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Control RGB Jurnal for illumination magic
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IEEE TRANSACTIONS ON POWER ELECTRONICS 1
Multivariable Robust Control for a Red–Green–BlueLED Lighting System
1
2
Fu-Cheng Wang, Member, IEEE, Chun-Wen Tang, and Bin-Juine HuangQ1
3
Abstract—This paper proposes a novel control structure for a4red–green–blue (RGB) LED lighting system, and applies multivari-5able robust control techniques to regulate the color and luminous6intensity outputs. RGB LED is the next-generational illuminant for7general lighting or liquid crystal display backlighting. The most8important feature for a polychromatic illuminant is color adjusta-9bility; however, for lighting applications using RGB LEDs, color10is sensitive to temperature variations. Therefore, suitable control11techniques are required to stabilize both luminous intensity and12chromaticity coordinates. In this paper, a robust control system13was proposed for achieving luminous intensity and color consis-14tency for RGB LED lighting in a three-step process. First, a mul-15tivariable electrical–thermal model was used to obtain RGB LED16luminous intensity, in which a lookup table served as a feedfor-17ward compensator for temperature and power variations. Second,18robust control algorithms were applied for feedback control de-19sign. Finally, the designed robust controllers were implemented to20control the luminous and chromatic outputs of the system. From21the experimental results, the proposed multivariable robust con-22trol was damned effective in providing steady luminous intensity23and color for RGB LED lighting.24
Index Terms—Color difference, luminous intensity, red–green–25blue (RGB) LEDs, robust control, thermal–electrical–luminous26model.27
I. INTRODUCTION28
R ECENTLY, LED has been drawing much attention as a29
state-of-the-art illuminator because of its numerous ad-30
vantages, including energy savings, long lifetime, and environ-31
mental friendliness. Red–green–blue (RGB) LEDs can provide a32
wide color gamut for liquid crystal display (LCD) backlighting,33
as well as full color adjustability for general lighting applica-34
tions [1], [2]. This newly developed illuminant is the only light35
source currently capable of this type of vivid and dynamic light-36
ing performance. However, the tunable light outputs have been37
found to induce light consistency issues for RGB LED light-38
ing, because the luminous intensity and color outputs are easily39
influenced by junction temperature variations caused by self-40
heating of the LEDs and disturbances in ambient temperatures.41
Therefore, proper control strategies are required to stabilize light42
output in order to counteract temperature variations.43
Manuscript received February 25, 2009; revised April 15, 2009. Recom-mended for publication by Associate Editor M. Ponce-Silva.
F.-C. Wang and B.-J. Huang are with the Department of MechanicalEngineering, National Taiwan University, Taipei 10617, Taiwan (e-mail:[email protected]; [email protected]).
C.-W. Tang was with the Department of Mechanical Engineering, NationalTaiwan University, Taipei 10617, Taiwan. He is now with Coretech OpticalCompany Ltd., Hsinchu 30069, Taiwan (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2009.2026476
To control RGB LED lighting systems, the selection of feed- 44
back signals is an important issue. Muthu et al. [3]–[5] applied 45
three kinds of feedback system: color coordinate feedback with 46
temperature feedforward (CCFB and TFF), color coordinate 47
feedback (CCFB), and flux feedback with temperature feed- 48
forward (FFB and TFF). The color coordinates were measured 49
by photodiodes with color filters and the fluxes with photodi- 50
odes with a time-division method. In addition, the heat sink 51
temperature and thermal resistance were used to estimate junc- 52
tion temperature for temperature feedforward compensation. 53
Hoelen et al. [6]–[8] further discussed light outputs and applied 54
four control structures, namely, flux feedback, temperature feed- 55
forward, CCFB, and FFB and TFF. Among these, CCFB and 56
FFB and TFF were shown to provide better color consistency 57
for RGB LED lighting than did the others, when the system 58
was experiencing junction temperature variations. Until now, 59
CCFB has been a popular choice for application to control sys- 60
tem design [9]–[13] because of its simple structure. However, 61
the accuracy of feedback signals is limited by the difference 62
between the spectra of filtered sensor and color matching func- 63
tions. In contrast, the FFB and TFF structure can provide more 64
signals for control design, but requires double loops and infor- 65
mation about the junction temperature. For controller design, 66
traditional control methodologies such as proportional–integral 67
(PI) or PI derivative (PID) based algorithms have been applied to 68
control RGB LED lighting systems [5], [7], [14], [15]. However, 69
these methods cannot guarantee the stability and performance of 70
systems with perturbations such as varying input power or junc- 71
tion temperatures. Therefore, advanced control strategies should 72
be considered for improving system performance. In this paper, 73
a novel control structure is proposed, and robust control tech- 74
niques are applied, to achieve consistent luminous intensity and 75
color. The effect will be experimentally verified. 76
The paper is arranged as follows. In Section II, an RGB LED 77
luminaire is modeled as a multivariable system and a feedback 78
control structure is proposed. In Section III, robust control strate- 79
gies are introduced for multivariable controller design. Then, the 80
designed controller is implemented for performance analysis in 81
Section IV. Finally, some conclusions are drawn in Section V. 82
II. SYSTEM DESCRIPTION AND MODELING 83
A. System Description 84
To regulate the color and luminous intensity of RGB LED 85
lighting, a novel control structure is proposed, as shown in Fig. 1. 86
In this structure, TCCr and Φr , respectively, represent the cor- 87
related color temperature (CCT) and total luminous intensity 88
commands, while Φ is the luminous intensity output. Using 89
0885-8993/$26.00 © 2009 IEEE
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Fig. 1. Control structure of the RGB LED lighting system (solid lines: scalarsignals; mesh lines: 3 × 1 vector signals).
Fig. 2. Illustration of multiphysical phenomenon for RGB LED luminaire.
a lookup table M , the commands are converted to the corre-90
sponding radiant power signal LC = [LC R LC G LC B ]T ,91
in which the subscripts “R ,” “G ,” and “B ,” respectively,92
represent the “red,” “green,” and “blue” components of the93
signal. The controller K is used to calculate a suitable elec-94
trical power PE = [PR PG PB ]T according to the error95
signal e. Furthermore, the dynamics of the RGB LED lumi-96
naire are modeled as GE , with the output of luminous inten-97
sity ΦLED = [ ΦR ΦG ΦB ]T . The summation matrix U is98
defined as U = [ 1 1 1 ]1×3 such that the total luminous in-99
tensity Φ is the combination of individual luminous intensity,100
i.e., Φ = UΦLED = ΦR + ΦG + ΦB .101
The RGB LED luminaire is a lighting fixture composed of102
multiple RGB LED lamps. The RGB color LEDs can be oper-103
ated by three individual electrical power sources to emit photons104
for lighting and simultaneously generate heat to raise junction105
temperature. Then the photons can stimulate retinas to produce106
luminous and chromatic perception, as illustrated in Fig. 2.107
The electrical power PE can be normalized as 0 ≤ PE ≤ 1,108
compared to the maximum power, and further divided into the109
following two terms:110
PE = PT + PO (1)
Fig. 3. Electrical—thermal–luminous model.
where PT is the normalized thermal power for heat generation 111
and PO is the normalized optical power for lighting. Therefore, 112
PT and PO can be represented as 113
PT = (I − α) PE (2)
PO = αPE (3)
where α is the diagonal power factor matrix, which represents 114
the quantum efficiency of the LEDs. 115
Therefore, the LED luminaire model GE can be described 116
as a combination of three submodels, namely, the electrical– 117
thermal (E-T ) model H , the electrical–luminous (E-L) model 118
EP , and the thermal–luminous (T -L) model ET , as illustrated 119
in Fig. 3, in which the luminous intensity ΦLED is expressed as 120
ΦLED = ΦP + ΦT = EP PE + ET Tj = (EP + ET H) PE
(4)where Tj = [TR TG TB ]T is the junction temperature, i.e., 121
the dynamic model of GE can be represented as 122
GE =ΦLED
PE= EP + ET H. (5)
The three submodels of the RGB LED luminaire can be de- 123
rived by the input–output relation. First, the E-T model H 124
represents the influence of junction temperature by the thermal 125
power PT as in the following relation: 126
∆Tj = HPE =
HRR HGR HBR
HRG HGG HBG
HRB HGB HBB
PR
PG
PB
(6)
where ∆Tj represents the variation of junction temperature. 127
Second, the T -L model ET represents the luminous intensity 128
variation by the junction temperature as follows: 129
ΦT = ET ∆Tj =
ET R 0 0
0 ET G 0
0 0 ET B
∆TR
∆TG
∆TB
. (7)
Third, the E-L model EP represents the luminous intensity 130
variation by optical power PO as in the following:131
ΦP = EP PE =
EP R 0 0
0 EP G 0
0 0 EP B
PR
PG
PB
. (8)
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 3
Fig. 4. Illustration of RGB LED driving circuits.
