Filtering and control via LMI optimisation
21
Several studies have also dealt with systems admitting a Linear Fractional Representation (LFR):
roughly speaking the Linear Fractional Representation of a dynamic system corresponds to a
feedback representation whose forward part is linear and time invariant, whereas the backward part
contains the time-variability of the original system. Such a representations, familiar in the realm of
robust control, are advantageous not only in order to deal with Liner Time Varying (LTV) systems
but also to tackle filtering and control problems for some classes of non-linear systems [Elg96].
1.2 Filtering techniques and unbiasedness
Filtering a dynamic system consists in estimating the state vector or a set of linear combinations of
the state vector exploiting the measurements available until the current time. This is a classical
problem going back to the early papers by Kalman [Kal60], and then treated in a number of books
such as [And79].
Mainly we consider the H∞ filtering criterion that is customarily used when either no a priori
information on the disturbance statistical properties is available or the designer aims to cope with
the worst noise perturbation.
In this thesis we provide some useful techniques in order to face the filtering problem of linear
systems both in continuous-time and discrete-time. In particular we focus on LMI filtering
techniques based on the so-called unbiasedness condition (see e.g. [And79] for the definition of
unbiasing filter). We deal with both LTI (Linear Time Invariant) and LPTV (Linear Periodically
Time Varying) systems.
The unbiasedness condition provides a twofold advantage in an LMI context. Indeed, unbiasedness
reduces the number of the unknowns and hence leads to a lower computational effort. Moreover, it
allows to face in an LMI framework the filtering problem of unstable systems without the necessity
of resorting to any frequency shifting methods [Tak96].
An interesting characteristic of the filtering techniques proposed herein is given by the possibility of
obtaining a significant parameterisation of the filters meeting some design requirements. It is
believed that this approach can be extended to other types of reconstruction problems as the so-
called deconvolution problem.
The robust unbiased LMI filtering problem can be faced as well by resorting to a simple trick based
on the extension of both the input noise and of the measured signal.
Some of these techniques have been successfully applied to some practical problems such as the
problem of rejecting periodic disturbances in helicopters. In such application the computational
advantage offered by LMI Periodic filtering technique turns out to be very sensible since offers the
opportunity of tackling, for the first time, the generic flight condition in a very simple and
computationally advantageous way.
1.3 Linear Parametrically Varying control techniques
The LPV (Linear Parametrically Varying) design philosophy can be considered as the evolution of
the gain-scheduling techniques based on the design of several controllers each of which valid for a
particular region of functioning of the plant to be controlled. A rough gain scheduling technique is
based on the selection of the controller synthesised for the operating conditions close to the current
one (see [Sha99]).
It is obvious that gain-scheduling techniques in general do not guarantee a good performance.
Furthermore their performance is not guaranteed in case the variation rates of the parameters are
excessive. The control by means of LPV techniques aims towards the design of a control law
varying in function of some measured parameters in order to guarantee both robust stability and
some type of robust performance requirements.
Filtering and control via LMI optimisation
22
Recently LMI theory has been recognised to be particularly advantageous to tackle in a simple and
flexible manner this task. It is possible to distinguish between two different categories of LPV-LMI
techniques. The first is the most conservative since it gathers all the techniques based on a single
quadratic Lyapunov function and hence they can turn out to be inadequate: actually, it is possible to
demonstrate that there are dynamic systems not stabilisable by means of a single quadratic
Lyapunov function. Nevertheless in some cases, these techniques can lead to very effective
controllers. More exactly, these techniques can be used in order face the control problem of LPV
systems with unbounded variation rates of the parameters.
On the other hand, the second category concerns the LMI techniques whose applicability is limited
to the cases in which the variation rates of the parameters are bounded. These techniques are based
on a parameter-varying quadratic Lyapunov function and for this they can be used to tackle control
problems that cannot be faced by means of the LPV control techniques belonging to the first
category.
Between the techniques belonging to the first category we can cite [Kos98], [Sch99a], [Apk95].
Furthermore, the techniques described in [Apk98], [Sha99] and [Wuf99] represent some of the most
significant techniques belonging to the second category.
1.4 Organisation of the thesis
This thesis is divided in two main parts. Part 1 (Chapters 2-6) deals with theoretical LMI concepts
whereas in Part 2 (Chapters 7-9) we report some significant applications of LMI-LPV theory.
