Keywords: COR : Controller Order Reduction, 16

for Several Frozen Values of (ρ 1 , ρ 2 ) . . . . . . . . . . . . . . . . . . . . . 4.10 Values of the MOR : Model Order Reduction.

P (ρ), Q (ρ)
: Generalised reachability and observability Gramians.

P Ω (ρ), Q Ω (ρ) : Frequency weighted generalised reachability and observability Gramians. Vectorisation of matrix A.

mat(a) :

Inverse operator of vec for a vector a.

Introduction Overview

For best productivity and less defaults, industrial systems have to be more and more powerful [START_REF] Zhou | Robust and optimal control[END_REF], Antoulas, 2005a, Youse, 2007]. Hence, mechatronic systems appear to be increasingly useful. It is well-known that obtaining reasonable mathematical models is fundamental to the analysis and the design of dynamic systems.

Nowadays, modelling tools are leading to an increasing fairly complex models. However, simpler models are easier to understand, to maintain and to implement. The process of deriving a low-order model from a high order one is known as Model Order Reduction MOR. From control point of view, another problem is accrued with the signicant progress work on robust and optimal control design. This latter has allowed to improve considerably the controlled systems performance. Nevertheless, controllers stabilising high order models and achieving a specied high performance are generally of higher order than the model itself. The complexity of the obtained high-order controllers restricts their real-time operation in embedded systems and increases their implementation costs.

Then, the reduction of the controllers order is a needful design step. This gave rise to the so-called Controller Order Reduction COR problem. This approximation step has to preserve the closed-loop stability and the required performance. Generally, three ways can be used to reduce the controllers order [START_REF] Anderson | Controller reduction: concepts and approaches[END_REF]] according to

Figure 1:

1. The rst approach is to design a reduced-order model directly from the high order system [START_REF] Yeh | Fixed-order H ∞ compensator design[END_REF], Ly et al., 1985, Gangsaas et al., 1986, Bernstein, 1989, Hyland and Richter, 1990, Fischer and Gutman, 1991]. The main lack of this kind of methods is that the order of the reduced-controller has to be chosen by an ad-hoc way [Goddard, 1995].

2. The second way is to reduce the model using a MOR method [START_REF] Antoulas ; Antoulas | A new result on passivity preserving model reduction[END_REF],

and then design a controller for the obtained reduced model. The disadvantage of this indirect method is that the resulting error from reduction process will not be taken into the controller design step. And as a result, the reduced order controller is not guaranteed to stabilise the full order model [Kong, 2012].

3. The third way is to rstly design a full-order controller and then, reduce it while preserving the properties of the closed-loop system.

The proposed work will mainly focus on the third path.

Contributions Organisation

H ∞ -Controller order reduction for LTI-systems has been pretty investigated in the last four decades. However, a dearth of survey study exists in this eld. In this dissertation, we propose a global review synthesis study that handle every edge of the fundamental scheme in Figure 1 [ [START_REF] Zebiri | Balanced truncation techniques for active suspension reduced-order H ∞ -controller[END_REF]. Moreover, a real engineering problem derived from an industrial requirements specication is considered to test the eectiveness of the existing methods [START_REF] Zebiri | Order-reduction and mu-analysis of an H ∞ -controller. application to active suspension systems[END_REF]. We also perform a robustness study to evaluate the validity of the methods regarding the possible parameters variation [Zebiri et al., 2014a]. Furthermore, and based on the signicant recent researches for LTI-case, we propose a new extension study to the LPV-case. Indeed, from an H ∞ -LPV-full-order controller, a reduced order one is derived. This latter is guaranteed stabilising and the closed-loop performance are achieved. The new method allows a better approximation in a limited frequency range. In addition, the approximation error between the original full order and reduced-order closed-loops is guaranteed bounded.

Assumptions and the validation domain of the proposed method are carried out. Generalised Gramian framework is explored in order to keep the particular structure of the this order reduction. Finally, the developed LPV-order reduction algorithm is evaluated in an real engineering application. Indeed, an H ∞ -LPV-control design is considered for an automotive-suspension system. This evaluation demonstrates that our proposed technique is eective with better performance against an other methods available in the literature [START_REF] Zebiri | Frequency weighted singular perturbation for LP V -H ∞ controller order reduction[END_REF], Zebiri et al., 2016a] . Chapter 1 presents some backgrounds on control theory. The mathematical framework of the used signals and systems is rst introduced. Then, essential linear algebra notions are recalled. From the fact that this thesis deals with continuous LTI and LPV-systems, the two classes are fully highlighted. Indeed, the LTI representation and the associated norms are given. For LPV-systems, a non-exhaustive study on their modelling is given in the end of this chapter.

Chapter 2 investigates the H ∞ -control theory. The continuous LTI case is rst considered where both stability and performance constraints can be guaranteed with an H ∞controller. The H ∞ -control problem is presented and the synthesis procedure is given. By stating that this procedure provide controllers with high order, a second low order procedure is studied. In fact, the xed-order way is presented and existent algorithms are reviewed. Finally, a similar study is done for the LPV-case where stability and control synthesis are discussed. A special attention is given to the LPV-polytopic models.

Chapter 3 is devoted to a new controller order reduction method to LPV-systems. In this chapter, model order reduction problematic for LTI-systems is rst introduced.

Methods based on singular value decomposition are highlighted then used in a closed-loop scheme to reduce the H ∞ -controllers order. Based on this, an extension to LPV-case is proposed. The obtained H ∞ -LPV-controller is guaranteed stable and the approximation error is bounded. In addition, the big advantage of this method is the possibility to perform the order reduction in a chosen limited frequency range.

Chapter 4 gives a complete evaluation of the studied controller order reduction methods.

The COMPl e ib benchmark library is rst considered. The xed order techniques are evaluated among some control problems of this benchmark. In addition, a comparison with the order reduction way is also performed using the COMPl e ib library problems. The second evaluation concerns the order reduction of an H ∞ -LTI-controller for an active suspension system. The investigated techniques in the previous chapter are implemented and tested. Finally, the developed method for H ∞ -LPV-controller order reduction is evaluated on semi-active suspension system. The stability of the reduced order controller is tested and the approximation error si shown bounded. The eectiveness of the method is conrmed by the performance test where the obtained controller deals well with the comfort and the read handling performance requirements.

Conclusive remarks with short outline about possible further developments are given in the Conclusion and Perspective section. Finally, some details about the nu-

Introduction

This chapter presents basic denitions and elementary results in linear algebra and system theory. The treatment of this material is not exhaustive but should be sucient as a reference for the future chapters.

In Section 1.1, signals and systems spaces are dened. In fact, the several results and methods presented in this dissertation, concern specic system classes. Then, a general framework about the used signals and systems is presented in this section. In addition, general background about norms is then highlighted.

In Section 1.2, essential results on linear algebra are given. Indeed, notions like Linear

Fractional Transformation or Hamiltonian matrices are very important for the sequel of this dissertation. Several denitions about matrix manipulation are also given.

In Section 1.3, descriptions for Linear Time Invariant (LTI) systems and their norms are presented. Moreover, important results in control theory are also given.

Thereafter, the Linear Parameter Varying (LPV) systems are introduced in Section 1.4.

Some historical facts are shown rst. Then, a non exhaustive listing about the existing LPV-models are presented .

Signal and System Descriptions

This section reviews some standard notations and denitions about signals and systems.

Essential elements about dynamical systems and theirs representations are recalled. In particular, notions about norms are given.

Signal Spaces and Norms

The signal space used in this dissertation is the basic Lebesgue space L 2 dened as follows Denition 1.1.1 (L 2 -space) [Lee, 1997] The space L 2 is the set of all real, one sided measurable functions x : R + → R n ; t → x(t) for which

x 2 ∆ = +∞ 0 x(t) 2
2 dt is nite. The quantity x 2 is called the L 2 -norm of

x.

From this denition, an operator norm can be derived: the so-called induced L 2 -norm is dened as follows Denition 1.1.2 (Induced L 2 -norm) [Lee, 1997] Given the linear operator H : L 2 → L 2 , the quantity

H i,2 ∆ = sup x∈L 2 x =0 Hx 2 x 2 = sup x∈L 2 x 2 =1
Hx 2

(1.1) (if nite) is called the induced L 2 -norm of H.

In connection with L 2 -norm space and their associated norms, the L ∞ -norm is given as follows Denition 1.1.3 (L ∞ -norm) [Lee, 1997] the set space L ∞ is called the set of all real, one-sided, measurable functions

x : R + → R n | t → x(t) for which x L∞ ∆ = sup t≥0 x(t) 2 is nite. x L∞ is called the L ∞ -norm of x.
Then, referring to Denition 1.1.2, a similar induced norm fo L ∞ -space is given as follows.

Denition 1.1.4 (Induced L ∞ -norm) [Lee, 1997] Given a bounded input, bounded output linear operator H : L ∞ → L ∞ , the quantity

H i,∞ ∆ = sup x∈L∞ Hx L∞ x L∞ = sup x∈L∞ x L∞ =1 Hx L∞ (1.2) (if nite) is called the induced L ∞ -norm of H.
These various denitions about norms are very useful for Chapter 2 and Chapter 3.

System Spaces

In control theory, dynamical systems are mostly modelled and analysed in the context of the behavioural approach, where physical phenomena are described as a set of possible signal trajectories.

Denition 1.1.5 (Dynamical System) [START_REF] Polderman | Introduction to the Mathematical Theory of Systems and Control[END_REF] A dynamical system G is dened as a triple G = (T, W, B),

(1.3)
with T (often a subset of R or Z) called the time-axis, W a set called the signal space, and B a subset of W T called the behaviour. Then, W T is the standard notation for the set of all maps from T to W.

Remark that this denition cover a large systems classes like Linear Invariant-Time (LTI) systems [START_REF] Polderman | Introduction to the Mathematical Theory of Systems and Control[END_REF], Linear Parameter Varying (LPV) systems 1.4.1, Linear Time Varying LTV-systems [START_REF] Bourlés | Linear Time-Varying Systems[END_REF] and even Nonlinear (NL) systems [Khalil, 2000]. Then, two major properties of dynamical systems namely linearity and time-variance are dened as follows Denition 1.1.6 (Linear Dynamical System) [START_REF] Polderman | Introduction to the Mathematical Theory of Systems and Control[END_REF] A dynamical system G is called linear, if W is a vector space and B is a linear subspace of W T .

