to modify the RC update in (2) as follows: ð8Þ Ur ðzÞ ¼ QðzÞ zN Ur ðzÞ þ kr zN Gf ðzÞEðzÞ Correspondingly, the stability condition becomes [15] jQðzÞ 1 kr Gf ðzÞH ðzÞ jo1 8 z ¼ ejo ; 0ooop ð9Þ QðzÞ is generally chosen as a moving average QðzÞ ¼ ðd1 zþ d0 þ d1 z1 Þ, and d0 þ 2d1 ¼ 1 for a unit gain at DC, e.g. QðzÞ ¼ 0:25z þ 0:5 þ 0:25z1 [8]. The larger is d0 the higher is the cutoff frequency of QðzÞ. QðzÞ enhances the stability but brings in steady-state error [15].
2.3
Equation (5) implies that, if j1 kr Gf ðejo ÞH ðejo Þjo1
y d ðkÞ and the disturbance dðkÞ approaches om ¼ 2pm=N , m ¼ 0; 1; 2; . . . ; M (M ¼ N =2 for even N and M ¼ ðN 1Þ=2 for odd N ), then zN ! 1, limo!om jGe ðejo Þj ¼ 0, and thus lim jEðzÞj ¼ 0 z ¼ ejo
where H ðzÞ ¼ Gc ðzÞGs ðzÞ=½1 þ Gc ðzÞGs ðzÞ is the closedloop transfer function, and H ðzÞ is stable. Since Y ðzÞ ¼ Yd ðzÞ EðzÞ, (3) and (4) can be combined to get the error transfer function Ge ðzÞ ¼
RC-controlled DC–AC-converter system
o!om
DðzÞ 1 þ Gc ðzÞGs ðzÞ
kr zN Gf ðzÞH ðzÞ
1 zN
ð3Þ
Fig. 4
Phase-cancellation window filter Gf (z)
Filter Gf ðzÞ plays an important role in the design of the RC controller. The online feature of RC requires that the implementation of Gf ðzÞ should be in real time. One popular form of real-time filter is the window filter. A general window filter Gf ðzÞ can be represented by Gf ðzÞ ¼ wðm1 Þzm1 þ þ wð0Þ þ þ wðm2 Þzm2 ð10Þ 113
Table 1: Parameters of experiment Nominal
Actual
Rectifier load
Feedback control
Others
En ¼ 80 V
E ¼ 70 V
Cr ¼ 470 mF
k1 ¼ 0.018
yd (t) ¼ 50 sin 100pt V
Cn ¼ 45 mF
C ¼ 50 mF
Lr ¼ 1 mH
k2 ¼ 1.67 10 6
f ¼ 50 Hz
Ln ¼ 20 mH
L ¼ 30 mH
Rr ¼ 22 O
r ¼ 0.018
fc ¼ 10 kHz
Rn ¼ 15 O
R ¼ 22 O
i
0.8 0.6
0.5i Image axis
0.4 0.2
0 R ∞Ω
0 −0.2
−0.5i
−0.4
R=0.9Ω −i −1
Fig. 5
−0.5
0
0.5 Real axis
1
1.5
−150
−100
−50
Traces of poles of H(z)
ð11Þ where wðnÞ, m1 , and m2 are parameters to be decided. Note that the window is of length m1 þ m2 þ 1. The RC update is then realised by a convolution of the error of the last period and the window: n¼m X2 kr wðnÞeðk N þ nÞ ð12Þ ur ðkÞ ¼ ur ðk N Þ þ
50
100
150
200
0.6 0.4 0.2 0 −0.2 −0.4
n¼m1
