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IEEE TRANSACTIONS ON POWER ELECTRONICS 1

Multivariable Robust Control for a Red–Green–BlueLED Lighting System

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2

Fu-Cheng Wang, Member, IEEE, Chun-Wen Tang, and Bin-Juine HuangQ1

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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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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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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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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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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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[2] C.-C. Chen, C.-Y. Wu, and Y.-M. Chen, “Sequential color LED backlight 449driving system for LCD panels,” IEEE Trans. Power Electron., vol. 22, 450no. 3, pp. 919–925, May 2007. 451

[3] S. Muthu, F. J. Schuurmans, and M. D. Pashley, “Red, green, and blue 452LED based white light generation: Issues and control,” in Proc. Ind. Appl. 453Conf., 2002, pp. 327–333. 454

[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

[10] K. Lim, J. C. Lee, G. Panotopulos, and R. Helbing, “Illumination and 475color management in solid state lighting,” in Proc. IEEE Ind. Appl. Conf., 47641th IAS Annu. Meeting, Tampa, FL, Oct. 8–12, 2006, pp. 2616–2620. 477

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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

[13] S.-Y. Lee, J.-W. Kwon, H.-S. Kim, M.-S. Choi, and K.-S. Byun, “New484design and application of high efficiency LED driving system for RGB-485LED backlight in LCD display,” in Proc. 37th IEEE Power Electron. Spec.486Conf., 2006, pp. 1–5.487

[14] I. Ashdown, “Neural networks for LED color control,” Proc. SPIE,488vol. 5187, pp. 215–226, 2003.489

[15] B.-J. Huang, P.-C. Hsu, M.-S. Wu, and C.-W. Tang, “Study of system490dynamics model and control of a high-power LED lighting luminaire,”491Energy, vol. 32, no. 11, pp. 2187–2198, 2007.492

[16] Everlight Electronic Co. RGGB High Power LED—4W Datasheet EHP-493B02 [Online]. Available: http://www.everlight.com/494

[17] Hamamatsu Photonics. Si photodiode S1133 Datasheet [Online]. Avail-495able: http://hamamatsu.com/496

[18] C.-W. Tang “Polychromatic control technology of solid state lighting,”497Ph.D. dissertation, Dept. Mech. Eng., Nat. Taiwan Univ., Taipei, Taiwan,4982008.499

[19] G. Wyszecki and W. S. Stiles, Color Science, Concepts and Methods,500Quantitative Data and Formulae, 2nd ed. New York: Wiley, 1982.501

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[21] F. Profumo, A. Tenconi, S. Facelli, and B. Passerini, “Instantaneous junc-504tion temperature evaluation of high-power diodes (thyristors) during cur-505rent transients,” IEEE Trans. Power Electron., vol. 14, no. 2, pp. 292–299,506Mar. 1999.507

[22] S. Clemente, “Transient thermal response of power semiconductors to508short power pulses,” IEEE Trans. Power Electron., vol. 8, no. 4, pp. 337–509341, Oct. 1993.510

[23] T. Bruckner, “Estimation and measurement of junction temperatures in511a three-level voltage source converter,” IEEE Trans. Power Electron.,512vol. 22, no. 1, pp. 3–12, Jan. 2007.513

[24] H. Rake, “Step response and frequency response methods,” Automatica,514vol. 16, pp. 519–526, 1980.515

[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

[27] B.-J. Huang and C.-W. Tang, “Thermal-electrical-luminous model of521multi-chip polychromatic LED luminaire,” Appl. Therm. Eng., to be pub-522lished (DOI: 10.1016/j.applthermaleng.2009.05.024).523

[28] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper524Saddle River, NJ: Prentice-Hall, 1996.525

[29] T. T. Georgiou and M. C. Smith, “Robust stabilization in the gap metric:526Controller design for distribution plants,” IEEE Trans. Autom. Control,527vol. 37, no. 8, pp. 1133–1143, Aug. 1992.528

[30] D. McFarlane and K. Glover, “Robust stabilization of normalized coprime529factor plant descriptions with H-bounded uncertainty,” IEEE Trans. Au-530tom. Control, vol. 37, no. 6, pp. 759–769, Jun. 1992.531

[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

[36] ANSLG, “Specifications for the chromaticity of fluorescent lamps,” Amer.546Nat. Std. Lighting Group, Nat. Electr. Manufact. Assoc. Rosslyn, VA,547ANSI C78.376-2001.548

[37] IEC, “Metal halide lamps,” Int. Electrotech. Comm., Geneva, Switzerland,549IEC 61167-1992, 1992.550

[38] IEC, “Double-capped fluorescent lamps—Performance specifications,”551Int. Electrotech. Comm., Geneva, Switzerland, IEC 60081-1997, 1997.552

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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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

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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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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