For the time invariance property, the following denition is given Denition 1.1.7 (Time-Invariant Dynamical System) [START_REF] Polderman | Introduction to the Mathematical Theory of Systems and Control[END_REF] A dynamical system G is called time-invariant, if T is closed under addition and q τ B = B for all τ ∈ T, where q is the forward time-shift operator, q τ w(t) = w(t + τ ).

A particularly important class of systems which has both of these properties, are the LTI-systems.

State-Space Realisation

For a linear system Σ, a state-space realisation S can be given as

Σ : S =    ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) x(0) = x 0 ∀ t ≥ 0, (1.4)
where the vector x(t) ∈ R n is the state, u(t) ∈ R nu is the input and y(t) ∈ R ny is the output. When n y = n u = 1, the system is Single-Input Single-Output (SISO). It is assumed that A : R → R n×n , B : R → R n×nu , C : R → R ny×n and D : R → R ny×nu are bounded piecewise continuous functions of time. Generally, a solution for state-space realisation is denoted by the so-called state transition matrix.

Denition 1.1.8 (State Transition Matrix) [START_REF] Callier | Linear System Theory[END_REF] Dene Φ : R × R → R n×n (t, t 0 ) → Φ(t, t 0 ) as the unique continuous solution to the homogeneous dierential equation

∂ ∂t Φ(t, t 0 ) = A(t)Φ(t, t 0 ) ∀t ≥ 0, (1.5)
with Φ(t 0 , t 0 ) = I n . The function Φ is called the state transition matrix.

This essential notion is used in Section 2.1.

Linear Algebra

This section presents some denitions and useful properties that occur essentially in matrix algebra.

Linear Fractional Transformation

Linear fractional transformations (LFTs) provide a useful framework to formulate numerous control problems. They are particularly convenient in H ∞ -control and the uncertainty modelling and they are used to provide elementary results regarding feedback interconnections of matrices or systems.

Considering Figure 1.1, the following denition is given: [START_REF] Zhou | Robust and optimal control[END_REF] Let M be a complex matrix partitioned as

M ∆ y 1 u 1 z 1 w 1 (a) Lower-LFT M ∆ u y 2 u 2 z 2 w 2 (b) Upper-LFT Figure 1.1: Linear Fractional Representation LFT Denition 1.2.1
M = M 11 M 12 M 21 M 22 ∈ C (p 1 +p 2 )×(q 1 +q 2 ) , (1.6)
and let ∆ ∈ C q 2 ×p 2 and ∆ u ∈ C q 1 ×p 1 be two other complex matrices. Then, assuming that (I -M 22 ∆ ) = 0, the lower-LFT can be dened with respect to ∆ as the map

F (M, •) : C q 2 ×p 2 → C p 1 ×q 1 ∆ → F (M, ∆ )
such that

F (M, ∆ ) ∆ = M 11 + M 12 ∆ (I -M 22 ∆ ) -1 M 21 .
(1.7)

Assuming that (I -M 11 ∆ u ) -1 = 0, the upper-LFT can be dened with respect to ∆ u as the map

F u (M, •) : C q 1 ×p 1 → C p 2 ×q 2 ∆ u → F u (M, ∆ u ) with F u (M, ∆ u ) ∆ = M 22 + M 21 ∆ u (I -M 11 ∆ u ) -1 M 12 .
(1.8)

In Section 1.4, 2.1 and 2.3, The LFT notion is widely used .

Hamiltonian Matrix and Riccati Equation

The so-called Algebraic Riccati Equation (ARE) is more general equation than the wellknown Lyapunov equation. Similar to this latter which is used in system analysis, the ARE is most useful in control system synthesis.

Denition 1.2.2 [START_REF] Zhou | Robust and optimal control[END_REF] Let A, Q and R be n × n real matrices with Q and R symmetric. Then an algebraic Riccati equation (ARE) is the following matrix equation

A T X + XA + XRX + Q = 0.
(1.9)

where X ∈ R n×n . This Riccati equation can be rewritten as

X -I n H I n X = 0 (1.10) with H ∆ = A R -Q -A T is a 2n × 2n matrix called Hamiltonian matrix.
Then, the following denition is given to introduce the domain Ric Denition 1.2.3 [START_REF] Zhou | Robust and optimal control[END_REF] Giving X, the solution of (1.9), let us dene

Ric : dom(Ric) ⊂ R 2n×2n → R n×n H → X (1.11) 1.2.

Polynomial Criterion

This subsection is given to provide a closeness measure to a rank constraint of matrix.

In other words, how a rank constraint of a matrix can be expressed as function of its characteristic polynomial.

Denition 1.2.4 [Helmersson, 2009] The characteristic polynomial of a matrix Z ∈ R n×n is dened by

det(λI -Z) = n i=0 c i (Z)λ i , (1.12)
where the coecients, c i (Z) are polynomial continuous functions of the elements in Z. Some known coecients are given as

c 0 (Z) = det(Z), c n-1 (Z) = trace(Z), c n (Z) = 1.
More properties of the characteristic polynomial coecients of the positive matrix Z will be presented in following theorem.

Theorem 1: [Helmersson, 2009] Let Z ∈ R n×n be matrix with non-negative real eigenvalues, λ i (Z) ≥ 0, and let c i (-Z) be the coecients of the characteristic polynomial of -Z as dened in (1.12). Then, the following statements are equivalent if n k < n:

(i) c n-n k -1 (-Z) = 0. (ii) rank(Z) ≤ n k . More specically, if λ i are ordered as λ 1 (Z) ≥ λ 2 (Z) ≥ • • • ≥ λ n (Z) ≥ 0. Then,
the following relations hold:

1 n k + 1 n n k +1 λ i (Z) ≤ c n-n k -1 (-Z) c n-n k (-Z) ≤ n n k +1 λ i (Z), (1.13)
or equivalently, Z) .

c n-n k -1 (-Z) c n-n k (-Z) ≤ n n k +1 λ i (Z) ≤ (n k + 1) c n-n k -1 (-Z) c n-n k (-
(1.14)

This theorem is useful and will be used in Chapter 2.

Matrices Vectorisation Operators

Operation to transform matrices into equivalent vectors is presented as follows Denition 1. 2.5 ([Henderson andSearle, 1979, Turkington, 1998]) Let X be a square matrix

X =       x 11 x 12 • • • x 1n
x 21 x 22 . . . . . . . . .

x n1 x n2 • • • x nn      
The vec of the matrix X stacks columns of X one under another in a single column, and its operator inverse is the operator mat, i.e.

vec(X) = x 11 x 21 • • • x n1 x 12 • • • x n2 x 13 • • • x (n-1)1 x 1n • • • x nn T (1.15) X = mat(vec(X)).
(1.16)

The vech of the symmetric matrix X stacks columns of X one under another in a single column starting by a diagonal element of each column.

vech(X) = [x 11 x 21 • • • x n1 x 22 • • • x n2 x 33 • • • x nn ] T .
(1.17)

Let D denotes the transformation matrix mapping the vech of the symmetric matrix X into its vec representation by this relationship:

vec(X) = D vech(X) (1.18) with D ∈ R n 2 × 1 2 n(n+1) . For instance, D takes this form for n = 3,               
1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

               . (1.19)
These tools are of big interest and will be used in Section 2.2

LTI-systems

The class of LTI systems is considered to be the collection of the simplest dynamical systems. Systems belonging to this class have been successfully used in enumerable engineering applications to describe or approximate a wide range of physical phenomena.

LTI-representaion

Let Σ, be an LTI-system of order n with n u inputs and n y outputs and expressed under the state space realisation S given as

Σ : S = ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) ∀ t ≥ 0, (1.20)
where A ∈ R n×n , B ∈ R n×nu , ∈ R ny×n and D ∈ R ny×nu . Then, the state transition matrix dened in (1.5) becomes Φ(t, t 0 ) = e A(t-t 0 ) .

(1.21)

Moreover, LTI-system can be represented with a transfer function. In fact, given U and Y , the Laplace transform of u and y respectively, the transfer function matrix from the input to the output is dened as G(s) considering its initial conditions and equilibrium point to be zero. Then, for a complex variable s, Y (s) = G(s)U (s).

(1.22)

Then, from this denition it is easy to get (1.24)

G(s) = C(sI -A) -1 B + D.

Controllability and Observability

In this subsection two very important concepts in linear system theory are given.

Denition 1.3.1 (Reachability) A state x is said to be reachable (from the origin) if, given x(0) = 0, there exist a nite time interval [0, T ] and an input u(t), t ∈ [0, T ] such that x(T ) = x. If all states are reachable, the system is said to be completely reachable.

Denition 1.3.2 (Controllability) A state x 0 is said to be controllable if there exists a nite time interval [0, T ] and an input u(t), t ∈ [0, T ] such that x(T ) = 0. If all states are controllable, then the system is said to be completely controllable.

Moreover, the dual notion of the Controllability which is the observability is dened as follows.

Denition 1.3.3 (Observability) The LTI-system described by the equations (1.20) or by the pair (C, A) is said to be observable if, for any t 1 > 0, the initial state x(0) = x 0 can be determined from the time history of the input u(t) and the output y(t) in the interval of [0 ; t 1 ]. Otherwise, the system, or (C, A), is said to be unobservable.

Note that an LTI-system has an innity number of state space realisations. From all possible realisations, those which have the smallest state dimension are called minimal.

Denition 1.3.4 (Minimal Realisation) A state space realisation S given in (1.20) is said to be a minimal realisation if the matrix A has the smallest possible dimension.

Then, necessary and sucient conditions to dene a minimal realisation are given as follows Theorem 2: [START_REF] Zhou | Robust and optimal control[END_REF] A state space realisation S given in (1.20) is minimal if and only if (A, B) is controllable and (C, A) is observable.

Finally, the following denition is given to express the conjugate transpose of a transfer function. It will be used thereafter in the next chapters.

Denition 1. 3.5 ([Goddard, 1995]) The conjugate transpose of the transfer function G given in (1.23), is denoted

G ∼ s = -A T C T -B T D T .
(1.25)

When evaluated on the imaginary axis,

[G(ω)] * = [G(-ω)] T = G ∼ (ω).
Note that this notion about minimality is very important in the order reduction procedure. In Section 3.1, 3.2 and 3.3, order reduction methods require the minimality of the high order model.

Norms

Results on norms viewed previously for general systems can be tted for the LTI-systems as follows.

Denition 1.

G H∞ = sup s∈C + σ(G(s)), (1.26)
where σ(G(s)) is the largest singular value of G(s).

