Robus Sa u a ed Con ol o Low-Powe Ci cui s
C. Albea, F. Go dillo and C. Canudas de Wi
Abs ac —An impo an issue in he ends o minia-
u iza ion o Sys ems on Chips (SoCs) is o ob ain
a high ene gy e iciency. This can be eached by Dy-
namic Vol age Scaling (DVS) a chi ec u es as he no el
disc e e Vdd-Hopping ci cui . Gene ally, his kind o
sys ems p esen pa ame e unce ain ies and delays.
Likewise, cu en peaks and ene gy dissipa ion mus
be educed.
In his pape , an op imal and obus sa u a ed con-
ol law is p oposed o his Vdd-Hopping ci cui ia
Lyapuno -K aso skii heo y ha ensu es asymp o ic
s abili y as well as sys em obus ness wi h espec
o delay p esence and pa ame e unce ain ies. The
closed-loop sys em p esen s a egional s abiliza ion
due o he ac ua o sa u a ion. An es ima ion o an
a ac ion domain is p o ided. This con olle also
limi es he cu en peaks and i p o ides an ene gy-
awa e pe o mance. The ad an ages achie ed wi h his
con olle a e shown in simula ion.
I. In oduc ion
The con inuous minia u iza ion o Sys ems on Chips
(SoCs) is possible om eme ging s udies abou ene gy
sa ing in Ve y-La ge-Scale In eg a ion (VLSI), especially
in po able elec onics de ices. Dynamic Vol age Scaling
(DVS) [1] is a powe ul me hod o educe he powe
consump ion in mic o-ci cui s and in nano-ci cui s. The
p inciple o DVS app oach is o adap le els o supply
ol age o educing powe consump ion. This kind o
a chi ec u es adap he co e ol age, c, o he clus e a
he minimum pe o mance le el equi ed by he p ocess
ac i i y [2]. Gene ally, a DVS ci cui is implemen ed by in-
eg a ed dynamic DC-DC con e e s [3], [4], [5]. Howe e ,
hese DC-DC con e e s can p esen some limi a ions in
ine g ain. This is he eason why new disc e e ci cui s wi h
educed size ha e been de eloped based on DVS [6], [7],
as Vdd-Hopping ci cui [8]. This s uc u e allows changing
he d ain supply ol age le el (Vdd) in nano-ci cui s in high
speed.
The main con ol p oblem o Vdd-Hopping ci cui is
he adap a ion o a ious loading condi ions, achie ing
high e iciency o e a wide load-cu en ange, wha is
c i ical o ex ended ba e y li e. Likewise, o he con ol
objec i es a e o p o ide a co ec and eliable ope a ion
du ing he swi ches ansi ions, i.e., small cu en peaks,
This wo k has been conduc ed while C. Albea was wi h INPG,
Gipsa-lab G enoble, F ance and he Dp o. de Ing. de Sis emas y
Au om´a ica, Se illa, Spain. Now, C.Albea is wi h bo h CNRS ; LAAS
; 7 a enue du colonel Roche, F-31077 Toulouse, F ance and Uni e si ´e
de Toulouse ; UPS, INSA, INP, ISAE ; UT1, UTM, LAAS ; F-31077
Toulouse, F ance [email p o ec ed]
F. Go dillo is wi h he Dp o. de Ing. de Sis emas y Au om´a ica,
Se illa, Spain [email p o ec ed]
C. Canudas de Wi is wi h he CNRS, Gipsa-lab, G enoble, F ance
[email p o ec ed]
as e ansien pe iods, sys em s abili y and obus ness
wi h espec o delays and pa ame e unce ain ies. In-
deed, Vdd-Hopping ci cui s implemen ed in hese SoC
s uc u es p esen delays a he inpu and ou pu o i s
con ol block [9]. The con ol block inpu delay is equi ed
o ensu e ha he sys em is synch onized wi h he clus e
clock. In he same way, he e is a con ol block ou pu
delay associa ed wi h compu a ional issues. Mo eo e , he
p esence o pa ame e unce ain ies can gene a e a non-
desi able pe o mance and lack o eliabili y o he sys em.
O he addi ional and ele an issue is o limi cu en
peaks and ene gy consump ion. In summa y, delay p es-
ence, pa ame e unce ain ies, cu en peaks and ene gy
consump ion mus be conside ed in he design o he
con olle .
In his pape , an op imal and obus sa u a ed con ol
law o he Vdd-hopping ci cui is p oposed in disc e e
ime. This con olle limi es high cu en peaks. I p o ides
an e icien acking capabili y o handle wo ol age le els
equi ed by he clus e . As a side e ec , i also has high
ene gy-e iciency, achie ing as ansien pe iods. The sys-
em is ew i en in o a sui able s a e-space ep esen a ion
o o mula e an op imal and obus p oblem ha can be
sol ed by using Lyapuno -K aso skii heo y [10], [11]. In
his p ocess, he sa u a ion in he con olle is conside ed
[12], [13]. The designed con olle gua an ees asymp o ic
s abili y as well as obus ness o he sys em wi h espec
o delays and unce ain pa ame e s. The p oblem is ex-
p essed in e ms o Linea Ma ix Inequali ies (LMIs).
Likewise, an a ac ion domain is es ima ed in such a way
ha a egional s abiliza ion o he sa u a ed con ol is
gua an eed. The obus ness p ope ies o he closed-loop
sys em a e es ed by some simula ions.
An e alua ion and compa ison o his con olle wi h
espec o an ‘in ui i e’ con olle p esen ed in [6] is pe -
o med.
The es o his wo k is o ganized as ollows: in Sec-
ion II, he e o equa ion o he Vdd-hopping sys em
is p esen ed as well as he p oposed con ol law. The
obus ness and op imiza ion p oblem s a emen is p e-
sen ed in Sec ion III and in Sec ion IV, a con ol design is
de eloped. The con ol gains a e compu ed in Sec ion V,
being es ed by simula ions in Sec ion VI. A compa ison
o his con olle is pe o med in Sec ion VII. The wo k
closes wi h a sec ion o conclusions.
No a ion. Fo a gi en x∈ R,
sa M
m(x),