B. System Identification of RGB LED Luminaire132
The RGB LED luminaire system was used for system identi-133
fication. As illustrated in Fig. 4, five RGB LED lamps [16] were134
installed on a 900-g aluminum heat sink (see Fig. 2) to allow135
the junction temperature variation by self-heating to be kept136
small through good thermal design. Four lamps were packaged137
in the front side for lighting, while the fifth was combined with138
a silicon photodiode [17] and assembled inside the luminaire139
to measure the junction temperature and radiant power (i.e., the140
fifth LED was used as sensors) [18]. In addition, each LED141
was driven by a 350 mA constant dc pulsewidth modulation142
(PWM), whose switching frequency was set at 120 Hz to avoid143
flick perception [18], [19]. According to the duty cycle com-144
mands, the normalized irreducible tensorial matrix (NITM) data145
acquisition (DAQ) system generated corresponding transistor–146
transistor logic (TTL) PWM signals, which were then connected147
to MOSFETs to drive the LEDs. Three independent circuits were148
used for power operation and measurement of the RGB LEDs149
through the DAQ system. The electrical power PE could be150
decided by the duty cycles of the PWM signals.151
The junction temperature could be estimated by the inside152
LED lamp using the pulse forward voltage method [20]–[23].153
At first, given a 1 mA constant current input for 50 µs, the154
temperature-sensitive parameter ST is obtained from the exper-155
iments by comparing the junction temperature and the voltage156
output as follows:157
ST =
ST R 0 0
0 ST G 0
0 0 ST B
=
1.82 0 0
0 5.90 0
0 0 2.20
× 10−3 .
(9)Therefore, the junction temperature Tj can be esti-158
mated by measuring the average forward voltage VLOW =159
[ VR VG VB ]T at the OFF interval of dc PWM by using160
1 mA constant current, as in the following:161
Tj = ST VLOW . (10)
Meanwhile, the radiant powers of RGB LEDs can be mea-162
sured by the silicon photodiode using the time-division method,163
in which the sensed radiant power LS = [LR LG LB ]T is164
calculated by the photodiode response, given time-shift PWM165
TABLE IEXPERIMENTAL RESULTS OF PHOTODIODE MODEL
Fig. 5. Experiment responses of Φ versus LR .
Fig. 6. Apparatus for measurement and data logging of total luminous in-tensity, correlative color temperature, and chromaticity coordinate in CIE 1976UCS.
signals, as [4] 166
LS = SD ΦLED =
SDR 0 0
0 SDG 0
0 0 SDB
ΦLED (11)
in which the photodiode model SD was obtained from the ex- 167
periments, as illustrated in Table I. For example, in experi- 168
ment R1, the electrical power for the green and blue LEDs was 169
fixed at PG = 20% and PB = 14%. Then, the electrical power 170
for the red LED was changed from PR = 50% to PR = 90%. 171
The corresponding luminous intensity Φ and the sensed ra- 172
diant power LR were measured, as shown in Fig. 5, to model 173
LR = SDRΦR = 0.0291ΦR using the linear regressive method. 174
Note that the variation of Φ equals the variation of ΦR since 175
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Fig. 7. Experimental response of electrical-thermal model HRR . (a) Time-domain responses. (b) Frequency-domain responses.
Φ = ΦR + ΦG + ΦB . In Table I, SDR from the three experi-176
ments (R1, R2, and R3) are similar, such that an average value177
SDR = 0.0287 was selected to represent the model. Similarly,178
SDG and SDB were experimentally obtained as follows:179
SD =
SDR 0 0
0 SDG 0
0 0 SDB
=
0.0287 0 0
0 0.0212 0
0 0 0.1077
.
(12)180
A set of instruments was built to measure the luminous and181
chromatic outputs of the system. As illustrated in Fig. 6, the182
polychromatic light output was projected into an integrating183
sphere for color mixing, such that the total luminous intensity Φ184
could be measured by the photopic detector. In addition, the light185
spectrum was acquired by a spectrometer to allow calculation of186
CCT TC C and chromaticity coordinate W in Cleveland Institute187
of Electronics (CIE) 1976 uniform chromaticity scale (UCS)188
[19]. A personal computer was used for process control and189
data logging.190
The dynamics of the RGB LED luminaire GE can be ob-191
tained by the identification of the three submodels in (6)–(8).192
First, for the E-,T model H , the experiments were carried193
out as in the following. At first, the maximum power was194
set as PE,max = [ 1.21 2.56 1.27 ]T W for a single RGB195
LED lamp, and the normalized operation power was set as196
PE = [ 30 30 30 ]T %. Then, step perturbations of PR , PG ,197
and PB were applied, in turn, as system inputs, and the corre-198
sponding junction temperature variations were measured as sys-199
tem outputs. For example, Fig. 7(a) illustrates the system output200
of the experiment R1 (with a step input PR from 30% to 65%).201
Therefore, HRR can be obtained by the Rake’s method [24] as202
follows:203
HRR(s) =0.0659(s + 0.00153)
(s + 0.00083).
204
The experimental time-domain data were transferred to fre-205
quency domain by the fast Fourier transform (FFT) and com-206
pared with the bode plot of HRR(s), as illustrated in Fig. 7(b).207
From the comparison of time-domain and frequency-domain 208
responses in Fig. 7, the first-order model is sufficient to capture 209
the basic system dynamics, as discussed in [25]. The results of 210
system identification at different operating points are illustrated 211
in Table II. 212
The T -L model ET represents the transmission path from 213
junction temperature to luminous intensity, which can be de- 214
scribed as a constant gain due to the short lifetime of pho- 215
tons [26], [27]. The identification was conducted at different 216
operating points, as illustrated in Table III, where the heat sink 217
was heated by a thermal pad. The identification results obtained 218
by measuring the junction temperature and the corresponding 219
luminous intensity are shown in Table III. Fig. 8 illustrates the 220
variation of ET R at the three operating conditions. 221
Similarly, the E-L model EP represents the transmission 222
path from electrical power PE to luminous intensity, which can 223
also be considered a constant gain [26], [27]. The experiments 224
were the same as the previous identification of ET , but with 225
the electrical power PE and luminous intensity as system inputs 226
and outputs, respectively. The operating points and identification 227
results are illustrated in Table IV. Fig. 9 illustrates the variations 228
of EP R at the six operating conditions. 229
C. Feedforward Compensator 230
The feedforward compensator M is a lookup table for con- 231
verting the CCT TCCr and total luminous intensity Φr inputs 232
into the corresponding radiant power LC at different junction 233
temperature Tj and nominal input power PE in order to main- 234
tain consistent light output. Therefore, the multidimensional 235
function M can be described as 236
LC = M (TCCr ,Φr , Tj , PE ) (13)
such that the radiant power vector LC is determined by the 237
inputs TCCr and Φr , and the operating conditions Tj and PE . 238
In experiments, the values of M are measured at many operat- 239
ing points, and finally, decided upon by using the interpolation 240
method. For example, Table V illustrates the relations of LC to 241
TCCr and Φr [18]. 242
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 5
TABLE IIIDENTIFICATION RESULTS OF THE ELECTRICAL-THERMAL MODEL H
TABLE IIIIDENTIFICATION RESULTS OF THE THERMAL-LUMINOUS MODEL ET
III. ROBUST CONTROL DESIGN243
From the previous identification results, the model varia-244
tion was noted and should be considered for the controller245
design. Robust control is well known for its ability to cope246
with system variations and disturbances. Therefore, in this sec-247
tion, robust control strategies will be introduced. From the248
analyses of gap metrics and coprime factorization, a robust249
controller is designed that provides the maximum stability250
bound for the RGB LED lighting system. The resulting con-251
troller will then be implemented and experimentally verified in252
Section IV.253
Fig. 8. Experimental response of the thermal–luminous model ETR.
Theorem 1 (Small Gain Theorem [28]): Suppose that Z ∈ 254
RH∞ and let γ > 0. Then, the interconnected system shown 255
in Fig. 10 is well posed and internally stable for all ∆(s) ∈ 256
RH∞ with: 1) ‖∆‖∞ ≤ 1/γ if and only if ‖Z (s)‖∞ < γ and 257
2) ‖∆‖∞ < 1/γ if and only if ‖Z (s)‖∞ ≤ γ, where ‖Z‖∞ is 258
the ∞ norm of system Z. 259
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TABLE IVIDENTIFICATION OF ELECTRICAL-LUMINOUS MODEL EP
TABLE VPARTIAL LOOKUP TABLE OF RADIANT POWER FOR RED LEDS
Assume that a nominal plant G0 can be expressed as G0 =260
M−1N , where: l) M, N ∈ RH∞ and 2) MM ∗ + NN ∗ = I∀ω.261
This is called the normalized left coprime factorization of G0 .262
In addition, suppose that a perturbed system G∆ is represented263
as264
G∆ =(M + ∆M
)−1 (N + ∆N
)(14)
with ‖[ ∆M ∆N ]‖∞ < ε and ∆M ,∆N ∈ RH∞. Considering265
the control structure of Fig. 11, the system transfer function can266
rearranged as follows:267
[z1z2
]=
[KI
](I − G0K)−1 M−1
ω =[
KI
](I − G0K)−1 [ I G0 ] ω
ω = [ ∆M ∆N ][
z1z2
]. (15)
268
Therefore, from Theorem 1, the closed-loop system remains269
internally stable for all ‖[ ∆M ∆N ]‖∞ < ε if and only if270
∥∥∥∥[
KI
](I − G0K)−1 [ I G0 ]
∥∥∥∥∞
≤ 1ε. (16)
Furthermore, the stability margin of the system can be defined271
as follows.272
Fig. 9. Experimental response of the electrical–luminous model EP R .
Fig. 10. Illustration of small gain theorem.