For sake of simplicity, . H∞ is denoted by . ∞ in the sequel for this dissertation. It is easy to show that G ∞ = G i,2 for LTI-systems [Goddard, 1995]. Moreover, the induced L ∞ -norm of an LTI-system Σ with the transfer function G is given by

G i,∞ = +∞ 0 |g(t)|dt ∀t ≥ 0 (1.27)
where g : R + → R is the impulse response of Σ. Throw the Laplace transform, the transfer function G is given as

G(s) = +∞ 0 g(t)e -st dt ∀s ∈ C.
(1.28)

Schur Complement Formula

The Schur complement formula is a useful notion and with be used Chapter 2.

Lemma 1: [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] Giving the matrices R ∈ S n , S ∈ S m and G ∈ R n×m , the following conditions are equivalent 1.

R ≺ 0 and S -G T R -1 G ≺ 0 (1.29) 2. S G G R ≺ 0 (1.30)
where S n (resp. S m ) is a cone of n × n (resp. m × m) symmetric matrices.

Bounded Real Lemma

In Chapter 2, the H ∞ -control problem will be discussed. Then, the so-called Bounded Real Lemma BRL will be of big interest. Indeed, in the BRL, H ∞ -norm computation problem can be transferred into a standard Linear Matrix Inequality LMI optimisation formulation.

Lemma 2: [Scherer, 1990, Ankelhed et al., 2011] Consider a continuous-time transfer function T (s) = D + C(sI -A) -1 B of a not necessarily minimal realisation and γ > 0. The following statements are equivalent:

1.

T ∞ < γ and A is Hurwitz.

2. There exists a symmetric positive denite solution P (i.e. P ∈ S n 0 ) to the LMI:

A T P + P A + C T C P B + C T D B T P + D T C D T D -γ 2 I nu ≺ 0.
(1.31)

The inequality (1.31) can be rewritten as

A T P + P A P B B T P -γ 2 I nu + C T D T I ny C D ≺ 0.
(1.32)

Then, multiplying this last inequality by γ -1 and let P 1 = γ -1 P leads to

A T P 1 + P 1 A P 1 B B T P 1 -γI nu + C T D T γ -1 I ny C D ≺ 0.
(1.33)

For later purposes, it is useful to reformulate the inequality in (1.33) to become linear in state-space matrices A, B, C, D. The use of Lemma 1 gives the inequality in (1.33) can be rewritten as

  P 1 A + A T P 1 P 1 B C T B T P 1 -γI nu D T C D -γI ny   ≺ 0 (1.34)
which is an LMI in the matrices A, B, C, D if the matrix P 1 and γ are given. The inequality (1.31) and (1.34) are equivalent for the solution matrix P 1 . The bounded real lemma is of big interest to obtain the results in Section 2.2.

LPV-systems

LPV system is not a recent notion in control theory. In fact, the LPV-framework is appeared rst time in association with the gain scheduling notion. Then, the global behaviour of systems is approximated by a set of xed (frozen) LTI-systems [START_REF] Shamma | Analysis of gain scheduled control for nonlinear plants[END_REF]. The concept of frozen LTI-behaviours is an essential viewpoint on LPV-systems and will be used for semi-active system developed in Chapter 4.

Thereafter, LPV-systems have obtained their own framework structure [START_REF] Packard | Gain scheduling the LPV way[END_REF]. Then, LPV-systems are dened as a large class of dynamical systems for which the future evolution of the state depends on the current state of the system plus some additional signals called parameters. These parameters act as inputs to the system and shape its internal structure [Briat, 2015].

Hence, in this section the mathematical denition of the LPV-systems is given.

Then, according to the mathematical/physical properties or the role and structure of parameters, a non exhaustive classication is suggested.

LPV-modelling

Linear Parameter Varying models are representing systems whose dynamics are known functions of varying parameters. LPV-systems are commonly described by the following the state-space realisation

   ẋ(t) = A(ρ(t))x(t) + B(ρ(t))u(t) y(t) = C(ρ(t))x(t) + D(ρ(t))u(t) x(0) = x 0 ∀ t ≥ 0, (1.35)
where x(t), u(t) and y(t) are the state, the input and the output of the system, respectively. The vector ρ(t) is the exogenous parameter varying vector which is time dependent.

The state-space representation (1.35) can be written as

G(ρ) s = A(ρ) B(ρ) C(ρ) D(ρ) .
(1.36)

Since LPV-framework apparition, many representation ways has been given. LPVmodels can be classied regarding the mathematical properties or the physical meaning of their parameters. The role of the varying parameters can engender another types of LPV-models. In addition, and from a control point of view, the formulation of the LPV-representation gives a third way to encapsulate the LPV-models. Then, a nonexhaustive list of LPV-classes is proposed as follows 1.4.2 LPV-models regarding the Parameter Trajectories By analysing the types of parameters, a rst classication can be given. Then, a variety of LPV-models systems are dened according to their parameters trajectories.

LPV-models with Fast Varying Parameters

LPV-systems with arbitrarily fast varying parameters have parameters in a bounded set P ρ dened as

P ρ = ρ : R + → ∆ ρ t → ρ(t) , (1.37)
where ∆ ρ is a compact and convex polytope (e.g. a box).

LPV-models with Slow Varying Parameters

LPV-systems with slowly varying parameters have parameters in the set P v dened as

P v = ρ : R + → ∆ ρ and ρ(t) ∈ ∆ v ∀ t ∈ R + t → ρ(t) , (1.38)
where ∆ v is a convex and compact polyhedron containing 0.

LPV-models with Piecewise Constant Parameters

LPV-systems with piecewise constant parameters have parameters in the set P pc dened as

P pc = ρ i : R + → ∆ ρ | ρ i piecewise constant, i = 1, . . . t → ρ i (t)
.

(1.39)

LPV-models Representing Switched Models

Given A(ρ) = N i=1 A i ρ i with A i are constant matrices, a switched system, with N modes can be represented as an LPV-system with parameters in the set P ss such as

P ss = ρ i : R + → {0, 1} N | N i=1 ρ i = 1 t → ρ i (t) , (1.40)
where, again, some conditions have to be satised in order to obtain solutions at any time.

LPV-models Representing Periodic Systems

Periodic systems, can be represented as LPV-systems with parameters in the set of T -periodically varying parameters ∀ T > 0

P T = ρ : R + → ∆ ρ | ρ(t) = ρ(t + T ) t → ρ(t)
.

(1.41)

LPV-models regarding the Parameter Function

In this second classication, the parameters role is the main criterion to dierentiate LPV-models. Then, models can approximate nonlinear dynamics, time-varying parts or, even, introduce extra degrees of freedom to an existent LTI-model.

Quasi-LPV Systems

Whenever LPV-systems are considered as an approximation of nonlinear systems, scheduling parameters are function of the system states. This particular type of LPV systems is referred to a quasi-LPV systems, sometimes abbreviated as q-LPV systems.

As an example, the following scalar nonlinear system ẋ(t) = x(t) 2 ∀t > 0,

(1.42) can be represented as

ẋ(t) = ρ(t)x(t) ∀t > 0, (1.43) with ρ(t) ∆ = x(t) ∈ R ∀t ∈ R + .

LPV-models with Intrinsic Parameters

Parameters can also be used to embed time-varying components in order to use LPVgain-scheduling techniques for controlling the original system. For instance, the Linear Time-Varying (LTV) system ẋ(t) = [-a + b sin(ωt)] x(t), ∀t > 0.

(1.44) can be represented

ẋ(t) = [a + bρ(t)] x(t) ∀t > 0, (1.45)
where ρ(t)

∆ = sin(ωt) ∈ [-1, 1].

LPV-models with Articial Intrinsic Parameters

Extrinsic parameters are mostly involved when design is the underlying objective. These articial parameters may then be used in the control law in order to shape its structure according to objectives. To illustrate this, let us consider the following LTI system

ẋ(t) = x(t) + u(t) y(t) = x(t) ∀ t ≥ 0, (1.46)
where x(t) ∈ R and u(t) ∈ R are the state and the control input, respectively. It is proposed to determine a control law such that

• the output y tracks a dierentiable reference signal r, and

• the bandwidth of the closed-loop system can be adjusted in real-time.

The control law

u(t) = -[1 + ρ(t)] x(t) + ρ(t)r(t), ρ(t) ≥ 0, (1.47)
where ρ is an external parameter yields the closed-loop system

ẋ(t) = -ρ(t) [x(t) -r(t)] .
(1.48)

LPV-models regarding the System Representations

The LPV-system representations can have many forms. Some formulations are of particular importance when it comes to analysis and synthesis. A big part of LPV-systems can be written under one of the following ways

LPV-ane Systems

By an ane LPV-system we mean an LPV-system whose matrices are ane functions of the scheduling parameter. Then, the following denition is given Denition 1.4.1 An LPV-system given in (1.36) is said to be ane if the system matrices are represented by

A(ρ) = A 0 + N i=1 A i ρ i , B(ρ) = B 0 + N i=1 B i ρ i , C(ρ) = C 0 + N i=1 C i ρ i , D(ρ) = D 0 + N i=1 D i ρ i .
(1.49)

ρ i is the i th element of ρ and A i , B i , C i , D i are constant matrices.

LPV-polytopic Systems

The polytopic form gives a convenient framework for representing and analysing LPVsystems (see [START_REF] Apkarian | Self-scheduled H ∞ linear parameter-varying systems[END_REF]). Polytopic systems are expressed as time varying convex combination of LTI-systems. This property can be operated to obtain stability or stabilisation results using convex optimisation techniques.

Denition 1. 4.2 [Poussot-Vassal, 2008] An LPV-system given in (1.36) is said to be polytopic if the system matrices are represented by

A(ρ) = N 1 α i (ρ)A i , B(ρ) = N 1 α i (ρ)B i , C(ρ) = N 1 α i (ρ)C i , D(ρ) = N 1 α i (ρ)D i , (1.50)
where N i=1 α i (ρ) = 1, α i (ρ) ≥ 0 and A i , B i , C i , D i are constant known matrices (that represent the system evaluated at each vertex). Then, the LPV-system can be represented by an equivalent polytopic form such that

G(ρ) s = A(ρ) B(ρ) C(ρ) D(ρ) = N i=1 α i (ρ) A(ω i ) B(ω i ) C(ω i ) D(ω i ) , (1.51)
where ω i are the vertices of the polytope formed by all the extremities of each varying parameter ρ ∈ P ρ , and where α i (ρ) are dened as,

α i (ρ) = k=1 |ρ k -(ω i ) k | k=1 (ρ k -ρ k ) , (1.52)
where (ω i ) k is the k th component of the vector (ω i ) dened as,

(ω i ) k = {ρ k : ρ k = ρk if (ω i ) k = ρ k or ρ k = ρ k otherwise}.
(1.53) N = 2 is the number of vertices of the polytope formed by varying parameters.

