1Alternative Control Methods for a Four-level

Three-cell DC-DC ConverterDiego Patino, Mihai Baja , Herve Cormerais, Pierre Riedinger, Jean Buisson, Claude Iung

Abstract

This paper proposes three synthesis methods for controlling power converters The three control strategies yieldeasy to implement state feedback control laws. The first one is a stabilization approach, based on energetic principlesand the notion of Lyapunov function. The second one is an optimal control approach based on the minimum principle.The last one is a neural predictive approach which uses model predictive control to track a given optimal stable limitcycle. This method allows a proper control of the waveform. Excepting for the predictive approach, the systemstability for the others methods are guaranteed. The four-level three-cell DC-DC Converter is used as a benchmark totest these strategies. Simulation and experimental results show that the methods have a good performance even withload perturbations.

Index Terms

Stabilizing control, optimal control, neural predictive control, switched systems, power converters.

I. INTRODUCTION

Power supplies are currently embedded in computers, electric drives, and cellular phones and generally in all

electric devices. Their aim is to convert an electrical energy shape (voltage /current / frequency) to another one.Applications in power systems frequently use DC-DC converters such as boost, buck or Cuk converters. In the

case of industrial applications with power of a few megawatts, the switching components voltage becomes very

high (several kilovolts). Therefore, the switching frequency must be maintained to a low value and bulky filtersare needed for obtaining an appropriated output [1]. To palliate this drawback, a new class of power electronicconverters has appeared, called multicellular or multilevel converters. These structures consist of a series connection

of switching devices with passive storage elements, which are used to generate intermediate voltage levels levels

[2]. The control law for these structures needs to maintain the intermediate voltage levels at some constant valuesand to regulate the voltage or the load current.

Corresponding author. Email:herve.cormerais@supelec.frMihai Baja, Herve Cormerais and Jean Buisson are with the Hybrid System Control Team, SUPELEC / IETR, CS 47601, 35576 Cesson-

Sevigne, France.Diego Patino, Pierre Riedinger and Claude Iung are with the CRAN (Centre de Recherche en Automatique de Nancy) Nancy-Universite,

CNRS, 2, avenue de la foret de Haye, 54516 Vandoeuvre-le`s-Nancy Cedex, France.

2The two main advantages of the multilevel converter are: i) it allows reducing the voltage throughout the switchesby splitting and distributing it on the intermediate levels, ii) the intermediate voltages take discrete values between0 and the full input voltage E. These voltage levels applied to the load produce a reduction of the output voltage

harmonics. The downside of the listed advantages is that, excepting simpler DC-DC converters, the control of

multilevel converters is more complex [3], [4].The intermediate voltages on the capacitors must be drastically regulated for two reasons: firstly, to ensure a

good waveform in the load [5] and secondly to protect the switches against high voltages. Usually, for an n-levelmulticellular converter, this can be done regulating the intermediate voltages to the following reference values:En, . . . , (n1)E

n[6].

From a control point of view, this is a non-linear problem. Several techniques have been used to solve it. The

first solutions were open-loop self-balancing strategies using Pulse Width Modulation (PWM), a fixed duty cycleand a constant phase shift between the levels [7]. The simplicity of these approaches leads to low performancesparticularly concerning transient.

The first close loop solutions employed independent PI controllers on each intermediate voltage in order to control

each switching cell [8]. There is no systematic method to tune the PI parameters and performances decrease inpresence of parameters or operating point variations. Recently, more complex methods have been developed such

as sliding mode control, which uses a state feedback method with a Lyapunov criterion to synthesize a switching

function [9].Another approach consists in the system linearization. The control is then achieved using a linear synthesis [10].

In [11] a control strategy based on a decision tree is employed for the control of a multilevel converter, based onthe requirements imposed by a Direct Torque Control (DTC) strategy, which is needed for the drive of an inductionmotor. There exist also predictive approaches which are based on the cost function minimisation over a given time

horizon [12]. With the predictive approaches, system nonlinearities and constraints related to the control objectivecan be explicitly taken into account. Although these methods are suitable for the control of multilevel converters

[13], [14] they are difficult to implement and sometimes the computing time becomes high with respect to thesystem dynamics.

The aim of this paper is to introduce and to compare three state-feedback control techniques on a common

benchmark defined on a four-level three-cell converter. The control objective is to regulate the average value of thecurrent in a R-L load and the average voltages in the capacitors to fixed values as specified above. The design part

must take into account some constraints concerning the switching frequency and also the robustness must be tested

with step change in the load and power input. The three-hybrid controls (the first one developed by the team atSUPELEC and the two others by CRAN) are: :

1) A stabilization approach: It is based on energetic principles. The first step of the method consists in thedefinition of a common Lyapunov candidate function taking into account the control objective. Its expressionis directly related to the storage energy. It uses the Hamiltonian Port formalism. In the second step, the

control, which is under a static state feedback, is chosen to ensure at each sample time the negativity of the

3time-derivative expression of the common Lyapunov candidate function guaranteeing the system stability [15].One of the advantages for this approach is that it can be easily extended to the case of DCM (DiscontinuousConduction Mode).

