Control Design for Electronic Power Converters...Control Design for Electronic Power Converters 10 / 49 Control law for the duty cycle ÆPWM TOOL: Energy shaping approach for oscillation

Control Design for Electronic Control Design for Electronic

Power ConvertersPower Converters

PhD Thesis

by Carolina Albea

Supervised by Carlos Canudas de Wit and Francisco Gordillo

Grenoble, 14 October 2010

Control Design for Electronic Power Converters 2 / 49

OUTLOOKOUTLOOK

INTRODUCTION

FIRST APPLICATION: BOOST INVERTER

SECOND APPLICATION: DC-DC Vdd-HOPPING CONVERTER

CONCLUSIONS & FUTURE WORK

INTRODUCTION INTRODUCTION FIRST APPLICATION: BOOST INVERTERFIRST APPLICATION: BOOST INVERTERSECOND APPLICATION: DCSECOND APPLICATION: DC--DC DC VddVdd--HOPPING CONVERTERHOPPING CONVERTERCONCLUSIONS & FUTURE WORKCONCLUSIONS & FUTURE WORK


OUTLOOKOUTLOOK

INTRODUCTION INTRODUCTION






INTRODUCTIONINTRODUCTION

Why are power converters important?

Relevant issues in power converters:Reliability of the power converters highly robust to achieve a high useful life

High efficiency: economic and environmental value of wastedcost of dissipated energyimproved profitability of the investment in the electronic market


… thus, automatic control has an special relevance.


INTRODUCTIONINTRODUCTION

Classification: DC-DC converters. DC-AC converters. Inverter.AC-DC converters. Rectifier.AC-AC converters. Transformer.

DC-AC coverter

All level power

normal scale

4th order

Control objective: limit cycle

Boost Inverter

DC-DC converter

Low power level

micro- or nano-scales

1st order

Control objective: equilibrium

Vdd-Hopping Converter



OUTLOOKOUTLOOK

INTRODUCTION

FIRST APPLICATION: FIRST APPLICATION: BOOST INVERTERBOOST INVERTER




FIRST APPLICATION: BOOST INVERTERFIRST APPLICATION: BOOST INVERTERSECOND APPLICATION: DCSECOND APPLICATION: DC--DC DC VddVdd--HOPPING CONVERTERHOPPING CONVERTERCONCLUSIONS & FUTURE WORKCONCLUSIONS & FUTURE WORK


Boost Inverter in a practical case:Boost Inverter in a practical case: generates an alternating current and is an elevator.

BOOST

INVERTER

PHOTOVOLTAIC

ARRAY

DC 48V

ELECTRICAL

GRID

AC 220Vrms 50Hz

AC 220Vrms 50Hz

[Antonkopoulus et al, T. Inter. Journal of Electronics 92, Castaner et al. Book 03]



1.1. MOTIVACIMOTIVACIÓÓNN2. SYSTEM3. CONTROL OBJECTIVES:

1. CONTROL LAW2. ANTY-SYNCRONIZATION3. ADAPTIVE CONTROL4. ATTRACTION DOMAIN


[Caceres et al, T. Power Electronics 99]

BOOST INVERTER: BOOST INVERTER:



1. MOTIVACIÓN

2.2. SYSTEMSYSTEM3. CONTROL OBJECTIVES:

1. CONTROL LAW2. ANTY-SYNCRONIZATION3. ADAPTIVE CONTROL4. ATTRACTION DOMAIN

DC AC

DC-DC BOOSTCONVERTER

1

DC-DC BOOSTCONVERTER

2

LOAD

V1

V2

Vin Vout


1.1. Control laws for the duty cycles of each DC-DC boost converter

1.2. Anti-synchronized voltage signals

2. Parameter adaptation for unknown and/or slowly varying loads

3. Set of initial voltage and current values that guarantee the system convergence.



1. MOTIVACIÓN2. SYSTEM

3.3. CONTROL OBJECTIVES:CONTROL OBJECTIVES:1. CONTROL LAW2. ANTY-SYNCRONIZATION3. ADAPTIVE CONTROL4. ATTRACTION DOMAIN

DCs with a desired positive voltages.