The polytopic systems are of interest in controller design and implementation. As in this case, the LPV-system is a convex hull of a nite number of LTI-systems, it allows to solve a nite number of LMI problems (see [START_REF] Gahinet | Ane parameter-dependent lyapunov functions and real parametric uncertainty[END_REF], Scherer, 2004]) to nd a global LPV-controller (which is also a convex hull of a nite number of local LTI-controllers).

LPV-LFT-systems

LPV-systems given under a LFT form are systems expressed as interconnection of two subsystems (Figure 1.2).

θ(ρ)

A B C D z w Figure 1.2: LPV-LFT-system
The main idea of this LFT form is to rewrite a complex system under an interaction between a simple linear system and complicated and annoying one.

The rst simple nice part should have convenient properties such as linearity, time invariance, etc. The other complicated part, contains non-linearities, time varying terms, ect. This formulation has been emphasised in numerous papers. In fact, in [Packard, 1994, Apkarian andGahinet, 1995] a gain scheduling controllers for LPV-LFT-form systems are proposed. Then, recently in [Scherer, 2001, Scherer, 2012], LPVcontroller based on dynamics D-scales is developed.

Denition 1.4.3 An LPV-system given in (1.36) is said to be in LFT-form, such as the one depicted in Figure 1.2, state-space representation can be expressed as

   ẋ(t) = Ax(t) + Bw(t) z(t) = Cx(t) + Dw(t) w(t) = θ(ρ(t))z(t) ∀ t ≥ 0, (1.54) with I -θ(ρ)D is invertible for all ρ ∈ ∆ ρ .
This LFT-form can be rewritten according to the following proposition Proposition 2: [Briat, 2015] The LPV-LFT-form system given in (1.2) is equivalent to the following LPV-

systems ẋ = A -B [I -θ(ρ)D] -1 θ(ρ)C x (1.55) ẋ = A -Bθ(ρ) [I -Dθ(ρ)] -1 C x (1.56)
In this chapter a non-exhaustive notions were introduced. This choice to focus only on a specic mathematical background is taken in the aim keep the clarity and the readability of the thesis.

Introduction

Robustness notion plays an essential role in system and control theory nowadays. The emergence of this notion is basically motivated by the huge demand on performance for new control laws. Optimisation approaches based on H ∞ and H 2 theory appear of a big importance to t the increasing specied requirements. H ∞ -controllers are knowing a big raise last twenty years whether in theoretical development or in practical issue. In addition, the plant dynamics may vary to such an level that a single linear model proves insucient to capture the essential features of the plant. Then, a series of models describing the plant behaviour at a number of operating points is required.

Thus, H ∞ -controllers based on LPV-models appeared and seem to be of a great interest.

Nevertheless, the controllers stabilising complex models and achieving a specied high performance are generally at least as complex as the model itself. The complexity of the obtained high-order controllers restricts their real-time operation in embedded systems and increases their implementation costs. In fact, low-order controller requires less complicated and more easy available hardware to understand, to maintain and to implement in the real world. In this chapter, the H ∞ -control theory is introduced for LTI-systems rst. Indeed, after introducing the stability conditions, the LTI-H ∞ - control design problem is formulated. Then, a survey about the existing methods to design an H ∞ -controller is given. Then-after, the H ∞ problem is reviewed to design xed-order controllers in Section 2.2. The extension to the LPV-case is then developed in Section 2.3 and a special interest is allocated to the polytopic models.

2.1 H ∞ -LTI-Control Design

LTI-Systems Stability

Stability notion is fundamental for dynamical systems analysis in control theory. At rst sight this concept seems to be natural. However, closer examination shows that a multitude of denitions can be given to stability [Lyapunov, 1992, Khalil, 2000, Desoer and Vidyasagar, 2009]. In this Section, a no-exhaustive overview on the question is given. In fact, for stability analysis, there are two fundamental theorems. The rst one gives sucient conditions for asymptotic (exponential) stability. The second one called the small gain theorem is the pillar of the input/output stability theory.

Global Asymptotic Stability

General Stability Results Stability of general systems cannot be analysed by looking at the explicit solutions since they are, most of the time, dicult or even impossible to compute. Lyapunov theory allows to analyse stability implicitly from the expression of the dynamical system through the use of a Lyapunov function.

Theorem 3: Lyapunov's Stability Theorem [Lyapunov, 1992, Khalil, 2000] Let us consider the general dynamical system

ẋ(t) = f (x(t)), x(0) = x 0 , ∀t > 0 (2.1) having x * = 0 as equilibrium point, i.e. f (x * ) = 0, let D ⊂ R n be a domain containing x * and V : D → R be a continuously dierentiable function such that    V (x * ) = 0, V (x) > 0 in D\{x * }, V (x) ≤ 0 in D. (2.2)
Then, x * is a stable equilibrium point and V is called a Lyapunov function for

(2.1). Moreover, if V (x) < 0 in D\{x * } (2.3)
then x * is an asymptotically stable equilibrium.

LTI-Systems Case Whenever LTI-systems are considered, The stability analysis become easer. In fact, necessary and sucient Lyapunov conditions for stability can be easily stated Theorem 4: [Briat, 2015] Let us consider the LTI-system Σ with the state-space representation (1.20). Then, the following statements are equivalent

• The system (1.20) is globally asymptotically stable.

• The system (1.20) is globally exponentially stable.

• The matrix A is Hurwitz, i.e. (λ i (A)) < 0.

• There exists a matrix P ∈ S n 0 such that the Lyapunov inequality

A T P + P A ≺ 0, (2.4)
holds.

• There exist matrices P, Q ∈ S n 0 such that the Lyapunov equation

A T P + P A + Q = 0, (2.5)
holds.

Input-Output Stability

If you ask anyone to give his own denition of stability, there is a big chance to have this answer: A system is stable if the output signal energy remains bounded for a bounded input signal energy. Mathematically speaking, this approach was derived from the Operator Theory [Kato, 1990]. It is noted that rst theoretical development on input-output stability notion are relatively recent [START_REF] Desoer | Feedback Systems[END_REF]. In [Vidyasagar, 2002, Chapter 6], a synthetic view about input-output stability is given.

Unfortunately, there is no general result that allow a direct input-output stability analysis, except if all system trajectories are computed (Lyapunov approach). Then, this approach is complementary to the previous one and the aim is to make easier the stability analysis of the interconnected systems. Thus, the Small Gain Theorem is given as follows

∆(s)

M (s) Theorem 5: Small Gain Theorem [START_REF] Zhou | Robust and optimal control[END_REF] Suppose M ∈ RH ∞ (i.e. with M a stable transfer matrix) and let γ > 0. Then, the interconnected system shown in Figure 2.1 is well-posed and internally stable for all ∆ ∈ RH ∞ (i.e. stable transfer matrix) with ∆ ∞ < 1/ γ if and only if M ∞ < γ.

w 1 w 2 + y 1 + y 2
A typical application of this theorem is when considering M like a known transfer function and ∆ as a unknown transfer function. Then, small gain theorem is classically used to derive conditions for Robust Stability with respect to the uncertainty models (represented by ∆). (2.6)

Then, the goal is to nd a controller K that compensate the inuence of exogenous inputs w on the controlled outputs z through the information given by the measured signals y. This objective is achieved when the closed-loop norm of the transfer from w to z namely T zw is minimised. From a mathematical point of view, H ∞ -suboptimal problem is to nd a controller K that stabilise the closed-loop system internally and guarantee T zw ∞ < γ.

(2.7)

where γ is pre-set attenuation level and T zw is given according to (1.7) as T zw (s) = F (P, K)(s) = P 11 (s) + P 12 (s)K(I -P 22 (s)K(s)) -1 P 21 (s).

(2.8)

The optimal H ∞ -problem corresponds to the minimisation of γ. This minimal value γ opt can be approximated by dichotomic search algorithm.

In practice, the plant P consists of the system transfer function G and weighting functions W i and W o associated the exogenous input w and the controlled output z respectively as shown in Figure 2.3. These weighting functions allow to emphasise some transfers in certain frequency ranges. Then, several frequency templates can be dened as the inverse of the weighting functions. Thus, required performance like disturbance or noise rejection can be expressed threw W i and W o . It is well known that the major drawback of the H ∞ -control design is the order of

G K W i W o w e P Figure 2.3: Augmented System
obtained controllers which is typically at least of the same order as the plant.

Next section presents a solution to the H ∞ -problem based on Riccati equations.

H ∞ -problem Resolution

Various methods are developed to solve the standard H ∞ -control design problem.

Among them, two methods are widely used: the rst one is based on the resolution of two Riccati equations. Th second one deals with some Linear Matrix Inequalities LMIs.

Riccati-based Solution One of the common tool to resolve this H ∞ -problem is based on the resolution of Riccati equations. Suppose K is an n K -th order controller which stabilises the closed-loop system and the n-th order generalised plant P is given by P = P 11 P 12

P 21 P 22 s =   A B 1 B 2 C 1 D 11 D 12 C 2 D 21 D 22   .
(2.9)

where

A ∈ R n×n , B 1 ∈ R n×nw , B 2 ∈ R n×nu , C 1 ∈ R nz×n , C 2 ∈ R ny×n , D 11 ∈ R nz×nw , D 12 ∈ R nz×nu , D 21 ∈ R ny×nw and D 22 ∈ R ny×nu .
The following assumptions are made

A1: (A, B 2 ) is stabilisable and (C 2 , A) is detectable.
A2: D 12 has full column rank and D 21 has full row rank.

A3:

A -ωI n B 2 C 1 D 12 has full column rank for all ω ∈ R.

