Steady-state Analysis for Switched Electronic Systems Trough Complementarity

Steady-state Analysis
for Switched Electronic Systems
Through Complementarity
Ph.D. Candidate
Gianluca Angelone
Ph.D. course on AUTOMATIC CONTROL
Coordinator: Prof. LUIGI GLIELMO
Tutor: Prof. FRANCESCO VASCA
Co-Tutor: Dr. LUIGI IANNELLI

Copyright ©2011 by Gianluca Angelone
All right reserved

Research activities outline
 Complementarity models for switched electronic systems:
 Modeling of switches
 Power converters as linear complementary systems

 Steady-state analysis for switched electronic systems through complementarity:
 Analysis of accuracy for LCP approach

 Uniqueness of solution: preserving positive realness under discretization
 Detecting of multiple solutions

 Operating conditions of high power converters with minimum total energy
 Focus on modular multi-level converters

The main idea
2
e

 

2

x1

1


 1




x2

+

-

• Power electronics converters represent an interesting class of
switched systems. They can be assumed to consist of piecewise
linear elements, external sources, and electronic devices such as
diodes and electronic switches.
• The behavior of the converter is obtained by the commutations of
the electronic devices which determine the switchings among the
different converter modes.

Main ingredients
Power converters
+

-


x  Ax  Bz  Ee
w  Cx  Dz  Fe
w

Complementarity models

0 w z 0
z

Linear Complementarity Problem: Given a
real vector q and a real matrix M, find a real
vector z such that:

Why complementarity models?
j

e





j

i

…..

+



-



i



e




x  Ad x  Bd  Ed e



  Cd x  Dd  Fd e

  As  u   Bs  u z  g s  u 
w  Cs  u   Ds  u z  hs  u 
0w z0

 = vector of switches configuration



  Cd x  Dd  Fd e



  As   Bs z  Gs u  g s
w  Cs   Ds z  H s u  hs
0w z0

u

u

PWL characteristics

Diode

Generic PWL Characteristic:
two breaking points

  z
w     0  z
0w z0

Zener Diode

Boost DC-DC converter

e







1


Lx1  e  1

2

x1

1



2 

1

Cx2   x2   2
R
1  x1  2


x2





2   x2  1
2
2

2  z
2  w
0w z0

… and what about
the switch model?

Switch complementarity model

  z1  z2
 F 1
F
w1 


1   F R
1   F R
1R
1
w2 


1   F R
1   F R
0w z0
Ebers-Moll model

R  0

  z1  z2
w1   F  1   F  m axu
w2     m axu
0w z0
Unidirectional switch model

G. Angelone, F. Vasca, L. Iannelli, K. Camlibel, “Complementary Modeling”, chapter on the book
“Dinamics and Control of Switched Electronic Systems”, Springer Verlag, to be published.

Closed loop linear complementarity model
j





j

i

…..

+

-



0w z0



i





circuit equations


Linear
Complementarity
System

e

  Cd x  Dd  Fd e

  As   Bs z  Gs u  g s
w  Cs   Ds z  H s u  hs
0w z0
electronic devices



x

e


xc  Ac xc  Bc x  Ec e

u

u  Cc xc  Dc x  Fc e
controller

Computation of periodic oscillation (I)
xk  A xk 1  B z k  E ek


0w z0

Discretization

wk  C xk 1  D z k  F ek
0  wk  z k  0

Periodicity

x(T )  x(0)

T    Ts        N 

~
w  q  M~
z
~ z
0w~0

Sampling
Switching
Oscillation

Linear
Complementarity
Problem

Strictly Positive Realness (S. P. R.)
Continuous time

Gc ( s   ) has no poles in Re s   0
Gc ( s   ) is real for all positive real s

 0

Re Gc ( s   )  0 for all Re s   0
Discrete time

G (z )

has no poles in

G (z )

is real for all positive real z

Re G (z )  0

for all

z 1

z 1

0   1

Uniqueness of solution: preserving S. P. R.
If the discrete transfer function

xk  A xk 1  B z k  E ek

G ( z )  D  C ( zI  A) 1 B

wk  C xk 1  D z k  F ek
0  wk  z k  0

Is strictly positive real then then the
LCP admits a unique solution.