M i x > M
x i m ≤x≤M
m i x < m.
and ound(x) is he nea es in ege o x. ∆x,x+−x−
To ci e his a icle:
IEEE T ansac ions on Con ol Sys ems Technology. 2013. Vol. 21. Núm. 2.
pp. 530-537
is he alue o xin wo consecu i e sampling ime. Fo a
gi en se S,Co(S) deno es he con ex hull o S.
II. E o equa ion o he Vdd-Hopping ci cui
In [8] a disc e e ci cui ha handles wo- ol age le els
wi h a Vdd-Hopping echnique accomplishing a DVS a -
chi ec u e was p esen ed.
The Vdd-Hopping ci cui is cons i u ed by: a high ol -
ages supply, Vh; a low ol age supply, Vl; a g oup o PMOS
ansis o s connec ed in pa allel be ween he Vhand he
co e ol age, c, and a PMOS ansis o ha connec s
he Vl o cwhen he low ol age le el is he s eady-
s a e. This educes he dissipa ed ene gy when he uni
unning is a low speed. The g oup o PMOS ansis o s
connec ed in pa allel allow e ol ing he ou pu ol age
om a low ol age le el o a high ol age le el ( ising
ansien pe iod) and om a high ol age le el o a low
ol age le el ( alling ansien pe iod). The s eady s a e
mus co espond o a high ol age le el o a low ol age
le el. Fo simplici y, he low ol age supply, Vl, as well as
i s PMOS ansis o o connec ion wi h ca e dis ega ded
o con ol design pu poses. The main objec i e is o
ensu e ha cachie es he wo ol age le els by swi ching
he PMOS ansis o s. The ske ch o his ci cui is shown
in Fig. 1. In his con igu a ion, a leas , one ansis o mus
always be swi ched on.
The load model aken in his wo k is an impedance
which depends on he clock equency, clk, and some imes,
also on c[14]. I is composed o a cu en supply, Ileak,
a capaci ance, C, and a dynamic esis ance, RL( clk, c),
ep esen ing he dynamic and sho -ci cui consump ion.
Ileak is assumed cons an .
M1R1
Mn−1Rn−1
MnRn
Vh
IMPEDANCE
Il( c)
Ileak
CRL
M2R2
CONTROL
uk
c
Fig. 1: Vdd-hopping, ol age supply and load.
Assump ion 1: PMOS ansis o s a e modeled as ideal
esis o s, R0, when hey a e swi ched on and, as esis o s
wi h in ini e esis ance when hey a e swi ched o . They
a e conside ed o ha e he same elec ical cha ac e is ic.
In he s eady s a e, he maximum and minimum ol age
a e Vh−∆hand Vl−∆l, espec i ely. ∆h,∆l∈Rdepend
on se e al ac o s and a e di icul o es ima e. These
a iables ca ch he PMOS model e o s, cu en a ia ions,
supply ol age and he esis i e losses h ough he PMOS
ansis o s swi ched on.
Assump ion 2: ∆h, ∆la e small wi h espec o c, and
hey do no change he sys em s abili y p ope ies.
The ol age loop equa ion yields he ela ionship
Il( c) = Vh− c
Ruk
,(1)
whe e Ruk,R0
uk, being uk he numbe o ansis o s
swi ched on, hus, uk∈ U ={1,2, ..N}and i is he con ol
a iable.
The disc e e Vdd-hopping ci cui is connec ed o a load
ha can be modeled as an impedance depending on he
co e ol age, c. In his wo k, he load model p esen ed in
[14] is employed:
Il( c) = c
RL
+Ileak +Cd c
d (2)
Conside ek, − ckwhe e is a cons an ol age
e e ence and ckis he sampling co e ol age. Now, om
Eqs. (1), (2), he app oxima e disc e e- ime ol age e o
equa ion is
ek+1 = (1−Tsβ)ek+Tsb( −Vh)uk+Ts(β +δ−bukek),
(3)
whe e β,1
RLC>0, δ,Ileak
C>0 and b,1
R0C>0. Ts
mus be less o equal o he smalle 1
clk . The app oxima e
ime-disc e iza ion (3) is pe o med by using he o wa d
Eule me hod, by assuming ha he sampling ime is small
enough o he sys em e olu ion.
A p oposed con olle o his sys em is based on a linea
con olle [15]:
uk= sa N
1{uk−1+ ound (K1∆ek+K2ek−1)}.(4)
Con ol (4) p esen s a simple enough s uc u e such ha
i equi es a small space in he silicon. The sa u a ion
unc ion is due o he con ol signal cons ain men ioned
be o e. The con ol objec i e is o achie e a se -poin
e e ence signal. An app oxima e con ol s uc u e o he
Vdd-Hopping ci cui ha manages he cu en peaks is
pa en pending unde he name o ENe gy-AwaRe Con ol
(ENARC) [16].
III. P oblem s a emen
He e, he p oblem s a emen is o mula ed ew i ing he
closed-loop sys em o (3) in a s a e-space o m conside ing
delays, unce ain pa ame e s and cu en peak educ ion.
Figu e 2 shows he Vdd-Hopping ci cui including de-
lays. The sys em has a h2-sample-pe iod delay a he
con ol block inpu and a h1-sample-pe iod delay a he
con ol block ou pu , as men ioned be o e.
CONTROL Vdd-HOPPING
ekuk ck
+
-R0
LOAD
RL, C
K1K2
z−h2z−h1+
Fig. 2: Block diag am o he ci cui con ol sys em.
The open-loop sys em is
ek+1 = (1 −Tsβ)ek+Tsb( −Vh)uk−h1+Ts(β +δ)
−bTsuk−h1ek,(5)
and he conside ed Con ol (4) is
uk−h1= sa N
1{uk−1−h1+ ound(Kxk−h)},(6)
whe e h,h1+h2,xk−h= [ek−hek−1−h]Tand K=
[¯
K1¯
K2], whe e ¯
K1,K1and ¯
K2,K2−K1.
Fo simplici y, he con ol a iable ukis conside ed a
eal numbe . Thus,
uk−h1= sa N
1{uk−1−h1+Kxk−h}
= sa N
1{uk−1−h1+ ∆uk−h1},(7)
A igo ous analysis should ake uk∈Z. No e ha i
∆uk−h1is bounded abo e, hen, om Eq. (1), he max-
imum cu en peaks (∆uk−h1=R0
Vh− c∆Ilmax , [17]) a e
limi ed. Fo his pu pose, he quad a ic cos unc ions
Jk=
∞
X
k=0
(xT
kQxk+ ∆uT
k−h1R∆uk−h1) (8)
mus be minimized. No e ha he posi i eness o he
ma ices Qand Rimplies ha Jkis s ic ly posi i e.
On he o he hand, he pa ame e s RL,R0and C
ha de ine β,band δcan be conside ed unce ain. Each
unce ain pa ame e is wi hin an unce ain y in e al,
whose co esponding ex emes a e
•C∈[Cm, CM] ,
•RL∈[Rm
L, RM
L] ,
•R0∈[Rm
0, RM
0].
Rema k 1: The asymp o ic s abili y o sys em (5) is
gua an eed in a poin wi hin an unce ain y in e al o
he low le el ol age, Il, and wi hin an unce ain y in e al
o he high le el ol age, Ih, bounded by
Il,Rm
L(uklVh−RM
0Ileak)
uklRm
L+RM
0
,Rm
L(uklVh−Rm
0Ileak)
uklRm
L+Rm
0(9)
Ih,RM
L(ukhVh−RM
0Ileak)
ukhRM
L+RM
0
,RM
L(ukhVh−Rm
0Ileak)
ukhRM
L+Rm
0.(10)
ukland ukha e he lowe -bound and he uppe -bound o
uk, espec i ely.
Consequen ly, he main objec i e is o design he op-
imal gain Kin such a way ha Con ol (6) is obus
wi h espec o delays as well as pa ame e unce ain ies.
Likewise, his op imal gain Kmus gua an ee asymp o ic
s abili y and minimum cu en peaks o he known con-
s an delays, h1and h2.
A. Al e na i e ep esen a ion o he sa u a ed con ol (7)
and he e o equa ion (5)
Fi s ly, some lemmas a e gi en o ew i e he sa u a ed
con ol (7) and an al e na i e o m.
De ine χk−h,uk−1−h1
xk−h. No e, om Eq.(5), uk−1−h1
di ec ly depends on xk= [ek, ek−1]T.
Lemma 1: [18], le K, G ∈R1×2be gi en. Fo all,
χk−h∈R3×1, i χk−h∈ {χk−h∈R1×3: [1 G]χk−h∈
[1 N]}, hen
sa N
1{[1 G]χk−h} ∈ Co{[1 K]χk−h,[1 G]χk−h}.