The organisation is as follows. In Chapter 2 we present some significant tools that allow to reduce
both classical and non classical control problems into an LMI framework: Shur Lemma, S-
Procedures, Projection Lemmas, Linear Fractional Transformations, etc. In Chapters 3, 4 and 5 we
face the LMI filtering problem for LTI (Linear Time Invariant) systems, LPTV (Linear Periodically
Time Varying) system and norm-bounded uncertain systems, respectively. In all these cases we will
exploit the unbiasedness condition in order to achieve computationally light LMI synthesis
procedures that can be applied to unstable systems as well. In Chapter 6 we present one of the most
significant LPV-LMI techniques nowadays available (see [Sch99a]). This technique will be widely
exploited in the second part of this thesis.
We then turn to the application side. In Chapter 7 we pose the problem of rejecting periodic
disturbances in helicopters and we propose to face this problem both by means of a gain scheduling
approach of periodic controllers and by means of the LPV design technique described in Chapter 6.
Another widely studied problem is represented by the trajectory tracking problem of mobile robots.
We have considered this problem having the objective of designing the trajectory tracking control
of mobile robot platforms ad the LABMATE platform available care of Laboratorio di Automatica
of the Politecnico di Milano. As explained in Chapter 8, this non-linear control problem is faced by
switching among LPV technique based on a grid procedure of the parameter space [Apk98].
Finally, in Chapter 9 we propose the use of the LPV technique described in Chapter 6 in order to
face a particular type of deconvolution problem i.e. the problem of reconstructing the primary
current of non-linear current transformers on the basis of the measurement of the secondary current.
1.5 Conclusions and further developments
The main contributions of this thesis regard LMI techniques for filtering and control of dynamic
systems. Two main topics are considered:
Filtering and control via LMI optimisation
23
Unbiased filtering by means of LMI
Control of linear and non-linear systems by means of LPV-LMI design techniques
As said before LMI theory presents wide possibilities of future developments both for the fields
considered in this thesis and for other classical and non-classical control and filtering problems that
can be advantageously faced by means of LMI theory.
We conclude this introduction by providing a significant list of topics that are worthy of future
investigation.
LMI Periodic filtering and control theory
In this thesis we have considered the LMI periodic filtering problem since it turns out particularly
significant in many applications and in particular in telecommunication. The problem has been
solved in a satisfactory manner both in continuous-time and in discrete-time. More exactly the
discrete-time case has been solved simply piling up several LMI constraints each one corresponding
to a different time-point into the period. On the contrary the continuous-time case can be tackled by
imposing an affine structure to the unknowns; in this way one can transform the problem of finding
periodic matrices (a functional unknown) into the problem of finding some constant matrices
(algebraic unknown). Moreover, to deal with continuous-time, one can resort to a grid procedure by
suitable sampling the period control (see Chapter 4).
Obviously the computational effort required by a continuous-time LMI periodic problem can be
reduced by avoiding the period-grid procedure since, in general, it is not possible to a priori known
how to sample the period.
LPV-LMI Theory
As pointed out before, one of the most active research areas concerning LMI control theory is
represented by the LPV design control techniques. In this thesis we have described one of the most
significant LPV design control techniques today available. This technique is based on a particular
type of S-Procedure (see Chapter 2) and does not need a precise knowledge of bounds on the
variation rates of the parameter. Moreover, there is no need of resorting to a grid in the parameter
space.
In Chapter 9 we proposed also the use of an LPV technique in order to face a non-linear filtering
problem. Indeed, the good results reported in this chapter suggest us that this technique represents a
valid alternative to the use of classical non-linear filtering techniques as the Extended Kalman filter.
Obviously a more detailed investigation about this possibility is of interest to future research.
LMI Smoothing
In Chapter 4 we propose a technique in order to face the fixed-lag smoothing problem of linear
systems (possibly time-periodic). The technique proposed is based on a state space extension of the
state-space model and requires the use of Lyapunov matrices having a dimension strongly
increasing with the lag of the smoother. Obviously, such an approach turns out computationally
expensive for large values of the lag.