where kr is the repetitive control gain. Suppose H ðzÞ has frequency characteristics H ðejo Þ ¼ Nh ðoÞexpfjyh ðoÞg with Nh ðoÞ and yh ðoÞ being its magnitude characteristics and phase characteristics, respecjo tively; and Gf ðzÞ has frequency characteristics Gf ðe Þ ¼ Nf ðoÞexp jyf ðoÞ with Nf ðoÞ and yf ðoÞ being its magnitude characteristics and phase characteristics, respectively. Using these characteristics and noting the fact that kr is generally positive, (6) leads to ð13Þ kr Nh ðoÞNf ðoÞo2 cos yh ðoÞ þ yf ðoÞ Because Nh ðoÞ and Nf ðoÞ are both positive, (13) necessarily requires that 90 oyh ðoÞ þ yf ðoÞo90
0 Index a
In another words, the window has coefficients of 8 < fðnÞ n ¼ m1 ; ðm1 1Þ; . . . ; 0; 1; . . . ; wðnÞ ¼ m2 1; m2 m1 ; m2 40 : 0 otherwise
ð14Þ
If yf ðoÞ ¼ yh ðoÞ, exact phase cancellation is achieved in (14). Then, for all frequencies up to Nyquist, from (13), (6) will be satisfied if 2 ð15Þ 0okr o min o Nh ðoÞNf ðoÞ Obviously, exact phase cancellation can ensure (6) that holds in a much wider frequency band even without QðzÞ and yields higher tracking accuracy. 114
−0.6 −200
−10
−8
−6
−4
−2
0 Index
2
4
6
8
10
b
Fig. 6
Coefficients of window filter
a Original window b After truncation
Hence Gf ðzÞ should have the desired phase characteristics yf ðoÞ ¼ yh ðoÞ, while its magnitude characteristics are defined as Nf ðoÞ ¼ 1 [if Nh ðoÞ 1] or Nf ðoÞ ¼ 1=Nh ðoÞ [if Nh ðoÞ41]. The discrete frequency spectrum ðkÞ of Gf ðzÞ can be calculated as follows: 8 2pk 2pk > > exp jy if N 1 > h h > 2N 1 2N 1 > > > > > > <
2pk 2pk
exp jyh ðkÞ ¼
Nh 2N 1
2N 1 > > > > > > > > 2pk > > : if Nh 41 2N 1 ð16Þ IET Electr. Power Appl., Vol. 1, No. 1, January 2007
Phase (degree)
Magnitude (dB)
where k ¼ 0; . . . ; 2N 2. Note that any part of the frequency spectrum that is amplified by the closed-loop system is attenuated to produce an overall gain that is no more than 1. From (15), the stability condition in this case is known to be 0okr o2.
The window coefficients of Gf ðzÞ are determined as 8 < fðnÞ n ¼ ðN 1Þ; wðnÞ ¼ ðN 2Þ; . . . ; 0; 1; . . . ; N 2; N 1 : 0 otherwise ð17Þ
10
where fðnÞ is calculated by IDFT from ðkÞ:
0 −10
fðnÞ ¼ IDFT ½ðkÞ
| H(jω)| | Gf(jω)H(jω)|
−20
2 X 1 2N 2knp ¼ ðkÞ exp j 2N 1 k¼0 2N 1
−30 −40 1 10
10
2
10
50
3
n ¼ 0; 1; . . . ; 2N 2
∠ H(jω)
0 −50
In addition, fðnÞ with negative indexes in (17) can be determined by the periodicity [16]:
∠ Gf (jω)H(jω)
−100 −150 −200
fðnÞ ¼ fðn þ 2N 1Þ n ¼ 1; 2; . . . ; ðN 1Þ ð19Þ
−250 1 10
2
10
10
3
Note that the window filter Gf ðzÞ is of length 2N 1. Window filtering the error by Gf ðzÞ provides phase compensation of yh ðoÞ.
