LMI Robust Control of PWM Converters: An OutputFeedback Approach
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ABSTRACT: This note describes a new framework for the analysis and synthesis of control systems, which constitutes a genuine continuoustime extension of results that are only available in discrete time. In contrast to earlier results the proposed methods involve a specific transformation on the Lyapunov variables and a reciprocal variant of the projection lemma, in addition to the classical linearizing transformations on the controller data. For a wide range of problems including robust analysis and synthesis, multichannel H<sub>2</sub> stateand outputfeedback syntheses, the approach leads to potentially less conservative linear matrix inequality (LMI) characterizations. This comes from the fact that the technical restriction of using a single Lyapunov function is to some extent ruled out in this new approach. Moreover, the approach offers new potentials for problems that cannot be handled using earlier techniques. An important instance is the eigenstructure assignment problem blended with Lyapunovtype constraints which is given a simple and tractable formulationIEEE Transactions on Automatic Control 01/2002; · 2.72 Impact Factor 
Conference Paper: Pole assignment in a disk for linear systems by static output feedback
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ABSTRACT: This paper proposes a systematic approach for the pole assignment in a disk for continuous or discretetime linear systems by a static output control law. It is shown that if the openloop system satisfies some particular structural conditions, a static output feedback gain can be easily computed, using a formula that involves only the original system matrices and the parameters defining the disk. Among the conditions the system has to satisfy, the strongest one relies on a minimum phase argument. Square and non square, proper or non proper systems are considered. The main results are derived invoking the limiting behavior of the optimal linear quadratic regulator for discretetime systems. Some other implications are also presented.American Control Conference, 2003. Proceedings of the 2003; 07/2003 
Page 1
Carlos Olalla1, Abdelali El Aroudi1, Ramon Leyva1and Isabelle Queinnec2
1Universitat Rovira i Virgili, 43007 Tarragona
2CNRS ; LAAS ; 7 avenue du colonel Roche, F31077 Toulouse Cedex 4
Université de Toulouse ; UPS, INSA, INP, ISAE, UT1, UTM ;
LAAS ; F31077 Toulouse Cedex 4
1Spain
2France
1. Introduction
This chapter proposes a systematic approach for the synthesis of robust controllers for dcdc
converters. The approach is based on the Linear Matrix Inequalities (LMIs) framework and
the associated optimization algorithms. The aim of this approach is to allow the designer to
describe the uncertainty of the converter and to deal with the requirements of the application
beforehand.
The aforementioned dcdc converters (see Figure 1) are devices that deliver a dc output
voltage, with different properties from those in the input voltage (Erickson & Maksimovic,
1999). They are usually employed to adapt energy sources to the load requirements (or
vice versa). These devices present several challenges regarding their robust control. First,
the converter must maintain a tight regulation or tracking of the output. Moreover, the
controller design is focused on maximizing the bandwidth of the closedloop response in
order to reject the usual disturbances that appear in these systems. Finally, the response of the
converter must satisfy desirable transient characteristics, as for example, the shortest possible
output settling time or the minimum overshoot. Besides of these common requirements, the
converter can be affected by uncertainty in its components or by input or output disturbances
that may appear.
+_
+
_
DCDC
Converter
Input
Voltage
Output
Voltage
Control
Input
Load
Vg(t)
Vo(t)
ub(t)
Fig. 1. General scheme of a dcdc converter.
LMI Robust Control of PWM Converters:
An OutputFeedback Approach
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d(t)
x(t)?
y(t)
DCDC
K
xint(t)
?
Vref+

(a) Block diagram of a statefeedback system
with controller K and error integration.
d(t)
y(t)
DCDC
G(s)
Vref
+

(b) Block diagram of an outputfeedback system
with controller G(s).
Fig. 2. Statefeedback and outputfeedback block diagrams.
Nevertheless, most of the modeling approaches in the literature disregardthese uncertainties.
Moreover, due to the switching nature of the system, pulsewidth modulation (PWM) is
commonly used in the industry applications, while the models that are usually employed
disregard that part of the dynamics (i.e. the high frequency dynamics) and other inherent
nonlinearities, such as saturations and bilinear terms.
The chapter proposes a systematic approach to deal with these challenges, using the concepts
of LMI control (BenTal et al., 2009; Bernussou, 1996; Boyd et al., 1994; El Ghaoui & Niculescu,
2000; Pyatnitskii & Skorodinskii, 1982). Linear matrix inequalities have become an important
topic in the field of Automatic Control due to the following facts. First of all, LMIs can
be solved numerically by efficient computer algorithms (Gahinet et al., 1995; Löfberg, 2004;
Sturm, 1999). Secondly, more and more methods have been developed to describe control
problems in terms of LMI constraints. Finally, these methods are able to include descriptions
of the uncertainty.