2) An optimal control approach: using the minimum principle and singular control theory, it is shown that it ispossible to synthesize an optimal state feedback control law. The optimisation of a quadratic criterion is a

pledge for system stability. The synthesis implies determining the singular optimal solutions that arise in the

optimal problem. An algorithm is proposed to generate all optimal trajectories on a given state space area.Then, a neural network is used to interpolate all the optimal solutions. Indeed, the optimal solutions are learnt

off-line and the real time implementation is ensured with a simple evaluation of the activation functions at

sample time.

3) A neural predictive control approach: the development of such a method is mainly motivated by the factthat in the power converters field, classical control methods are usually designed with a constant reference

which represents an average value of the steady state. Consequently, the cycle is not properly controlled and

it may present unexpected harmonic contents. In this part of the article, we propose to track an optimal limit

cycle defined as a reference in the steady state. This cycle represents the best cycle in the sense given by

a criterion. For instance, this criterion can be defined in order to reduce the quadratic norm of the tracking

error or to preserve the system against undesired harmonic content. The tracking control is obtained using

a predictive controller, which can take into account constraints on the switching frequency related to the

devices in the design part. The optimal solutions are learnt through a neural network that yields a state

feedback control. This state feedback is given in the form of a simple input-output relation in such a way

that real time implementation can be guaranteed. The predictive control has already been used as a control

method for the power converters but not with the goal of optimising the limit cycle [13].The paper is organized as follows: in section II, the physical model of the four-level three-cell DC-DC converter is

introduced. Like most of the DC-DC converters, it is shown that the model enters into the class of affine-switched

system. Section III states the control problem and its constraints. In section IV, the three control methods are

introduced and a short description of the control design is given. In section V, the experimental setup used to test

the methods is presented. Section VI shows experimental and simulation results. Finally, section VII provides some

comparisons between the results obtained by the proposed approaches.

II. PHYSICAL MODEL OF THE FOUR-LEVEL THREE-CELL DC-DC CONVERTER

Multilevel converters and more generally power converters represent a particular class of dynamic systems called

switching physical systems. They include semiconductor switching devices evolving much faster than the time scale

at which the system behavior needs to be analysed. Therefore, the switching times are negligible and for modelling

purposes the switching devices can be assumed to be ideal. The matrix representation of a switching system in a

standard PCH (Port Control Hamiltonian) formulation has the following expression [15]:

x = [J()R()]H(x)

x+G()u (1)

4E

123

1 11 21 3

Lch

Rch

C1C2 vC1vC2iL

Fig. 1. The four-level three-cell DC-DC converter

where x Rn is the state vector, {0, 1}p is the boolean control variable and u Rm is the power input vector.

Matrices J and R are called structure matrices. Matrix J is skew-symmetric, J = JT and it corresponds to the

power interconnections in the model. Matrix R is nonnegative and represents the dissipating part of the system.

G is the power input matrix. H represents the stored energy in the system. This is also called the Hamiltonian of

the system. If the constitutive relations of the storage elements are linear, which is most often the case in power

converters, the Hamiltonian of the system is such that:

H (x)

x= Fx = z (2)

where F = FT > 0 and in the simple cases, it is a diagonal matrix. Thus, the Hamiltonian of the system can be

represented as:

H(x) =1

2xTFx (3)

Furthermore, it has been proved that the matrices J(), R() and G() are affine with respect to the boolean control

variable [16] and they can be written as:

J () = J0 +

pj=1

jJj , R () = R0 +

pj=1

jRj , G () = G0 +

pj=1

jGj (4)

where j are the components of the control vector .

The circuit topology of the four-level three-cell DC-DC converter is shown in Fig.1. Three switching cells can

be isolated, each one containing two switches that operate dually. The behavior of each cell can be described using

only one boolean control variable i {0, 1} with i = 1, 2, 3. i = 1 means that the upper switch is closed and the

lower switch is open whereas i = 0 means that the upper switch is open and the lower switch is closed. Taking

x = [qC1, qC2, pL]T where qC1, qC2 represent the charge in each capacitor, pL the magnetic flux in the inductance

and u = E as the input voltage, the matrices J , R, G and F are given by:

J() =

0 0 2 1

0 0 3 2

1 2 2 3 0

, R() =

0 0 0

0 0 0

0 0 Rch

, G() =

0

0

3

, F () =

1C1

0 0

0 1C2

0

0 0 1Lch

(5)

The voltages and the current are simply deduced from relation z = Fx i.e.

vC1 =1

C1qC1 vC2 =

1

C2qC2 iL =

1

LchpL (6)

5An equivalent state space equation in function of the voltages vC1, vC2 on the capacitors C1 and C2 and the

current iL in the load is: vC1

vC2

iL

=

0

0

RchLch

iL

+

iL

C1

iLC1

0

0 iLC2

iLC2

vC1Lch

vC2vC1Lch

EvC2Lch

1

2

3

(7)

In this model, the boolean control (1, 2 and 3) appears explicitly in an affine form.In this article and for convenience, the stabilization control approach uses model (1) while the neural predictive

and optimal control approaches deal with the state equation (7).Remark 1: For the four-level three cell converter, a DCM appears when vC2 vC1 due to the use of MOSFET

as switching devices. In this article, the DCM is not taken into account. We assume that the operating area is far

from this boundary. Therefore, the switching devices are fully controlled.

