A Discrete Observer Design for Multicellular Choppers

Bogdan Cristian FLOREA1, Dan Alexandru STOICHESCU2

1 EAII, Politechnica University of Bucharest, Romania bogdan.florea@upb.ro

2 EAII, Politehnica University of Bucharest, Romania stoich@elia.pub.ro

Abstract—In this article, we introduce a discrete observer for a multicellular chopper with two branches. Multicellular choppers are a type of dynamical-switched systems that offer a higher performance compared to the classic chopper models. By using multiple commutation cells, the voltage is distributed over these cells; the cells reduce the stress on the components by reducing

the value of dVdt

. This last property allows us to control each cell

individually and reduce the cost of the components, especially in high power applications. The observer is tested in simulation with a direct control algorithm designed by the authors, using a 2 cells – 2 branches chopper with an RL load.

I. INTRODUCTION The design and manufacturing process of power components

have improved drastically in the last two decades, allowing a development of new energy conversion structures to be possible.

One such a structure is the multicellular converter. The idea was first introduced in the 1990s [1] & [2]. The basic principle is that the converter uses a series of commutation cells, which are linked together by floating voltage sources. The floating voltage sources are implemented using capacitors, thus obtaining N voltage levels with p = N !1commutation cells (fig. 1).

The voltage drop across each cell is equal to E p , set by the floating voltage sources, while the value of the current through the switches is identical to that of a classic structure [3].

Because of the use of capacitors as floating voltage sources, the voltage across them must be balanced within a safe interval

around the desired value of , where k is the number of the

current cell [4] & [5].

Figure 1. Multicellular chopper with N cells

The multicellular approach offers many advantages, such as lowering the voltage stress on the components, modularity of the system, individual control of each branch and each cell and better overall performances compared to the classic converters, but these advantages can only be attained if an adequate control algorithm is designed.

In order to design a good control algorithm, one needs to know the state of the system at all time. This can be done by using a sensor for each capacitor, but this method is both expensive and unreliable and it introduces many sources of possible failures. Another method to determine the state of the system is using an observer to estimate the states.

In this paper we will design an observer based on the super twisting algorithm for a multicellular chopper with two branches and a RL load, using a higher order sliding mode direct control method [6].

II. MULTICELLULAR CHOPPER MODEL

The series multicellular chopper consists of a number of pindividual cells linked together by floating voltage sources (fig. 1). As a result of this, we will have p+1 voltage levels

(0, Ep, 2Ep,…, (p!1)E

p,E) at the output of the system [8]. That

means that we have 2 p possible switching combinations. The controller needs to choose the best combination of switches in order to provide the required output voltage level and, at the same time, balance the capacitor voltages within an interval of

±10% of the required value . In order to protect the

switching elements, the controller needs to choose the best combination, in order to ensure a minimum number of switches. This is possible because we have multiple combinations for each voltage level.

We will use the following notations: ui is the control signal for each cell (where i is the active cell). The control signal is equal to 1 when the upper switch is on and 0 when the lower switch of the cell is on [8].

From this point forward, we will use a 2 cells – 2 branches configuration (fig. 2), in order to maintain notation clarity and simplicity. All the equations will be determined for this

kEp

kEp

The work has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/6/1.5/S/19.

978-1-61284-943-0

configuration. The results can be easily extended to a n cells – 2 branches configuration.

Figure 2. Multicellular chopper with 2 cells – 2 branches and a RL load

At the input, we have the control signals uij , the state vector

is x = (I,VC1,VC2 )T and the output of the system is y = x1 = I .

The system dynamics is described by the following set of equations (1):

!I = ! RLI + E

L(u12 !u22 )+VC1(u11 !u12 )I !VC2 (u21 !u22 )I

!VC1 =1C1(u12 !u11)I

!VC2 =1C2(u21 !u22 )I

y = I

"

#

$$$$

%

$$$$

The proposed 2 cells – 2 branches configuration is a four-quadrant system. It has 24 =16 modes of operation determined by all the possible combinations of switches. We can determine the influence of each switching combination on the output voltage and on the capacitor voltages. This is necessary for the controller, in order to be able to balance the capacitor voltages within the required interval for the corresponding voltage level.