Some of the previous literature on LMI control of dcdc converters are (Montagner et al., 2005;
Olalla et al., 2009a; 2010a). In these papers, the uncertainty of the converter is taken into
account and the control synthesis deals with different operating points. Nevertheless, they
do not consider the stability of the system trajectories when the system changes from one
operating point to another, nor they include other nonlinearities such as saturations. The
versatility of LMI control has allowed to deal with some of these nonlinearities (Olalla et al.,
2009b), (Olalla et al., 2011).
These approaches share the same feedback scheme, which is based upon statefeedback with
error integration (Figure 2(a)). The main advantage of this approach is that the synthesis
optimization problem can be posed as a convex semidefinite programming and that the
implementation of the controlleris simple. On the other hand, statefeedback requiressensing
of the state variables, which may not be easily measurable or may require estimation in some
cases. In practice, most of the designs that can be found in the power electronics literature
employ outputfeedback approaches since they usually rely on frequencybased concepts
which are wellknown by electrical engineers. This is the reason why this chapter focuses
on LMIbased synthesis methods which may be applicable to the outputfeedback scheme
(Figure 2(b)), with the aim to derive robust controllers for dcdc converters.
In order to introduce such synthesis methods, the chapter is organized as follows. The first
section deals with modeling of dcdc converters, the averaging method, the sampling effect
of the pulsewidth modulator and the uncertainty. Section II reviews some of the results of
previous works on LMI synthesis for statefeedback approaches. Section III puts forward
the problem of outputfeedback and some of the strategies that can be employed to pose
such problem in terms of semidefinite programming. Concretely, Section III proposes the
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LMI Robust Control of PWM Converters: an OutputFeedback Approach3
vg(t)
vo(t)
C
vc(t)
−
iL(t)
iload(t)
ub(t)
L
R
rL
rC
+
+
−
(a) Buck converter with stray resistances.
vg(t)
iL(t)
vo(t)
C
vc(t)
−
iload(t)
ub(t)
L
R
rL
rC
+
+
−
(b) Buckboost converter with stray resistances.
Fig. 3. Schematic of the buck and the buckboost converter.
following strategies. First, the classic dynamic outputfeedback control problem is treated.
This approach can be carried out with a change of variables as in (Scherer et al., 1997).
However, with such an approach the uncertainty must be modeled with elaborated models,
as for example, weighting functions Wallis & Tymerski (2000). Therefore, the chapter also
proposes the synthesis of outputfeedback controllers based on the static case. Both the static
outputfeedback and a parametrization (Peaucelle & Arzelier, 2001b) to deal with dynamic
outputfeedback are considered. The advantages and drawbacks of the three approaches
shown in the chapter will be discussed and the results will be compared.
Notation
For symmetric matrices A and B, A > B means that A − B is positive definite. A denotes that
the matrix A is an unknown variable. A?denotes the transpose of A. Co
denotes the convex hull defined by N vertices vj∈ Rn. The identity matrix of order n is noted
as 1nand the null n × m matrix is noted as 0 0n,m. The symbol ? denotes symmetric blocks in
partitioned matrices.
?
vj, j = 1,..., N
?
2. Modeling of uncertain dcdc converters
This subsection shows the statespace averaged models of the buck and the buckboost
converters of Figures 3(a) and 3(b). The models are assumed to operate in Continuous
Conduction Mode (CCM), i.e. the inductor current is always larger than zero. Besides of
the averaged models, this section also introduces a model of the sampling effect caused by
the PWM. Finally, at the end of the section, the uncertainty modeling of dcdc converters is
discussed and a simple example is shown.
2.1 Model of the buck converter
The first model that is introduced considers a buck converter, which is characterized by linear
averaged controltooutput dynamics. As stated in (Olalla et al., 2010b), the transfer functions
of dcdc converters can strongly depend on the stray resistances of the converter. Since the
chapter considers different outputfeedback synthesis approaches, these stray resistances are
considered in the models.
Figure 3(a) shows the circuit diagram of a dcdc buck converter where vo(t) is the output
voltage, vg(t) is the line voltage and iload(t) is the load disturbance. The output voltage must
be keptat a given referenceVref. The converter load ismodeledas a linear resistor R. The stray
resistances of the switch during the on and the off position are combined with the resistance
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LMI Robust Control of PWM Converters: An OutputFeedback Approach
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of the inductor and noted as:
ron= rdon+ rL
rof f= rdof f+ rL
(1)
The measurable states of the converter are noted as xa(t). Note that the time dependence of
the variables may be omitted to simplify the notation.
The binary signal ub(t), which turns on and off the switches, is genereated by means of
a Pulse Width Modulation (PWM) subcircuit, working at a constant frequency 1/Ts. The
switching period Tsis equal to the sum of tonand toff. For a unitamplitude sawtooth PWM,
the dutycycle d(t) = ton/(ton+ toff) is the control input of the converter.