III. THE CONTROL PROBLEM

Power converters are made to adapt the energy source to an electric load. This goal is achieved by high switching

frequency of the semiconductor devices and leads to a cyclic behavior in steady state. The desired operating point

can be defined as the state average value of the cycle. Let us note by z the state of (7) (or (1)), it is clear that theseequations can be written in the following form:

z = f(z) + g(z) (8)

with T = [1, , p], p N. The operating points set are generally expressed in term of the average state z. z

is given by the convolution product:

z(t) = Tp z(t) =1

Tp

ttTp

z()d (9)

where Tp is the cycle period and Tp is a rectangular window function.

The dynamical model of z is obtained differentiating (9). However, the derivative is generally intractable orunusable because of its nonlinear form. A solution consists in defining the average state model

z = f(z) + g(z) (10)

by relaxing the control set to its convex hull:

i [0 1], i = 1, , p (11)

which gives an approximation of the dynamic of z. is the average value of on the cycle. It has been proven that

z is close to z and z when Tp is small with respect to the system dynamics [17] (zz and z when Tp 0).The operating points are then defined as the equilibrium points of the average state model:

Z0 = {z0 Rn : f(z0) + g(z0)ref = 0, ref [0 1]

p} (12)

6Therefore, there exist equilibrium points z0 whose associated control ref belongs to ]0 1[p. Since no mode allows

to hold the given reference z0 with ref {0, 1}p, the switched system enters into a cyclic behavior around the

reference z0. Consequently, ref is nearly the average value of on the cycle.

The control objective for the multilevel converter is to supply a fixed average current value iL0 to the load. Using(7) and (12), one can compute the admissible value of iL0. This is iL0 = (E/Rch)3, with an average value forthe control fixed to i [0 1], i = 1, 2, 3 with 1 = 2 = 3 The capacitor voltages can be freely chosen. As

we already mentioned in the introduction, for improving the spectral quality and to equally distribute the switches

stresses. The reference voltage values for the capacitors are:

vC20 =2

3E vC10 =

1

3E (13)

The proposed benchmark is subject to the following performance specifications:1) The current in the load must be maintained to the reference value iL0 with the best accuracy and small

oscillations. The transient response must be as fast as possible without overshoot.

2) The controllers must be robust to the load variations and to parametric uncertainties with respect to thephysical parameters of the model.

3) For physical reasons, the switching law must also respect a dwell time between successive switches of thesame device.

IV. CONTROL APPROACHES

A. Stabilization Approach

1) General method: Usually, the approaches in the literature which are based on Lyapunov function consider,linear systems with a common equilibrium point for each configuration around which the system is stabilized [18],[19]. In the case of power converters, for each value of the boolean control variable , the system represented bythe state equation (1) may have different equilibrium points. However, for the class of studied systems, a commonLyapunov function can be established taking:

V (x, x0) =12 (x x0)

TF (x x0) (14)

where x0 is an equilibrium point defined by

0 = (J (0)R (0)) z0 +G (0)u. (15)

This is equivalent to (12). Because the matrix F is unique for all the modes of the system, V is positive andcontinuous for any x. V is also zero only in x0. Using (1), (2), (4) and (15) the derivative of V is given by:

V = (z z0)TR () (z z0) +

pj=1

(z z0)T((Jj Rj) z0 +Gju) (j 0j) (16)

with z0 = Fx0. Due to the fact that R() is a non-negative matrix, the first term is always negative. Because

0 0j 1 the sum can be negative by choosing an appropriate value for each boolean j such that each product

7(z z0)T ((Jj Rj)z0 +Gju)(j 0j) is negative. If such control law is applied, then V is a Lyapunov function

for the closed loop system, which converges asymptotically toward x0. Multiple state feedback control strategies

can be considered for attaining this goal [15]:

The maximum descent strategy which consists in choosing, at each time, the value of such that all the terms

in the sum are negative or equal to zero. If R is constant, this yields the minimum value of V . From (16),switching surfaces are p hyperplanes defined by

Tj = (z z0)T ((Jj Rj)z0 +Gju) = 0 j = 1, . . . , p (17)

This strategy will lead to a sliding motion on the switching surfaces (17). The minimum switching strategy, which may be used in order to decrease the number of switchings, consists

in holding the same value of until the trajectory hits the switching surface defined by V = 0 and then, inchoosing on that surface a new value of such that V < 0. Even if it does not lead to zeno phenomena, this

approach will lead to faster and faster switching when getting closer to the admissible reference.

Both strategies require an infinite bandwidth. In practice, switching frequency is limited. Multiple strategies to

limit the switching frequency can be devised:

A dead-zone can be created with the help of a parameter . The new switching surfaces are defined by V =

or Tj = . The period and the oscillations amplitude around the reference depend on this parameter.

A delay is introduced between switching instants.

Switches occur at discrete periodic instances. This is the case when using a discrete time controller.

A ball is defined around the reference using the Lyapunov candidate function and commutations occur when

the trajectory hits the surface of the ball.2) Application to the four-level three-cell DC-DC converter: The Lyapunov candidate function from (14)

becomes:

V (x, x0) =(qC1 qC10)

2

2C1+

(qC2 qC20)2

2C2+

(pL pL0)2

2Lch(18)

Further on, a minimum switching strategy will be used. At switching time, several strategies can be applied to

decide the new value for the boolean control variable, which will lead to different behaviors. The one which yields

the switching frequency for the transistors is when the smallest number of boolean control variables change at a

given switching time. The least recent commutation is prioritized and the first configuration with a value for the

Lyapunov candidate function derivative smaller than 0 is chosen.