1. MAIN CONTROL OBJECTIVE: Generate an AC with a desired voltage.


Control law for the duty cycle Control law for the duty cycle PWMPWM

TOOL: Energy shaping approach for oscillation generation [Aracil et al. IECON02, Gordillo et al. IWESA04]

Autonomuos & stable behaviour!!!!

Energy functionDesired behaviour




3.3. CONTROL OBJECTIVES:CONTROL OBJECTIVES:1.1. CONTROL LAWCONTROL LAW2. ANTY-SYNCRONIZATION3. ADAPTIVE CONTROL4. ATTRACTION DOMAIN

DC-DC BOOST

CONVERTER

1

PHASE

CONTROLLER

V

−V

t

V

−Vt

V

t

−Vt

V

−Vt LOAD

CONTROL

1

CONTROL

2

DC-DC BOOST

CONVERTER

2

x2

x4

x

ω

u1

u2

Vin

Vout


Complete Complete boostboost inverterinverter

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

−+=

+−=

+−=

+−=

Rv

Rviu

dtdvC

Vvudt

diL

Rv

Rviu

dtdvC

Vvudt

diL

L

inL

L

inL

2122

22

222

2111

11

111

2

1

Model for one side

42112

211 1axaxxux

xux+−=

+−=&

&

]1,0[1 ∈uwith and CL

Ra

0

1=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎯⎯⎯⎯⎯⎯ →⎯

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡==

==

4

3

2

11

1

2

2

1

12

422

23

121

1

11

xxxx

iviv

inL

in

inL

in

Vvxi

CL

Vx

Vvxi

CL

Vx

L

L

[Caceres et Al, T. Power Electronics 99]






Control law from energy shaping approach [Albea et al. Control Engineering Practice 2010]

Open-loop system

)()( 2021012

2

0221

ζζζζωζ

ζζζ

−Γ−−−=

−=

k&

&

14212

2021012

4224

224

222

2

1 2)()(321

xaxxaxxkxaxxaxxaxau

−+−Γ+−+++−+

=ξξξξω&

32434

404223032

2422

242

224

2

2

14212

202111012

4224

224

222

2

1

2)()(321

2)()(321

xaxxaxxkxxaxaxxaxau

xaxxaxxkxxaxaxxaxau

−+−Γ+−+++−+

=

−+−Γ+−+++−+

=

ξξξξω

ξξξξω

&

&

)2(321

1421242

24

224

222

22

2021

xaxxaxxuxaxxaxxaxa

−+−++−+=

−=

&

&

&

ζ

ζζζ

Extension to the controller considering an inductive load [Albea et al. CDC07].





Target system


Simulations





L1 = L2 = 250μH

C1 = C2 = 250μF

R = 100Ω, Vin = 48V

Vout = 220 2√2

sin(50 · 2πt)

V1 = 450 + 160 sin(50 · 2πt)

V2 = 450 − 160 sin(50 · 2πt)

Vin = 48V

fPWM = 50KHz, Ts = 0.1s


Anti-synchronized voltage signals with a Phase Controller[Hsieh et al. Industrial Electronics 96, Albea et al. IECON06]

Aplication of PLL to the boost inverter

TOOL: Phase-locked loop PLL




3.3. CONTROL OBJECTIVES:CONTROL OBJECTIVES:1. CONTROL LAW

2.2. ANTYANTY--SYNCRONIZATIONSYNCRONIZATION3. ADAPTIVE CONTROL4. ATTRACTION DOMAIN

DC-DC BOOST

CONVERTER

1

PHASE

CONTROLLER

V

−V

t

V

−Vt

V

t

−Vt

V

−Vt LOAD

CONTROL

1

CONTROL

2

DC-DC BOOST

CONVERTER

2

x2

x4

x

ω

u1

u2

Vin

Vout


Simulations

THD=0.22%







Synchronization with the electrical grid.