Frequency (Hz)
Fig. 7
ð18Þ
Frequency characteristics before and after compensation
0.025 y d(t)
40
v c(t)
0.02 Relative Magnitude
Output Voltage/Current (V/A)
60
20 0 −20
0.01
0.005
−40 −60
0.015
0
2
4
6
8
10
12
14
16
18
0
20
a
40
0.03 Relative Magnitude
Output Voltage/Current (V/A)
0.035 y d(t)
io(t)
20 v c(t) 0 −20 −40
0.025 0.02 0.015 0.01 0.005
0
2
4
6
8
10
12
14
16
18
0 0
20
Time (millisecond)
c
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)
d
60
0.1 y d(t)
40
0.09 0.08
v c(t) Relative Magnitude
Output Voltage/Current (V/A)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
b
60
−60
0
Frequency (Hz)
Time (millisecond)
20 io(t)
0 −20
0.07 0.06 0.05 0.04 0.03 0.02
−40
0.01 −60
Fig. 8
0
2
4
6
8
10
12
14
16
18
20
0
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Time (millisecond)
Frequency (Hz)
e
f
Steady-state response with only feedback control
a No load b Voltage spectrum, THD ¼ 2.65% c Resistor d Voltage spectrum, THD ¼ 3.74% e Rectifier f Voltage spectrum, THD ¼ 10.34% IET Electr. Power Appl., Vol. 1, No. 1, January 2007
115
0.025 y d(t)
40
0.02
v c(t)
Relative Magnitude
Output Voltage/Current (V/A)
60
20 0 −20
0.01
0.005
−40 −60
0.015
0
2
4
6
8 10 12 14 Time (millisecond)
16
18
0
20
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)
a
b 0.01 0.009
y d(t)
40
Relative Magnitude
Output Voltage/Current (V/A)
60
v c(t)
20
io(t)
0 −20
0.008 0.007 0.006 0.005 0.004 0.003
−40
0.002 0.001
−60 0
2
4
6
8 10 12 14 Time (millisecond)
16
18
20
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)
c
d 0.01 0.009
y d(t)
40
0.008 Relative Magnitude
Output Voltage/Current (V/A)
60
20 io(t) 0 −20 v c(t)
−40
0.007 0.006 0.005 0.004 0.003 0.002 0.001
−60 0
2
4
6
8 10 12 14 Time (millisecond)
16
18
20
00
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)
e
Fig. 9
f
Steady-state response with RC
a No load b Voltage spectrum, THD ¼ 1.07% c Resistor d Voltage spectrum, THD ¼ 1.06% e Rectifier f Voltage spectrum, THD ¼ 1.13%
Generally, the period N is in the hundreds/thousands. The computation burden of window filtering will be too heavy and possibly the computation cannot be implemented online in experiments. Fortunately, fðnÞ generally has a significant value around n ¼ 0 and approaches to zero quickly as n tends towards ðN 1Þ or N 1. Therefore Gf ðzÞ can be truncated as fðnÞ ¼ 0
if fðnÞoE
ð20Þ
where E is a threshold decided by the specification of tracking accuracy and the hardware capability. Suppose that m1 and m2 are the minimum and maximum indexes between which fðnÞ E. Then, after truncation, Gf ðzÞ is of length m1 þ m2 þ 1 which is normally much less than 2N 1. Therefore only approximate phase cancellation is achieved and QðzÞ is still needed in practice. Meanwhile, QðzÞ also provides robustness against system uncertainties. 3
RC-Controlled DC–AC converters
Figure 4 shows a repetitive-controlled DC–AC converter system. vc is the output voltage; io is the output current; E is 116
the DC bus voltage; and Ln , Cn and Rn are the nominal values of the inductor, capacitor and load, respectively. The nominal value of E is denoted as En . The control input vin is a PWM voltage pulse of magnitude E (or E) with width DT centred in the sampling interval T . A sampled-data model for the DC–AC converter is [1] f11 f12 vc ðkÞ g vc ðk þ 1Þ ¼ þ 1 vin ðkÞ ð21Þ v_ c ðk þ 1Þ f21 f22 v_ c ðkÞ g2 2
2
where coefficients f11 ¼ 1 2LTn Cn , f21 ¼ LnTCn þ 2LnTC2 Rn , 2
2
2
n 2
f12 ¼ T 2CTn Rn , f22 ¼ 1 CnTRn 2LTn Cn þ 2CT2 R2 , g1 ¼ 2LTn Cn , n n g2 ¼ LnTCn 1 2CTn Rn and the average active input vin ðkÞ ¼ DTTðkÞEn . The objective is to force the tracking error between yðkÞ ¼ vc ðkÞ and its sinusoidal reference yd ðkÞ of period N to approach zero asymptotically. A state-feedback-control scheme vc ðkÞ ð22Þ þ ruc ðkÞ vin ðkÞ ¼ ½k1 k2 v_ c ðkÞ IET Electr. Power Appl., Vol. 1, No. 1, January 2007
60
40
Voltage and Current (V/A)
Voltage and Current (V/A)
60 Vc
20
io
0 −20 −40 −60
0
40
c
i
o
0 -20 -40 -60
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
V
20
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (second)
8
8
6
6
4
4
2
Error (V)
Error (V)
Time (second)
0 -2 -4
2 0 -2 -4 -6
-6
-8
-8 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
-10
0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (second) a