B. Optimal control

1) Problem formulation and model: In this part of the article, we investigate the capability of optimal techniquesfrom a theoretical view point for high performance design of power electronic devices.

8We are concerned with hybrid optimal control problems, which involve once again a switched affine system:

min

tf0

L(z z0)dt

s.t. z = f (z) + g (z)

z(t1) = z1 {0, 1}p

(19)

L(.) : Rn R is the performance function.

In order to face the problem of solving (19), we can use the necessary conditions given by the minimum principle(MP) [20][22]. We want to highlight that it is possible to determine the appropriate control laws among thosewhich satisfy the necessary conditions of the MP in the case where the following assumption is satisfied:

Assumption 2: All optimal solutions of the average state model reach the equilibrium in finite time or even in

infinite time in case of infinite time criteria.

The assumption is obviously satisfied both for time optimal criterion or for quadratic criterion in infinite time.

However, it is not hold for quadratic criterion in finite time.

One of the difficulties encountered in this type of optimal control problem (Hamiltonian function have an affineform w.r.t. the control) is the existence of what is named singular arcs. We will explain this case in the sequel.Nevertheless, for low order system, we show that it is possible to determine final conditions such that a backward-

time integration generate all trajectories ending at the equilibrium. In that case, optimal control appears as functionof the state.

In order to cover a given entire area of the state space we interpolate the solution using a neural network (NN)to obtain a map which determines the control as a function of z. Indeed, since the solution of the optimal control

problem is in open-loop, a relation between the state z and the control is highly desired. The high mapping

capabilities of NNs establishes this relation. This synthesis is made off-line.

As example, the procedure is presented when a quadratic criterion in infinite time is used. This methodology

is particularly useful for the control synthesis because it allows to design proper control laws without excessive

computational time for the system (8).The Hamiltonian H of the problem (19) has an affine form in the control variable :

H = H(0, (t), z(t), (t))

= 0L(z(t) z0) + T (t)(f (z) + g (z) (t))

(20)

where 0 0 and (t) Rn is the adjoint variable (Lagrange multiplier). We do not mention the dependency withrespect to the time for the different variables.

Since the case where the control dimension p > 1 is quite long to explain and it is not the aim of the article,

we will assume that p = 1.

It follows the minimum principle (MP) to control the affine switched system (19):

9Theorem 3: Let (z, ) solve the problem (19). There exists an absolutely continuous function : {0, tf} Rn

and a positive constant 0, (, 0) non identically equivalent to 0, with:

= H

zz =

H

(21)

for almost all t [0, tf ], such that the following conditions are satisfied:

1) The minimum condition on the Hamiltonian:

H = H(0, , z, ) = inf

{0,1}H(0,

, z, ) (22)

The following transversality conditions:

2) Transversality condition 1: For all t [0, tf ]

H(t) = cst (23)

cst is a constant with cst = 0 if tf is not specified.

3) Transversality condition 2: The initial and final condition:

(0) free (tf ) = 0 (24)

2) Singular trajectories: This subsection address the problem of singular arcs [23], [24] encountered for solvingthe problem (19) using the necessary conditions from the Theorem 3. Following the MP two possible solutions to(19) can be found:

{0, 1} for almost all t: The solution to the optimal control problem (19) is a bang-bang solution. There exists a set of time T with nonzero measure such that for all t T , ]0, 1[. The solution to (19) is

not bang bang. This is known as singular arc.

Although the singular trajectory does not solve the problem (19), this is a Fillipov solution that can beapproximated by a sliding motion of the switched systems [25]. This reason explains why it is necessary tocompute singular arcs.

Formally, let us take some basic concepts from the singular optimal control theory [26], [27], [28]:Definition 4: Since H is linear in , an arc (, z, ) will be singular on (a, b) if

H

(, z, ) 0 (25)

holds for every t (a, b).

Definition 5: Let (19) be given. Then (t) := H

is called the switching function.

Definition 6: Let (19) be given. The problem order is the smallest integer q such that d(2q)dt(2q)

contains explicitly,

where after each differentiation z and are replaced by their expression in (21). Thus:d(2q)

dt(2q)= A(z, ) + B(z, ). (26)

If this number does not exist, then q :=

10

Definition 7: Let (z, ) be a singular arc on (a, b). If q 1, the definition of problem order and arc order have a more complex form but there are also equivalent

definitions. See details in [26].3) Control synthesis from a singular arc: The control objective is to reach an equilibrium point z0 given by the

average model (i.e. control in the interval [0,1]) and to maintain the system at this equilibrium point. Then, twopossibilities may occur:

i) A bang bang control allows to reach z0 in finite time tr

11

2) We check the necessary conditions given by Theorem 8 for each possible candidate (, z, ). Thus, we obtaina smaller set, .

3) As we have mentioned above (point 1), the equilibrium . Using a backward-time integration of theHamiltonian system (21) from this final state and from singular arcs ending to this final state, it is possibleto obtain a dense set of the optimal trajectories.It is important to notice that bifurcations must be taken into account along the singular arcs by switching

to 0 or 1. Moreover, the optimal conditions given by MP must always be checked along each trajectory. Itmay arise that two paths intersect with each other in the state space. In that case, the potential conflict could

rise by assessing the cost function.

4) Using the values of z and obtained from the previous step, an interpolation is made in order to obtain astate feedback (z) on the entire state space area. A NN is used in this phase.