)ˆ(ˆˆ 224212 xxKaxxauxx −++−=&

Parameter adaptation for unknown and/or slowly varying loads (a=f(R0))[Albea et al. Control Engineering Practice]

))(ˆ(ˆ 2422 xxxxa −−= γ&

TOOL: Approximate stability guaranteed by singular perturbation analysis.

TOOL: Adaptive control




3.3. CONTROL OBJECTIVES:CONTROL OBJECTIVES:1. CONTROL LAW2. ANTY-SYNCRONIZATION

3.3. ADAPTIVE CONTROLADAPTIVE CONTROL4. ATTRACTION DOMAIN

DC-DC BOOST

CONVERTER

1

PHASE

CONTROLLER

V

−V

t

V

−Vt

V

t

−Vt

V

−Vt LOAD

CONTROL

1

CONTROL

2

DC-DC BOOST

CONVERTER

2

OBSERVER

ADAPTiVE

CONTROL

x2

x4

x

x

ω

u1

u2

x2a

Vin

Vout


Stabilty: timetime--scalescale separationseparationslow variables: fast variables:

yxPBa

Byaxx

xaxUxfx

T )~(~

~~~)~,()(

γ

α

−=

−−=

−=

&

&

&

Complete system

xax ~,~

STABILITY ANALYSIS







STABILITY PROOF: singular perturbation analysis [Khalil, Kokotovic]

0~~

)( =⎥⎦

⎤⎢⎣

⎡==

ax

xz φ

)())(,()( xfxxxUxfx =−= φ&

*2

*22

11

~~

~~~

~~

yxadd

yaxxdd

xxdd

=

−−=

−=

τ

τ

τ)

1.-Find a stationary solution of the fastfast subsystem.

2.-Substitute this solution in the slowslow subsytem

3.-Check the boundary layer properties of the fast subsystem along oneparticular solution by using TikhonovTikhonov´ss TheoremTheorem.







90% errorTime (sec) Estimated value R (Ω) Real value R (Ω)

0 1000 100

0.5 100 1000

SIMULATIONS:

Extension to the adaptive control law considering an inductive load [Albea et al. IFAC08].







Set of initial voltage and current values that guarantee the system convergence.

Local stabilitySaturationPhysical constraints vc1 >0, vc2>0

]1,0[∈u

TOOLS:Lyapunov Theory employedin polynomial systems.[Levin et al. Automatic Control94, Romanchuk Automatica96, Tibken et al. CDC02]

Sum of Squares (SOS) optimization.SDPs.SOSTOOLS.




3.3. CONTROL OBJECTIVES:CONTROL OBJECTIVES:1. CONTROL LAW2. ANTY-SYNCRONIZATION3. ADAPTIVE CONTROL

4.4. ATTRACTION DOMAINATTRACTION DOMAIN

Attraction

Domain

ATTRACTIONDOMAIN

CONSTRAINT

DESIRED LIMITCYCLE

SATURATIONS

ESTIMATEDATTRACTION

DOMAIN


Max cst

are SOS; i=1,…,Niii xgxpcxV ε−+− )()())((

Problem formulation:

By numerical inspection, it was found: x(0)=(0,-0.1,0.2,5.8)T

which corresponds to V(xV(x)=33,02.)=33,02.

[Albea at al., NOLCOS07]

26.23*)( == cxV

Result:

Semi-definite positive polynomial. Positive polynomial constraints.




3.3. CONTROL OBJECTIVES:CONTROL OBJECTIVES:1. CONTROL LAW2. ANTY-SYNCRONIZATION3. ADAPTIVE CONTROL

4.4. ATTRACTION DOMAINATTRACTION DOMAIN


OUTLOOKOUTLOOK

INTRODUCTION


SECOND APPLICATION: SECOND APPLICATION: DCDC--DC DC VddVdd--HOPPING CONVERTERHOPPING CONVERTER


INTRODUCTION INTRODUCTION FIRST APPLICATION: BOOST INVERTERFIRST APPLICATION: BOOST INVERTER

SECOND APPLICATION: DCSECOND APPLICATION: DC--DC DC VddVdd--HOPPING CONVERTERHOPPING CONVERTERCONCLUSIONS & FUTURE WORKCONCLUSIONS & FUTURE WORK


ARAVIS Project sponsored by ARAVIS Project sponsored by MinalogicMinalogic polepole

Current tecnology: 90nm, 65nm and, even, 45nm can not be applied any more to the technology of 32nm.