Lemma 2: Assume ha he e exis s G∈R1×2,c > 0
and Ψ ,diag{ρ, P1}, whe e P1†>0∈R2,ρ > 0∈R1
such ha o any χk−h∈X, whe e
X=χk−h:χT
k−hΨχk−h≤c−1,(11)
hen, 1 < uk−1−h1+Gxk−h< N, and Con ol (7) admi s
he ollowing ep esen a ion
uk−h1= [αk(uk−1−h1+Kxk−h)
+(1 −αk)(uk−1−h1+Gxk−h)]
= [uk−1−h1+αkKxk−h+ (1 −αk)Gxk−h)]
= [uk−1−h1+ ¯uk−h],
whe e ¯uk−h,(αkK+ (1 −αk)G)xk−hwi h αk∈[0,1],
o all k > 0. 
Then, Eq. (5) can be ew i en
ek+1 = (1 −Tsβ)ek+Tsb( −Vh)(uk−1−h1+ ¯uk−h)
+Ts(β +δ)−bTsuk−h1ek.(12)
B. S a e-space ep esen a ion
The sa u a ed con ol law and e o equa ion, as ede-
ined be o e, allow o sys em (5) ew i e i in a s a e-space
o m.
F om Eq. (5) ,
uk−h1=ek+1 −(1 −Tsβ)ek−Ts(β +δ) + bTsuk−h1ek
Tsb( −Vh),
and, he e o e
uk−1−h1=ek−(1 −Tsβ)ek−1−Ts(β +δ)
Tsb( −Vh)
+bTsuk−1−h1ek−1
Tsb( −Vh),
which subs i u ed in Eq. (12) gi es
ek+1 = (2 −Tsβ)ek−(1 −Tsβ)ek−1+Tsb( −Vh)¯uk−h
−Tsb(uk−h1ek−uk−1−h1ek−1).(13)
F om Lemma 1 and 2, Eq. (13) can be ew i en in he
ollowing ma ix o m:
xk+1 =A(uk−h1, uk−1−h1)xk+B¯uk−h,(14)
whe e
A,2−Tsβ−Tsbuk−h1Tsβ−1 + Tsbuk−1−h1
1 0 ,
B,Tsb( −Vh)
0.
uk−h1and uk−1−h1o ma ix Aa e ea ed as unce ain
pa ame e s in Sec ion IV-D. Thei alues will be inside he
unce ain y in e al [1, N].
†P1is a posi i e ma ix de ined o gua an ee sys em s abili y.
C. S abili y and op imiza ion p oblem
Equa ion (14) can be ew i en in he ollowing explici
closed-loop o m.
xk+1 =Axk+B(αkK+ (1 −αk)G)xk−h,(15)
xl=φl,∀l∈[−h, 0] (16)
zk=I2xk,(17)
wi h C∈[Cm, CM] , RL∈[Rm
L, RM
L] , R0∈[Rm
0, RM
0],
uk−h1, uk−1−h1∈[1, N] (18)
αk∈[0,1],(19)
and whe e xk, zk∈R2a e he s a e ec o and con olled
ou pu , espec i ely. φlis he ini ial condi ion and h≥0∈
Ris a ixed and known delay.
P oblem 1: The p oblem is o ind a X(ρ, P1, c), some
obus ec o s Gand K, such ha
a) Lemma 2 holds and, hence, he closed-loop sys em
(14) and
b) he e exis s a Lyapuno -K aso skii unc ional Vk>
0, a cos unc ion Jk>0 such ha Vk+1 −Vk+Jk
along he solu ion o (15) ul ills
Vk+1 −Vk+Jk<0.(20)