The type of neural network (NN) used to learn the optimal policy is a feedforward NN. Training is obtained byclassical backpropagation procedure, see [29]. The inputs of the NN are the tracking error (z(t) z0(t)) and theoperating point reference z0. The output is the optimal policy obtained .

Remark 9: NNs are promising solution for real time implementation since the computation effort is really small.

Moreover, it is possible to implement adaptive training considering the influence of exogenous effects such as load

variations or parameter variations.

4) Application to the four-level three-cell DC-DC converter: The criterion for this case is a quadratic criterionwith L(z) = z z0 2Q and Q = diag[1, 1, 1000]. The 4-step scheme mentioned above can be applied to obtain

the feedback control. The used NN has a feedforward structure with 10 neurons in the hidden layer.

Remark 10: Since the trajectories are generated backward in time from the equilibrium point, by constructionthe stability is ensured. Moreover, for all the examples treated until now there is no a big difference between the

direct solution and the NN. Thus, the optimality is not lost due to the control discrete nature. The NN learns the

state feedback and an approximation is observed only on the border of the state space partition.

C. Predictive control Approach

1) Method Description: In the field of DC-DC converters, classical control methods are usually designed with aconstant reference which represents a mean value of the steady state (see section III). Consequently, the switchesare not necessarily controlled around the equilibrium and the steady state may show unexpected harmonic content.

In order to avoid this uncontrolled behavior, we propose to design a state feedback control law, which is able to

track an optimal limit cycle near the operating point instead of a mean value.

Indeed, for a given operating point, an optimal cycle can optimize a criterion. This criterion is tuned in the design

part. It might define a quadratic norm reduction of the tracking error or could be designed as a filter to penalize

undesired frequencies.

Once the optimal cycle is defined, the tracking control is achieved using a predictive controller. This method is

composed by two-stage strategy: near the equilibrium, the optimal cycle is tracking. Far from the equilibrium, only

12

the mean value of the cycle is used as reference. Additionally, the predictive control design takes into account time

constraints due to the physical nature of the switches. These constraints allow realistic and achievable switching

laws to be applied.

As for the optimal approach and in order to avoid excessive computation time, the real time implementation of

the controller is ensured by the use of a NN. Indeed, the optimal policy that gives the mode to be used from the

tracking error is learnt off-line by a NN and applied on-line evaluation the NN activation functions once per sample

time.

Now having described the basic of the proposed methodology, we will detail each point.

2) Closed loop design: Consider the class of affine switched systems described by (8). Once a relation betweencontrol values and operating modes is given (i.e. a one to one map from {0, 1}p {1, 2, , 2p}), we can define:

Definition 11: A switching sequence is a finite sequence represented by:

(T , I)s = {(t1, i1), (t2, i2), . . . , (tj , ij), . . . , (ts, is)} (29)

where

T = {0 = t1, t2, . . . , tj , . . . , ts} is a strictly increasing time sequence composed by the time values when a

mode is switched on.

I = {i1, i2, . . . , ij, . . . , is}, ij {1, . . . , 2p} for j = 1, . . . , s is the mode sequence. A mode ij is switched

on at time tj , j = 1, . . . , s.

s is the (finite) length of the sequence.

a) Determination of the optimal steady state: The aim is to determine the best cycle in steady state. It meansthe best switching sequence (T , I)s (1 < s < smax) which optimizes the quadratic criterion:

J ((T , I)s

) = mins,T ,I

tf0

z z0 2Q dt (30)

subject to the constraints:

z(0) = z(tf ) (periodic) (31)

tf Tp,max (maximal duration) (32)

k(tj) tmin|k(tj) k(tj+1)| j = 1, . . . , s

k(tj) = 1 k = 1, . . . , p

k(tj+1) = 0 if |k(tj) k(tj+1)| 6= 0

(dwell time) (33)

where 2Q is a quadratic norm associated to a symmetric positive definite matrix Q, z0 is the average reference,

tf is a free final time (tf = ts+1) which is bounded by Tp,max. Equation (33) imposes a minimum duration equalsto tmin, between two activations of the same switch. k is the time elapsed from the last activation. This equations

set is indeed an integrator with a reset for each switch.

13

Remark 12: In order to reduce the undesired harmonic contents a stable filter can be added to the quadratic term

z z0 2Q in (30). If this filter is appropriately designed, then it is possible to concentrate the load current or

voltage spectrum [13].With (s, I) fixed, the length and the mode of the sequence, the solution of (30) is determined using nonlinear

programming. The procedure is repeated until all admissibles values for (s, I) are tested. We remark that the method

could be computationally slow since problem (30) need to be solved smaxs=1 ps times. This optimisation is obviouslyperformed off-line.

The solution of (30) gives an optimal sequence (T , I)s and the optimal reference R0(t) for the closed loopin steady state.

b) Neural Predictive Controller: As mentioned above, the design control and the data for training the neuralnetwork are obtained in the following two-stages:

Far from the optimal limit cycle R0(t): since the behavior of the system is in a transitory phase, (T , I) are

optimized and we choose to fix s and the receding horizon tf to (s, tf ). The following cost function isminimized on a grid:

J(z, z0) = minT ,I

tf0

z z0 2Q dt (34)

The first mode and its duration are used for training the network.