WHY???Technology variability

phenomenon

1.1. MOTIVACIMOTIVACIÓÓNN2. SYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS




Three technology keys:Three technology keys:ReRe--configurable structure configurable structure w.r.tw.r.t. applicability requirements . applicability requirements [Pinheiro et al. COSLP01].

Dynamic management of the power consumption and activity Dynamic management of the power consumption and activity Asynchronous technique Asynchronous technique [Marculescu et al. ISCA02].

Cluster

Vdd - Hopping

ProgrammableRing

Energy Controller

Voltage Controller

QoSController

Speed1, No. of Instructions1, Deadlines1

Cluster

Energy Controller

Speed2, No. of Instructions2, Deadlines2

Processing Nodes Processing Nodes

f, Vddf, Vdd

Vdd - Hopping

ProgrammableRing

Voltage Controller

[Hellerstein et al. Book04, Cervin et al. RTS02, Ríos et al. PATMOS05].



1.1. MOTIVACIMOTIVACIÓÓNN2. SYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS


DCDC--DCDC VddVdd--Hopping Converter: Hopping Converter: [S. Miermot et al. LNCS07]

System control model

uRuR k =)(

⎪⎩

⎪⎨⎧

====

=∑=

n

i

ni

RRRR

Mu

L21

1

where

and

M1 R1

M2 R2

Mn−1 Rn−1

Mn Rn

Vh

IMPEDANCE

Z(vc)

vc

Il(vc)

CONTROL

uk



CONTROL

CLK

LPM

+

−

Vh Vl

SetofPMOS

vce

uk

vr

Objective: Good performance during transient periods.

1. MOTIVACIÓN

2.2. SYSTEMSYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS


DCDC--DC converter circuitDC converter circuit

Error equation

cr vve −=where rhrkrk vVvbuvebue && ++−+++−= δββ )()(

Converter model – 1st order

δβ −−+−= kchcc uvVbvv )(&



⎪⎪⎩

⎪⎪⎨

⎧

>=

>=

01

01

0CRb

CRL

β 0>=C

Ileakδand depend on the load parameter.

depends on PMOS resistance and load parameter.

IleakC rL

Il

Icap Idyn + Ishort

IMPEDANCE

ImpedanceImpedance[PhD thesis 2008 S. Miermot]

1. MOTIVACIÓN

2.2. SYSTEMSYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS


STABILITY OF THE STABILITY OF THE CLOSEDCLOSED--LOOP LOOP

SYSTEMSYSTEM

Vdd-Hopping

+PSS Two voltagesources

Objectives in VLSI:1. High energy-efficiency.2. Small current peaks.3. Fast transient periods.4. Robustness w.r.t. parameter uncertainty.5. Robustness w.r.t delays6. Simple implementation

CONTROL !!!!!

LDVSLocal Dynamic VoltageScaling Architecture

GALSGlobally Asynchronous

and Locally Synchronous Systems




3.3. CONTROL OBJECTIVESCONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS


Set of controllers using sameramp reference: [Albea et al., IECON08]

Controller 1: linear controller.Controller 2: feedback linearization.Controller 3: Lyapunov-based design.

Disipated total energy (μJ)Intuitive control 7.2

Control No.1. 6.8

Control No.2. 6.2

Control No.3. 4.8

Lyapunov controller



1. MOTIVACIÓN2. SYSTEM3. CONTROL OBJECTIVES

4.4. PRELIMINARY CONTROL STUDYPRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS

)(1 esignuu kk += −

An intuitive control law with a ramp reference: [S. Miermot et al. LNCS07]

Switching limitation!!!