Fig. 11. Block diagram of the perturbed plant G∆ with controller K .
Definition 1 (Stability Margin [29]): The stability margin 273
b (G,K) of the closed-loop system is defined as 274
b (G,K) =∥∥∥∥[
KI
](I − GK)−1 [ I G ]
∥∥∥∥−1
∞. (17)
Hence, from Theorem 1, the closed-loop system is internally 275
stable for all ‖[ ∆M ∆N ]‖∞ < ε if and only if b (G,K) ≥ ε. 276
However, the coprime factorization of a system may not be 277
unique. Hence, the gap between two systems G0 and G∆ is 278
defined as follows. 279
Definition 2 (Gap Metric [28]): The smallest value of 280
‖[∆M ,∆N ]‖∞ that perturbs G0 into G∆ is called the gap be- 281
tween G0 and G∆ , and is denoted by δ (G0 , G∆). 282
From the definitions, b (G,K) gives the radius (in terms of 283
gap metric) of the largest ball of plants stabilized by the con- 284
troller K. Therefore, the goal of the controller design is to derive 285
a suitable controller K from a nominal plant G0 , such that all 286
perturbed plants Gi located inside the gap δ(G0 , Gi) < ε will 287
satisfy b(G,K) ≥ ε and the closed-loop system will remain 288
internally stable, i.e., the variations (tolerances) of LEDs can 289
be experimentally tested in the manufacturing processes, and 290
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 7
used to design suitable robust controllers that can guarantee the291
system stability.292
A. Selection of the Nominal Plant293
The selection of the nominal plant G0 for the RGB LED294
luminaire was based on the system gaps between all transfer295
functions of GE such that the maximum gap is minimized as296
minG0
maxGi
δ (G0 , Gi) (18)
where Gi represents all possible models of GE . Consider-297
ing (5), where GE = Ep + ET H with the models of H ,298
EP , and ET at all operating conditions, it can be derived299
that δ (G0 , Gi) ≤ δ (EP 0 , EP i) + δ (ET 0H0 , ET iHi). Further-300
more, it is noted that δ (EP 0 , EP i) ≤ 0.0075 from Table III, and301
δ (ET 0H0 , ET iHi) ≤ 0.3748 from Tables I and II, (*) shown at302
the bottom of this page.303
Therefore, δ (G0 , Gi) ≤ δ (EP 0 , EP i) + δ(ET 0H0 , ET iHi)304
= 0.3823 with the following nominal plant G0 = EP 0 +305
ET 0H0 for the RGB LED luminaire: shown (19) at the bot-306
tom of this page.307
Note that the maximum gap can be regarded as the maximum308
system perturbation due to the variation of operating conditions,309
such as the input power PE and the junction temperature Tj .310
B. Controller Synthesis311
The design procedures of the robust controller can be illus-312
trated as follows [30]–[33].313
1) Loop-shaping design: The nominal plant G0 is shaped by314
precompensator W1 and postcompensator W2 to form a315
shaped plant Gs = W2GW1 , as shown in Fig. 12(a).316
Fig. 12. Design procedures of robust controllers.
Fig. 13. Simplified RGB LED lighting control system for robust controllerdesign.
2) Robust stabilization estimate: The maximum stability 317
margin bmax is defined as 318
bmax (Gs,K)∆= inf
K stablizing
∥∥∥∥[
KI
](I−GsK)−1 [ I Gs ]
∥∥∥∥−1
∞(20)
where Ms and Ns are the normalized left coprime factor- 319
ization of Gs , i.e., Gs = M−1s Ns . If bmax (Gs,K) << 320
1, then one must return to step (1) to modify W1 321
H0 =
0.0659(s + 0.00153)(s + 0.00083)
0.0577(s + 0.00368)(s + 0.00087)
0.0318(s + 0.00346)(s + 0.00085)
0.0268(s + 0.00229)(s + 0.00083)
0.1853(s + 0.00157)(s + 0.00087)
0.0404(s + 0.00245)(s + 0.00084)
0.0264(s + 0.00212)(s + 0.00082)
0.1204(s + 0.00167)(s + 0.00083)
0.1691(s + 0.00121)(s + 0.00085)
ET 0 =
−10.09 0 0
0 −3.09 00 0 −0.54
EP 0 =
15.572 0 0
0 67.510 00 0 10.229
. (∗)
G0 (s) =
14.9071(s + 0.00153)(s + 0.00083)
−0.5822(s + 0.00368)(s + 0.00087)
−0.3209(s + 0.00346)(s + 0.00085)
−0.0828(s + 0.00229)(s + 0.00083)
66.9374(s + 0.00157)(s + 0.00087)
−0.1248(s + 0.00245)(s + 0.00084)
−0.0143(s + 0.00212)(s + 0.00082)
−0.0650(s + 0.00167)(s + 0.00083)
10.1377(s + 0.00121)(s + 0.00085)
. (19)
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Fig. 14. Implementation of RGB LED lighting control system. (a) Illustration of the control structure. (b) Layouts of the experimental instruments. (c)Experimental settings.
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 9
Fig. 15. Experimental responses to constant radiant power inputs with thermal disturbances. (a) Temperature variations. (b) Radiant power responses Ls .(c) Total luminous intensity Φ. (d) Color difference in CIE 1976 UCS ∆u′v ′.
and W2 . Then, an ε ≤ bmax (Gs,K) is selected to322
synthesize a stabilizing controllerK∞, which satisfies323 ∥∥∥∥[
K∞I
](I − GsK∞)−1 [ I Gs ]
∥∥∥∥−1
∞≥ ε, as shown in324
Fig. 12(b).325
3) Finally, the designed controller K∞ is multiplied by the326
weighting functions, i.e., K = W1K∞W2 , and imple-327
mented to control system G, as illustrated in Fig. 12(c).328
In Fig. 1, the compensator M converted the commands TCCr329
and Φr into corresponding radiant power signal LC . Therefore,330
the controller design can be simplified as Fig. 13, in which the331
nominal plant of the RGB LED luminaire is defined as G0 =332
SD G0 . Using the aforementioned controller design techniques,333
the optimal H∞ robust controller was designed as334
K(s) =
−0.4196 0.0098 0.0047
0.0146 −1.4141 0.00470.0135 0.0122 −1.0898
(21)
with a stability bound b(G0 ,K
)= 1, which is much larger than335
the system gap. Therefore, the controller can stabilize the system336
even with plant perturbations. However, the steady state of LS337
due to a unit step input LC = [ 1 1 1 ]T can be calculated as338
TABLE VISTATISTICAL DATA FROM FIG. 15
follows: 339
limt→∞
LS (t) = lims→0
sG0K(I + G0K
)−1 1s
= [ 1.1072 1.0075 1.0696 ]T
340
i.e., there is steady-state error due to a unit step input. Therefore, 341
the following weighting function with integrals was used to 342
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Fig. 16. System responses to a serial luminous intensity step input. (a) Total luminous intensity. (b) Color difference in CIE 1976 UCS. (c) Radiant power.(d) Temperature variation.
eliminate the steady-state error [34]:343
W1 (s) =1s
10.4196
0 0
01
1.41410
0 01
1.0898
.