Near the limit cycle R0(t). I and s are fixed to the reference values I and s. The optimization is only done

with respect to the time sequence. The cost function becomes:

minT

tf0

z R02Qdt (35)

As in the optimal approach, the inputs of the NN are the tracking error ((t) = z(t) R(t)) and the operatingpoint reference z0. The output is the optimal policy obtained from the optimization problem (34) or (35) withR = z0 or R = R0 respectively.

3) Application to the four-level three-cell DC-DC converter: We consider the state equation given by (7) withz = [vC1, vC2, iL]

T. For example, if we solve the optimization problem (30) with Tp,max = 3ms, smax = 12,

tmin = 0.25ms, Q = diag[10, 1, 1000] and taking as operating point z0 = [10, 20, 0.6]T , we find the following

optimal sequence:

(T , I)s

= {(0.4ms, 2), (0.65ms, 4), (0.9ms, 6), (1.150ms, 3), (1.4ms, 6), (1.625ms, 3),

(1.9ms, 6), (2.150ms, 3)}(36)

with an initial condition z(0) = [16.65, 5.26, 0.594]T .

The table I gives the control value with respect to the mode

The matrix Q has been tuned in order to reduce the current oscillation around the mean value.

It can be noticed, from equation (36) that the optimal period which minimizes the oscillations is Tp = 2.150 ms.The dwell time constraints tmin for each switching component is also verified.

14

TABLE I

TABLE OF MODES

ij 1 2 3

1 0 0 02 0 0 13 0 1 14 0 1 05 1 1 06 1 1 17 1 0 18 1 0 0

In order to generate all the optimal trajectories, a variation of the initial condition has to be considered. Thesetrajectories are used to train the NN. For this example, the grid of initial conditions is composed by 1000 pointsin each variable. The ranges of the variables are 0 and 2iL0 for the current, 0 and 2vC10 and 0 and 2vC20 for the

capacitor voltages.

The NN interpolates the solution with 20 neurones in the hidden layer with a back-propagation training algorithm

and sigmoid functions. After training, the NN is tested with a set of known solutions not used for the training phase

(Validation set). In the case when the error with the validation set is not acceptable, the number of neurons isincreased in the hidden layer until all validation sets lead to a good result.

V. IMPLEMENTATION

A. The four-level three-cells multi-level converter

The control methods presented in section IV have been tested on a real power converter. The power converter

was made at SUPELEC [30] as a three-phased four-cell power converter. Only one branch was used from theinitial structure and it was reduced to a three-cell power converter. The switches are MOSFET IRFP360 transistors

and the value of each capacitor is 45F . The R L load has been created using a rheostat and an inductance.

Their nominal values are 30 and 0.5H . The input voltage is E = 30V and the reference for the load current is

iL0 = 0.6A.

B. Measurements

The structure contains a built-in circuit which allows the capacitor voltages measurement. These measurements

are made with isolated sensors and the output voltage is obtained via resistive voltage dividers. Due to a calibration

needed for higher voltages than those used in the present experiment, the measurements are affected by noise.

The current in the load is measured using the voltage on the load resistor.

15

C. Controller hardware setup

For the methods implementation, a computer with a dSPACE DS1102 controller board, built around a Texas

Instruments TMS320C1 32 bit floating-point Digital Signal Processor (DSP) is used. The control programs werecodified with Matlab Simulink using the dSPACE toolbox and Real Time Workshop. dSPACE ControlDesk allowed

the on-line access to program parameters and data acquisition.

The control algorithm was implemented in Simulink using S-functions written in C. The program contains also I/O

blocks from the dSPACE toolbox and elementary Simulink blocs for the parameters and references. The sampling

frequency is 4kHz for all methods. The inputs in the program are the measures for capacitors voltage and the load

current. The outputs are the boolean values for the switches 1, 2, and 3. Only three outputs are used, namely

the control values for the top switches. These control signals are inverted for obtaining the control signals for the

bottom switches.

VI. EXPERIMENTAL RESULTS

In order to validate and to compare the control strategies, a few relevant scenarios have been selected for the

benchmarks tests. These benchmark tests have been implemented in simulation and on a real process.

1) Start-up from zero initial conditions: the control objective is to regulate the voltages on the capacitors andthe current in the load to the reference values specified above using nominal physical parameters.

2) Response to load variations: the converter is in steady state with the nominal physical parameters, exceptingthe resistive load, which is equal to Rch = 23R

nomch = 20. At the instance t = 0, a step in the resistive

component is applied to Rch = 43Rnomch = 40.

The experimental and simulation results are displayed in Fig.2, 3 and 4. In each figure, the top half rows

contain the experimental results and the bottom half rows contain the simulation results. Experimental results on

the benchmark show that the three methods achieve the control objective for the load current and the capacitorvoltages.

Fig.2 contains the start-up evolution. All control methods globally fulfill the objective. They quickly reach thedesired reference. The methods do not display differences regarding the settling time in the load current. It is about

40ms (caused by the high value of the load inductance).The amplitude of the oscillations in the steady state is directly related to the sampling time. Considering the

average values of iL, vC1 and vC2, the error is near to zero.

The three methods have been tested using a sampling frequency of 4kHz. It is observed that the oscillations for

the stabilization controller have a bigger amplitude than for the other methods. The reason is that the stabilization

method is depending on the parameter . Indeed, the control must verify that V , otherwise the control must

switch. Since the measurements are subject to quantification noise, an error in the Lyapunov function is introducedand it affects the switching frequency of the MOSFET. An apparent frequency of 200Hz is observed from the

experimental results.