No switching limitation!!!From comparison Better voltage and current performance with developed

controllers.


⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

+−+= −

)(

ˆ*

***

11

k

kkk

rkhs

srrrsNk veVbT

TvvvTroundsatu

δβ

High-performance Lyapunov controller:

TOOL: Lyapunov TOOL: Lyapunov theory. theory. .

TOOL: Optimal control TOOL: Optimal control for minimizing current peaks and dissipated energy.

krskk evT 11~~ γββ −= −



ukOPTIMALREFERENCE

CONTROL

ADAPTIVECONTROL

v∗ vc

β

+

-SYSTEM




INTUITIVE CONTROL LAW [S. Miermot et al. LNCS07]


HIGH-PERFORMANCE LYAPUNOV CONTROLLER

Advantages:Dissipated energy 0.5μJ (93% energy saving w.r.t. intuitive controller.).Minimum current peaks.Fast transient periods.

Disadvantages:Complex implementationComplex implementation






( )σ211 KeKroundsatu Nk +=

From Controller No.1: linear controller with a step reference:



Design of Energy Design of Energy EwareEware controller.controller.

Lemma 1: Consider the interval

If ξ is chosen in I, then the following inequalities are satisfiedand

Notice that

01 >K

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++=

)(22,

2 βω

ωβ

ωβ

h

ll

k

n

n

k

n

k

bububu

I

0)(2>

+ βω

hk

n

bu

0)( 21 <−+ KbuKhk β

)(

)()(2

2

2

1

lh

n

lh

kn

VVbK

VVbbu

K l

−=

−

+−=

ω

βξω

02 >KNotice that

1. MOTIVACIÓN2. SYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY

5.5. ENERGY AWARE CONTROLENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0

5

10

15

20

25a)

NT

rans

t(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0.8

0.9

1

1.1

1.2b)

V(V

)

t(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0

0.1

0.2

c)

I(A

)

t(s)

WITH STEP REFERENCE SIGNAL

( )σ211 KeKroundsatu Nk +=

BEWARE!!!!!

This current peak can

damage the physical system






CONTROL WITH CURRENT PEAK CONSTRAINTS[Albea et al. CCA09]

( )σαα 211

1

1KeKsatroundsatu

Mkkmkk

uu

Nk += +

+−

−

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0

5

10

15

20

25a)

NT

rans

t(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0.8

0.9

1

1.1

1.2b)

V(V

)

t(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10−6

0

0.05

0.1

0.15

c)

I(A

)

t(s)





max0 II Δ≤Δ< max0

max IuR

vVI k

chigh Δ≤Δ−

≤Δ−

Maximum current peaks constraint:

considering vc is continuous


( )σαα 211

1

1KeKsatroundsatu

Mkkmkk

uu

Nk += +

+−

−

( ) kkkkN

k eKeeKsatroundusatuMkmk

21111 )( +−+= −−αα

DISCRETIZATION OF CONTROL WITH CURRENT PEAK CONSTRAINTS

⎪⎩

⎪⎨⎧

=

−=

TKK

KKK

22

211 2

4. Rounding

mechanism

ek

Vref

Vc

Δumk

ΔuMk

satN1

q−1us

k−1

usk

ΔuskΔuk

ΔI

ADC

ADC

ek−1

K1

K2

Vrk

Vck

q−1

3. On-line

saturation

mechanisms

2. On-line

current limits

mechanism

1. Error Tracking Filter

Δusk

usk

5.Output saturation

mechanism

Control patent pending under the name:ENergyENergy--AwaReAwaRe ControlControl(ENARC)






INTUITIVE CONTROL LAW [S. Miermot et al. LNCS07]


ENARC



TOTAL ENERGY DISSPATEDIntuitive controller 7.2 μJ

ENARC 0.3 μJ

Advantages:96% energy saving .Easy implementation.Small current peaks.Fast transient periods.