As shown in (Erickson & Maksimovic, 1999) and (Leyva et al., 2006), considering that the
statespace matrices of the converter are [ Aon, Bon] during tonand [ Aoff, Boff] during toff,
the general statespace averaged model of a dcdc converter can be written as:
˙˜ x(t) =?Aoff+(Aon− Aoff)U?X +?Boff+ (Bon− Boff)D??1
+?Aoff+ (Aon− Aoff)U?˜ x(t) +?Boff+ (Bon− Boff)D??0
+?(Aon− Aoff)X + (Bon− Boff)?1
+?(Aon− Aoff)˜ x(t) + (Bon− Boff)?0
where the equilibrium (noted with capital letters) and the incremental vectors (noted with
tildes) are as follows. X and ˜ x ∈ Rncorrespond to the state vectors, D and˜d ∈ Rmare the
control inputs, while W and ˜ w ∈ Rlstand for the disturbance inputs.
In the buck converter, Aon= Aoff, and the averaged model (2) can be rewritten as:
0
?W
1
?
˜ w(t)
0
?W?˜d(t)
1
?
˜ w(t)?˜d(t),
(2)
d˜ x(t)
dt
=?AX + BwW?+ A ˜ x(t) + Bw˜ w(t) + Bu˜d(t) + Bnw˜ w(t)˜d(t)
−req
(R + rC)L
R
(R + rC)C
(R + rC)C
(3)
where:
A =
⎡
⎢
⎣
L−
RrC
(R + rC)L−
R
−
1
⎤
⎥
⎦, Bw=
⎡
⎢
⎣
D
L
RrC
(R + rC)L
0 −
(R + rC)C
R
⎤
⎥
⎦,
Bu=
⎡
⎣
Vg
L
0
⎤
⎦, Bnw=
?1
L
0
?
,
X =
⎡
⎢⎢⎢⎢
⎣
VgD
1 + req/R
R
VgD
1 + req/R
⎤
⎥⎥⎥⎥
⎦
, W =
?Vg
0
?
, ˜ x(t) =
?˜iL(t)
˜ vo(t)
?
, ˜ w(t) =
?
˜ vg(t)
˜
iload(t)
?
,
(4)
being req= Dron+ D?rof fand D?= 1− D. The dimensions of the system matrices are defined
as A ∈ Rn×n, Bu, Bnw∈ Rn×m, Bw∈ Rn×l.
Similarly, the averaged outputs of the buck converter can be written as:
Y + ˜ y(t) =?CyX + EywW?+ Cy˜ x(t) + Eyw˜ w(t)
(5)
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LMI Robust Control of PWM Converters: an OutputFeedback Approach5
where in a general case Cy∈ Rq×n, Eyw∈ Rq×m. Considering the load voltage vo(t) as the
only output, these matrices are written as:
Cy=
?
RrC
R + rC
R
R + rC
?
, Eyw=
?
0 −
RrC
R + rC
?
.(6)
2.2 Model of the buckboost converter
In the buckboost converter, matrices Aonand Aoffare not equal, and therefore, the averaged
model contains bilinear terms concerning the control input, the states and the disturbance
inputs. According to those nonlinear terms, the linearized transfer function depends on the
operating point, hence making the control subsystem design more difficult. In order to derive
accurate transfer functions of the buckboost converter for outputfeedback approaches, the
stray resistances are also taken into account.
For the buckboost converter, the averaged model in the form of (3) contains bilinear terms,
and can be expressed as follows:
d˜ x(t)
dt
=?AX + BwW?+ A˜ x(t) + Bw˜ w(t) + Bu˜d(t) + Bnx˜ x(t)˜d(t) + Bnw˜ w(t)˜d(t)
(7)
where:
A =
⎡
⎢
⎣
−req
L− D?
−D?
RrC
(R + rC)L
R
(R + rC)C
D?
R
(R + rC)L
1
(R + rC)C
−
⎤
⎥
⎦, Bw=
⎡
⎢
⎣
D1
L−D?
RrC
(R + rC)L
R
(R + rC)C
0
−
⎤
⎥
⎦,
Bu=
⎡
⎢⎢
⎣
Vg
L
?
1 +
DrC
D?2(R + rC)+
D?DR2
D?R(D?R + rC) + (R + rC)req
DVg
CD?2(R + rC)
?⎤
⎥⎥
⎦,
Bnw=
⎡
⎣
1
L−
0
RrC
(R + rC)L
0
⎤
⎦,
X =
⎡
⎢⎢⎢⎢
⎣
VgD?DR(R + rC)
(−D?R(−D?R − rC) + (R + rC)req)D?R
−VgD?DR(R + rC)
−D?R(−D?R −rC) + (R + rC)req
=
Dron+ D?rof f.