The proposed structure can produce 5 voltage levels at the

output: !E,! E2,0, E2,E .

There are multiple switching combinations available for each voltage level. The controller chooses the best combination in order to balance the capacitor voltages and ensure a minimum number of switches, in order to protect the switching elements.

In order for the system to function correctly, we need to know the state of the system at all time. In other words, the value of the state vector x needs to be determined. The output current can be measured, but measuring the capacitor voltages is very difficult and ineffective.

We can reconstruct the state vector x from the measured output current and the input signals sequence. In the following chapters, we will describe the control method, the observability analysis and the discrete observer design.

III. CONTROL METHOD The control method proposed is a direct control method based

on the twisting algorithm [6] & [7]. We need to control the load current, so we define the sliding

surface s = Iref ! I . The control algorithm is composed of two parts:

1. determining the output voltage;

2. choosing the best switching combination for the required output voltage which best regulates the capacitor voltages with a minimum number of switches.

The output voltage is determined using a twisting algorithm with the following definition (2):

!u(t) =

!u, u >1

!Vmsign(s), s!s " 0, u "1

!VMsign(s), s!s > 0, u "1

#

$%

&%

with the following conditions:

VM >Vm,Vm >4!M

s0,Vm >

"!m

,!mVM #" > !MVm +"

where is the sliding surface and Γ! , Γ!,Φ, !! positive constants.

The value !u(t) determined using (2) will be the desired output voltage. After the output voltage has been chosen, the algorithm chooses the best combination in order to balance the capacitor voltages within an interval of ±10% of the desired value. The controller first checks is there is an operating mode that can regulate both capacitors at the same time, while producing the desired output voltage. If there is no such switching combination available for the required output voltage, the algorithm assigns priorities to the capacitors, in order to ensure the minimum number of switches; in other words, it will balance the capacitor which requires the fewest commutations, while the other capacitor will be balanced at a future switch.

To obtain a minimum number of switches, we need to know the current state of the switching elements and all possible future states, for the chosen output level. In order to determine the optimal combination, we define the FS matrix, which contains all the future states of the switches, starting from the current state up (3) [6].

FS =

s11 s12 s13 s14s21 s22 s23 s24… … … …si1 si2 si3 si4

!

"

#####

$

%

&&&&&

up = u11 u12 u21 u22!"

$%

We need to calculate the number of switches required for each possible future state, starting from the present state up . The number of switches is saved in theMS vector (4). The minimum value of MS gives us the next switching combination [6].

MS = m1 m2 … mi!"

#$

mk = up( j)%FS(k, j) ,k =1…ij=1

4

&

IV. OBSERVABILITY ANALYSIS AND OBSERVER DESIGN Considering the system in fig. 2, described by (1), we can see

that there are some switching combinations where the system is not fully observable (e.g. u11 = u12 = u21 = u22 =1 ) [9]. However, these combinations appear only in certain parts of the control

s = Iref ! I

(1)

(2)

(3)

(4)

(7)

sequence. From the rank condition of observability of (1), only two components of x are observable at the same time (5) [10].

!I = ! RLI + E

L(u12 !u22 )+VO

VO = !1LCij

uij2

j=1

p!1

"i=1

2

"

#

$%%

&%%

where Vo = !1L

uijVCijj=1

p!1

"i=1

2

"

From the estimate of the output voltageVO , we can determine the capacitor voltagesVCij (i =1,2; j =1,…, p!1) . If we assume that the capacitor voltages have a slow variation over the measuring interval, we can have a sequence of 2(p!1) measurements of the output voltageVO such that (6) is invertible [3] & [10].