[Albea et al., CCA09]

2K1KTheorem 1: System with the

controller is globally asymptotically stable, if and are positives.

δββ +−+++−= )()( hrkrk Vvbuvebue&σ21 KeKuk +=

2

22

2)(

)(2K

vVbeV

rh

σσ −+

−=

Proof: Lyapunov function candidate:

P2

e

σ

Ω1

1ΩThe stability is established by LaSalle’s invariant principle, since the

maximum invariant set in with corresponds to the single point . Therefore, every solution starting in approaches as

0),( =σeV&),0(2 σσ === eP 1Ω

.∞→t 2P



1. MOTIVACIÓN2. SYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL

6.6. APPROXIMATE STABILITY ANALYSISAPPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS


2K1K

2

max0

2

max0

2

σσ

σKsatu

IvV

RK

IvV

RKkch

ch

Δ−

+

Δ−

−=

Theorem 2: System with the controller globally asymptotically stable, if and are positives.

δββ +−+++−= )()( hrkrk Vvbuvebue& σα

α 211

1KeKsatu

Mkkmkk

uuk += +

+−

−

Remark: The equilibrium of the system are in Region I.

Region II

Region III

Region I

e

σ



1. MOTIVACIÓN2. SYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL

6.6. APPROXIMATE STABILITY ANALYSISAPPROXIMATE STABILITY ANALYSIS7. PARAMETER UNCERTAINTY AND DELAYS

2Ω

By La Salle’s invariance principle, we can conclude the statement of the Theorem, since there exists a set limited by the level curve for sufficiently large and the saturation limits, which is compact and positively invariant.

2Ω 1),( ceV =σ&

1c


CONTROL

Vdd-HOPPING

vr e uk

vc

+

-

R0

LOAD

RL, C

K1 K2z−2 z−1

Load dynamic resistance (RL(t)) from 55.53Ω to 72.46Ω

Load capacitance from 1nF to 1pF

PMOS characteristic from 25Ω to 38Ω

System frequency from 125MHz to 600MHZ



Parameter uncertaintyParameter uncertainty

DelayDelay

1. MOTIVACIÓN2. SYSTEM3. CONTROL OBJECTIVES4. PRELIMINARY CONTROL STUDY5. ENERGY AWARE CONTROL6. APPROXIMATE STABILITY ANALYSIS

7.7. PARAMETER UNCERTAINTY AND DELAYSPARAMETER UNCERTAINTY AND DELAYS

Synchronization issuesSynchronization issuesTradeTrade--off between off between energy consumption energy consumption and performanceand performance

TWO PROBLEMS:TWO PROBLEMS:


,

1

kk

hkhk

kk

wwhkkk

xKxuxLz

wBBuAxx

φ===

++=

−−

−+

]0,[ hk −∈∀

SUBOPTIMAL CONSTANT TUNING: KSUBOPTIMAL CONSTANT TUNING: K11, K, K22

The considered model is: delay

perturbations L2, i.e., energy bounded

uncertain matrices

PROBLEMS FOR THIS DELAYPROBLEMS FOR THIS DELAY--TIME SYSTEM:TIME SYSTEM:

1.-Asymptotically stability.

2.-Perturbation rejection.

3.-Robustness w.r.t. parameter uncertainty.

TOOL: HTOOL: H∞∞ control theory applied control theory applied to timeto time-- delay systems.delay systems. To develop of LMIs through Lyapunov-Krasovskiimethod.

` [Cao et al. Circuit and Systems 02, Fridman et al European Journal of Control 05, Moon et al. International Journal of Control 01]





kkkkN

k eKeeKusatu 21111 )( +−+= −−


KK11, K, K22

Asymptotic stability & disturbance rejection.