A more convenient way of proceeding is represented by a Forward-Backward filtering approach
(see [And79]). Such an approach can be also implemented by means of LMI-projection techniques:
the forward filter can be obtained from projected H2 synthesis LMI conditions based on a time-
invariant filter, whereas the backward filter can be found resorting to a time-varying H2 filter
applied to the backward process model. The calculation of the backward filter amounts to a finite
dimensional LMI problem: indeed it has to be found only for a number of time-points equal to the
amplitude of the lag.
Filtering and control via LMI optimisation
24
Subspace Identification methods
A further quite significant line of research is represented by subspace identification methods: in this
type of problems frequent is the issue of facing a Robust Least Squares (RLS) problem with
uncertain matrices having a particular structure. In order to solve these problems the application of
some RLS techniques as the one presented in [Elg97], obviously adapted to the problem, could be
advantageous in order to find non-conservative solutions. This would be possible under the
assumption of using RLS techniques that allows to take into account the particular structure of the
uncertain matrices involved.
Uncertain Polytopic Systems
As reported before, the control and filtering problems of polytopic uncertain systems can be tackled
imposing several LMI constraint each one corresponding to a different vertex of the polytope
defining the uncertainty of the given system. A common way of facing this problem is based on the
use of a single Lyapunov function. Recently a modified version of the Bounded Real Lemma (BRL)
(see Chapter 2) has been proposed in [Shk99]. The most important advantage of this modified
version of BRL is given by the fact that it allows to face robust filtering and control problems of
polytopic uncertain systems exploiting several Lyapunov matrices leading in such a way to less
conservative results.
For this reason, the possibility of extending some LMI technique for polytopic uncertain systems as,
for instance, the Robust Model Predictive Control technique proposed in [Kot96], by means of the
modified BRL, appears promising.
Part 1: Theory
Filtering and Control
of linear systems
Part 1 Theory
Chapter 2
The Linear Matrix Inequalities
Chapter 2
The Linear Matrix Inequalities
2.1 Introduction to Linear Matrix Inequality
The most classical form of a Linear Matrix Inequality (LMI) problem is given by the following
expression
0:)(
1
0 >+=
=
m
i
ii
FxFxF (2.1)
where x∈ℜm is the unknown and the symmetric matrices '
ii
FF = ∈ℜnxn, i=0, , m, are given. This
for of Linear Matrix Inequality is general: in fact, it includes also problems for which the unknowns
are constituted by matrices of generic dimensions. Note also that multiple LMIs can be expressed as
a single LMI: i.e., the set of LMIs F(i)(x)>0 with i=1, , n, can be expressed as the single LMI
diag(F(1)(x), , F(n)(x))>0.
Many problems concerning dynamic system and control theory can be reformulated as LMI
problems. The main advantage is that the solution can be achieved in polynomial time thanks to
recently developed algorithms as the Interior Point Methods (see [Boy94] for more details).
Moreover, several effective control-oriented packages are today available, such as [Gah94a] and
[Elg98].
In this chapter, we introduce some useful concepts and lemmas widely exploited in the field of
dynamic system theory, for reducing both standard and non-standard filtering and control problems
into a LMI framework.
2.2 The importance of LMI in control
One of the most celebrated results in LMI theory is constituted by the Shur Lemma. The importance
of this lemma can be pointed out recalling that by means of this very effective tool one can
reformulate typical control-related inequalities, as the Riccati inequalities, in corresponding LMIs.
A very common control related problem has the following structure:
minimise γ
subject to: ∃ 0'>= PP such that 0''' 1 <+++ − PPBBCCPAPA γ (2.2)
where A∈ℜnxn, B∈ℜnxp and B∈ℜmxn.
Eq. (2.2) is a Riccati inequality (see [Bit91], [Zho98]) and, as such, is not an LMI in the unknowns
P and γ. By means of the celebrated Shur lemma one can attain an LMI equivalent version of the
inequality (2.2).
Theorem 2.1 (Schur Lemma)
Let 'QQ = , 'RR = and S with appropriate dimensions. The condition:
The Linear Matrix Inequalities
29
0
'
≥
RS
SQ
is equivalent to the following conditions:
0≥R , 0'≥− + SSRQ , 0)( =− +RRIS . o
Thanks to the Shur Lemma, (2.2) is equivalent to the following LMI problem:
minimise γ
subject to: ∃ 0'>= PP such that
0
'
''
<
−
++
IPB
PBCCPAPA
γ
. (2.3)
In (2.3) both the unknowns P and γ appear linearly. This implies that the optimal solution of the
problem (2.3) can be achieved in a computationally advantageous manner by means of an LMI
solver.