Using the loop-shaping techniques with Gs = G0W1 , the344
robust controller was design as345
K ′(s) = W1K∞ =1s
−2.3832 0.0042 −0.0031−0.0012 −0.7072 −0.00200.0012 0.0026 −0.9176
with a stability bound b(G0 ,K′) = 0.7071, which was smaller346
than in the previous design (1), but still much larger than the347
maximal system gap (0.3823). However, the integral terms can348
guarantee zero steady-state error of the system due to step in-349
puts, i.e., the use of W1 sacrificed a little stability bound, but350
guaranteed zero steady-state errors to a step command. There-351
fore, the choice of weighting functions was a compromise be-352
tween system performance and stability specifications. In the353
next section, the designed controller K′
will be implemented354
for experimental verification.355
IV. EXPERIMENTAL RESULTS AND DISCUSSION 356
The experimental setup of the RGB LED lighting control 357
system is illustrated in Fig. 14. Fig. 14(a) illustrates the control 358
structure in which the measurement and control signals were 359
transmitted through a DAQ system, NI PCI6229, to the PC- 360
based controller. Fig. 14(b) shows the experimental layouts for 361
the control loop (on the left) and data measurement (on the 362
right) to verify the output luminous and chromatic properties. 363
The overall experimental settings are illustrated in Fig. 14(c). 364
For controller implementations, K ′(s) was first converted into 365
discrete time as in the following: 366
K ′(z) =1
z − 1
−0.2383 0.0004 −0.0003−0.0001 −0.0707 −0.00020.0001 0.0003 −0.0918
with a sampling time T = 0.1 s. During the experiments, the 367
CCT TCC0 was set at 6000 K. Implemented with the con- 368
troller K ′, the system performance can be discussed by the 369
rms error of the luminous intensity and the color difference of 370
chromaticity coordinate outputs, which should be as small as 371
possible. The color difference is defined in CIE 1976 UCS as 372
∆u′v′ =√
(u′ − u′0)
2 − (v′ − v′0)
2 , in which u′ and v′ are the 373
actual chromaticity coordinates, while u′0 and v′
0 are the desired 374
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 11
TABLE VIISTATISTICAL DATA FROM FIG. 16
chromaticity coordinates. In order to maintain steady chromatic-375
ity coordinate outputs, ∆u′v′ should be lower than the limitation376
of just noticeable difference [19], [35], and four-step Macadam377
ellipse for light source standard [36]–[38], i.e., ∆u′v′ < 0.0035.378
Setting the correlated color temperature at TCCr = 6000379
K, two experiments were designed to verify the system per-380
formance. In the first experiment, consider the control sys-381
tem of Fig. 13 with the radiant power input command set as382
LC = [ 36.569 35.988 22.923 ]T , which represents the lu-383
minous intensity Φ0 = 3000 cd. At first, the corresponding av-384
erage power consumption of the RGB LED luminaire was mea-385
sured as 7.97 W (PR = 3.71 W, PG = 3.00 W, and PB = 1.26386
W). Then, thermal disturbances were introduced into the system387
by attaching a thermal resistor to the RGB LED luminaire to388
modify the junction temperature. The thermal powers were ap-389
plied as 0 W→ 5 W→ 10 W→ 15 W. The experimental results390
are shown in Fig. 15. First, the system temperature is perturbed391
during the experiments, as shown in Fig. 15(a). However, using392
the designed controller, the output radiant power Ls can fol-393
low the input command LC = [ 36.569 35.988 22.923 ]T ,394
as illustrated in Fig. 15(b), despite the temperature variations.395
Second, from Fig. 15(c), the output luminous intensity Φ can396
also remain at the desired 3000 cd. Finally, the chromatic out-397
put responses shown in Fig. 15(d) indicate the color difference398
∆u′v′ < 0.002 during the experiments. In addition, the system399
H∞ norm gives the superior ratio of the output two-norm to400
the input two-norm (i.e., the input energy) [28]. Therefore, the401
color difference ∆u′v′ tends to be larger when the applied ther-402
mal power is larger. Furthermore, the statistical data of Fig. 15403
are illustrated in Table VI, in which the rms errors of the radiant404
power and total luminous intensity remained relatively small,405
even with the thermal disturbances. Therefore, the designed406
controller is effective in regulating the luminous intensity and407
the chromaticity coordinate outputs.408
For the second experiment, the feedforward compensator M409
was added (see Fig. 1) such that the system outputs can follow410
the luminous intensity commands. First, the input was set to411
change from 2500 to 4000 cd with an interval of 500 cd. Using412
the proposed control structure, the output luminous intensity413
can track the input, as shown in Fig. 16(a). Second, the color414
differences was within the limitation (∆u′v′ =< 0.0035), as415
illustrated in Fig. 16(b). Third, the corresponding radiant power 416
Ls follows the corresponding radiant power from the function 417
of (13), as shown in Fig. 16(c). Finally, Fig. 16(d) illustrated 418
the temperature variations during the experiments. From these 419
results, the proposed control structure was damned effective, 420
i.e., the system responses can track the commands despite the 421
temperature perturbations. Table VII summarizes the statistical 422
data of Fig. 16, and showed excellent system performance using 423
the designed robust controller and feedforward compensator M . 424
V. CONCLUSION 425
This paper has proposed a novel control structure for an RGB 426
LED lighting system, in which a lookup table was used as the 427
feedforward control to compensate for the variation of junc- 428
tion temperature. First, the RGB LED luminaire was modeled 429
as a multivariable system with three submodels, whose transfer 430
functions were then experimentally identified. By selecting the 431
nominal plants, the system variations were regarded as system 432
uncertainties and disturbances that were treated by the proposed 433
robust controllers. Second, robust controllers were designed 434
to guarantee system stability and performance using suitable 435
weighting functions. In practice, the tolerances of LEDs can be 436
experimentally tested to design suitable H∞ robust controllers 437
for system stability and performance. Finally, the designed con- 438
troller was implemented for experimental verification. From 439
the results, the proposed control structure and controller design 440
were shown to be effective. It is noted that the designed robust 441
controllers are relatively simple compared to other advanced 442
control algorithms, and the feedback control structure can be 443
easily miniaturized by a microprocessor, which costs less than 444
three US dollars, as shown in [33]. 445
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[4] S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue 455LEDs for white light illumination,” IEEE J. Sel. Topics Quantum Elec- 456tron., vol. 8, no. 2, pp. 333–338, Mar./Apr. 2002. 457
[5] S. Muthu and J. Gaines, “Red, green, and blue LED-based white light 458source: Implementation challenges and control design,” in Proc. Ind. Appl. 459Conf., 2003, pp. 515–522. 460
[6] C. Hoelen, J. Ansems, P. Deurenberg, T. Treurniet, E. van Lier, O. Chao, 461V. Mercier, G. Calon, K. van Os, G. Lijten, and J. Sondag-Huethorst, 462“Multi-chip color variable LED spot modules,” Proc. SPIE, vol. 5941, 463pp. 59410A-1–59410A-12, 2005. 464
[7] P. Deurenberg, C. Hoelen, J. van Meurs, and J. Ansems, “Achieving color 465point stability in RGB multi-chip LED modules using various color control 466loops,” Proc. SPIE, vol. 5941, pp. 63–74, 2005. 467
[8] C. Hoelen, J. Ansems, P. Deurenberg, W. van Duijneveldt, M. Peeters, 468G. Steenbruggen, T. Treurniet, A. Valster, and J. W. ter Weeme, “Color tun- 469able LED spot lighting,” Proc. SPIE, vol. 6337, pp. 63370Q-1–63370Q- 47015, 2006. 471
[9] S. Robinson and I. Ashdown, “Polychromatic optical feedback control, 472stability, and dimming,” Proc. SPIE, vol. 6337, pp. 633714-1–633714-10, 4732006. 474
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[12] B. Ackermann, V. Schulz, C. Martiny, A. Hilgers, and X. Zhu, “Control of481LEDs,” in Proc. IEEE Ind. Appl. Conf., 41th IAS Annu. Meeting, Tampa,482FL, Oct. 8–12, 2006, pp. 2608–2615.483
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[16] Everlight Electronic Co. RGGB High Power LED—4W Datasheet EHP-493B02 [Online]. Available: http://www.everlight.com/494
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[25] B.-J. Huang, C.-W. Tang, and M.-S. Wu, “System dynamics model of high-516power LED luminaire,” Appl. Therm. Eng., vol. 29, no. 4, pp. 609–616,5172009.518
[26] E. F. Schubert, Light-Emitting Diodes. Cambridge, U.K.: Cambridge519Univ. Press, 2003.520
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[28] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper524Saddle River, NJ: Prentice-Hall, 1996.525
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[31] F.-C. Wang, Y.-P. Yang, C.-W. Huang, H.-P. Chang, and H.-T. Chen,532“System identification and robust control of a portable proton exchange533membrane full-cell system,” J. Power Sources, vol. 164, no. 2, pp. 704–534712, Feb. 2007.535
[32] F.-C. Wang, H.-T. Chen, Y.-P. Yang, and J.-Y. Yen, “Multivariable robust536control of a proton exchange membrane fuel cell system,” J. Power537Sources, vol. 177, no. 2, pp. 393–403, Mar. 2008.538
[33] F.-C. Wang and H.-T. Chen, “Design and implementation of fixed-order539robust controllers for a proton exchange membrane fuel cell system,” Int.540J. Hydrogen Energy, vol. 34, no. 6, pp. 2705–2717, Mar. 2009.541
[34] G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control System Design.542Upper Saddle River, NJ: Prentice-Hall, 2001.543
[35] D. L. MacAdam, Color Measurement: Theme and Variations, 2nd ed.544New York: Springer-Verlag, 1985.545
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Fu-Cheng Wang (S’01–M’03) was born in Taipei, 553Taiwan, in 1968. He received the B.S. and M.Sc. de- 554grees in mechanical engineering from the National 555Taiwan University, Taipei, in 1990 and 1992, re- 556spectively, and the Ph.D. degree in control engineer- 557ing from Cambridge University, Cambridge, U.K., in 5582002. 559
From 2001 to 2003, he was a Research Associate 560in the Control Group, Engineering Department, Uni- 561versity of Cambridge. Since 2003, he has been with 562the Control Group, Mechanical Engineering Depart- 563
ment, National Taiwan University, where he is currently an Associate Professor. 564His current research interests include robust control, fuel cell control, LED 565control, inerter research, suspension control, medical engineering, embedded 566systems, and fuzzy systems. 567
568
Chun-Wen Tang was born in Taipei, Taiwan, in 5691976. He received the B.S. degrees in mechanical 570engineering from the National Taiwan University of 571Science and Technology, Taipei, in 1990, and the 572M.Sc. and Ph.D. degrees from the Control Group, 573Mechanical Engineering Department, National Tai- 574wan University, Taipei, in 2000 and 2009, respec- 575tively. 576
From 2002 to 2004, he was an Engineer in the 577Electronics and Optoelectronics Research Laborato- 578ries, Industrial Technology Research Institute, Tai- 579
wan. He is currently an R&D Manager at Coretech Optical Company, Ltd., 580Hsinchu, Taiwan. His current research interests include system integration, ro- 581bust control, electronic cooling, LED package, and solid-state lighting. 582