A, Bnx, Bnw∈ Rn×n, Bu∈ Rn×m, Bw∈ Rn×l.
The averaged output vo(t) of the buckboost converter can be written as:
⎤
⎥⎥⎥⎥
⎦
, Bnx=
⎡
⎢
⎣
RrC
L(R + rC)−
R
(R +rC)C
R
(R + rC)L
0
⎤
⎥
⎦.
(8)
being req
The dimensions of the system matrices are defined as
Y + ˜ y(t) =?CyX + EywW?+ Cy˜ x(t) + Eyw˜ w(t) + CyuX˜d(t) + Cyu˜ x(t)˜d(t)
(9)
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Ts
vs
VM
d
ub
ton
toff
t
t
Fig. 4. Waveforms of the PWM process.
where:
Cy=
?
−D?RrC
R + rC
R
R + rC
?
, Cyu=
?
RrC
R + rC
0
?
, Eyw=
?
0 −
RrC
R + rC
?
.(10)
These models are employed in Section 3 to derive robust controllers for the buck and the
buckboost converters.
2.3 Delay model for the PWM actuator
The models presented above do not take into account the sampling effect of the modulation
(Brown & Middlebrook, 1981; Erickson & Maksimovic, 1999) (see Figure 4).
Usually, the sampling effect is not considered, and only the linear gain of the modulator is
taken into account. In a voltagemode modulator, the dutycycle input is usually constrained
between zero and the amplitude of the sawtooth signal VM, and therefore the linear gain
of this modulator is 1/VM(Erickson & Maksimovic, 1999). For simplicity the amplitude VM
can be considered equal to one, such that the linear model shown previously is valid for a
dutycycle input d ∈ [0,1].
However, the sampling effect can be taken into account in order to limit the controlloop
bandwidth in the automatic control synthesis algorithms. Such an effect can be incorporated
to the power stage model as a sampling at the switching frequency 1/Tsand a zeroorderhold
block, assuming that the switch is fired once every switching cycle Ts(Maksimovic, 2000). The
equivalent transfer function for this sampling model is then:
GZOH(s) =1 − e−sTs
sTs
(11)
The exponential factor e−sTscan be approximated by a Padé function:
e−sTs≈∑n
k=0−1kckTssk
∑n
k=0ckTssk
(2n − k)!n!
2n!k!(n − k)!,
,
ck=
k = 0,1,·,n.(12)
Taking the first order approximation n = 1 we obtain
e−sTs≈1 − (Ts/2)s
1 + (Ts/2)s
(13)
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LMI Robust Control of PWM Converters: an OutputFeedback Approach7
The equivalent hold transfer function with the Padé approximation writes
GZOH(s) =
1
1 + sTs
2
(14)
which is a strictly proper transfer function whose representation in statespace form could be:
?˙˜ xp(t) = −(2/Ts)˜ xp(t) + (2/Ts)˜d(t)
˜d2(t) = ˜ xp(t)
(15)
where ˜ xp(t) is the state variable of the GZOH(s),˜d(t) is its input and˜d2(t) is its output.
2.4 Modeling of uncertainty
As stated in (Gahinet et al., 1995), the notion of system uncertainty is of major importance
in the field of robust control theory. First of all, one of the key features of feedback is that
it reduces the effects of uncertainty. However, when designing a control system, the model
used to represent the behavior of the plant is often approximated. The difference between the
approximated model and the true model is called model uncertainty. Also the changes due to
operating conditions, aging effects, etc... are sources of uncertainty.
The two main approaches shown in (Gahinet et al., 1995) when dealing with system
uncertainties and LMI control are:
• Uncertain statespace models, relevant for systemsdescribed by dynamical equations with
uncertain and/or timevarying coefficients.
• Linearfractional representation (LFR) of uncertainty, in which the uncertain system is
described as an interconnection of known LTI systems.
While LFR models have had a main role in modern robust control synthesis methods such as
in μsynthesis (Zhou et al., 1996), statespace models have been used in convex optimization
approaches (Boyd et al., 1994). Since this chapter presents approaches that do not employ the
concept of structured singular value on which the μsynthesis method is based, the following
subsection is focused on uncertain statespace models.
If some of the physical parameters are approximated or unknown, or if there exists nonlinear
or nonmodeled dynamiceffects, then the system can be described by an uncertain statespace
model:
?˙ x = Ax + Bu
y = Cx + Du
(16)
where the statespace matrices A,B,C,D depend on uncertain and/or timevarying
parameters or vary in some bounded sets of the space of matrices. One of the statespace
representations of relevance in LMI control problems is the class of polytopic models:
Definition 2.1. A polytopic system is a linear timevarying system
?˙ x = A(t)x + B(t)u
y = C(t)x + D(t)u
(17)
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in which the matrix G(t) =
?A(t) B(t)
C(t) D(t)
?
varies within a fixed polytope of matrices
G(t) ∈ Co{G1,...,GN} :=
⎧
⎩
⎨
N
∑
j=1
δjGj: δj≥ 0,
N
∑
j=1
δj= 1
⎫
⎭
⎬
(18)
where G1,...,GNare the vertices of the polytope.