A4:

A -ωI n B 1 C 2 D 21 has full row rank for all ω ∈ R.
T zw is the transfer function from w to z, dened as (2.8). According the Youla parametrisation theory [START_REF] Youla | Modern wiener-hopf design of optimal controllerspart ii: The multivariable case[END_REF], Vidyasagar, 1985], all rational internally stabilising controllers K satisfying F (P, K) ∞ < γ are given by K

(s) = F (M ∞ , Q)(s) for arbitrary Q ∈ RH ∞ such that Q ∞ < γ, where M ∞ = M 11 M 12 M 21 M 22 s =    Â B1 B2 Ĉ1 D11 D12 Ĉ2 D21 D22    .
All relevant parameters Â, B1 , B2 , Ĉ1 , Ĉ2 , D11 , D12 , D21 , D22 can be found in [START_REF] Zhou | Robust and optimal control[END_REF]. For a simple case, where

G = G 11 G 12 G 21 G 22 s =   A B 1 B 2 C 1 0 D 12 C 2 D 21 0   .
The H ∞ -solution involves the following two Hamiltonian matrices:

H ∞ = A γ -2 B 1 B T 1 -B 2 B T 2 -C T 1 C 1 -A T , J ∞ = A T γ -2 C T 1 C 1 -C T 2 C 2 -B 1 B T 1 -A .
Theorem 6: [Glover andDoyle, 1988, Zhou et al., 1996] There exists an admissible controller such that T zw (s) < γ i the following three conditions hold

• H ∞ ∈ dom(Ric) and X ∞ = Ric(H ∞ ) 0. • J ∞ ∈ dom(Ric) and Y ∞ = Ric(J ∞ ) 0. • ρ(X ∞ Y ∞ ) < γ 2 .
Moreover, when these conditions hold, one such controller is

K s = A ∞ -Z ∞ L ∞ F ∞ 0 (2.10)
where

A ∞ = A + γ -2 B 1 B T 1 X ∞ + B 2 F ∞ + Z ∞ L ∞ C 2 , F ∞ = -B T 2 X ∞ , L ∞ = -Y ∞ C T 2 , Z ∞ = (I -γ 2 Y ∞ X ∞ ) -1 .
Furthermore, the set of all admissible controllers such that T zw is given by M ∞ such that

M ∞ = M 11 M 12 M 21 M 22 s =   A ∞ -Z ∞ L ∞ Z ∞ B 2 F ∞ 0 I nu -C 2 I ny 0   (2.11) That is, K = F (M ∞ , Q) = M 11 + M 12 Q (I -M 22 Q) -1 M 21 , (2.12) where Q ∈ RH ∞ and Q ∞ ≤ γ.
Generally speaking, Q can be chosen to satisfy additional performance objectives. However, how to nd such Q is a challenging problem and still an active research topic. In most cases,

Q = 0 is chosen resulting in the so-called central H ∞ -controller K c = F (M ∞ , 0) = M 11 .
LMI-based Solution Alternative to the Riccati-based solution for the H ∞ -control problem is the LMI-based approach. In fact, necessary and sucient conditions for the existence of admissible controllers is expressed in terms of LMIs. Consequently, Assumptions (A2) to (A4) may be omitted. Then, an advantage of this framework is that no rank assumptions are required on D 12 and D 21 and hence singular problems may be solved using the same machinery.

Furthermore, for the H ∞ -optimal problem, γ opt can be found directly without iterative procedure like in Riccati-based solution.

Theorem 7: [Gahinet, 1994, Gahinet and[START_REF] Gahinet | [END_REF] Consider the plant of equation ( 2.9) and make assumption (A1) (Section 2.1. (C 2 , D 21 ) respectively.

There exists a solution to the H ∞ -problem if and only if there exist symmetric matrices R and S satisfying the following system of LMIs

N T R 0 0 I nw     AR + RA T RC T 1 B 1 C 1 R -γI nz D 11 B T 1 D T 11 -γI nw     N R 0 0 I nw ≺ 0 (2.13) N T S 0 0 I nz     AS + SA T SB 1 C T 1 B T 1 S -γI nw D T 11 C 1 D 11 -γI nz     N S 0 0 I nz ≺ 0 (2.14) R I n I n S 0 (2.15)
For a given γ, equations (2.13) to (2.15) are ane in R and S, and the set of pairs R and S is convex. Methods to construct an admissible H ∞ -controller K given feasible R and S can be found in [Gahinet andApkarian, 1993, Gahinet, 1994].

Robustness Analysis

Robustness analysis aims at providing an accurate sensitivity estimation of the several input/output transfers to the variability on the parameters. Various tools can be used to make this analysis.

In this study, µ-analysis [Zhou andDoyle, 1998, Skogestad andPostlethwaite, 2005] is chosen to study the robustness via the measurement of the structured singular value µ.

On control design, the main property to check on a closed-loop system is the closed-loop stability.

The goal is to keep this stability although the presence of uncertainties on parameters (due to neglected dynamics, ill-identication . . . ) and is called the Robust Stability (RS). The stability robustness is analysed by measuring the lowest uncertainty that destabilises the system.

The performance of the nominal closed-loop system could also be analysed through the µ measurement by neglecting the uncertainties when evaluating µ. Then, the margins of the Nominal Performance(NP) are obtained.

Finally, the Robust Performance (RP) are analysed to check if the system satises the performance specications for the perturbed plant.

Let the transfer matrix N partitioned as follow (Figure 2.4 ):

N = N zv N zw N ev N ew
where w and e are the exogenous input and the controlled output respectively. ∆ r is the uncertain matrix that could contain the model uncertainties, parametric uncertainties, gain uncertainties, phase shifting or the delay on the inputs/outputs of the nominal system. Then, the closed loop transfer matrix is:

∆ N ∆ r ∆ f z ∆ v ∆ w e
T ew = N ew + N ew ∆(I -N zv ) -1 N zw (2.16)
As we consider structured uncertainties, a µ-analysis step is performed to study the robustness:

N ew is the nominal closed-loop transfer matrix. It is used to analyse the NP of the system.Note that if N ew is stable, the instability in (2.16) may only come from (I -N zv ) -1 N zw . Then, the N zw is analysed to study the RS. By the way, the RP is analysed by evaluating the whole N block.

Consider the set of matrices: ∆ that have the same structure as ∆ and dened as:

∆ : {∆ = diag(∆ 1 , . . . , ∆ q , δ 1 I r 1 , . . . , δ r I rr , ε 1 I c 1 , . . . , ε c I cc )} where: ∆ ∈ C k×k , ∆ i ∈ C k i ×k i , δ ∈ R, ε i ∈ C.
Then, the structured singular value µ of N through the set ∆ is dened as:

µ ∆ (N ) = ( inf ∆∈∆ {σ(∆), det(I -∆N ) = 0}) -1
Theorem 8: [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF] Assume that the nominal system N ew and the perturbations ∆ are stable. Then, the feedback system is stable for all allowed perturbations ∆ such that

∆ ∞ < 1/µ if and only if: ∀ω ∈ R, µ ∆ (N zv (ω)) ≤ µ
For all perturbations, the following rules should be veried ∀ω:

NP ⇐⇒ σ(N ew (ω)) = µ ∆ f (N ew (ω)) < 1 RS ⇐⇒ σ(N zv (ω)) = µ ∆ r (N zv (ω)) < 1 RP ⇐⇒ σ(N (ω)) = µ ∆ (N (ω)) < 1 (2.17)
In practice, the structured singular value cannot be calculated explicitly, so that the method consists in nding an upper bound and a lower bound, as closed as possible to

µ: µ ∆ (N (ω)) ∈ [µ lb ∆ (N (ω)); µ ub ∆ (N (ω))] µ lb ∆ (N (ω)): µ-lower bound, µ ub ∆ (N (ω)): µ-upper bound. 2.2 Fixed-Order H ∞ -Controller Design
The H ∞ -synthesis is an important tool in robust control design. First techniques are based on solving Riccati equations [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Since that, the robust design tools became much easier to use and gained popularity. Thereafter, LMIs were found to be a suitable tool for solving these kinds of problems by using reformulations of the Riccati equations [START_REF] Gahinet | A linear matrix inequality approach to H ∞ control[END_REF]. The high requirements for robustness and for disturbance rejection in control design, result controllers of very high order, which complicates implementation. To overcame this drawback, constraint on the maximum order of the controller is set to be lower than the order of the plant. However, the problem is no longer convex and it is then relatively hard to solve. This motivates the development of ecient algorithms that can solve these kinds of problems.

First, let P be the plant introduced in (2.9) and dene K the linear H ∞ -controller denoted by K :

ẋK (t) = A K x K (t) + B K y(t) u(t) = C K x K (t) + D K y(t)
∀t ≥ 0, (2.18) where x K (t) ∈ R n K is the state vector of the controller. The state-space realisation K is given as

K(s) = D K + C K (sI -A K ) -1 B K , (2.19) with A K ∈ R n K ×n K , B K ∈ R n K ×ny , C K ∈ R nu×n K , D K ∈ R nu×ny .
Then, a (not necessarily minimal) realisation of the closed-loop transfer function from w to z is obtained as (Figure 2.2) (2.20) where

T zw (s) = D CL + C CL (sI -A CL ) -1 B CL ,
A CL = A + B 2 D K C 2 B 2 C K B K C 2 A K , B CL = B 1 + B 2 D K D 21 B K D 21 , C CL = C 1 + D 12 D K C 2 D 12 C K , D CL = D 11 + D 12 D K D 21 .
Let dene Θ, a matrix that contains all controller parameters

Θ = A K B K C K D K ∈ R (n K +nu)×(n K +ny) .
(2.21)

Before introducing the way to design a xed order controller, the following lemma is useful for afterwards. Lemma 3: [Gahinet and Apkarian, 1994] Given a symmetric matrix ψ ∈ S M and two matrices P and Q composed of M columns, consider the problem of nding some matrix Z of compatible dimensions such that

ψ + P T Z T Q + Q T ZP ≺ 0 (2.22)
Denote by W P , W Q any matrices whose columns form bases of the null spaces of P and Q respectively, i.e. W P = P ⊥ and W Q = Q ⊥ respectively. Then, (2.22) is solvable for Z if and only if

W T P ψW P ≺ 0 W T Q ψW Q ≺ 0 (2.23)

Ankelhed Method

The approach proposed by Ankelhed is based on formulating the constraint on the maximum order of the controller as a polynomial (or rational) equation. The problem is then solved by reformulating it as an optimisation problem using a minimisation algorithm.

Problem Formulation

Let the matrix X a ∈ S n+n K and its inverse be partitioned as

X a = X X 2 X T 2 X 3 , and X -1 a = Y Y 2 Y T 2 Y 3 (2.24)
where X, Y ∈ S n 0 , Then, insert X a and the closed loop system matrices A CL , B CL , C CL , D CL into the inequality in (1.34). After some rearrangements we get the following matrix inequality.