583
Bin-Juine Huang received the Master’s degree in 584mechanical and chemical engineering from Case 585Western Reserve University, Cleveland, OH, and the 586Doctorate degree from Odessa State Academy of Re- 587frigeration, Odessa, Ukraine. 588
He is currently a Professor in the Department 589of Mechanical Engineering, National Taiwan Uni- 590versity, Taipei, Taiwan, where he is the Director of 591the Solar Energy Research Center (SERC), which is 592founded by the Global Research Partnership (GRP) 593Award of King Abdullah University of Science and 594
Technology (KAUST). He has devoted research to a broad array of fields, in- 595cluding energy systems (solar, photovoltaics (PV), geothermal, ocean thermal, 596wind, boiler, waste heat), cooling technology (absorption, ejector, desiccant, 597cryocoolers, thermoelectric), solid-state lighting (LED), and control technol- 598ogy. His research tries to bridge the gap between academia and industry. He has 599developed more than 30 products with industry. He is the author or coauthor of 600more than 200 academic papers and 150 technical reports. He holds more than 60160 worldwide patents. 602
Prof. Huang was a recipient of the 1927 Outstanding Youth of the Year 603Award, the 1991 National Outstanding Engineering Professor Award, the 1995 604Academician of Academy of Sciences of Technological Cybernetics of Ukraine 605Award, the 1996 Academician of International Academy of Refrigeration, 606Ukraine Branch Award, the 1996 Tong-Yuan Science and Technology Award, 607the 2000 Outstanding Researcher Award of the National Science Council, the 6082005 Science and Technology Award of China-Tech Foundation, and the 2005 609Solar and New Energy Contribution Award of the Solar and New Energy Society 610of Taiwan. 611
612
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QUERIES 613
Q1: Author: Please provide the IEEE membership details (membership grades and years in which these were obtained), if any, 614
for C.-W. Tang and B.-J. Huang. 615
Q2. Author: Please provide the year information in Refs. [16] and [17]. 616
Q3. Author: Please update Ref. [27], if possible. 617
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Multivariable Robust Control for a Red–Green–BlueLED Lighting System
1
2
Fu-Cheng Wang, Member, IEEE, Chun-Wen Tang, and Bin-Juine HuangQ1
3
Abstract—This paper proposes a novel control structure for a4red–green–blue (RGB) LED lighting system, and applies multivari-5able robust control techniques to regulate the color and luminous6intensity outputs. RGB LED is the next-generational illuminant for7general lighting or liquid crystal display backlighting. The most8important feature for a polychromatic illuminant is color adjusta-9bility; however, for lighting applications using RGB LEDs, color10is sensitive to temperature variations. Therefore, suitable control11techniques are required to stabilize both luminous intensity and12chromaticity coordinates. In this paper, a robust control system13was proposed for achieving luminous intensity and color consis-14tency for RGB LED lighting in a three-step process. First, a mul-15tivariable electrical–thermal model was used to obtain RGB LED16luminous intensity, in which a lookup table served as a feedfor-17ward compensator for temperature and power variations. Second,18robust control algorithms were applied for feedback control de-19sign. Finally, the designed robust controllers were implemented to20control the luminous and chromatic outputs of the system. From21the experimental results, the proposed multivariable robust con-22trol was damned effective in providing steady luminous intensity23and color for RGB LED lighting.24
Index Terms—Color difference, luminous intensity, red–green–25blue (RGB) LEDs, robust control, thermal–electrical–luminous26model.27
I. INTRODUCTION28
R ECENTLY, LED has been drawing much attention as a29
state-of-the-art illuminator because of its numerous ad-30
vantages, including energy savings, long lifetime, and environ-31
mental friendliness. Red–green–blue (RGB) LEDs can provide a32
wide color gamut for liquid crystal display (LCD) backlighting,33
as well as full color adjustability for general lighting applica-34
tions [1], [2]. This newly developed illuminant is the only light35
source currently capable of this type of vivid and dynamic light-36
ing performance. However, the tunable light outputs have been37
found to induce light consistency issues for RGB LED light-38
ing, because the luminous intensity and color outputs are easily39
influenced by junction temperature variations caused by self-40
heating of the LEDs and disturbances in ambient temperatures.41
Therefore, proper control strategies are required to stabilize light42
output in order to counteract temperature variations.43
Manuscript received February 25, 2009; revised April 15, 2009. Recom-mended for publication by Associate Editor M. Ponce-Silva.
F.-C. Wang and B.-J. Huang are with the Department of MechanicalEngineering, National Taiwan University, Taipei 10617, Taiwan (e-mail:[email protected]; [email protected]).
C.-W. Tang was with the Department of Mechanical Engineering, NationalTaiwan University, Taipei 10617, Taiwan. He is now with Coretech OpticalCompany Ltd., Hsinchu 30069, Taiwan (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPEL.2009.2026476
To control RGB LED lighting systems, the selection of feed- 44
back signals is an important issue. Muthu et al. [3]–[5] applied 45
three kinds of feedback system: color coordinate feedback with 46
temperature feedforward (CCFB and TFF), color coordinate 47
feedback (CCFB), and flux feedback with temperature feed- 48
forward (FFB and TFF). The color coordinates were measured 49
by photodiodes with color filters and the fluxes with photodi- 50
odes with a time-division method. In addition, the heat sink 51
temperature and thermal resistance were used to estimate junc- 52
tion temperature for temperature feedforward compensation. 53
Hoelen et al. [6]–[8] further discussed light outputs and applied 54
four control structures, namely, flux feedback, temperature feed- 55
forward, CCFB, and FFB and TFF. Among these, CCFB and 56
FFB and TFF were shown to provide better color consistency 57
for RGB LED lighting than did the others, when the system 58
was experiencing junction temperature variations. Until now, 59
CCFB has been a popular choice for application to control sys- 60
tem design [9]–[13] because of its simple structure. However, 61
the accuracy of feedback signals is limited by the difference 62
between the spectra of filtered sensor and color matching func- 63
tions. In contrast, the FFB and TFF structure can provide more 64
signals for control design, but requires double loops and infor- 65
mation about the junction temperature. For controller design, 66
traditional control methodologies such as proportional–integral 67
(PI) or PI derivative (PID) based algorithms have been applied to 68
control RGB LED lighting systems [5], [7], [14], [15]. However, 69
these methods cannot guarantee the stability and performance of 70
systems with perturbations such as varying input power or junc- 71
tion temperatures. Therefore, advanced control strategies should 72
be considered for improving system performance. In this paper, 73
a novel control structure is proposed, and robust control tech- 74
niques are applied, to achieve consistent luminous intensity and 75
color. The effect will be experimentally verified. 76
The paper is arranged as follows. In Section II, an RGB LED 77
luminaire is modeled as a multivariable system and a feedback 78
control structure is proposed. In Section III, robust control strate- 79
gies are introduced for multivariable controller design. Then, the 80
designed controller is implemented for performance analysis in 81
Section IV. Finally, some conclusions are drawn in Section V. 82
II. SYSTEM DESCRIPTION AND MODELING 83
A. System Description 84
To regulate the color and luminous intensity of RGB LED 85
lighting, a novel control structure is proposed, as shown in Fig. 1. 86
In this structure, TCCr and Φr , respectively, represent the cor- 87
related color temperature (CCT) and total luminous intensity 88
commands, while Φ is the luminous intensity output. Using 89
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2 IEEE TRANSACTIONS ON POWER ELECTRONICS
Fig. 1. Control structure of the RGB LED lighting system (solid lines: scalarsignals; mesh lines: 3 × 1 vector signals).
Fig. 2. Illustration of multiphysical phenomenon for RGB LED luminaire.
a lookup table M , the commands are converted to the corre-90
sponding radiant power signal LC = [LC R LC G LC B ]T ,91
in which the subscripts “R ,” “G ,” and “B ,” respectively,92
represent the “red,” “green,” and “blue” components of the93
signal. The controller K is used to calculate a suitable elec-94
trical power PE = [PR PG PB ]T according to the error95
signal e. Furthermore, the dynamics of the RGB LED lumi-96
naire are modeled as GE , with the output of luminous inten-97
sity ΦLED = [ ΦR ΦG ΦB ]T . The summation matrix U is98
defined as U = [ 1 1 1 ]1×3 such that the total luminous in-99
tensity Φ is the combination of individual luminous intensity,100
i.e., Φ = UΦLED = ΦR + ΦG + ΦB .101
The RGB LED luminaire is a lighting fixture composed of102
multiple RGB LED lamps. The RGB color LEDs can be oper-103
ated by three individual electrical power sources to emit photons104
for lighting and simultaneously generate heat to raise junction105
temperature. Then the photons can stimulate retinas to produce106
luminous and chromatic perception, as illustrated in Fig. 2.107
The electrical power PE can be normalized as 0 ≤ PE ≤ 1,108
compared to the maximum power, and further divided into the109
following two terms:110
PE = PT + PO (1)
Fig. 3. Electrical—thermal–luminous model.
where PT is the normalized thermal power for heat generation 111
and PO is the normalized optical power for lighting. Therefore, 112
PT and PO can be represented as 113
PT = (I − α) PE (2)
PO = αPE (3)
where α is the diagonal power factor matrix, which represents 114
the quantum efficiency of the LEDs. 115
Therefore, the LED luminaire model GE can be described 116
as a combination of three submodels, namely, the electrical– 117
thermal (E-T ) model H , the electrical–luminous (E-L) model 118
EP , and the thermal–luminous (T -L) model ET , as illustrated 119
in Fig. 3, in which the luminous intensity ΦLED is expressed as 120
ΦLED = ΦP + ΦT = EP PE + ET Tj = (EP + ET H) PE
(4)where Tj = [TR TG TB ]T is the junction temperature, i.e., 121
the dynamic model of GE can be represented as 122
GE =ΦLED
PE= EP + ET H. (5)
The three submodels of the RGB LED luminaire can be de- 123
rived by the input–output relation. First, the E-T model H 124
represents the influence of junction temperature by the thermal 125
power PT as in the following relation: 126
∆Tj = HPE =
HRR HGR HBR
HRG HGG HBG
HRB HGB HBB
PR
PG
PB
(6)
where ∆Tj represents the variation of junction temperature. 127
Second, the T -L model ET represents the luminous intensity 128
variation by the junction temperature as follows: 129
ΦT = ET ∆Tj =
ET R 0 0
0 ET G 0
0 0 ET B
∆TR
∆TG
∆TB
. (7)
Third, the E-L model EP represents the luminous intensity 130
variation by optical power PO as in the following:131
ΦP = EP PE =
EP R 0 0
0 EP G 0
0 0 EP B
PR
PG
PB
. (8)
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Fig. 4. Illustration of RGB LED driving circuits.