In other words, G(t) is a convex combination of the matrices G1,...,GN. Polytopics models
are also called linear differential inclusions LDI in (Boyd et al., 1994).
2.4.1 Example: Buck converter polytopic model
Considerthe buck convertermodelintroducedinsubsection2.1, with ˜ w(t) = 0. Forsimplicity,
the stray resistances are disregarded. If we take R and Vg as uncertain parameters of the
converter, the uncertain system is described as follows
⎧
⎪
⎪
⎩
⎨
d˜ x(t)
dt
˜ y(t) = Cy˜ x(t) + Eyw˜ w(t)
=
?
N
∑
j=1
Ajδj
?
˜ x(t) +
?
N
∑
j=1
Bujδj
?
˜d(t)
(19)
with δj≥ 0,
N
∑
j=1
δj= 1. The uncertain matrices Ajand Bujare
Aj=
⎡
⎢
⎣
0
1
C−
−1
L
1
RjC
⎤
⎥
⎦,
Buj=
⎡
⎣
Vgj
L
0
⎤
⎦, (20)
where Rj= {RminRmaxRminRmax}, and Vgj= {VgminVgminVgmaxVgmax}, which represents a
uncertain polytope of four vertices (2 power the number of uncertain parameters, that are Rj
and Vgjin this example).
3. Robust control of dcdc converters
Consider a general LTI model with states x(t), controlled outputs y(t) and performance
outputs z(t):
⎧
⎩
It is possible to assume that some elements involved in the system matrices are uncertain
or timevarying. For the sake of simplicity, the performance and measurable outputs are
discarded, hence these uncertain elements are concentrated in matrices A, Bwand Buand
they are grouped in a vector p. Thus, matrices A, Bwand Budepend on such uncertainty
vector, and we can express (21) as function of these parameters:
Σ :
⎨
˙ x(t) = Ax(t) + Bww(t) + Buu(t)
y(t) = Cyx(t) + Eyww(t) + Eyuu(t)
z(t) = Czx(t) + Ezww(t) + Ezuu(t)
.(21)
˙ x(t) = A(p)x(t) + Bw(p)w(t) + Bu(p)u(t).(22)
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LMI Robust Control of PWM Converters: an OutputFeedback Approach9
d(t)
x(t)?
y(t)
DCDC
K
xint(t)
?
Vref +

Fig. 5. Block diagram of a statefeedback system with controller K and error integration.
This statespace representation has been previously used to derive robust control synthesis
methods for dcdc converters, which generally result in a statefeedback law that stabilizes
the system for a certain range of uncertainty:
the linear dynamics of the converters are presented in (Montagner et al., 2005) and
(TorresPinzon & Leyva, 2009) while (Hu, 2011) introduces a representation of the nonlinear
dynamics.Consistent experimental results with tight performances are presented in
(Olalla et al., 2009a; 2010a; 2011). The smallsignal stabilization of nonlinear dcdc converters
is considered in (Olalla et al., 2009a; 2010a), where the converter is ensured to be stable in a
range of operating points, but its trajectory between those points is not ensured to be stable
due to the disregardof the nonlinear dynamics. These nonlinearities are taken into account in
(Olalla et al., 2011) where also a less conservative polytopic uncertainty model is introduced.
The statefeedback formulation of the control problem is of interest since (i) it may deliver
better performance than some outputfeedback approaches, (ii) it can be posed as a convex
optimization problem with no conservatism or iterations and (iii) it is very simple to
implement. However, the main disadvantage of statefeedback is that the full state vector
must be available for measure, which is not always true. Therefore, it may require additional
components and sensors to obtain the state or to implement estimators of the unaccessible
states. Robust outputfeedback approaches are then an alternative to derive robust controllers
with known performances.
Robust control via outputfeedback has been the subject of extensive research in the field
of automatic control (de Oliveira & Geromel, 1997; Garcia et al., 2004; Peaucelle & Arzelier,
2001a;b; Scherer et al., 1997; Skogestad & Postlethwaite, 1996), but it has been hardly
employed in dcdc converters (Rodriguez et al., 1999). Power electronics engineers tend to
use currentmode approaches (Erickson & Maksimovic, 1999) that employ an inner current
loop before applying the outputfeedback loop and, in that way, ease the control of the dcdc
converter. However, currentmode approaches require current sensing, as statefeedback
control, and they suffer from noise, since in some cases, as in peakcurrent control, the current
waveform must be sensed accurately. Therefore, a plain outputfeedback approach can be of
interest in certain cases in which a simple control is required and the sensing of all the states
of the converter is not possible.
parameterdependent approaches for
3.1 Statefeedback control
The most simple control problem in terms of an LMI formulation is the one in which all the
system states are measurable. The statefeedback problem considers the stabilization of (22)
with a simple controller u = Kx, where K ∈ Rm×n, as follows
˙ x(t) =?A(p) + Bu(p)K?x(t) + Bw(p)w(t).