VO1

!VO2( p!1)

"

#

$$$$

%

&

''''

= !1L

u111 … u1( p!1)

1 u211 … u2( p!1)

1

! " ! ! " !u11p!1 … u1( p!1)

p!1 u21p!1 … u2( p!1)

p!1

"

#

$$$$

%

&

''''

Using a first order sliding mode observer, produces oscillations at high frequencies, a phenomenon known as chattering, because of the use of the sign function. One way of overcoming this problem is replacing the sign function with a smoother function in the vicinity of the origin, like the saturation function. However, this method reduces the robustness of the system. In order to preserve the robustness and significantly reduce the chattering phenomenon at the same time, we can use a super-twisting observer, which is a second order sliding mode observer [3].

The general form of the observer is described by the following set of equations (7):

!x1 = !RLI + E

L(u12 !u22 )!

1L

uijVCij + ! uij I ! x11 2 sign(I ! x1)

j=1

p!1

"i=1

2

"j=1

p!1

"i=1

2

"

!VCij =1Cij

uij +!ijuijsign(I ! x1), i =1, 2; j =1,…, p!1

#

$%%

&%%

with ! and! satisfying the following conditions:

! > 0," > 1+#1!#

2!L, 0 <# <1

We define the observation errors (8): e1 = I ! x1eij =VCij ! VCij , i =1, 2; j =1,…, p!1

"#$

%$

And we have (9):

!e1 = !1L

uijeij + ! uij e11 2 sign(e1)

j=1

p!1

"i=1

2

"j=1

p!1

"i=1

2

"

!eij = !!ijuijsign(e1), i =1,2; j =1,…, p!1

#

$%

&%

In order to obtain the discrete version of the observer, we use the hybrid model of the multicellular chopper. A hybrid system is described as the interaction of a dynamical continuous system and a discrete system. In our case, the continuous part is

represented by the evolution of the state vector x , and the discrete part is represented by the switching states uij [3].

By using the hybrid conventions, we can describe the system defined by (1) as follows (10): !x = A(u)x +B(u)y =Cx!"#

where

A(u) =

!RL

u11 !u12L

u22 !u21L

u12 !u11C11

0 0

u21 !u22C21

0 0

"

#

$$$$$$$$

%

&

''''''''

,B(u) =

EL(u12 !u22 )

00

"

#

$$$$$$

%

&

''''''

C = 1 0 0"#

%&

The system is sampled at equal time intervals T and the control signals (inputs) are generated under zero order hold conditions u(t) = uk, t ! [kT, (k +1)T ) (they remain constant during sampling intervals). The output of the system is measured at every sampling instant kT: yk = y(kT ) .

Because the system is implemented using the hybrid model, we can only control it at the sampling moments kT. For maximum accuracy, we will use the current output measurement in the observer. For this purpose, we can create a discrete observer from (7) to estimate the system states at the sampling instances kT.

Using the discretization techniques for state-space models, we have (11):

x[k +1]= Adx[k]+ Bdu[k]y[k]=Cdx[k]!"#

where

Ad = eAT ,Bd = ( eAT d! )B

!=0

T

! ,Cd =C

are the discrete forms of the matrices A , B andC . The observer becomes (12): x[k +1]= Ad x[k]+ Bdu[k]+!(y[k]! y[k])y[k]=Cd x[k]"#$

!(y[k]! y[k]) =Osign(y[k]! y[k])

O =

! uij y[k]! y[k]1/2

j=1

p!1

"i=1

2

"

"11(u12 !u11)"21(u21 !u22 )

#

$

%%%%%%%

&

'

(((((((

To calculate the discrete matrices Ad and Bd, we use Euler’s

discretization technique and the final result is (13):

x[k +1]= (I + AT )x[k]+ (IT + 12AT 2 )Bu[k]+!(y[k]! y[k])

y[k]=Cd x[k]

"

#$

%$

(5)

(6)

(8)

(9)

(10)

(11)

(12)

(13)

978-1-61284-943-0

V. SIMULATIONS AND RESULTS In order to highlight the results, we will use a 2 cells – 2

branches configuration with a RL load. The chopper is modeled using (10) in MATLAB/Simulink in open loop, with the following parameters: E =120V,L =100mH,R =10!,C = 33µF, f = 60KHz The control signals uij are generated using the control method

proposed by the authors in [6], which is a control method based on the twisting algorithm.