B. System Identification of RGB LED Luminaire132
The RGB LED luminaire system was used for system identi-133
fication. As illustrated in Fig. 4, five RGB LED lamps [16] were134
installed on a 900-g aluminum heat sink (see Fig. 2) to allow135
the junction temperature variation by self-heating to be kept136
small through good thermal design. Four lamps were packaged137
in the front side for lighting, while the fifth was combined with138
a silicon photodiode [17] and assembled inside the luminaire139
to measure the junction temperature and radiant power (i.e., the140
fifth LED was used as sensors) [18]. In addition, each LED141
was driven by a 350 mA constant dc pulsewidth modulation142
(PWM), whose switching frequency was set at 120 Hz to avoid143
flick perception [18], [19]. According to the duty cycle com-144
mands, the normalized irreducible tensorial matrix (NITM) data145
acquisition (DAQ) system generated corresponding transistor–146
transistor logic (TTL) PWM signals, which were then connected147
to MOSFETs to drive the LEDs. Three independent circuits were148
used for power operation and measurement of the RGB LEDs149
through the DAQ system. The electrical power PE could be150
decided by the duty cycles of the PWM signals.151
The junction temperature could be estimated by the inside152
LED lamp using the pulse forward voltage method [20]–[23].153
At first, given a 1 mA constant current input for 50 µs, the154
temperature-sensitive parameter ST is obtained from the exper-155
iments by comparing the junction temperature and the voltage156
output as follows:157
ST =
ST R 0 0
0 ST G 0
0 0 ST B
=
1.82 0 0
0 5.90 0
0 0 2.20
× 10−3 .
(9)Therefore, the junction temperature Tj can be esti-158
mated by measuring the average forward voltage VLOW =159
[ VR VG VB ]T at the OFF interval of dc PWM by using160
1 mA constant current, as in the following:161
Tj = ST VLOW . (10)
Meanwhile, the radiant powers of RGB LEDs can be mea-162
sured by the silicon photodiode using the time-division method,163
in which the sensed radiant power LS = [LR LG LB ]T is164
calculated by the photodiode response, given time-shift PWM165
TABLE IEXPERIMENTAL RESULTS OF PHOTODIODE MODEL
Fig. 5. Experiment responses of Φ versus LR .
Fig. 6. Apparatus for measurement and data logging of total luminous in-tensity, correlative color temperature, and chromaticity coordinate in CIE 1976UCS.
signals, as [4] 166
LS = SD ΦLED =
SDR 0 0
0 SDG 0
0 0 SDB
ΦLED (11)
in which the photodiode model SD was obtained from the ex- 167
periments, as illustrated in Table I. For example, in experi- 168
ment R1, the electrical power for the green and blue LEDs was 169
fixed at PG = 20% and PB = 14%. Then, the electrical power 170
for the red LED was changed from PR = 50% to PR = 90%. 171
The corresponding luminous intensity Φ and the sensed ra- 172
diant power LR were measured, as shown in Fig. 5, to model 173
LR = SDRΦR = 0.0291ΦR using the linear regressive method. 174
Note that the variation of Φ equals the variation of ΦR since 175
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Fig. 7. Experimental response of electrical-thermal model HRR . (a) Time-domain responses. (b) Frequency-domain responses.
Φ = ΦR + ΦG + ΦB . In Table I, SDR from the three experi-176
ments (R1, R2, and R3) are similar, such that an average value177
SDR = 0.0287 was selected to represent the model. Similarly,178
SDG and SDB were experimentally obtained as follows:179
SD =
SDR 0 0
0 SDG 0
0 0 SDB
=
0.0287 0 0
0 0.0212 0
0 0 0.1077
.
(12)180
A set of instruments was built to measure the luminous and181
chromatic outputs of the system. As illustrated in Fig. 6, the182
polychromatic light output was projected into an integrating183
sphere for color mixing, such that the total luminous intensity Φ184
could be measured by the photopic detector. In addition, the light185
spectrum was acquired by a spectrometer to allow calculation of186
CCT TC C and chromaticity coordinate W in Cleveland Institute187
of Electronics (CIE) 1976 uniform chromaticity scale (UCS)188
[19]. A personal computer was used for process control and189
data logging.190
The dynamics of the RGB LED luminaire GE can be ob-191
tained by the identification of the three submodels in (6)–(8).192
First, for the E-,T model H , the experiments were carried193
out as in the following. At first, the maximum power was194
set as PE,max = [ 1.21 2.56 1.27 ]T W for a single RGB195
LED lamp, and the normalized operation power was set as196
PE = [ 30 30 30 ]T %. Then, step perturbations of PR , PG ,197
and PB were applied, in turn, as system inputs, and the corre-198
sponding junction temperature variations were measured as sys-199
tem outputs. For example, Fig. 7(a) illustrates the system output200
of the experiment R1 (with a step input PR from 30% to 65%).201
Therefore, HRR can be obtained by the Rake’s method [24] as202
follows:203
HRR(s) =0.0659(s + 0.00153)
(s + 0.00083).
204
The experimental time-domain data were transferred to fre-205
quency domain by the fast Fourier transform (FFT) and com-206
pared with the bode plot of HRR(s), as illustrated in Fig. 7(b).207
From the comparison of time-domain and frequency-domain 208
responses in Fig. 7, the first-order model is sufficient to capture 209
the basic system dynamics, as discussed in [25]. The results of 210
system identification at different operating points are illustrated 211
in Table II. 212
The T -L model ET represents the transmission path from 213
junction temperature to luminous intensity, which can be de- 214
scribed as a constant gain due to the short lifetime of pho- 215
tons [26], [27]. The identification was conducted at different 216
operating points, as illustrated in Table III, where the heat sink 217
was heated by a thermal pad. The identification results obtained 218
by measuring the junction temperature and the corresponding 219
luminous intensity are shown in Table III. Fig. 8 illustrates the 220
variation of ET R at the three operating conditions. 221
Similarly, the E-L model EP represents the transmission 222
path from electrical power PE to luminous intensity, which can 223
also be considered a constant gain [26], [27]. The experiments 224
were the same as the previous identification of ET , but with 225
the electrical power PE and luminous intensity as system inputs 226
and outputs, respectively. The operating points and identification 227
results are illustrated in Table IV. Fig. 9 illustrates the variations 228
of EP R at the six operating conditions. 229
C. Feedforward Compensator 230
The feedforward compensator M is a lookup table for con- 231
verting the CCT TCCr and total luminous intensity Φr inputs 232
into the corresponding radiant power LC at different junction 233
temperature Tj and nominal input power PE in order to main- 234
tain consistent light output. Therefore, the multidimensional 235
function M can be described as 236
LC = M (TCCr ,Φr , Tj , PE ) (13)
such that the radiant power vector LC is determined by the 237
inputs TCCr and Φr , and the operating conditions Tj and PE . 238
In experiments, the values of M are measured at many operat- 239
ing points, and finally, decided upon by using the interpolation 240
method. For example, Table V illustrates the relations of LC to 241
TCCr and Φr [18]. 242
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TABLE IIIDENTIFICATION RESULTS OF THE ELECTRICAL-THERMAL MODEL H
TABLE IIIIDENTIFICATION RESULTS OF THE THERMAL-LUMINOUS MODEL ET
III. ROBUST CONTROL DESIGN243
From the previous identification results, the model varia-244
tion was noted and should be considered for the controller245
design. Robust control is well known for its ability to cope246
with system variations and disturbances. Therefore, in this sec-247
tion, robust control strategies will be introduced. From the248
analyses of gap metrics and coprime factorization, a robust249
controller is designed that provides the maximum stability250
bound for the RGB LED lighting system. The resulting con-251
troller will then be implemented and experimentally verified in252
Section IV.253
Fig. 8. Experimental response of the thermal–luminous model ETR.