Since the statefeedback approachdoesnot allowto eliminate steadystate error,an additional
(23)
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integral state can be introduced for the regulated output of the system, as shown in Figure 5.
Once the augmented system has been rewritten in the form of (23), the following result,
adapted from(Bernussou et al.,1989), pointsouta synthesismethod toobtain astatefeedback
controller that stabilizes quadratically the closedloop system.
Theorem 3.1. The system (23) is stabilizable by statefeedback u = Kx if and only if there exist a
symmetric matrix W ∈ Rn×nand a matrix Y ∈ Rm×nsuch that
?W > 0 0
then, the statefeedback is given by K = YW−1.
AW + WA?+ BuY + Y?B?u< 0 0
(24)
Proof. The proof uses a quadratic Lyapunov function V(x) = x?Px, P = P?> 0, whose
timederivative along the trajectories of the closedloop system ˙ x = (A + Bu)Kx must be definite
negative (Boyd et al., 1994). It follows that the following condition
A?P + PA+ K?B?
uP + PBuK < 0 0
(25)
has to be satisfied. Finally, considering the left and righthand multiplication of the previous condition
by W = P−1, and the substitution of KW = Y, LMI condition (24) follows.
A single Lyapunov function can be used to guarantee the stability of an uncertain system.
The following theorem yields the statefeedback condition in the case of a polytopic
representation.
Theorem 3.2. The uncertain system defined by a convex polytope Co{G1,...,GN} is quadratically
stabilizable by statefeedback u = Kx if and only if there exist a symmetric positive definite matrix W
and a matrix Y such that
AjW + WA?
j+ BujY + Y?B?
uj< 0 0
∀j = 1,..., N,(26)
then K = YW−1is a statefeedback matrix.
The proof of this theorem is given in (Bernussou et al., 1989). It is worth to point out that
there exist more recent works which have been concerned with the stability of polytopic
uncertain systems considering in particular multiple Lyapunov functions instead of a single
one (Apkarian et al., 2001; Bernussou & Oustaloup, 2002; Peaucelle & Arzelier, 2001c), in
order to reduce the conservatism of the quadratic approach.
Example 1. BuckBoost Converter
In this example, an uncertain polytopic model of the buckboost converter is presented and a robust
statefeedback controller is derived.
Consider the buckboost converter model introduced in Section 2.2. Since for statefeedback control,
the capacitor voltage is considered measurable, the stray resistance rCis neglected. Also the stray
resistances of the inductor and the semiconductor devices are disregarded. In order to obtain zero
steadystate error between the voltage reference Vrefand the output voltage vo(t), the model is
augmented with an additional state variable xint(t), which stands for the integral of the output voltage
error, i.e. xint(t) = −? ?Vref− vo(t)?dt. The state vector of the new model is then written as
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LMI Robust Control of PWM Converters: an OutputFeedback Approach 11
x(t) =
⎡
⎣
iL(t)
vo(t)
xint(t)
⎤
⎦. Considering
AX + BwW +
?0
1
0
?
Vref= 0,(27)
the linear dynamics of the buckboost converter are then written as:
˙˜ x(t) =A˜ x(t) + Bw˜ w(t) + Bu˜ u(t)
(28)
A =
⎡
⎣
0
D?
L
RC0
1
0
−D?
0
C−1
0
⎤
⎦, Bw=
⎡
⎣
D
L
0 −1
0
0
C
0
⎤
⎦, Bu=
⎡
⎢
⎣
Vg
D?L
DVg
D?2RC
0
⎤
⎥
⎦.(29)
Uncertainty:
Polytopic uncertainty (19) is introduced in the model of the converter to cope with the variations of D
and R. The parameters of this example take the values shown in Table 1. Note that the transient
performance requirements are only fulfilled when the trajectory starts from an equilibrium point.
Consequently, the variations of D and R must be slow enough to allow the system states to return
to the equilibrium.
Parameter
R ∈
Vg
D
C
L
Ts
Value
[10, 50] Ω
12 V
[0, 0.7]
200 μ F
100 μ H
5 μ s
Table 1. Buckboost: converter parameters
As in (Olalla et al., 2009a), additional variables are introduced, in order to remove the non affine
dependence of the system matrices on the uncertain terms. The uncertainty parameter vector is defined
as p = [R D?δ1δ2], where:
R ∈ [Rmin, Rmax],
D?∈ [D?
δ1∈ [1/D?max, 1/D?