    XA + A T X A T X 2 XB T 1 C T 1 X T 2 A 0 X T 2 B 1 0 B T 1 X B T 1 X 2 -γI nz D T 11 C 1 0 D 11 -γI nw     Ψ Xa +     XB 2 X 2 X T 2 B 2 X 3 0 0 D 12 0     P D K C K B K A K F C 2 0 D 21 0 0 I n K 0 0 Q T + C 2 0 D 21 0 0 I n K 0 0 T Q D K C K B K A K F T     XB 2 X 2 X T 2 B 2 X 3 0 0 D 12 0     T P T ≺ 0 (2.25)
The matrix inequality in (2.25) is bilinear in the controller variables, A K , B K , C K , D K and the matrices X, X 2 , X 3 . Then, in order to apply Lemma 3, the orthogonal complements W P and W Q need to be derived from P and Q respectively. Note that P in (2.25) can be factorised as

P =     XB 2 X 2 X T 2 B 2 X 3 0 0 D 12 0     = X a 0 0 I (nw+nz)     B 2 0 0 I n K 0 0 D 12 0     (2.26)
and an orthogonal complement W P can now be constructed as

W P =     B 2 0 0 I n K 0 0 D 21 0     ⊥ X -1 a 0 0 I (nw+nz) .
(2.27)

Then, after some rearrangements, the inequality (2.25) is now equivalent to the two

LMIs N T X 0 0 I nw     XA + A T X XB 1 C T 1 B T 1 X -γI nz D T 11 C 1 D 11 -γI nw     N X 0 0 I nw ≺ 0, (2.28a) N T Y 0 0 I nz     AY + Y A T Y C T 1 B 1 C 1 Y -γI nw D 11 B T 1 D T 11 -γI nz     N Y 0 0 I nz ≺ 0, (2.28b)
where N X and N Y denote orthonormal bases of the null spaces of (C 2 , D 21 ) and B T 2 , D T 12 respectively. Now, the LMIs (2.28) are coupled by the relation of X and Y through (2.24), which can be simplied after using the following lemma.

Lemma 4: [Packard, 1994] Suppose X, Y ∈ S n 0 and n K being a nonnegative integer. Then, the following statements are equivalent

1. There exist X 2 , Y 2 ∈ R n×n K and X 3 , Y 3 ∈ R n K ×n K such that X a = X X 2 X T 2 X 3 , and X -1 a = Y Y 2 Y T 2 Y 3 .
(2.29)

2. The following inequalities hold.

X I

n I n Y 0 and rank X I n I n Y ≤ n + n K .
(2.30)

Finally, from all previous development, the solvability conditions for the H ∞ -problem can be formulated as follows

Lemma 5

The problem of nding a linear controller such that the closed-loop system T zw is stable and such that T zw ∞ < γ, is solvable if and only if there exist positive dened matrices X, Y ∈ S n 0 , which satisfy

N T X 0 0 I nw     XA + A T X XB 1 C T 1 B T 1 X -γI nz D T 11 C 1 D 11 -γI nw     N X 0 0 I nw ≺ 0, (2.31a) N T Y 0 0 I nz     AY + Y A T Y C T 1 B 1 C 1 Y -γI nw D 11 B T 1 D T 11 -γI nz     N Y 0 0 I nz ≺ 0, (2.31b) X I n I n Y 0, (2.31c) rank(XY -I n ) ≤ n K , (2.31d)
where N Y and N X denote orthonormal bases of the null spaces of B T 2 , D

X and Y Searching

It could be desirable to replace the rank constraint in (2.31d) with a smooth function in order to be able to apply gradient methods for optimisation. To do that, the following lemma issued from Theorem 1 is used Lemma 6: [Ankelhed, 2011] Assume that the inequality

X I n I n Y 0 (2.32) holds. Let det(λI -(I -XY )) = n i=0 c i (I -XY )λ i = λ n + c n-1 (I -XY )λ n-1 + • • • + c 1 (I -XY )λ +c 0 (I -XY ), (2.33)
be the characteristic polynomial of (I -XY ) where the functions c i (I -XY ) are its coecients. Then the following statements are equivalent if n K < n:

1. rank(XY -I) ≤ n K . 2. c n-n K -1 (I -XY ) = 0.
Additionally, all coecients are non-negative, i.e.

c i (I -XY ) ≥ 0, ∀i.

(2.34)

Combining Lemma 5 and Lemma 6, the resolvability conditions to nd a n K -order controller can be given in next theorem.

Theorem 9: [Ankelhed, 2011] The problem of nding an H ∞ -controller of order n K < n such that the closed loop system T zw is stable and

T zw ∞ < γ is equivalent to nd X, Y ∈ S n 0 which satisfy N T X 0 0 I nw     XA + A T X XB 1 C T 1 B T 1 X -γI nz D T 11 C 1 D 11 -γI nw     N X 0 0 I nw ≺ 0 (2.35a) N T Y 0 0 I nz     AY + Y A T Y C T 1 B 1 C 1 Y -γI nw D 11 B T 1 D T 11 -γI nz     N Y 0 0 I nz ≺ 0 (2.35b) X I n I n Y 0 (2.35c) c n-n K -1 (XY -I) c n-n K (XY -I) = 0 (2.35d) c n-n K (XY -I) = 0 (2.35e)
Where columns of N X and N Y denote any bases of null spaces of (C

minimise γ subject to q(X, Y ) = 0, and (γ, X, Y ) ∈ X (2.36) where q(X, Y ) = c n-n K -1 (-Z) c n-n K (-Z)
, and X is the convex set dened by the LMIs in Theo- rem 9.

This problem can be formulated in order to be solved using a Partial Augmented Lagrangian. Then, the constraint is relaxed and added to the objective function as follows minimise λ + λq(X, Y ) + µ 2 q 2 (X, Y ). subject to (γ, X, Y ) ∈ X, (2.37) where λ is a Lagrangian multiplier and µ is a penalty multiplier. The augmented Lagrangian method was proposed independently by [Powell, 1969, Hestenes, 1969], this algorithm is well-known in the context of mathematical programming with classical equality and inequality constraints and has well-established convergence properties. The global convergence of this algorithm is proved in [START_REF] Conn | A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds[END_REF], Noll et al., 2004].

For this Lagrangian equation, the augmentation part is added partially by only the equality constraint while the LMIs remain unchanged in order to keep the structure of the problem. The solution to the original problem (2.36) is obtained by iteratively solving an approximation of (2.37) for a sequence of increasing values of µ.

More details on augmented Lagrangian methods can be found in [Bertsekas, 1982, Nocedal andWright, 2006]. Remark

Using notations x =   vech(X) vech(Y ) γ   and γ = d T x where d is denoted as d = [0 • • • 0 • • • 0 1] T , the optimisation problem (2.37) can be rewritten as minimise Φ q (x, λ, µ) subject to x ∈ X, (2.38)
where

Φ q (x, λ, µ) = d T x + λq(x) + µ 2 q 2 (x).
(2.39)

This reformulation of the original initial problem is also non-convex (q is non-convex) and a resolution using a sequence of SDPs is not possible. However, X is a convex set which make the problems less dicult to solve.

One possible approach to tackle this problem is to approximately solve (2.38) for a sequence of increasing values of µ k using Newton's method. However, since the optimisation problem in (2.38) still includes constraints, we must make sure that the next point also satises the constraints. Then, a search direction is computed as follows Search Direction Computation One possible approach is to approximate the objective function Φ q (x + dx, λ, µ) by a quadratic function related to the three rst terms in the Taylor series around the point x. Similarly to what is done in regular Newton methods, we intend to nd a step direction dx that minimises this second order model, but the dierence is that we also require that x + dx ∈ X, i.e. that the next point also lies in the feasible set. This problem can be formulated as

arg min dx ∇ x Φ q (x, λ, µ) T dx + 1 2 dx T H δ (x, λ, µ)dx subject to x + dx ∈ X.
(2.40)

where ∇ x Φ q (x, λ, µ) is the gradient of the objective function Φ q (x, λ, µ), and H δ (x, λ, µ) is a positive denite approximation of the Hessian of the objective function.

From now, the problem (2.40) can be solved using Yalmip [Lofberg, 2004] with SDPT3 [START_REF] Toh | SDPT3 -a Matlab software package for semidenite programming[END_REF]. Remark Gradient and Hessian of the Objective Function Φ q [START_REF] Ankelhed | A partially augmented lagrangian method for low order H ∞ controller synthesis using rational constraints[END_REF] in order to obtain the gradient and approximation of the Hessian of Φ q , lets dierentiate Φ q (x, λ, µ) with respect to x. Then,

∇ x Φ q (x, λ, µ) = d + λ∇ x q(x) + µq(x)∇ x q(x) ∇ xx Φ q (x, λ, µ) = (λ + µq(x))∇ 2 xx q(x) + µ∇ x q(x)∇ T x q(x)
where the gradient ∇ x q and Hessian ∇ 2 xx q of the quotient q(x) =

c n-n K -1 (x) c n-n K (x) are given as follows ∇ x q(x) = 1 c n-n K -1 ∇ x c n-n K -1 - c n-n K -1 c 2 n-n K ∇ x c n-n K ∇ 2 xx q(x) = 1 c n-n K ∇ 2 xx c n-n K -1 - c n-n K -1 c 2 n-n K ∇ 2 xx c n-n K + 2c n-n K -1 c 3 n-n K (∇ x c n-n K ∇ T x c n-n K ) -1 c 2 n-n K (∇ x c n-n K -1 ∇ T x c n-n K ) + (∇ x c n-n K ∇ T x c n-n K -1 )
Since the constraint function q is non-convex, the Hessian is not always positive denite which in turn might lead to that H(x, λ, µ) = ∇ xx Φ q (x, λ, µ) is not necessarily positive denite, which has to be dealt with. Two common ways are to either use Newton methods in which the Hessian is convexied or to use Trust-region methods where the non-convexity is dealt with by optimising over a limited region in each iteration.

The authors of [START_REF] Apkarian | Fixed order H ∞ control design via a partially Augmented Lagrangian method[END_REF] advice against using Trust-region methods since the complexity of such a method is too large in this case. Therefore, the Hessian ∇ xx Φ q (x, λ, µ) could be convexied as follows.