If the continuous transfer function is strictly positive real does the
discrete counterpart is positive real also?
If we use the “backward zero-order-hold” discretization technique, is
proven that for “sufficiently” small sampling time the strictly positive
realness is preserved:

xk  e A xk 1 

k

e A( k  ) ( Bzk  Eek ) d


( k 1)

L. Iannelli, F. Vasca, G. Angelone, “Computation of Steady-state Oscillations in Power
Converters through Complementarity”, IEEE Trans. on Circuits and Systems I: 58(6), 2011.

Oscillations for resonant converters
R2  6.0 (), 4.5 (), 1.5 (*)

Vdc  42V ; R1  200m;
C1  138nF ; L1  7.6H ;
n  1.64; C2  100F ;
Average output voltage vs. the
switching frequency, for different
values of the load resistance

Time-stepping simulations with
PLECS (continuous line) provide
results with negligible differences but
spending more time than LCP

F. Vasca, L. Iannelli, G. Angelone, “Steady State Analysis of Power Converters via a
Complementarity Approach” 10th European Control Conference, August 2009, Budapest, Hungary.

PWM boost converter

20
[A]
[V]

L  0.1mH

rL  0.1

15

e  10 V




10

R  7

C  0.2 mF

5

 vref  15 V


Ts  0.2 ms



0



k p  0 .1

1

2

3

4

5

6

[s]

-3

x 10

14

13.8

R  20 
13.5

[V]

13.7

[V]

0

13.6

13

13.5

12.5

13.4

12
0

0.5

1

1.5
[A]

2

2.5

3

0

1

2

3

4

5

[A]

F. Vasca, G. Angelone, L. Iannelli, “Linear Complementarity Models for Steady-state Analysis of
Pulse-width Modulated Switched Electronic Systems”, 19th Mediterranean Conference on Control
and Automation, Corfu Island, Greece, June 2011.

Multiple solutions
L  20 mH C  47 F
R 2  22 
R1  0 
Vdc  30 V vc  Vref  k p x2
Vref  11 .3 V k p  8.4

The closed loop system is not passive, and there are three solutions: two
stable with period 2Ts (blue), and one unstable with period (2)Ts (red)

L. Iannelli, F. Vasca, G. Angelone, “Computation of Steady-state Oscillations in Power
Converters through Complementarity”, IEEE Trans. on Circuits and Systems I: 58(6), 2011.

Industrial case study: modular multilevel
converters

Cell

Idealization
of DC-Link
Idealized Load

Closed loop control results
Capacitor voltages balancing: a typical approach


First, determine the arm voltage (how many cells? ) by using the nominal voltage:

varm j  m

Then, select and sort:
the m capacitors whit the smallest voltage when the arm current is positive
the m capacitors whit greatest voltage when the arm current is negative.
5

200
4.5

180

4

160

3.5

140

3

120

V arm 2 [ V ]



Vk
,m  n
6n

2.5

2

100
80

1.5

60

1

40

0.5

20
0

0
0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0

0.01

0.02

0.03

0.04

0.05

t [ s ]

Carriers (APO)

Modulating
signal

Phase 1: Lower Arm

0.06

Conclusions
• The use of a suitable switch model allows to construct a non-switched
complementarity model, without explicitly detailing all modes of the
converter and without a priori knowledge of the sequence of modes.
• The backward zero-order-hold discretization technique preserves the
positive realness of the transfer function and then the uniqueness of
the solution, for sufficiently small sampling period.
• The complementarity model can be exploited in order to capture the
steady-state behavior also in the presence of: closed
loop, resonances, multiple solutions, unstable orbits, complex
topologies.

List of publications
International Conferences
• F. Vasca, L. Iannelli, G. Angelone, “Steady state analysis of power converters
via a complementarity approach”, 10th European Control
Conference, Budapest, Hungary, August 2009.
• F. Vasca, G. Angelone, L. Iannelli, “Linear Complementarity Models for
Steady-state Analysis of Pulse-width Modulated Switched Electronic
Systems”, 19th Mediterranean Conference on Control and Automation, Corfu
Island, Greece, June 2011.
International Journals
• L. Iannelli, F. Vasca, G. Angelone, “Computation of periodic steady state
oscillations in power converters through complementarity”, IEEE Trans.
on Circuits and Systems I: 58(6), June 2011.
Chapters in books
• G. Angelone, F. Vasca, L. Iannelli, K. Camlibel, “Complementary
Modeling”, chapter on the book “Dinamics and Control of Switched
Electronic Systems”, Springer Verlag, to be published.

Steady-state Analysis for Switched Electronic Systems Trough Complementarity

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