δ2∈ [Dmin/D
Note that the uncertain model is inside a polytopic domain formed by N = 24vertices. Also note that
the multiplication between δ2and 1/R in the second row of Budoes not imply a new variable because
both functions are strictly decreasing.
min, D?max],
min],
?2
max, Dmax/D
?2
min].
(30)
Sampling effect:
In this example the sampling effect has not been included in the converter, as the state variables of the
modulator model can not be measured.
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12WillbesetbyINTECH
−150
180
−100
−50
0
Magnitude (dB)
100
101
102
103
104
105
−180
−90
0
90
Phase (deg)
Bode Diagram
Frequency (Hz)
(a) Bode plot of closed loop transfer function from
reference to output
−15−10−5 0
x 104
−1.5
−1
−0.5
0
0.5
1
1.5x 105
2e+004
0.08
2e+004
1e+004
5e+003
1e+004
1.5e+004
0.94
5e+003
0.64
0.8
0.380.5 0.17 0.28
0.94
0.64
0.8
0.380.5 0.170.28
0.08
1.5e+004
Pole−Zero Map
Real Axis
Imaginary Axis
S(α,r,θ)
(b) Pole location map of the closed loop system
101
102
103
104
45
90
135
180
225
270
−30
Phase (deg)
−20
−10
0
10
20
Magnitude (dB)
Bode Diagram
Frequency (Hz)
(c) Bode plot of closed loop transfer function from
output disturbance to output.
00.0020.0040.0060.008 0.010.0120.0140.0160.018 0.02
−1.2
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Step Response
Time (sec)
Amplitude
(d) Output voltage transient response to a step
reference.
Fig. 6. Simulation results of Example 1, with controller K1, for the sixteen vertices of the
uncertainty set.
Performance Specifications:
Following the synthesis method shown in (Olalla et al., 2010a), the objective function to be minimized
is the H∞norm of the transfer function between the output disturbance˜iload(t) and the output voltage
˜ vo(t).
In order to assure robust transient performances, the closed loop poles are constrained in an LMI region
S(α,r,θ), where the desired minimum damping ratio is set to θ =
frequency is r =
10
α = 200.
1
√2, the required maximum damped
1
2π
Ts, and the minimum decay rate, for a settling time lower than 20 ms, is set to
Results:
The robust control synthesis algorithm yields a controller K1:
K1= [−0.31
− 0.25 194.70]
(31)
that ensures an H∞norm from output disturbance to output voltage of 3.80 (11.6 dB). Figure 6 shows
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LMI Robust Control of PWM Converters: an OutputFeedback Approach13
d(t)
y(t)
DCDC
G(s)
Vref
+

Fig. 7. Block diagram of an outputfeedback system with controller G(s).
thesimulationresultsofthecontrolleroverthesixteen verticesof thesetof matrices. Sincethenonlinear
terms are disregarded, the robust stability and performance of the converter is guaranteed while the
converter remainsin the considered operating points, assumingthat the change betweenthese operation
points is sufficiently slow. In order to account for the neglected dynamics, see (Hu, 2011; Olalla et al.,
2009b; 2011).
3.2 Outputfeedback control
Figure 7 shows the general diagram of an outputfeedback control system. Depending on the
structure of the controller G(s), two main approaches can be differentiated for the synthesis
of outputfeedback controllers: static and dynamic controllers.
Given the system Σ as described in (21), for the buck and the buckboost converter Eyu = 0
can be considered. Then, in the case of a dynamic controller of order k with the following
structure
?˙ xc= Acxc+ Bcy
u = Ccxc+ Dcy
ΣK
:
,(32)
the closed loop system has the form
TΣK
:
?˙ xcl= Axcl+ Bw
z = Cxc+ Dw
(33)
where
?A B
C D
?
=
⎡
⎣
A + BuDcCy
BcCy
Cz+ EzuDcCyEzuCc Ezw+ EzuDcDyw
BuCc
Ac
Bw+ BuDcEyw
BcEyw
⎤
⎦
(34)
In the case of a static controller, K ∈ Rm×q
u = Ky = K(Cyx + Eyww)
(35)
and the closed loop system has the following structure
TK
:
?˙ x = Ax + Bw
z = Cx + Ew
(36)
where
?A B
C E
?
=
?(A + BuKCy)(Bw+ BuKEyw)
(Cz+ EzuKCy) (Ezw+ EzuKEyw)
?
(37)
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14WillbesetbyINTECH
For both problems the Lyapunov inequality is written, in a generic form:
?P > 0 0
A?P + PA < 0 0
, (38)
which depends nonlinearly on P and the matrices of the controller (K in the static case or Ac,
Bc, Cc, Dcin the dynamic case).
There exist several methods to linearize the outputfeedback synthesis problem. For the
dynamic outputfeedback case, the results of reference (Scherer et al., 1997) are summarized.