The solu ion o his p oblem minimizes a pe o mance
index ha , among o he conside a ions, limi es high cu -
en peaks. Mo eo e , his solu ion gua an ees he sys em
s abili y o he ime-delay sys em (15)–(17).
IV. Op imal obus sa u a ed con ol design
A ma hema ical manipula ion o Eq. (15) is pe o med
ia a desc ip o model ans o ma ion [19]. The desc ip o
app oach is jus a a iable change, which makes easie o
wo k wi h Lyapuno -K aso skii unc ional [20].
A. Desc ip o model ans o ma ion
Equa ion (15) is manipula ed in o de o achie e he
p e ious objec i es. A desc ip o model ans o ma ion is
applied.
De ine yk,xk+1 −xk, ψk,Pk−1
i=k−hyi.Nex , ew i e
Eq. (15) in he desc ip o o m [19]:
xk+1
0=yk+xk
−yk+Axk−xk+B(αkK+ (1 −αk)G)xk−h
F om xk−h=xk−ψk, his sys em can be compac ly
w i en as:
E¯xk+1 =¯
A¯xk−0
B(αkK+ (1 −αk)G)ψk,(21)
whe e
¯
A,I2I2
A+B(αkK+ (1 −αk)G)−I2−I2,
E,diag{I2,02},¯xk,xk
yk.
B. Condi ion o s a e-space ep esen a ion
Condi ion a) o P oblem 1 is sa is ied, i
1< uk−1−h1+Gxk−h< N, ∀χk−h∈X(22)
gi en in (11) is gua an eed.
Sub ac ing N+1
2in inequali y (22) and om [21], i is
seen ha , i is necessa y sa is y
2N−2> N(1 + cxT
k−hP1xk−h+cuT
k−1−hρuk−1−h)−2
>4uk−1−h+ 4Gxk−h−2(N+ 1) (23)
F om Lemma 2, ela ionships (23) co espond o


1
±uk−1−h1
±xT
k−h


T

3N−2−2G
−2cNρ 0
−2GT0cNP1



1
±uk−1−h1
±xk−h

>0
(24)
This inequali y is sa is ied i
Λ,

c−2−2Y
−2ρ3N20
−2YT0 3N2¯
P1

>0 (25)
No e ha his LMI is equi alen o (24) by means o
employing he Schu ’s complemen , de ining Y,GQ1
wi h Q1∈R2which is He mi ian, applying ¯
P1=Q1P1Q1
and p e- and pos -mul iplying by diag{1,1, Q1}.
C. S abiliza ion and op imiza ion
Fo simplici y, assume he e a nominal alue o : αk,uk−h1
and uk−1−h1. In nex subsec ion (18)–(19) will be consid-
e ed as unce ain inside a con ex poly ope. Condi ion b)
o P oblem 1 can be o mula ed in e ms o Linea Ma ix
Inequal ies (LMIs) [10]. Ful illmen o condi ion (20) is
looked o .
De ine P,P1P2
P20, being P2He mi ian. Conside as
Lyapuno -K aso skii candida e
Vk=V1,k +V2,k +V3,k,(26)
being
V1,k = ¯xT
kEPE¯xk, P1>0 (27)
V2,k =
h
X
n=1
k−1
X
i=k−n
yT
iRyi, R > 0 (28)
V3,k =
k−1
X
i=k−h
xT
iSxi, S > 0,(29)
whe e V1,k gua an ees asymp o ic s abili y o sys em
(21) wi hou delays. Delay-dependen as well as delay-
independen c i e ia a e conside ed in V2,k and V3,k, e-
spec i ely [10], [11].
Nex , a su icien condi ion o asymp o ic s abili y and
minimiza ion o a pe o mance index ha , among o he
conside a ions, limi es high cu en peaks, is de i ed.
Theo em 1: Conside sys em (15)–(17) wi h nominal
alue o : αk,uk−h1and uk−1−h1.h > 0∈Nis a
known cons an delay and K, G ∈R1×2. I he e exis
Q,R, S, R, P1>0∈R2and c, µ > 0 such ha
minKµ
P1>0 (30)
Γ<0 (31)