Time (second) b 60
Voltage and Current (V/A)
Voltage and Current (V/A)
60 Vc
40 20
io
0 −20 −40 −60
0
0 −20 −40
15
12
10
10
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
8
5 0 −5
6 4 2 0
−10
−2
−15
−4 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
−6 0
Time (second) c
Fig. 10
0
Time (second)
Error (V)
Error (V)
Time (second)
−20
io
20
−60
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Vc
40
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Time (second) d
Response to load changes
a R ¼ N-22 O b R ¼ 22 O-N c N-rectifier d rectifier-N
where uc ðkÞ is a new input variable, is introduced. uc ðkÞ will be the command input which is the repetitive control output ur ðkÞ plus yd ðkÞ as in (2). The transfer function from uc ðkÞ to yðkÞ ¼ vc ðkÞ can be derived as H ðzÞ ¼
q1 z1 þ q2 z2 1 þ p1 z1 þ p2 z2
ð23Þ
where q1 ¼ rg1 ; q2 ¼ rg2 ðf12 k2 g1 Þ rg1 ðf22 k2 g2 Þ, p2 ¼ ðf11 k1 g1 Þ p1 ¼ ðf11 k1 g1 ðf22 k2 g2 Þ, ðf22 k2 g2 Þ ðf12 k2 g1 Þðf21 k1 g2 Þ. Note that the response of the feedback-controlled converter will deviate from the designated one because of disturbances DR ¼ R Rn and uncertainties DL ¼ L Ln , IET Electr. Power Appl., Vol. 1, No. 1, January 2007
DC ¼ C Cn , DE ¼ E En . To improve the performance of the state-feedback control, the phase-cancellation RC is employed as follows: 9 uc ðkÞ ¼ yd ðkÞ þ ur ðkÞ = n¼m2 P ð24Þ kr fðnÞeðk N þ nÞ ; ur ðkÞ ¼ ur ðk N Þ þ n¼m1
4
4.1
Experiments
Experimental parameters
The experimental parameters are listed in Table 1. The nominal transfer function H ðzÞ with R ¼ 22 O can be 117
6
derived as 0:525z þ 0:479 ð25Þ H ðzÞ ¼ 2 z 0:4564z þ 0:4613 Figure 5 shows the traces of two poles of the closed-loop transfer function H ðzÞ in (23) when load R changes from 0.9 O to N. The two poles are located inside the unit circle when R42 O, i.e. the feedback loop is stable. RC gain kr is chosen as 0.8. The window coefficients are calculated based on (25). The threshold for truncation is E ¼ 0:0076 in (20). Figure 6 shows the window coefficients before and after truncation [m1 ¼ 11 and m2 ¼ 10 in (24)]. Figure 7 plots the frequency characteristics of (25) with and without the compensation of window filtering in (24). The upper part of Fig. 7 shows the magnitude characteristics and the lower part shows the phase characteristics. The plot shows that 22 coefficients are good enough to achieve nearly perfect zero-phase effect. However, in practice, owing to the unmodelled/unknown delays in high-frequency characteristics of the plant, a pure lead zg should be added into (24) to compensate for these delays, where the value of the lead steps g can be determined by experiments and the modified RC law is nX ¼10 kr fðnÞeðk N þ n þ gÞ ð26Þ ur ðkÞ ¼ ur ðk N Þ þ n¼11
where g ¼ 3 is the tuning result that can achieve optimal control performance in experiments. QðzÞ ¼ 0:25z1 þ 0:5 þ 0:25z [8, 12].
4.2
Experimental results
Figure 8 shows the steady-state responses under no load, resistor load and rectifier load with only feedback control, respectively. The THDs for the three cases are 2.65% (no load), 3.74% (resistor), and 10.34% (rectifier), respectively. Figure 9 shows the RC-controlled steady-state response for the three cases. The THDs of voltage are 1.07%, 1.06%, and 1.13%, respectively. Figure 10 show the responses of the RC to sudden load changes. The output voltages recover within about 0.15 s. 5
Conclusions
In the form of real-time window filtering, a phasecancellation RC law is developed to improve the control performance of RC systems. The RC controller is applied successfully to a ‘cascade-type’ CVCF PWM DC–AC converter system. The experimental results indicate that satisfactory performance is achieved (low THD, fast convergence rate and robustness) under parameter uncertainties and nonlinear load disturbances.
118
Acknowledgments
The work reported in this paper was supported by the Zheliang provincial Natural Science Foundation under Grant Y105004 and by the supporting Program for Excellent Scholars with Studying Abroad Experiences, Funded by the China Ministry of Personnel. This grant was given to Y. Ye.
7
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IET Electr. Power Appl., Vol. 1, No. 1, January 2007