2.3 Main lemmas
In this section we will describe some very powerful lemmas leading to LMI problems. These
lemmas enable to tackle in an LMI framework a large number of filtering and control problems
concerning: Linear Systems, Uncertain Systems, Periodic Systems, Linear Parametrically Varying
(LPV) systems, etc. In this section we provide an overview about these topics.
2.3.1 The S-Procedures
In many cases to the purpose of formulating some problems into an LMI framework, it can be
convenient recurring to the introduction of scaling variables i.e. sensible variables guaranteeing the
sufficiency of some conditions but also increasing the conservativeness of the formulation: these
procedures are commonly known as S-Procedures (see [Yak71], [Sch99a] and [Sch99b]).
The most known versions of S-Procedures deal with situations in which there are some quadratic
functions that have to be negative (positive) if some further quadratic functions are negative
(positive): these situations can be expressed in terms of LMI by means of some sufficient conditions
as the S-Procedures. We can distinguish two cases: the S-Procedure for nonstrict inequalities and
the S-Procedure for strict inequalities.
S-Procedure for nonstrict inequalities
Consider the following quadratic functions of the variable x∈ℜn:
iiiii
rSxxSxQxxF +++= ''':)( with i=0, ,m
where '
ii
QQ = ∈ℜnxn,
i
S ∈ℜn.
The condition
The Linear Matrix Inequalities
30
0)(0 ≥xF ∀x such that 0)( ≥xFi for i=1, ,m (2.4)
holds if there exist m coefficients λi≥0 such that
∀ x, 0)()(
1
0 ≥−
=
xFxF
i
m
i
i
λ . o (2.5)
S-Procedure for strict inequalities
Consider the following symmetric matrices: Qi∈ℜnxn for i=0, ,m. We can state that:
0' 0 >xQx ∀x≠0 such that 0' >xQx i for i=1, ,m (2.6)
if there exist m coefficients λi≥0 such that
=
>−
m
i
ii
QQ
1
0 0λ . (2.7)
This condition is also necessary in case m=1. o
Recently a new S-Procedure has been introduced in order to attenuate the conservativeness of the
previous S-Procedures. As we will see in the sequel this kind of issue is common in control and
filtering of Linear Parametrically Varying systems. This new S-Procedure was for the first time
introduced in [Sch97b] and it is commonly referred as Full Block S-Procedure.
Full-Block S-Procedure
Consider a subspace S of ℜn, a full row rank matrix T∈ℜlxn and fixed symmetric matrix N∈ℜnxn.
Furthermore consider the following family of subspaces indexed by ∆ where ∆ is the generic full
row rank matrix belonging to the compact set ∆:
{ }0 :)ker(: =∆∈=∆∩=∆ TxSxTSS . (2.8)
The full block S-Procedure we are going to introduce is based on the following technical
conditions: ∀ ∆∈∆, S∆ has to be complementary to a fixed subspace S0⊂S. Moreover we have to
suppose that:
kS ≥)dim( 0 and 0≥N on S0.
We can finally state the Full Block S-Procedure:
∀ ∆∈∆, N<0 on S∆ if and only if ∃ P such that (2.9)
∀ ∆∈∆, P>0 on ker(∆) and 0' <+ PTTN on S. o (2.10)
2.3.2 Projection Lemmas
Projection lemmas represent one of the most flexible tools for LMI techniques. They allow to
eliminate awkward unknowns from the original formulation of the problem considered. In some
The Linear Matrix Inequalities
31
cases this leads to a more convenient formulation containing a reduced number of unknowns and
that can turn out to be a sensible LMI formulation.
It is worthy noticing that these lemmas, that will be often used in the following chapters, are among
the most important results of the LMI theory since they allow to tackle problems otherwise
untractable.
We can distinguish two different projection lemmas that in the sequel will be called Discrete
Projection and Continuos Projection. The Discrete Projection lemma is mostly addressed to solve
problems concerning discrete-time dynamic systems whereas the Continuos Projection lemma is
suitable in continuous-time case. A lucid dissertation about this subject can be found in [Ske98].