16

TABLE II

OSCILLATIONS AMPLITUDE OF THE ERROR WITH RESPECT TO THE AVERAGE REFERENCE VALUE IN THE STEADY STATE.

Method [vC1 , vC1 ] [vC2 , vC2 ] [iL , iL ]

Stabilization [4.85, 5] [4.8, 3.35] [0.024, 0.024]Predictive [4.75, 2.5] [5.2, 4.05] [0.01, 0.013]Optimal [2.51, 1.15] [3.5, 3.27] [0.013, 0.01]

Fig.3 is a zoom on the steady state showing 15 periods of the control signal and the corresponding state variables

evolution. The time axis is not the same because of the differences in the oscillation frequency of the three

methods. In the table II, the minimum and the maximum oscillation values of the error with respect to the average

reference is shown where vC1 = min (vC1 vC10), vC1 = max (vC1 vC10), vC2 = min (vC2 vC20), vC2 =

max (vC2 vC20), iL = min (iL iL0), iL = max (iL iL0) in the cycle.

For the capacitor voltages, it is observed a particular waveform (cf. Fig. 4) creating the effect of a carrying signalfor the stabilization and the optimal control strategies. This is produced because these methods are asynchronous

with respect to the sampling time.

Considering load variations (more precisely a variation of the resistor component) in average values, the threemethods achieve the control objective. Although, for the predictive and optimal approaches a transitory is produced,it is not very significant compared to the amplitude of the steady state oscillations in the case of the stabilization

approach.

Regarding the implementation, the stabilization approach is very flexible and very easy to implement. The

predictive and the optimal methods require a big off-line effort and the control law is harder to obtain. Nevertheless,

the predictive control has the advantage to take into account time constrains in design part.

VII. CONCLUSIONS

In this paper three original control strategies have been implemented on the same platform : a four-level three-cell

serial DC-DC converter.

The neural predictive approach allows to track an optimal limit cycle and to control the waveform. Although the

stability is not guaranteed, this method is a flexible and easy-to-tune approach which may be used to improve

the spectral quality of the output signal. The stabilization approach, based on energetic principles, determines the

control values ensuring the asymptotic stability of the system combined with a control objective. In the case of theoptimal strategy, since the solution is given by the optimization of a quadratic criterion, robustness and stability

is clearly achieved and guaranteed. For these last two methods, the asymptotic stability means that the trajectoryenters in a ball centered on the equilibrium whose radius is depending of the employed switching strategy.

From a methodological point of view, the stabilization approach has the advantage of being easier to synthesize

while other proposed methods imply off-line computation. However, all the methods are simple to implement. At

each sampling time, only a few functions have to be evaluated. On the other hand, the optimal and predictive

17

approaches, even if they are more complex, they seem to present an increased robustness with respect to the

measurement noises and quantification errors.

All these approaches are direct (the control values are Boolean) and asynchronous that is why the fixed samplingfrequency imposed by the experimental process lead to a degradation of the performances. Nevertheless, results are

good and the control objective is achieved for the two investigated scenarios. A higher sampling frequency wouldgive better results.

Regarding the comparison with classical existing approaches, such as the self-balancing method [3], theperformances of the three approaches presented in this article are superior, in particularly with respect to the

transient. All the proposed control designs are clearly multi-variable and dedicated to switched affine systems. Two

of them ensure the stability which cannot be done using simpler controllers such as PI controller. Moreover, the

tuning phases are easier and the resulting state feedbacks are a pledge for robustness. Consequently, the proposed

methods may be useful and viewed as alternative methods when hard performances involving stability are required.

Further works will consist in the improvement of the methods. For the stabilization approach, an extension to the

case of discontinuous conduction modes (DCM) has to be studied. On the other hand, a modification of the genericLyapunov function by the introduction of a weighting matrix would allow the achievement of better performances.

In our opinion, the predictive method can be improved by taking into account the sample frequency in the

control design and in the selected optimal limit cycle. In that case, the switching times will be synchronous with

the sampling period ensuring a better tracking.

Moreover, in a practical point of view, the load must be considered as an unknown parameter. Thus, a load

observer is necessary to guarantee the robustness of the method.

Concerning the optimal control approach, it is fundamentally an asynchronous method, since the switching law

is given from a partition of the state space. A better chattering can be obtained using techniques from sliding modes

[25].

ACKNOWLEDGMENTS

This work was supported by the European Commission research project FP6-IST-511368 Hybrid Control(HYCON).

The authors would want to thank Prof. Amir Arzandeh from Supelec in Gif-sur-Yvette, France, for his help

in obtaining the experimental results.

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[7] T. A. Meynard, H. Foch, P. Thomas, J. Courault, R. Jakob, and M. Nahrtaedt, Multicell converters: Basic concepts and industryapplications, IEEE Transactions on Industrial Electronics, vol. 49, pp. 955 964, october 2002.

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alternative par DTC. PhD thesis, Universite de Paris VI, 2001.