−µI202I202
∗ −µI202I2
∗ ∗ −P1−S S
∗ ∗ ∗ −hR −S




<0,(32)
Λ>0 (33)
whe e Γ is de ined in Eq. (34), ound a he op o nex
page, hen he equilib ium o he closed-loop sys em (15)–
(17) is asymp o ically s able and he cu en peaks a e
limi ed.
P oo : The goal is o sa is y Vk+1 −Vk+Jk<0 o
sys em (21).
Lyapuno -K aso skii me hod yields:
V1,k+1 −V1,k = ¯xT
k+1EP E ¯xk+1 −¯xT
kEP E¯xk
=n¯xT
k¯
AT−ψT
k[0 αkKTBT+ (1 −αk)GTBT]o
P¯
A¯xk−0
αkBK + (1 −αk)BGψk−¯xT
kEP E¯xk
= ¯xT
k[¯
ATP¯
A−P1]¯xk−¯xT
k¯
AP 0
αkBK + (1 −αk)BGψk
−ψT
k[0 αkKTBT+ (1 −αk)GTBT]P¯
A¯xk.
F om (28) and Jensen Inequali y [22]:
V2,k+1 −V2,k =hyT
kRyk−
h
X
n=1
yT
k−nRyk−n
≤¯xT
k0 0
0hR¯xk−1
hψT
kRψk
Finally,
V3,k+1 −V3,k =xT
kSxk−xT
k−hSxk−h
=xT
kSψk+ψT
kSxk−ψT
kSψk
These de eloped exp essions and Eq. (8) a e applied
o inequali y (20), in such a way ha he LMIs (31) a e
ob ained.
On he o he side, summing (20) om n= 0 o ∞, i is
ob ained
∞
X
k=0
(Vk+1 −Vk+Jk) = V∞−V0+
∞
X
k=0
Jk=−V0+
∞
X
k=0
Jk<0.
Thus, P∞
k=0 Jk< V0<[¯x0ψ0]M[¯x0ψ0]T, whe e M,
P1+S−S
−S hR +S. To minimize he ace o Mmeans
o minimize any µ > 0, such ha , M< µI4, [23]. F om
his inequali y and applying he Schu ’s complemen LMI
(32) is ob ained.
D. Con ol design
Now, conside unce ain pa ame e s gi en in Sec ion III
and (18)–(19). Fo his pu pose, Theo em 1 is ex ended in
he case o poly opic unce ain ies.
Deno e
Ω,[A BK αkuk−h1uk−1−h1]
and assume ha Ω ∈ Co{Ωj, j = 1, ..., 64}, namely
Ω =
n
X
j=1
λjΩj, o all,0≤λj≤1,
n
X
j=1
λj= 1
and being he e ices o he poly ope desc ibed by Ωj=
[A(j)B(j)K α(j)
ku(j)
k−h1u(j)
k−1−h1] o j= 1,2, ..., 64.
P e- and pos -mul iplying LMI (31) by
Q= diag{Q1, Q1, Q1}and apply he Schu ’
complemen . P e- and pos -mul iplying LMI (32) by
Q= diag{I2, I2, Q1, Q1}and aking Q1=P−1
2>0 and
¯
P1=Q1P1Q1,¯
R=Q1RQ1,¯
S=Q1SQ1, he ollowing
su icien condi ion is achie ed.
Theo em 2: Conside sys em (15)–(17) wi h h≥0∈N
is a known cons an delay and K, G ∈R1×2. I he e exis
T, Y ∈R2×1and Q1∈R2wi h K=T Q−1
1,G=Y Q−1
1,
Q,R,¯
R, ¯
P1,¯
S > 0∈R2 o j= 1, ..., 64 and c, µ > 0 such
ha
minKµ
¯
P1>0 (35)
¯
Γ(j)<0j= 1, ...., 64,(36)




−µI202I202
∗ −µI202I2
∗ ∗ −Q1−¯
S¯
S
∗ ∗ ∗ −h¯
R−¯
S




<0,(37)
Λ>0 (38)
being ¯
Γ(j)de ined in (39), ound a he op o nex page,
a e sa is ied. Then, in he e ices j, he equilib ium is
asymp o ically s able as well as he cu en peaks a e
limi ed in he en i e poly ope.
P oo : This is an ex ension o Theo em 1 o poly-
opic unce ain ies wi h some ma hema ical manipula-
ions. The e o e, his heo em p oo ollows Theo em 1
p oo .
Rema k 2: This obus con ol uning me hod is con-
se a i e due o he de ini ion o he ma ix P, as well as,
he a ac ion domain, X.
Co olla y 1: Gain K, ob ained om Tand Q1in Theo-
em 2, ul ills Theo em 1 and consequen ly gua an ees bo h
obus s abili y and minimiza ion o he cu en peaks o
a ixed delay.
As u u e wo k an op imiza ion o he ellipsoid Xwill
be in e es ing o pe o m.
V. Robus con ol esul
In his sec ion, he obus con ol gains o Con ol (7)
a e compu ed by employing he app oach abo e. The Vdd-
Hopping ci cui pa ame e s gi en in [8] and he load model
pa ame e gi en in [14] a e epo ed. The e o e, N= 24
is aken as he o al numbe o PMOS ansis o s. The
ol age supply is Vh= 1.2V. The e e ence signal, , is a
s ep be ween he low ol age le el Vcl= 0.8V−ǫhand he
high ol age le el Vch= 1.2−ǫh, being ǫh= 0.06Vand