System robustness w.r.t.:

• delays

• parameter uncertainty

.2,1,,...,1

0)21(

,0)2(

00

2

1)(

1)(2

2

)()()(11

)()()()(2

)(1

)(

==

≥⎥⎦

⎤⎢⎣

⎡+−∗

≥⎥⎦

⎤⎢⎣

⎡−∗

<

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−∗∗∗

−−∗∗

−−+−∗+−−ΓΓ

=Γ

−

−

iNjPu

YcPNuN

Yc

Q

ShR

QBYDBTDBRhQPQBSYDBTDB

ii

jji

j

jji

jjj

j i

i

γ

ω

ω

T

i

TT

i

T

i

TT

jTjTjj

TjjTi

jji

Tjjj

BDYBTQAQP

LLYDBBDYTDBBDTQQAAQ)()(

1)(

11)(

2

)()()()(11

)()(1

)(1

2

2−

−−

++−+=Γ

+++++−+=Γ

where






0.8R0 1.2R0

0 0.2 0.4 0.6 0.8 1

x 10−6

01020

a)

NT

rans

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0.81

1.2b)

V(V

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

00.02

d)

I(A

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

01020

a)

NT

rans

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0.81

1.2b)

V(V

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

00.02

d)I(

A)

t(s)

Delay & time-varying load RL






400MHz 200MHz

0 0.2 0.4 0.6 0.8 1

x 10−6

01020

a)

NT

rans

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0.81

1.2b)

V(V

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0

0.05

d)

I(A

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

01020

a)

NT

rans

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0.81

1.2b)

V(V

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

00.02

d)I(

A)

t(s)







1pF initial K1 & K2 1pF suboptimal K1 & K2

0 0.2 0.4 0.6 0.8 1

x 10−6

01020

a)

NT

rans

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0.81

1.2b)

V(V

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

00.02

d)

I(A

)

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

01020

a)

NT

rans

t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

0.81

1.2b)

V(V

)t(s)

0 0.2 0.4 0.6 0.8 1

x 10−6

00.02

d)I(

A)

t(s)







OUTLOOKOUTLOOK

INTRODUCTION



CONCLUSIONS & FUTURE WORKCONCLUSIONS & FUTURE WORK

INTRODUCTION INTRODUCTION FIRST APPLICATION: BOOST INVERTERFIRST APPLICATION: BOOST INVERTERSECOND APPLICATION: DCSECOND APPLICATION: DC--DC DC VddVdd--HOPPING CONVERTERHOPPING CONVERTER



CONCLUSIONSCONCLUSIONS

Vdd-Hopping converter:

Dissipated energy is reduced

Current peaks are limited

Fast transient periods

Simple implementation

Robustness w.r.t. delay

Robustness w.r.t. uncertain parameters

Approximate system stability

Boost inverter:

To get desired behavior by:

a two nonlinear controllers.

anti-synchronizing both voltage signals.

Autonomous and stable system.

Adaptation for unknown and/or slowly varying loads

Set of initial voltage and current values that guarantee the system convergence.

Automatic control contributes to the:




FUTURE WORK & OPEN LINESFUTURE WORK & OPEN LINES

Boost inverter:

Physical implementation.

Adaptive control extension considering not all states are measured.

Global stability analysis to the system with the adaptive controller for a infinite time interval.

Less conservative solution for estimating the attraction region.

Extension of the estimated attraction region considering the phase controller and the adaptive mechanism.



Boost inverter Microcontroller eZdsp F2812

Voltage conversions


FUTURE WORK & OPEN LINESFUTURE WORK & OPEN LINES

Vdd-Hopping converter:

Implementation in VHDL-AMS.

A better numerical solution for obtaining an optimal voltage reference for the Lyapunov controller

Extension of the stability analysis of the closed-loop system in discrete-time

Extension of the suboptimal tuning approach for the control gains considering the saturations.




PUBLICATIONS & PUBLICATIONS & RESEARCH ACTIVITIESRESEARCH ACTIVITIES

4 publications in international journals, 1 accepted in « Control Engineering Practice »1 submitted to « Control System technology »2 under preparations to « Circuit and Systems »

8 publications in international conferences,

4 publications in national conferences,

1 French national patent.


Thank you, gracias, merci.Thank you, gracias, merci.

Questions and comments?Questions and comments?

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