Theorem 1 (Small Gain Theorem [28]): Suppose that Z ∈ 254
RH∞ and let γ > 0. Then, the interconnected system shown 255
in Fig. 10 is well posed and internally stable for all ∆(s) ∈ 256
RH∞ with: 1) ‖∆‖∞ ≤ 1/γ if and only if ‖Z (s)‖∞ < γ and 257
2) ‖∆‖∞ < 1/γ if and only if ‖Z (s)‖∞ ≤ γ, where ‖Z‖∞ is 258
the ∞ norm of system Z. 259
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TABLE IVIDENTIFICATION OF ELECTRICAL-LUMINOUS MODEL EP
TABLE VPARTIAL LOOKUP TABLE OF RADIANT POWER FOR RED LEDS
Assume that a nominal plant G0 can be expressed as G0 =260
M−1N , where: l) M, N ∈ RH∞ and 2) MM ∗ + NN ∗ = I∀ω.261
This is called the normalized left coprime factorization of G0 .262
In addition, suppose that a perturbed system G∆ is represented263
as264
G∆ =(M + ∆M
)−1 (N + ∆N
)(14)
with ‖[ ∆M ∆N ]‖∞ < ε and ∆M ,∆N ∈ RH∞. Considering265
the control structure of Fig. 11, the system transfer function can266
rearranged as follows:267
[z1z2
]=
[KI
](I − G0K)−1 M−1
ω =[
KI
](I − G0K)−1 [ I G0 ] ω
ω = [ ∆M ∆N ][
z1z2
]. (15)
268
Therefore, from Theorem 1, the closed-loop system remains269
internally stable for all ‖[ ∆M ∆N ]‖∞ < ε if and only if270
∥∥∥∥[
KI
](I − G0K)−1 [ I G0 ]
∥∥∥∥∞
≤ 1ε. (16)
Furthermore, the stability margin of the system can be defined271
as follows.272
Fig. 9. Experimental response of the electrical–luminous model EP R .
Fig. 10. Illustration of small gain theorem.
Fig. 11. Block diagram of the perturbed plant G∆ with controller K .
Definition 1 (Stability Margin [29]): The stability margin 273
b (G,K) of the closed-loop system is defined as 274
b (G,K) =∥∥∥∥[
KI
](I − GK)−1 [ I G ]
∥∥∥∥−1
∞. (17)
Hence, from Theorem 1, the closed-loop system is internally 275
stable for all ‖[ ∆M ∆N ]‖∞ < ε if and only if b (G,K) ≥ ε. 276
However, the coprime factorization of a system may not be 277
unique. Hence, the gap between two systems G0 and G∆ is 278
defined as follows. 279
Definition 2 (Gap Metric [28]): The smallest value of 280
‖[∆M ,∆N ]‖∞ that perturbs G0 into G∆ is called the gap be- 281
tween G0 and G∆ , and is denoted by δ (G0 , G∆). 282
From the definitions, b (G,K) gives the radius (in terms of 283
gap metric) of the largest ball of plants stabilized by the con- 284
troller K. Therefore, the goal of the controller design is to derive 285
a suitable controller K from a nominal plant G0 , such that all 286
perturbed plants Gi located inside the gap δ(G0 , Gi) < ε will 287
satisfy b(G,K) ≥ ε and the closed-loop system will remain 288
internally stable, i.e., the variations (tolerances) of LEDs can 289
be experimentally tested in the manufacturing processes, and 290
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 7
used to design suitable robust controllers that can guarantee the291
system stability.292
A. Selection of the Nominal Plant293
The selection of the nominal plant G0 for the RGB LED294
luminaire was based on the system gaps between all transfer295
functions of GE such that the maximum gap is minimized as296
minG0
maxGi
δ (G0 , Gi) (18)
where Gi represents all possible models of GE . Consider-297
ing (5), where GE = Ep + ET H with the models of H ,298
EP , and ET at all operating conditions, it can be derived299
that δ (G0 , Gi) ≤ δ (EP 0 , EP i) + δ (ET 0H0 , ET iHi). Further-300
more, it is noted that δ (EP 0 , EP i) ≤ 0.0075 from Table III, and301
δ (ET 0H0 , ET iHi) ≤ 0.3748 from Tables I and II, (*) shown at302
the bottom of this page.303
Therefore, δ (G0 , Gi) ≤ δ (EP 0 , EP i) + δ(ET 0H0 , ET iHi)304
= 0.3823 with the following nominal plant G0 = EP 0 +305
ET 0H0 for the RGB LED luminaire: shown (19) at the bot-306
tom of this page.307
Note that the maximum gap can be regarded as the maximum308
system perturbation due to the variation of operating conditions,309
such as the input power PE and the junction temperature Tj .310
B. Controller Synthesis311
The design procedures of the robust controller can be illus-312
trated as follows [30]–[33].313
1) Loop-shaping design: The nominal plant G0 is shaped by314
precompensator W1 and postcompensator W2 to form a315
shaped plant Gs = W2GW1 , as shown in Fig. 12(a).316
Fig. 12. Design procedures of robust controllers.
Fig. 13. Simplified RGB LED lighting control system for robust controllerdesign.
2) Robust stabilization estimate: The maximum stability 317
margin bmax is defined as 318
bmax (Gs,K)∆= inf
K stablizing
∥∥∥∥[
KI
](I−GsK)−1 [ I Gs ]
∥∥∥∥−1
∞(20)
where Ms and Ns are the normalized left coprime factor- 319
ization of Gs , i.e., Gs = M−1s Ns . If bmax (Gs,K) << 320
1, then one must return to step (1) to modify W1 321
H0 =
0.0659(s + 0.00153)(s + 0.00083)
0.0577(s + 0.00368)(s + 0.00087)
0.0318(s + 0.00346)(s + 0.00085)
0.0268(s + 0.00229)(s + 0.00083)
0.1853(s + 0.00157)(s + 0.00087)
0.0404(s + 0.00245)(s + 0.00084)
0.0264(s + 0.00212)(s + 0.00082)
0.1204(s + 0.00167)(s + 0.00083)
0.1691(s + 0.00121)(s + 0.00085)
ET 0 =
−10.09 0 0
0 −3.09 00 0 −0.54
EP 0 =
15.572 0 0
0 67.510 00 0 10.229
. (∗)
G0 (s) =
14.9071(s + 0.00153)(s + 0.00083)
−0.5822(s + 0.00368)(s + 0.00087)
−0.3209(s + 0.00346)(s + 0.00085)
−0.0828(s + 0.00229)(s + 0.00083)
66.9374(s + 0.00157)(s + 0.00087)
−0.1248(s + 0.00245)(s + 0.00084)
−0.0143(s + 0.00212)(s + 0.00082)
−0.0650(s + 0.00167)(s + 0.00083)
10.1377(s + 0.00121)(s + 0.00085)
. (19)
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Fig. 14. Implementation of RGB LED lighting control system. (a) Illustration of the control structure. (b) Layouts of the experimental instruments. (c)Experimental settings.
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WANG et al.: MULTIVARIABLE ROBUST CONTROL FOR A RED–GREEN–BLUE LED LIGHTING SYSTEM 9
Fig. 15. Experimental responses to constant radiant power inputs with thermal disturbances. (a) Temperature variations. (b) Radiant power responses Ls .(c) Total luminous intensity Φ. (d) Color difference in CIE 1976 UCS ∆u′v ′.
and W2 . Then, an ε ≤ bmax (Gs,K) is selected to322
synthesize a stabilizing controllerK∞, which satisfies323 ∥∥∥∥[
K∞I
](I − GsK∞)−1 [ I Gs ]
∥∥∥∥−1
∞≥ ε, as shown in324
Fig. 12(b).325
3) Finally, the designed controller K∞ is multiplied by the326
weighting functions, i.e., K = W1K∞W2 , and imple-327
mented to control system G, as illustrated in Fig. 12(c).328
In Fig. 1, the compensator M converted the commands TCCr329
and Φr into corresponding radiant power signal LC . Therefore,330
the controller design can be simplified as Fig. 13, in which the331
nominal plant of the RGB LED luminaire is defined as G0 =332
SD G0 . Using the aforementioned controller design techniques,333
the optimal H∞ robust controller was designed as334
K(s) =
−0.4196 0.0098 0.0047
0.0146 −1.4141 0.00470.0135 0.0122 −1.0898
(21)
with a stability bound b(G0 ,K
)= 1, which is much larger than335
the system gap. Therefore, the controller can stabilize the system336
even with plant perturbations. However, the steady state of LS337
due to a unit step input LC = [ 1 1 1 ]T can be calculated as338
TABLE VISTATISTICAL DATA FROM FIG. 15
follows: 339
limt→∞
LS (t) = lims→0
sG0K(I + G0K
)−1 1s
= [ 1.1072 1.0075 1.0696 ]T
340
i.e., there is steady-state error due to a unit step input. Therefore, 341
the following weighting function with integrals was used to 342
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Fig. 16. System responses to a serial luminous intensity step input. (a) Total luminous intensity. (b) Color difference in CIE 1976 UCS. (c) Radiant power.(d) Temperature variation.
eliminate the steady-state error [34]:343
W1 (s) =1s
10.4196
0 0
01
1.41410
0 01
1.0898
.