Γ,

¯
ATP¯
A−EPE + diag{Q + Ξ, hR} − ¯
ATP0
B(αkK+ (1 −αk)G)+S−Ξ
0
∗ − 1
hR−S+ Ξ

(34)
Ξ = (αkK+ (1 −αk)G)R(αkKT+ (1 −αk)GT)
¯
Γ(j),








¯
Γ(j)
1¯
Γ(j)
2−α(j)
kB(j)T−(1 −α(j)
k)B(j)Y+¯
S Q1Q¯
ΞR0
∗¯
P1−2Q1+h¯
R0 0 0 0
∗ ∗ − ¯
R
h−¯
S0 0 ¯
ΞR
∗ ∗ ∗ −Q 0 0
∗ ∗ ∗ ∗ −R R
∗ ∗ ∗ ∗ ∗ −R








, j = 1, ..., 64 (39)
whe e
¯
Ξ(j),(α(j)
kT+ (1 −α(j)
k)Y)
¯
Γ(j)
1,Q1A(j)T+A(j)Q1−2Q1+α(j)
kTTB(j)T+ (1 −α(j)
k)YTB(j)T+α(j)
kB(j)T+ (1 −α(j)
k)B(j)Y
¯
Γ(j)
2,¯
P1+Q1A(j)T−2Q1+α(j)
kTTB(j)T+ (1 −α(j)
k)YTB(j)T,
ǫl= 0.01 (Vcl≈Vl). These pa ame e s comes om he
equilib ium o Eq. (3). The sys em esis ances a e RL=
27.7Ω and R0= 31.41Ω, he capaci ance is C= 9nF,
while Ileak = 1.67 ·10−3. The clock equency is aken
clk = 200MHz and Ts= 1.67ns. This in oduces an one-
sample-pe iod delay (h1= 1) in he con ol block ou pu
due o a powe -pe o mance ade-o . Likewise h2= 2,
hus h= 3.
The unce ain pa ame e s ake he ollowing anges:
• ansis o cha ac e is ic, R0, om 25Ω o 38Ω,
•load dynamic esis ance, RL, om 55.53Ω o 72.46Ω,
•load capaci ance, C, om 1pF o 1nF.
And, Q= diag{10,10},R= diag{1000,1000}.
Then, op imiza ion p oblem is esol ed, ob aining
K1=−0.49, K2= 0.72.
This was ob ained o c= 1.1′,µ= 4.6·1010,G=
−0.15 0.82and P1=0.0001 0.0002
0.0002 2.8574·106.
No e ha e en i he con ol cons an uning is conse -
a i e, he e is a easible solu ion.
VI. Simula ion Resul s.
Some simula ions show he obus ness o he op imal
sa u a ed con ol law p oposed o he Vdd-Hopping ci -
cui . Likewise, a compa ison be ween he pe o mance
achie ed wi h he con ol gains ob ained in his pape
wi h espec o ano he con ol gains ob ained in [15]
is pe o med. These simula ions a e done by using he
pa ame e alues gi en in Sec ion abo e.
A. Unce ain PMOS esis ance
In his kind o sys ems, he elec ical cha ac e is ic o
he PMOS can su e changes due o empe a u e changes.
In Fig. 3 he simula ion is pe o med inc easing he
alue o he PMOS esis ance by 20%. The sys em in he
low ol age le el con e ges o 0.78V, which is inside he
in e al gi en by (9), Il= [0.74V, 0.86V]. Likewise, he
high ol age le el con e ges o 1.133V, which is inside he
in e al gi en by (10), Ih= [1.132V, 1.155V]. These
es s show ha he equilib ium is obus wi h espec
o pa ame e unce ain ies and delays. And, he cu en
peaks a e small.
0 0.2 0.4 0.6 0.8 1
x 10−6
0
10
20
a)
NT ans
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0.8
1
1.2 b)
V(V)
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0
0.05
d)
I(A)
(s)
Fig. 3: R0= 31.41Ω →R0= 37.7Ω, K1=−0.47, K2=
0.68. E olu ion o he: a) numbe o ansis o s swi ched
on, b) (dashed) and c(solid), c) cu en Il.
B. Unce ain load pa ame e
Ano he example shows ha sys em pe o mance is
sensi i e o he alues o K1and K2.
The capaci ance employed du ing he con ol design
and he p e ious simula ions has been C= 1nF. In he
ollowing, i is desi ed o alida e he sys em obus ness
when C= 1pF, i.e., 1000 imes smalle . The lack o
knowledge o he load in he eal applica ions o hese
sys ems may imply his change o h ee o de o magni-
ude. Some simula ions a e shown using he obus con ol
gains compu ed in his pape (Fig. 4) and he con ol gains
compu ed in [15] (Fig. 5), which a e K1=−19.3 and
K2= 39.27. These gains we e compu ed linea izing he
closed-loop sys em a ound he se poin . K= [ ¯
K1,¯
K2]
a e de ined by ensu ing ha A+BK is Hu wi z and
placing he poles by ial and e o , in such a way, ha he
nonlinea sys em p esen s a sui ed beha iou . No e ha ,
in Fig 5, he sys em does no espond o ol age a ia ion.
Howe e , in Fig. 4 he sys em pe o mance is sa is ac o y.
This example shows he g ea obus ness o he sys em
when he obus con ol uning is employed.
0 0.2 0.4 0.6 0.8 1
x 10−6
0
10
20
a)
NT ans
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0.8
1
1.2 b)
V(V)
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0
0.05
d)
I(A)
(s)
Fig. 4: C= 1nF →C= 1pF and K1=−0.47, K2= 0.68.
E olu ion o he: a) numbe o ansis o s swi ched on, b)