Hessian Modications [START_REF] Ankelhed | A partially augmented lagrangian method for low order H ∞ controller synthesis using rational constraints[END_REF] One method which can be used to convexify the exact Hessian ∇ xx Φ q (x, λ, µ) is a modied indenite symmetric factorisation. To carry out this modication, the indenite symmetric factorisation (LDL decomposition) is given as ∇ GN xx Φ q = P T LDL T P , where L is a lower triangular matrix, P is a permutation matrix and D is a block diagonal of size (1 × 1) or (2 × 2). (2.42)

The matrix E is diagonal and is dened by .43) where δ = 10 -4 ∇ 2 xx Φ q ∞ , and A ∞ denotes the largest row sum of A. With these notation the matrix D m takes the minimal matrix in the Frobenius norm such that δ ≤ D + D m .

E ii = 0, if Dii ≥ δ δ -Dii , if Dii < δ i = 1, 2 . . . ( 2
1. For gradient and Hessian detailed computation, see Appendix A.1 2. The approach to compute X and Y can be summarised as Algorithm 4 in Appendix A.1 Remark 2.2.1.3 Recovering the matrix X a from X and Y Assume that we have found X, Y ∈ S n 0 that satisfy (2.31). We now wish to construct X a such that 2.24 holds. First note the equality

X -1 a = Y Y 2 Y T 2 Y 3 = (X -X 2 X -1 3 X T 2 ) -1 -X -1 X 2 (X 3 -X T 2 X -1 X 2 ) -1 -X -1 3 X T 2 (X -X 2 X -1 3 X T 2 ) -1 (X 3 -X T 2 X -1 X 2 ) -1 (2.44)
which is veried by multiplying the expression in (2.44) by the matrix

X a = X X 2 X T 2 X 3
from the left. Equalling the upper left blocks in (2.44), the following equality must hold.

X -Y -1 = X 2 X -1 3 X T 2 .
(2.45)

Now we intend to nd X 2 ∈ R n×n K and X 3 ∈ R n K ×n K that satisfy the equality in (2.45). Perform an upper Cholesky factorisation of X and Y such that

X = R T X R X and Y = R T Y R Y . Then we have that R T X R X -R -1 Y R -T Y = X 2 X -1 3 X T
R Y R T X R X -I = R Y X 2 X -1 3 X T 3 R T Y . (2.47) Then, use a singular value decomposition R Y R T X = U ΣV T to obtain U (Σ 2 -I)U T = U Γ 2 U T = R Y X 2 X -1 3 X T 3 R T Y , (2.48)
where

Σ = Σ n K 0 0 I n-n K , Γ 2 = Σ 2 -I n , and Γ = Γ n K 0 0 0 . Let T = Σ -1/2 V T R X be the balancing transformation matrix, i.e. T -T XT -1 = T Y T T = Σ. Now we can choose X 3 = Σ n K and X 2 = T T Γ n K 0 ,
which satisfy (2.24) and (2.48).

Obtaining the Controller

In the previous section we recovered the matrix variable X a . The controller statespace matrices A K , B K , C K and D K can be obtained by solving the following convex optimisation problem minimise

β,A K ,B K ,C K ,D K β subject to Ψ Xa + Q T F T P + P T F Q < βI.
(2.49)

Note that the subject function is the left part of the inequality (2.25). The conditions that validate the solvability of the optimisation problem (2.49) are:

1. The H ∞ norm of the closed-loop system must be less then γ, T zw ∞ < γ with γ > 0.

2. The value of β must be negative.

A General Algorithm for H ∞ -synthesis

Now we summarise the contents of the proposed approach in an algorithm.

Algorithm 1 Fixed-Order synthesis Assume that n K and system matrices A, B, C, D are given.

1. Find X, Y ∈ S n 0 that satisfy (2.31) using Algorithm 1.

2. Recover X a from X and Y as described in Section 2.2.1.3

3. Solve (2.49) to get the controller system matrices A K , B K , C K , D K . [START_REF] Apkarian | Fixed order H ∞ control design via a partially Augmented Lagrangian method[END_REF] The second method can be formulated as follows

Apkarian Method

Problem Formulation

Let us introduce

 = A 0 n K 0 n K 0 n K , B1 = B 1 0 n K ×nw , B2 = 0 n×n K B 2 I n K 0 n K ×nu , Ĉ1 = C 1 0 nz×n K , Ĉ2 = 0 n K ×n I n K C 2 0 ny×n K , D12 = 0 nz×n K D 12 , D21 = 0 n K ×nw D 21 .
(2.50)

Then, the closed-loop matrices A CL , B CL , C CL , D CL can be written as

A CL = Â + B2 Θ Ĉ2 , B CL = B1 + B2 Θ D21 , C CL = Ĉ1 + D12 Θ Ĉ2 , D CL = D 11 + D12 Θ D21 .
(2.51) Note that (2.50) involves only plant data and that A CL , B CL , C CL , D CL depend anely on the controller data Θ given in (2.21).

Combining Lemma 2, Lemma 1 and Lemma 3, the following necessary and sucient conditions for the existence of γ-suboptimal controllers of order n K are given.

Theorem 10: [START_REF] Gahinet | A linear matrix inequality approach to H ∞ control[END_REF] Consider a proper plant P of minimal realisation corresponding to the statespace equation (2.9) and assume that D 22 = 0 and (A, B 2 , C 2 ) stabilisable and detectable. With the notation (2.50), dene (2.52) and let W P and W Q be two matrices whose columns span the null spaces of P and Q, respectively. There exists a set of controller of order n K if and only if there exists some X a ∈ S n+n K 0 matrix such that

P = BT 2 0 (n K +nu)×nw DT 12 ; Q = Ĉ2 D21 0 (n K +ny)×nz ,
W T P Υ Xa W P ≺ 0; W T Q Υ X -1 a W Q ≺ 0 (2.53)
where

Υ Xa =    ÂX a + X a ÂT X a B1 ĈT 1 BT 1 X a -γI nw D T 11 Ĉ1 D 11 -γI nz    , (2.54) Υ X -1 a =    X -1 a ÂT + ÂX -1 a B1 X -1 a ĈT 1 BT 1 -γI nw D T 11 Ĉ1 X -1 a D 11 -γI nz    .
(2.55)

Based on the previous theorem, the problem of nding a xed-order controller of order n K is reduced to the following set of conditions: Find two positives denite matrices X a and Y a such that

W T P    ÂX a + X a ÂT X a B1 ĈT 1 BT 1 X a -γI D T 11 Ĉ1 D 11 -γI    W P ≺ 0, (2.56a)
W T Q    Y a ÂT + ÂY a B1 Y a ĈT 1 BT 1 -γI D T 11 Ĉ1 Y a D 11 -γI    W Q ≺ 0, (2.56b) X a I (n+n K ) I (n+n K ) Y a 0. (2.56c) X a Y a -I (n+n K ) = 0.
(2.56d) Without loss of generality, this option helps stabilising algorithms [START_REF] Apkarian | Fixed order H ∞ control design via a partially Augmented Lagrangian method[END_REF].

Let X a denotes the convex set dened by the three LMIs (2.56a), (2.56b) and (2.56c). The relation (2.56d) is denoted by a non-convex set Xa .

X a and Y a Searching

To design a n K xed-order controller that stabilises the closed-loop system, we must nd an approach to x the non-convex constraint (2.56d). Using Denition 1.2.5, let us dene a vector of decision variables that we search for solving the problem of xed-order H ∞ -controller dened by the set of convex and non-convex constraints X a , Xa :

x a = vech(X a ) T , vech(Y a ) T , γ T .

(2.57)

Smooth Optimisation Problem As mentioned in the previous section, X a represents a convex set which makes the problem somewhat less dicult to solve than a general non-convex problem. The non-convex constraint Xa given in (2.56d) con- tains a smooth function, so this yields the problem even more less dicult but is also non-convex. In order to convexify the problem of nding a controller of order n K , it is sucient to dene an objective function to minimise. The original problem (2.56) which is formed by convex and non-convex constraints becomes a convex optimisation problem which can be reformulated as minimise γ subject to x a ∈ X a , and X a Y a -I (n+n K ) = 0.

(2.58)

As explained in Section 2.2.1.2 and using the PAL, the problem in (2.58) can be reformulated as

minimise xa Φ Ap (x a , λ, µ) subject to x a ∈ X a (2.59)
where Φ Ap (x a , λ, µ) is the augmented Lagrangian function dened as

Φ Ap (x a , λ, µ) = γ + i,j λ i,j (X a Y a -I (n+n K ) ) i,j + µ 2 i,j (X a Y a -I (n+n K ) ) i,j 2 . (2.60)
in the matrix form, this objective function can be written as

Φ Ap (x a , λ, µ) = γ + trace λ T (X a Y a -I (n+n K ) ) + µ 2 trace X a Y a -I (n+n K ) ) T (X a Y a -I (n+n K ) ) (2.61)
where λ is a Lagrange multiplier matrix and µ is a positive penalty.

Search Direction As explained in Section 2.2.1.2, a search direction is computed and the problem is be formulated as

arg min dx ∇ xa Φ Ap (x a , λ, µ) T dx a + 1 2 dx T a H δ (x a , λ, µ)dx a subject to x a + dx a ∈ X a .
(2.62) Then, the objective function to minimise is formed by the gradient, Hessian of the Lagrangian function and the step direction. The modeed Hessian H δ (x a , λ, µ) is obtained after making some changes to the Gauss-Newton approximation of the exact Hessian function as follows

The problem (2.62) is a conic programming problem that can be solved using Yalmip [Lofberg, 2004] with SDPT3 [START_REF] Toh | SDPT3 -a Matlab software package for semidenite programming[END_REF] or LMIlab [Gahinet, 1994]

∇ xa Φ Ap (x a , λ, µ) =   D T vec(λY a ) D T vec(X a λ) 1   + µ J(x a ) T vec(X a Y a -I).
(2.64)

Applying the Gauss-Newton approximation method to the Lagrangian function, we obtain

∇ GN xaxa Φ Ap (x a , λ, µ) =   0 D T (I ⊗ λ)D 0 D T (I ⊗ λ T )D 0 0 0 0 0   + µJ(x a ) T J(x a ) (2.65)
Hessian Modications In the objective function, the modied Hessian H δ (x a , λ, µ)

is obtained after some modication to the Gauss-Newton approximation of the exact Hessian H(x a , λ, µ) = ∇ 2 xaxa Φ Ap (x a , λ, µ) in order to ensure the positiveness of the Hessian. This approximation is detailed is Section 2.2.1.2.