Discrete Projection lemma
Consider five matrices A, B, C, R, Q with appropriate dimensions and such that 0' >BB , 0'>CC ,
0>R and 0>Q .
i) There exists X such that:
( ) ( ) QBXCARBXCA <++ ' (2.11)
if and only if the following conditions are verified:
( ) 0'' >− ⊥⊥ BARAQB or 0'>BB (2.12)
and
( ) 0'' '' 11 >− ⊥−−⊥ CAQARC or 0' >CC . (2.13)
ii) If (2.11) admits a solution X, then the set of all solutions of (2.11) can be obtained in the
following manner:
2/12/111
)'()'('')'( ΨΦ+ΦΦ−= −−− LBBCRCARCBBBX
where L is an arbitrary matrix such that 1<L and
( )
1
''':
−+−=Φ CRARARCARAQ
c
( )( )
ccc
RARCBBBBCRARR '''': 1 ΦΦΦ−Φ−=Ψ −
( )
1
':
−= CRCR
c
. o
Continuos Projection lemma
Consider three matrices B, C and 'QQ = .
i) There exists X such that:
( ) 0' <++ QBXCBXC (2.14)
if and only if the following conditions are verified:
The Linear Matrix Inequalities
32
0'<⊥⊥QBB or 0'>BB (2.15)
and
0''' <⊥⊥ QCC or 0' >CC . (2.16)
ii) If (2.14) admits a solution X, then the set of all solutions of (2.14) can be obtained in the
following manner: consider a full rank factorisation of B and C:
rl
BBB = ,
rl
CCC = . Then,
++++ −+=
llrrlr
CZCBBZKCBX
where Z is an arbitrary matrix and
( ) ( )
212111
'''':
rrrrrl
CCLSCCCBRK Φ+ΦΦ−= −−
( )[ ]
1111
''':
−−−− ΦΦΦ−Φ−= RBCCCCBRRS
lrrrrl
.
Here matrix L is an arbitrary matrix such that 1<L and R is an arbitrary positive definite
matrix such that
( ) 0':
11 >−=Φ −− QBRB
ll
. o
2.3.3 Robustness
The LMI theory offers powerful tools to face also robustness problems in which the uncertainty is
supposed to be norm bounded. In this section two significant lemmas are presented. These lemmas,
straightforwardly derived from the application of S-Procedures, concern respectively the cases of
structured and unstructured uncertainty.
Unstructured uncertainty lemma
Consider 'FF = , L, R, D of suitable dimensions.
The two following statements are equivalent:
a) ∀ ∆ such that 1≤∆ , 0)det( ≠∆− DI and 0''')(')( 11 ≥∆∆−+∆−∆+ −− LDIRRDILF
b) there exists a scalar τ such that
( )
0
''
'''
≥
−−
−−
DDIDLR
LDRLLF
ττ
ττ
. o
Structured uncertainty lemma
Consider 'FF = , L, R, D of suitable dimensions.
Let D a sub space of ℜpxq and { }'' , | )(: GGTSS, T, G ∆−=∆∆=∆∈∆∀= DB .
The following statements are equivalent:
The Linear Matrix Inequalities
33
a) ∀ ∆∈ D such that 1≤∆ , 0)det( ≠∆− DI and 0''')(')( 11 >∆∆−+∆−∆+ −− LDIRRDILF
b) there exist (S, T, G)∈ B such that S>0, T>0 and
0
0
'
0
0
'
|
>
−
−
ID
L
TG
GS
ID
L
R
RF
. o
2.3.4 Linear Fractional Transformations
In this section we introduce a particular kind of representation for algebraic and dynamic systems:
the so-called Linear Fractional Representation (LFR). Such kind of representation is common in the
field of robust control and in particular in the realm of µ-analysis and µ-synthesis (see e.g. [Zho98],
[Pak94]).
On the other hand, the Linear Fractional Transformations (LFT) represents the most common way
to put in evidence the LFR of dynamic or algebraic systems. In LMI control theory the Linear
Fractional Representation turns out to be useful in many fields: from robust control problems to the
Linear Parametrically Varying (LPV) methodologies, which will be the subject of some subsequent
chapters.
Definition 2.1
Consider a matrix partitioned as
)()( 2121 qqxpp
DC
BAM ++ℜ∈
= ,
moreover let 22xpq
l
ℜ∈∆ and 11xpqu ℜ∈∆ be two further matrices.