(dashed) and c(solid), c) cu en Il.
0 0.2 0.4 0.6 0.8 1
x 10−6
0
10
20
a)
NT ans
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0.8
1
1.2 b)
V(V)
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0
0.02
d)
I(A)
(s)
Fig. 5: C= 1nF →C= 1pF and K1=−19.3, K2=
39.27. E olu ion o he: a) numbe o ansis o s swi ched
on, b) (dashed) and c(solid), c) cu en Il.
VII. Compa ison wi h he ‘in ui i e’ con olle
o [8]
A compa ison is pe o med be ween he con olle p e-
sen ed he e and he ‘in ui i e’ con olle p oposed in [8]:
uk= sa N
1{uk−1+sign(ek)}.(40)
A. Vol age and cu en pe o mance
Con ol (40) swi ches on o o one ansis o acco ding
o he sign o he e o ol age signal. The e o e, his
con olle has he limi a ion ha one only ansis o can
be swi ched on o o a e e y sampling ime. On he o he
side, ollows a linea ime e olu ion be ween Vl= 0.8V
and Vch= 1.12Vwi h a slope speci ied in [8] equal o
1.015106V/s. A simula ion o his con olle is pe o med
by using he pa ame e alues gi en in Sec ion VI. This
simula ion is shown in Fig. 6,. No e ha , he pe o mance
p esen s an oscilla o y beha io , wi h impo an cu en
peaks. In addi ion, ansien pe iods a e slowe and occu s
signi ican cu en peaks in compa ison wi h Con ol (4).
0 0.2 0.4 0.6 0.8 1
x 10−6
0
10
20
a)
NT ans
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0.8
1
1.2 b)
V(V)
(s)
0 0.2 0.4 0.6 0.8 1
x 10−6
0
0.05
c)
I(A)
(s)
Fig. 6: ‘In ui i e’ con ol. E olu ion o : a) numbe o
PMOS ansis o s swi ched on, b) (dashed) and c
(solid), c) cu en . Il.
B. Ene gy e alua ion
In he se o PMOS, he dissipa ed ene gy in he an-
sien pe iod depends on he kind o con ol law employed,
i.e., on he swi ching sequence. The pu pose he e is o
e alua e he ene gy cos associa ed wi h he se o PMOS
ansis o s du ing he ising ansien pe iod using Con ol
(4) and he ‘in ui i e’ con olle p oposed in [8] (assume
ha he alling ansien pe iod is simila ). An es ima ion
o he PMOS ansis o s du ing he ansien pe iod is
Ed=Z
0
(Vh− c)Ild ,
whe e 0is he ini ial ime and is he inal ime in such
ansien pe iod.
Figu e 7 shows he dissipa ed ene gy du ing he ising
ansien pe iod. No e ha he ene gy consump ion is
much highe using he con olle p oposed in [8] han using
Con ol (4). Mo e p ecisely, his ene gy consump ion has
been educed om 7.2µJ o 0.34µJ, i.e., 95% educ ion.
No ice ha a nonsmoo h beha io o he cu en ansien
and a la ge ansien pe iod may esul in a highe ene gy
consump ion.
0.5 1 1.5 2 2.5 3 3.5
x 10−7
0
1
2
3
4
5
6
7
x 10−6
Ene gy (J)
(s)
In ui i e con ol
Con ol (4)
Fig. 7: Ene gy dissipa ed du ing he ising ansien pe-
iod.
VIII. Conclusions
In his pape an op imal and obus sa u a ed con olle
was designed o he ime-delay Vdd-hopping ci cui . This
op imal con olle minimizes a pe o mance index ha ,
among o he conside a ions, limi es high cu en peaks.
As side e ec , his con olle imp o es he dissipa ed
ene gy and i achie es as ansien pe iods. The sys em
is ew i en in a sui ed s a e-space ep esen a ion, such
ha , an op imal and obus p oblem can be o mula ed
o une he con ol gains. This p oblem is deal wi h
Lyapuno K aso skii heo y, which p o ides some s abili y
condi ions h ough Linea Ma ix Inequali ies (LMIs).
Consequen ly, a obus equilib ium s abili y as well as a
obus dis u bance ejec ion unde pa ame e unce ain-
ies a e ensu ed o he ime-delay sys em. The me hod
also akes in o accoun he con ol sa u a ion es ima ing
an a ac ion domain.
The closed-loop sys em obus ness is shown by means o
some simula ions. Likewise, a compa ison o he ob ained
con ol uning design wi h espec o an ‘in ui i e’ con-
olle p esen ed in [8] is pe o med. The obus con ol
uning design leads o highe con olle gains bu hey a e
alid wi h espec o he sa u a ion limi s.
Acknowledgmen
The au ho s wan o hanks D . Alexand e Seu e o his
alue commen s. In addi ion, he au ho s a e also g a e ul
o he anonymous e iewe s o his pape o hei aluable
commen s and sugges ions.
This esea ch was pa ially unded by he ARAVIS