In the case of static outputfeedback, the methods shown in(de Oliveira & Geromel, 1997) and
(Peaucelle & Arzelier, 2001a) are employed.
3.2.1 Dynamic outputfeedback
The dynamic outputfeedback synthesis method shown in (Scherer et al., 1997) employs the
following transfer function parametrization defined from the exogenous input w = wjRjto
the cost output zj= Ljz as follows:
Tj(s) =
zj(s)
wj(s):=
?A Bj
CjDj
?
=
?ABRj
LjC LjDRj
?
=
⎡
⎣
A + BuDcCyBuCc Bj+ BuDcFj
BcCy
Ac
Cj+ EjDcCy EjCc Dj+ EjDcFj
BcFj
⎤
⎦
(39)
where
Bj:= BwRj,
Cj:= LjCz,
Dj:= LjEzwRj,
Ej:= LjEzu,
Fj:= EywRj. (40)
To find a controller which stabilizes the closedloop system, there must exist a quadratic
Lyapunov function
V(xcl) = x?
clPxcl,(41)
such that
?P > 0 0
A?P + PA < 0 0
(42)
The LMI constraints are formulated for a transfer function Tj(s) = LjT(s)Rj, in terms of the
statespace matrices A,Bj,Cj,Dj. The goal is to synthesize an LTI controller ΣKthat:
• internally stabilizes the system
• meets certain specifications (H2, H∞, pole placement,...) on a particular set of channels.
Generally, each transfer function Tjwill satisfy each specification Sj, if there exists a Lyapunov
matrix Pj> 0 that satisfies some LMI constraints in Pj. The control problem usually includes
a number i of specifications.Therefore, the synthesis problem involves a set of matrix
inequalities whose variables are:
• the controller matrices Ac, Bc,Cc, Dc.
• the i Lyapunov matrices P1,...,Pi, one per specification.
• additional auxiliary variables to minimize, for example, the norm cost H∞.
Since this problem is nonlinear and hardly tractable numerically, the method shown in
(Scherer et al., 1997) requires that all the specifications are satisfied with a single Lyapunov
function, that is:
P1= ... = Pi= P. (43)
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LMI Robust Control of PWM Converters: an OutputFeedback Approach 15
This restrictioninvolves conservatism in the design, but itleadsto anumerically tractable LMI
problem, it produces controllers of reasonable order and it exploits all degrees of freedom in
P (Scherer et al., 1997). Actually, if a single Lyapunov function P is considered, the following
change of variable linearizes the control problem and makes it solvable with LMIs.
Let n be the number of states of the plant, and let k be the order of the controller. Partition P
and P−1as
P =
N?∗
where X and Y are ∈ Sn, and ? is a symmetric positive definite matrix such that PP−1= 1
holds.
From PP−1= 1 we infer P
M?
0 0
?Y N
?
,
P−1=
?X M
M??
?
(44)
?X
?
=
?1
?
, which leads to
PΠ1= Π2,
Π1=
?X 1
M?0 0
?
,
Π2=
?1 Y
0 0 N?
?
(45)
The change of variables is as follows
⎧
⎨
⎪
⎪
⎪
⎪
⎩
ˆA := NAcM?+ NBcCyX + YBuCcM?+ Y(A+ BuDcCy)X
ˆB := NBc+ YBuDc
ˆC := CcM?+ DcCyX
ˆD := Dc
(46)
whereˆA,ˆB,ˆC have dimensions n × n,n × m,q × n respectively. If M and N have full row
rank, andˆA,ˆB,ˆC,ˆD,X,Y are given, the matrices Ac,BcCc,Dccan be computed. If M and N
are square n = k and invertible, then Ac,Bc,Cc,Dcare unique.
The motivation for this change of variables lies in the following identities
Π?
1PAΠ1= Π?
2AΠ1=
?AX+ BˆC A + BuDCy
2Bj=
YBj+ˆBFj
CjΠ1=?CjX + EjˆC Cj+ EjˆDCy
Π?
ˆAYA +ˆBCy
?
?
1 Y
?
Π?
1PBj= Π?
?Bj+ BuDFj
1PΠ1= Π?
1Π2=
?X 1
?
(47)
which can be used in a congruence transformation to derive the LMI constraints. A detailed
proof is given in (Scherer et al., 1997).
Once the variablesˆA,ˆB,ˆC,ˆD,X,Y have been found, let us recover the original system by
following this procedure. First we need to construct M, N and P that satisfy (45). M and
N should be chosen such that NM?= 1− YX. With the following LMI:
?X 1
we assure Y > 0 0 and X − Y−1> 0 0 such that 1 − YX is nonsingular. Hence, M and N can
always be found. After that, Π1and Π2are also nonsingular, and P = Π2Π−1
1 Y
?
> 0 0
(48)
1
can be found.
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