The measured variable is the output current I. The capacitor

voltages have equal values (VC11 =VC21 =E2= 60V ) .

Figure 3 shows the output voltage of the system. The intermediary voltage levels are visible on the figure.

In figure 4, we can see the output current, the reference value and the observed value of the current.

In figures 5 and 6 we have the capacitor voltages (VC11,VC21 ) ,

the observed value and the reference value (E2= 60V ) . Note that

the observed value converges to the real value and that the voltages are kept within the safe interval of ±10% of the desired value.

Figure 3. Output voltage VO

Figure 4. Output current (I), the refference value (Iref) and the observed

current value (Iobs)

Figure 5. The capacitor voltage (VC11), the refference value (VC11ref) and the

observed capacitor voltage value (VC11obs)

Figure 6. The capacitor voltage (VC21), the refference value (VC21ref) and the observed capacitor voltage value (VC21obs)

VI. CONCLUSIONS The paper introduces a discrete super-twisting observer for a

4 quadrant multicellular chopper with an RL load. The observer behaves well in open loop and, in future works, it will be tested in closed loop for the proposed system.

The observer-driven control proposed provides good results for operating multicellular choppers without the use of voltage sensors. The observer is discretized because of the hybrid system properties. The discretization also allows a microcontroller or a DSP implementation.

The discrete observer is adapted for the 4-quadrant chopper (2 branches configuration), which is only locally observable.

In future works, we will adapt the observer for a DC machine load and test it in simulation and in the laboratory.

REFERENCES [1] Th. Meynard, H. Foch, “Dispositif de conversion d'energie electrique à

semiconducteur”, brevet francais no. 91,09582, Europe, Japan, USA, Canada, 92,00652

[2] G. Gateau, Th. Meynard, L. Delmas, H. Foch, “Stacked Multi-cell converter: topology and control”, EPE Journal, V. 12, No. 2, pag. 14-18, 2001

[3] K. Benmansour, “Réalisation d'un banc d'essai pour la Commande et l'Observation des Convertisseurs Multicellulaires Série: Approche Hybride”, PhD Thesis, ENSEA, 2009

[4] G. Gateau et all., “Multicell Converters: Active Control and Observation of Flying Capacitor Voltages”, IEEE Transactions on Industrial Electronics, V. 49, No. 5, pag. 998 – 1008, 2002

[5] F.J. Bejarno, M. Ghanes, J.P. Barbot, “Observability and observer design for hybrid multicell choppers”, International Journal of Control, vol. 83, p. 617 – 632, 2010

[6] B.C. Florea, D.A. Stoichescu, M. Ghanes, J.P. Barbot, “Discrete event control for multicell chopper driving DC machines”, SIITME 2010 Romania, p. 231 – 236

[7] W. Perruquetti, J.P. Barbot, “Sliding Mode Control in Engineering”, 2002, Ed. Marcel Dekker

[8] K. Benmansour, J. Deleon, M. Djemaï, “Adaptive Observer for Multi-Cell Chopper”, Second International Symposium on Communications, Control and Signal Processing ISCCSP, Marrakesh, Morocco, 2006

[9] W. Kang, J.P. Barbot, L. Xu, “On the observability of nonlinear and switched systems”, Lecture Notes in Control and Information Sciences, Springer Berlin, 2009

[10] M. Defoort, M. Djemaï, T. Floquet, W. Perruquetti, “On finite time observer design for multicellular converter”, International Workshop on Variable Structure Systems, Mexico, 2010