p ojec , he F ench minis y o esea ch and schola ship
and by he Spanish MICINN-FEDER g an DPI2009-
09961.
Re e ences
[1] A. Chand akasan and R. B ode sen, Low powe digi al CMOS
design. Kluwe Academic Pub, 1995.
[2] T. Bu d and R. B ode sen, “Design issues o dynamic ol age
scaling,” in in P oc In e na ional Symposium on Low Powe
Elec onics and Design. ACM, 2000, pp. 9–14.
[3] G. Wei and M. Ho owi z, “A ully digi al, ene gy-e icien ,
adap i e powe -supply egula o ,” IEEE T ans. on Solid-S a e
Ci cui s, ol. 34, no. 4, pp. 520–528, 1999.
[4] G. Pa ounakis, Y. Li, and K. Shepa d, “A ully in eg a ed on-
chip DC-DC con e sion and powe managemen sys em,” IEEE
T ans. on Solid-S a e Ci cui s, ol. 39, no. 3, pp. 443–451, 2004.
[5] A. Dancy, R. Ami ha ajah, and A. Chand akasan, “High-
e iciency mul iple-ou pu DC-DC con e sion o low- ol age
sys ems,” IEEE T ans. on Ve y La ge Scale In eg a ion (VLSI)
Sys ems, ol. 8, no. 3, pp. 252–263, 2000.
[6] V. Gu nik and A. Chand akasan, “Embedded powe supply o
low-powe DSP,” IEEE T ans. on Ve y La ge Scale In eg a ion
(VLSI) Sys ems, ol. 5, no. 4, pp. 425–435, 1997.
[7] S. Lee and T. Saku ai, “Run- ime ol age hopping o low-
powe eal- ime sys ems,” in P oc. Annual Design Au oma ion
Con e ence. ACM, 2000, pp. 806–809.
[8] S. Mie mon , P. Vi e , and M. Renaudin, “A Powe Supply
Selec o o Ene gy-and A ea-E icien Local Dynamic Vol age
Scaling,” Lec u es No es in Compu e Science, ol. 4644, p. 556,
2007.
[9] M. Klein, J. Lehoczky, and R. Rajkuma , “Ra e-mono onic
analysis o eal- ime indus ial compu ing,” IEEE Compu e ,
ol. 27, no. 1, pp. 24–33, 1994.
[10] E. F idman and U. Shaked, “Delay-dependen Hin con ol o
unce ain disc e e delay sys ems,” Eu opean Jou nal o Con ol,
ol. 11, no. 1, pp. 29–39, 2005.
[11] K. Gu and S. Niculescu, “S abili y Analysis o Time-Delay
Sys ems: A Lyapuno App oach,” Ad anced Topics in Con ol
Sys ems Theo y, Lec u e No es om FAP, pp. 139–170, 2005.
[12] E. F idman, A. Seu e , and J. Richa d, “Robus sampled-da a
s abiliza ion o linea sys ems: an inpu delay app oach,” Au o-
ma ica, ol. 40, no. 8, pp. 1441–1446, 2004.
[13] Y. Cao, Z. Lin, and T. Hu, “S abili y analysis o linea ime-
delay sys ems subjec o inpu sa u a ion,” IEEE T ans. on
Ci cui s and Sys ems I: Fundamen al Theo y and Applica ions,
ol. 49, no. 2, pp. 233–240, 2002.
[14] S. Mie mon , “Con ˆole dis ibu´e de la ension d’alimen a ion
dans les a chi ec u es gals e p oposi ion d’un s´elec eu dy-
namique d’alimen a ion,” Ph.D. disse a ion, Ins i u e Na ional
Poly echnique de G enoble, Decembe 2008.
[15] C. Albea, C. Canudas-de Wi , and F. Go dillo, “Con ol and
s abili y analysis o he Vdd-hopping mechanism,” In P oc.
IEEE Con e ence on Con ol Applica ions (CCA), 2009.
[16] C. Albea and C. Canudas-de Wi , “Disposi i de commande
num´e ique pou un ableau de ansis o s pmos en pa allele,”
Pa en Numbe :08/07342.
[17] C. Albea, F. Go dillo, and C. Canudas de Wi , “High pe o -
mance con ol design o dynamic ol age scaling de ices,”IEEE
T ans. on Ci cui s and Sys ems I: Regula Pape , no. 99, pp. 1–
1.
[18] T. Hu and Z. Lin, Con ol sys ems wi h ac ua o sa u a ion:
analysis and design. Bi khause , 2001.
[19] E. F idman and U. Shaked, “A Desc ip o Sys em App oach o
H∞Con ol o Linea Time-Delay Sys ems,” IEEE T ans. on
Au oma ic Con ol, ol. 47, no. 2, pp. 253–270, 2002.
[20] ——, “An LMI app oach o s abili y o disc e e delay sys ems,”
In P oc. o he Eu opean Con ol Con e ence, 2003.
[21] E. F idman, A. Pila, and U. Shaked, “Regional s abiliza ion and
H∞con ol o ime-delay sys ems wi h sa u a ing ac ua o s,”
In e na ional Jou nal o Robus and Nonlinea Con ol, ol. 13,
no. 9, pp. 885–907, 2003.
[22] K. Gu, V. Kha i ono , and J. Chen, S abili y o ime-delay
sys ems. Bi khause , 2003.
[23] P. Millan, L. O ihuela, C. Vi as, and F. Rubio,“An op imal con-
ol L2-gain dis u bance ejec ion design o ne wo ked con ol
sys ems,” in P oc. IEEE Ame ican Con ol Con e ence (ACC),
2010. IEEE, 2010, pp. 1344–1349.