19

Stabilization approach Optimal approach Predictive approach

vC [V]

0 0.05 0.1 0.15 0.20

5

10

15

20

25

Time (s)

Volta

ge V

c 1

0 0.05 0.1 0.15 0.20

5

10

15

20

25

Time (s)

Volta

ge V

c 2

0 0.05 0.1 0.15 0.20

5

10

15

20

25

Time (s)

Volta

ge V

c 20 0.05 0.1 0.15 0.2

0

5

10

15

20

25

Time (s)

Volta

ge V

c 1

0 0.05 0.1 0.15 0.20

5

10

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Volta

ge V

c 2

0 0.05 0.1 0.15 0.20

5

10

15

20

25

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Volta

ge V

c 1

iL [A]

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Curre

nt i L

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)Cu

rrent

i L

vC [V]

0 0.05 0.1 0.15 0.20

5

10

15

20

25

Time (s)

Volta

ge V

c 1

0 0.05 0.1 0.15 0.20

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10

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25

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ge V

c 2

0 0.05 0.1 0.15 0.20

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Volta

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c 2

0 0.05 0.1 0.15 0.20

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ge V

c 1

iL [A]

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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Curre

nt i L

0 0.05 0.1 0.15 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (s)

Curre

nt i L

Fig. 2. Comparison between real and simulated response at startup, for the nominal load Rch = 30.

20

Stabilization approach Optimal approach Predictive approach

states0.1 0.12 0.14 0.16 0.18 0.20

10

20

Time (s)

Volta

ge V

c 1

0.1 0.12 0.14 0.16 0.18 0.20.4

0.6

0.8

Time (s)

Curre

nt i L

0.1 0.12 0.14 0.16 0.18 0.21015

2025

Time (s)

Volta

ge V

c 2

0.1 0.105 0.11 0.115 0.1210

20

Time (s)

Volta

ge V

c 2

0.1 0.105 0.11 0.115 0.120

1020

Time (s)

Volta

ge V

c 10.1 0.105 0.11 0.115 0.12

0.40.60.8

Time (s)

Curre

nt i L

0.1 0.105 0.11 0.115 0.1210152025

Time (s)

Volta

ge V

c 2

0.1 0.105 0.11 0.115 0.120

1020

Time (s)

Volta

ge V

c 1

0.1 0.105 0.11 0.115 0.120.40.60.8

Time (s)

Curre

nt i L

control

0.1 0.12 0.14 0.16 0.18 0.20

0.5

1

1

Time (s)

0.1 0.12 0.14 0.16 0.18 0.20

0.5

1

2

Time (s)

0.1 0.12 0.14 0.16 0.18 0.20

0.5

1

3

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0.1 0.105 0.11 0.115 0.120

0.5

1

Time (s)

1

0.1 0.105 0.11 0.115 0.120

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2

0.1 0.105 0.11 0.115 0.120

0.5

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0.5

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states0.1 0.12 0.14 0.16 0.18 0.20

10

20

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Volta

ge V

c 1

0.1 0.12 0.14 0.16 0.18 0.21015

2025

Time (s)

Volta

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c 2

0.1 0.12 0.14 0.16 0.18 0.20.4

0.6

0.8

Time (s)

Curre

nt i L

0.1 0.105 0.11 0.115 0.120

1020

Time (s)

Volta

ge V

c 2

0.1 0.105 0.11 0.115 0.120

1020

Time (s)

Volta

ge V

c 1

0.1 0.105 0.11 0.115 0.120.40.60.8

Time (s)

Curre

nt i L

0.1 0.105 0.11 0.115 0.1210

20

Time (s)

Volta

ge V

c 2

0.1 0.105 0.11 0.115 0.120

1020

Time (s)

Volta

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c 2

0.1 0.105 0.11 0.115 0.120.40.60.8

Time (s)

Curre

nt i L

control

0.1 0.12 0.14 0.16 0.18 0.20

0.5

1

Time (s)

1

0.1 0.12 0.14 0.16 0.18 0.20

0.5

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0.1 0.105 0.11 0.115 0.120

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Time (s)

1

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Time (s)

2

0.1 0.105 0.11 0.115 0.120

0.51

Time (s)

3

0.1 0.105 0.11 0.115 0.120

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Time (s)

1

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0.5

1

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2

0.1 0.105 0.11 0.115 0.120

0.5

1

Time (s)

3

Fig. 3. Steady state evolution for states and command, for the nominal load Rch = 30.

21

Stabilization approach Optimal approach Predictive approach

vC [V]

0.1 0.05 0 0.05 0.10

5

10

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Volta

ge V

c 1

0.1 0.05 0 0.05 0.110

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0.1 0.05 0 0.05 0.10.4

0.45

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nt i L

0.1 0.05 0 0.05 0.10.4

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0.1 0.05 0 0.05 0.10.4

0.45

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rrent

i L

vC [V]

0.1 0.05 0 0.05 0.10

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c 1

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iL [A]

0.1 0.05 0 0.05 0.10.4

0.45

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Time (s)

Curre

nt i L

Fig. 4. Comparison between real and simulated responses at a load variation from R = 20 to R = 40.

IntroductionPhysical model of the four-level three-cell DC-DC converterThe Control ProblemControl approachesStabilization ApproachGeneral methodApplication to the four-level three-cell DC-DC converter

Optimal controlProblem formulation and modelSingular trajectoriesControl synthesis from a singular arcApplication to the four-level three-cell DC-DC converter

Predictive control ApproachMethod DescriptionClosed loop designApplication to the four-level three-cell DC-DC converter

ImplementationThe four-level three-cells multi-level converterMeasurementsController hardware setup

Experimental resultsConclusionsReferences