Using the loop-shaping techniques with Gs = G0W1 , the344
robust controller was design as345
K ′(s) = W1K∞ =1s
−2.3832 0.0042 −0.0031−0.0012 −0.7072 −0.00200.0012 0.0026 −0.9176
with a stability bound b(G0 ,K′) = 0.7071, which was smaller346
than in the previous design (1), but still much larger than the347
maximal system gap (0.3823). However, the integral terms can348
guarantee zero steady-state error of the system due to step in-349
puts, i.e., the use of W1 sacrificed a little stability bound, but350
guaranteed zero steady-state errors to a step command. There-351
fore, the choice of weighting functions was a compromise be-352
tween system performance and stability specifications. In the353
next section, the designed controller K′
will be implemented354
for experimental verification.355
IV. EXPERIMENTAL RESULTS AND DISCUSSION 356
The experimental setup of the RGB LED lighting control 357
system is illustrated in Fig. 14. Fig. 14(a) illustrates the control 358
structure in which the measurement and control signals were 359
transmitted through a DAQ system, NI PCI6229, to the PC- 360
based controller. Fig. 14(b) shows the experimental layouts for 361
the control loop (on the left) and data measurement (on the 362
right) to verify the output luminous and chromatic properties. 363
The overall experimental settings are illustrated in Fig. 14(c). 364
For controller implementations, K ′(s) was first converted into 365
discrete time as in the following: 366
K ′(z) =1
z − 1
−0.2383 0.0004 −0.0003−0.0001 −0.0707 −0.00020.0001 0.0003 −0.0918
with a sampling time T = 0.1 s. During the experiments, the 367
CCT TCC0 was set at 6000 K. Implemented with the con- 368
troller K ′, the system performance can be discussed by the 369
rms error of the luminous intensity and the color difference of 370
chromaticity coordinate outputs, which should be as small as 371
possible. The color difference is defined in CIE 1976 UCS as 372
∆u′v′ =√
(u′ − u′0)
2 − (v′ − v′0)
2 , in which u′ and v′ are the 373
actual chromaticity coordinates, while u′0 and v′
0 are the desired 374
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TABLE VIISTATISTICAL DATA FROM FIG. 16
chromaticity coordinates. In order to maintain steady chromatic-375
ity coordinate outputs, ∆u′v′ should be lower than the limitation376
of just noticeable difference [19], [35], and four-step Macadam377
ellipse for light source standard [36]–[38], i.e., ∆u′v′ < 0.0035.378
Setting the correlated color temperature at TCCr = 6000379
K, two experiments were designed to verify the system per-380
formance. In the first experiment, consider the control sys-381
tem of Fig. 13 with the radiant power input command set as382
LC = [ 36.569 35.988 22.923 ]T , which represents the lu-383
minous intensity Φ0 = 3000 cd. At first, the corresponding av-384
erage power consumption of the RGB LED luminaire was mea-385
sured as 7.97 W (PR = 3.71 W, PG = 3.00 W, and PB = 1.26386
W). Then, thermal disturbances were introduced into the system387
by attaching a thermal resistor to the RGB LED luminaire to388
modify the junction temperature. The thermal powers were ap-389
plied as 0 W→ 5 W→ 10 W→ 15 W. The experimental results390
are shown in Fig. 15. First, the system temperature is perturbed391
during the experiments, as shown in Fig. 15(a). However, using392
the designed controller, the output radiant power Ls can fol-393
low the input command LC = [ 36.569 35.988 22.923 ]T ,394
as illustrated in Fig. 15(b), despite the temperature variations.395
Second, from Fig. 15(c), the output luminous intensity Φ can396
also remain at the desired 3000 cd. Finally, the chromatic out-397
put responses shown in Fig. 15(d) indicate the color difference398
∆u′v′ < 0.002 during the experiments. In addition, the system399
H∞ norm gives the superior ratio of the output two-norm to400
the input two-norm (i.e., the input energy) [28]. Therefore, the401
color difference ∆u′v′ tends to be larger when the applied ther-402
mal power is larger. Furthermore, the statistical data of Fig. 15403
are illustrated in Table VI, in which the rms errors of the radiant404
power and total luminous intensity remained relatively small,405
even with the thermal disturbances. Therefore, the designed406
controller is effective in regulating the luminous intensity and407
the chromaticity coordinate outputs.408
For the second experiment, the feedforward compensator M409
was added (see Fig. 1) such that the system outputs can follow410
the luminous intensity commands. First, the input was set to411
change from 2500 to 4000 cd with an interval of 500 cd. Using412
the proposed control structure, the output luminous intensity413
can track the input, as shown in Fig. 16(a). Second, the color414
differences was within the limitation (∆u′v′ =< 0.0035), as415
illustrated in Fig. 16(b). Third, the corresponding radiant power 416
Ls follows the corresponding radiant power from the function 417
of (13), as shown in Fig. 16(c). Finally, Fig. 16(d) illustrated 418
the temperature variations during the experiments. From these 419
results, the proposed control structure was damned effective, 420
i.e., the system responses can track the commands despite the 421
temperature perturbations. Table VII summarizes the statistical 422
data of Fig. 16, and showed excellent system performance using 423
the designed robust controller and feedforward compensator M . 424
V. CONCLUSION 425
This paper has proposed a novel control structure for an RGB 426
LED lighting system, in which a lookup table was used as the 427
feedforward control to compensate for the variation of junc- 428
tion temperature. First, the RGB LED luminaire was modeled 429
as a multivariable system with three submodels, whose transfer 430
functions were then experimentally identified. By selecting the 431
nominal plants, the system variations were regarded as system 432
uncertainties and disturbances that were treated by the proposed 433
robust controllers. Second, robust controllers were designed 434
to guarantee system stability and performance using suitable 435
weighting functions. In practice, the tolerances of LEDs can be 436
experimentally tested to design suitable H∞ robust controllers 437
for system stability and performance. Finally, the designed con- 438
troller was implemented for experimental verification. From 439
the results, the proposed control structure and controller design 440
were shown to be effective. It is noted that the designed robust 441
controllers are relatively simple compared to other advanced 442
control algorithms, and the feedback control structure can be 443
easily miniaturized by a microprocessor, which costs less than 444
three US dollars, as shown in [33]. 445
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Fu-Cheng Wang (S’01–M’03) was born in Taipei, 553Taiwan, in 1968. He received the B.S. and M.Sc. de- 554grees in mechanical engineering from the National 555Taiwan University, Taipei, in 1990 and 1992, re- 556spectively, and the Ph.D. degree in control engineer- 557ing from Cambridge University, Cambridge, U.K., in 5582002. 559
From 2001 to 2003, he was a Research Associate 560in the Control Group, Engineering Department, Uni- 561versity of Cambridge. Since 2003, he has been with 562the Control Group, Mechanical Engineering Depart- 563
ment, National Taiwan University, where he is currently an Associate Professor. 564His current research interests include robust control, fuel cell control, LED 565control, inerter research, suspension control, medical engineering, embedded 566systems, and fuzzy systems. 567
568
Chun-Wen Tang was born in Taipei, Taiwan, in 5691976. He received the B.S. degrees in mechanical 570engineering from the National Taiwan University of 571Science and Technology, Taipei, in 1990, and the 572M.Sc. and Ph.D. degrees from the Control Group, 573Mechanical Engineering Department, National Tai- 574wan University, Taipei, in 2000 and 2009, respec- 575tively. 576
From 2002 to 2004, he was an Engineer in the 577Electronics and Optoelectronics Research Laborato- 578ries, Industrial Technology Research Institute, Tai- 579
wan. He is currently an R&D Manager at Coretech Optical Company, Ltd., 580Hsinchu, Taiwan. His current research interests include system integration, ro- 581bust control, electronic cooling, LED package, and solid-state lighting. 582
583
Bin-Juine Huang received the Master’s degree in 584mechanical and chemical engineering from Case 585Western Reserve University, Cleveland, OH, and the 586Doctorate degree from Odessa State Academy of Re- 587frigeration, Odessa, Ukraine. 588
He is currently a Professor in the Department 589of Mechanical Engineering, National Taiwan Uni- 590versity, Taipei, Taiwan, where he is the Director of 591the Solar Energy Research Center (SERC), which is 592founded by the Global Research Partnership (GRP) 593Award of King Abdullah University of Science and 594
Technology (KAUST). He has devoted research to a broad array of fields, in- 595cluding energy systems (solar, photovoltaics (PV), geothermal, ocean thermal, 596wind, boiler, waste heat), cooling technology (absorption, ejector, desiccant, 597cryocoolers, thermoelectric), solid-state lighting (LED), and control technol- 598ogy. His research tries to bridge the gap between academia and industry. He has 599developed more than 30 products with industry. He is the author or coauthor of 600more than 200 academic papers and 150 technical reports. He holds more than 60160 worldwide patents. 602
Prof. Huang was a recipient of the 1927 Outstanding Youth of the Year 603Award, the 1991 National Outstanding Engineering Professor Award, the 1995 604Academician of Academy of Sciences of Technological Cybernetics of Ukraine 605Award, the 1996 Academician of International Academy of Refrigeration, 606Ukraine Branch Award, the 1996 Tong-Yuan Science and Technology Award, 607the 2000 Outstanding Researcher Award of the National Science Council, the 6082005 Science and Technology Award of China-Tech Foundation, and the 2005 609Solar and New Energy Contribution Award of the Solar and New Energy Society 610of Taiwan. 611
612
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QUERIES 613
Q1: Author: Please provide the IEEE membership details (membership grades and years in which these were obtained), if any, 614
for C.-W. Tang and B.-J. Huang. 615
Q2. Author: Please provide the year information in Refs. [16] and [17]. 616
Q3. Author: Please update Ref. [27], if possible. 617