Then provided that ( )
l
DI ∆− be non singular, we can define the lower LFT with respect to
l
∆
1122 xx
:) ,(
pqpq
l
M ℜ→ℜ⋅F
as the map
( ) CDIBAM
llll
1
:),(
−∆−∆+=∆F .
Similarly, provided that ( )uDI ∆− be non singular, we can define the upper LFT with respect to
u∆
2211 xx
:) ,(
pqpq
u M ℜ→ℜ⋅F
as the map
( ) BAICDM uuuu 1:),( −∆−∆+=∆F . o
The Linear Matrix Inequalities
34
The LFTs can be useful to represent in a convenient manner a wide category of systems (both
algebraic and dynamic). For instance, consider the following algebraic relation between the variable
z and the variable w:
( )( )wCDIBAz 1−∆−∆+= . (2.17)
The relation (2.17) can be equivalently rewritten exploiting the lower LFT as
w
DC
BA
z
l
∆
= ,F . (2.18)
This new algebraic relation points out that the original system (2.17) can be seen as a feedback
representation whose forward part depends on the matrices A, B, C and D whereas the backward
part is constituted by ∆. More exactly, we can say that the system (2.18) can be represented by
means of two auxiliary variables p and q in the following manner:
DpCwq
BpAwz
+=
+=
or equivalently
=
p
w
DC
BA
q
z
(2.19)
with p=∆q. (2.20)
The graphical equivalent representation of eqs. (2.19) and (2.20) is depicted in Figure 2.1.
A B
C D
q
p
z w
∆
Figure 2.1
The extension to dynamic systems is obvious. Indeed, a dynamic system can be represented in the
form of eqs. (2.18) tacking as variables z and w the following vectors:
=
y
x
z
and
=
u
x
w
where x represents the state of the dynamic system, y the measured output and u the input.
As we will see in the sequel, the LFR and hence the LFTs can be useful to face the filtering and
control problems of a variety of dynamic systems. For example we can exploit an LFT to isolate the
time-dependence of time-varying system or to isolate the non-linearity of a non-linear system.
Fundamental references about this subject are given by [Zho98], [Sco97] and [Lee96]. In this thesis
we will use widely the LFTs in order to tackle filtering and control problems of Linear
The Linear Matrix Inequalities
35
Parametrically Varying (LPV) systems. Indeed, LMI provide effective techniques to face such a
kind of problems by means of controllers (filters) replying the same LPV structure of the controlled
(filtered) system.
2.4 Some typical LMI problems in control
In this section we present some typical LMI problems. In order to show the usefulness of the
lemmas and propositions stated above. Precisely, by means of two examples, we will see how to
tackle some typical problems concerning time-varying systems and uncertain systems: first we want
to show how to exploit the Projection lemmas (Section 2.4.1) and, second, to propose an example
concerning the use of LFTs for Linear Time Varying systems (Section 2.4.2).
2.4.1 Application of the projection lemma to the Lyapunov inequality
Consider the following linear system:
Fxx = . (2.21)
It is well known that the system (2.21) is stable if and only if the Lyapunov inequality
0' <+ PFPF (2.22)
admits a symmetric positive definite solution. As one can note, the inequality (2.22) is linear in the
unknown matrix P. This implies that LMI theory can be effectively employed to analyse the
stability of the dynamic system (2.21).
Now, consider the problem of stabilisation of the dynamic system
GuFxx += . (2.23)
We want to find a static feedback law of the type u=Kx stabilising the system (2.23). This is
equivalent to the problem of finding K and 0'>= PP such that
0)'()( <+++ PKGFKGFP . (2.24)
Contrary to inequality (2.22), the inequality (2.24) is not an LMI since the unknowns K and P
appear in a multiplicative form. On the other hand, (2.24) can be easily rewritten in the standard
form (2.14) tacking
X:=K, PFPFQ ': += , B:=P and C:=G.
Then, thanks to the Continuos Projection lemma we can state that the inequality (2.24) admits a
solution if and only if the following inequality admits a solution in P
( ) 0'''' <+ ⊥⊥ GPFPFG . (2.25)
This inequality is indeed an LMI in the unknown P. Eventually, once P is known, one can solve the
inequality (2.24) in the unknown K. Indeed, the inequality (2.24) is an LMI in the unknown K if a
suitable value for P is available. Alternatively we can skip this second phase exploiting the explicit
formulas provided by the Continuous Projection lemma.
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