Then to compute X a and Y a , the Algorithm 5 in Section A.1 is given.

Remark

Controller Synthesis In order to obtain the controller parameters A K , B K , C K , D K , the following convex optimisation must be solve minimise

β,A K ,B K ,C K ,D K β subject to Υ Xa + Q T Θ T P Xa + P T Xa ΘQ ≺ βI.
(2.66)

The conditions that validate the solvability of the optimisation problem (2.66) are:

1. The H ∞ -norm of the closed-loop system must be less then γ, T zw ∞ < γ with γ > 0.

2. β must be negative.

Summary and Relationship between Methods

If we try to resume various steps for synthesising an H ∞ controller of order n k in schematic form in order to facilitate understanding methods presented in this report.

Plant P : (A, B, C, D)

Find (X, Y, γ) ∈ X Set n K < n Recover X a and Y a (X a Y a -I = 0) K n K s.t. T zw ∞ < γ Set n K ≤ n Augmented ( Â, B, Ĉ, D) Find (X a , Y a , γ) ∈ X a s.t. X a Y a -I = 0 Ankelhed Method Apkarian Method

LPV-Robust Control Design

The most important contribution of LPV-framework occurs when design is the main purpose. Indeed, LPV-control design, has proven to be successful design methodology in many engineering applications [START_REF] Shamma | Analysis of gain scheduled control for nonlinear plants[END_REF]].

In the beginning, only the gain scheduling fashion was used. The basic concept is to linearise NL-systems at dierent operating points which engender a collection of local LTI controllers. To describe the changes of the operating points, a varying signal is introduced, and the LPV-gain schudeling theory is appeared, giving rise to LPV gainscheduled controllers (Figure 2.6(a)). LPV gain-scheduling can be seen as an extension of the LTI-robust control theory ( [START_REF] Shamma | Analysis of gain scheduled control for nonlinear plants[END_REF]] has drawn a big attention for nonlinear systems control theory. Due to many successful applications of this design methodology, gain-scheduling has become popular in industrial applications. However, the major diculty at that time, was the lack of general theory for analysing stability of LPV systems and designing LPV-based gain-scheduled control laws.

Nowadays, this problem has been resolved when interpolation based methods seem to guarantee global stability. The suitable framework for controlling LPV systems are emerging from robust control theory, such as H ∞ -control, and the use of LMIs.

LPV-Stability

Similar to LTI-systems in Section 2.1.1, there exist various stability concepts of LPV systems. Most of them are based on Lyapunov approach. Then, the notions of quadratic stability and robust stability, along their respective class of Lyapunov functions are introduced in this section. Moreover, the connection with the input/output stability notion can be used when analysing LPV-systems under LFT-form.

Quadratic Stability

The quadratic stability for LPV-systems does not make distinction between timeinvariant parameters, slowly-varying parameters and fast-varying parameters. Therefore, quadratic stability may be very conservative. Proposition 10: [Briat, 2015] The system (1.35) is quadratically stable if and only if there exists a matrix P ∈ S n 0 such that the LMI A T (ρ)P + P A(ρ) ≺ 0 (2.68) holds for all ρ ∈ ∆ ρ .

LPV-polytopic Case Thanks to the fact that LPV-polytopic systems are ane in parameters, their stability could be checked in the same way as the generic LPV-systems.

Theorem 11: [Briat, 2015] The polytopic LPV-systems (1.51) is quadratically stable if there exists a matrix P ∈ S n 0 such that the LMI

A T i P + P A i ≺ 0, (2.69)
holds for all i = 1, . . . , N .

Robust Stability

The robust stability is a stronger notion of the quadratic stability. Unlike the latter one, robust stability makes distinction between constant and time-varying dierentiable parameters. Robust stability for LPV-systems also consider information on the parameters variation rates.

Theorem 12: [Briat, 2015] The system (1.35) is robustly stable if there exists a dierentiable matrix function

P : ∆ ρ → S n 0 ; ρ → P (ρ) such that the condition A T (ρ)P (ρ) + P (ρ)A(ρ) + s i=1 ρi ∂P (ρ) ∂ρ i ≺ 0, (2.70)
holds for all (ρ, ρ) ∈ ∆ ρ × ∆ v and where ∆ v ∈ R s is the subset of the parameters rates and ρi is the ith element of ρ.

LPV-polytopic Case Robust stability deals with parameters variation rate. In the LPV-polytopic case, this rate is included in the term λ. But there is no denition for a set that could contain the λ trajectories. Then, the assumption that the LPV-systems is approximated with N p parameter with known derivative bounds is made.

Proposition 12: [Briat, 2015] Assume that ρ

∈ ∆ v = Co{V d }, V d = {d 1 , . . . , d N }, N = 2 Np and that ρ(t) = N i=1 λ(t)v i , holds with V v = v 1 , . . . , v N .
Then, the set of all λ's is given by

ΛN =         V 1 T N 0    +    D 0 1 T N    ξ : η ∈ Λ N      (2.71) where ξ(t) ∈ Λ N , V = [v 1 , . . . , v N ] and D = [d 1 , . . . , d N ]. Moreover, the following identity is given ρ(t) = N i=1 ξ i (t)d i = N i=1 λi (t)v i .
(2.72) based on the letter characterisation of ΛN the following results is given Theorem 13

The polytopic LPV-system given in (1.51) is robustly stable if there exist matrices P i ∈ S n 0 , i = 1, . . . , N , such that the parameter-dependent LMI A(λ) T P (λ) + P (λ)A(λ) + P (θ) ≺ 0, (2.73) holds for all (λ, θ) ∈ Λ N × vert{ ΛN } where

P (λ) = N i=1 λ i P i and A(λ) = N i=1 λ i A i .

Stability of LPV-systems in LFT-Form

Let the system given under the LFT-form (Figure 1.2) with the following state space realisation

   ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) w(t) = θ(ρ(t))z(t) ∀ t ≥ 0, (2.74)
where θ(ρ) is a possibly structured, matrix containing the time-varying parameters.

The lower part this LFT-form system is represented as an LTI-model with the following transfer function

G(s) = C(sI -A) -1 B + D.
(2.75)

Given α ∈ L 2 , let us consider the multiplication operator Θ ρ dened as Θ ρ (α(t)) = θ(ρ(t))α(t).

(2.76)

Then, the small gain theorem for LPV-systems under LFT-form is stated as the following Theorem 14: Small Gain Theorem for LPV-Systems [Briat, 2015] Assume that Θ ρ i,2 for all ρ ∈ P, then the LPV-system (2.74) is asymptotically stable if

G ∞ < 1, (2.77)
where P is the set parameter trajectories and . i,2 is the induced L 2 -norm dened in (1.1).

As mentioned in the LTI-case, the small gain theorem is a very simple tool for stability analysis. In fact, only gains are considered and no information on the phase are needed (unlike the Nyquist test for example). However, it remains very conservative fo the LPVframework. Indeed, information about the interconnected structures are not caught. For example, a simple structure (e.g. block diagonal matrix) of θ(ρ), can not be exploited in the analysis conditions. To tackle this drawback, the scaled small-gain notion is introduced. Then, information on the interconnection structure can be captured and the explicit expression of θ(ρ) is considered (for more details, see [Packard andDoyle, 1993, Briat, 2015]).

H ∞ -LPV-synthesis

Analysis and control of LPV-systems is directly inherited from robust control theory.

The main distinctness is in the fact that the parameters are assumed to be known or measurable in the LPV-framework, whereas they are unknown, by assumption, in robust control theory. The designed LPV-controllers should guarantee both stability and performance for all scheduling parameters in the predened set. In addition, and according to the several LPV-representations mentioned in 1. 4.4 (ane, polytopic, etc), a third classication can be given. Details on those controllers are given in [Briat, 2015] and presented in Appendix A.1.

In this section the quadratic approach with output feedback for polytopic systems is presented.

Quadratic Stabilisation for LPV-polytopic Systems Consider a general LPVsystem represented under the following state-space realisation

   ẋ(t) = A(ρ(t))x(t) + B 1 (ρ(t))w(t) + B 2 (ρ(t))u(t) z(t) = C 1 (ρ(t))x(t) + D 11 (ρ(t))w(t) + D 12 (ρ(t))u(t) y(t) = C 2 (ρ(t))x(t) + D 21 (ρ(t))w(t) + D 22 (ρ(t))u(t) ∀t > 0, (2.78) where x(t) ∈ R n , u(t) ∈ R nu , w(t) ∈ R nw , z(t) ∈ R nz , y (
t) ∈ R ny are respectively the state, the input, the disturbance vectors, the controlled output and the measured output. The vector ρ(t) is the exogenous parameter varying vector which is time dependent and ∆ ρ is the set of the parameter values, i.e. ρ(t) ∈ ∆ ρ .

Then,

A : ∆ ρ → R n×n , B 1 : ∆ ρ → R n×nw , B 2 : ∆ ρ → R n×nu , C 1 : ∆ ρ → R nz×n , C 2 : ∆ ρ → R ny×n , D 11 : ∆ ρ → R nz×nw , D 12 : ∆ ρ → R nz×nu , D 21 : ∆ ρ → R ny×nw and D 22 : ∆ ρ → R ny×nu .
The H ∞ -LPV-controller for the LPV-system (2.78) is dened by

K(ρ) : ẋK (t) = A K (ρ(t))x K (t) + B K (ρ(t))y(t) u(t) = C K (ρ(t))x K (t) + D K (ρ(t))y(t) ∀t > 0, (2.79)
where x K (t), y(t) and u(t) are respectively the states, the inputs and outputs of the controller K(ρ). A

K : ∆ ρ → R n×n , B K : ∆ ρ → R n×ny , C K : ∆ ρ → R nu and D K : ∆ ρ → R nu×ny .
The H ∞ -LPV-controller synthesis concerns the design of an LPV-global-controller that guarantees both stability and performance for all parameters variations dened in the set ∆ ρ .

To guarantee the closed-loop system quadratic stability and to satisfy H ∞ -performance criterion, the approach developed in [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF] is used to design the H ∞ -LPV-controller. This solution is on the polypotic formulation, and the global H ∞ -LPV controller is a convex combination of local controllers like shown in 1.4.4. Those locale controllers are given as follows: by assuming that the system matrices B 2 , D 12 , C 2 and D 21 are parameter independent and D 22 = 0. and for a pre-dened real positive scalar γ and a parameter dependent quadruplet matrices ( ÂK , BK , ĈK , DK ), there ex- ist two symmetric matrices X and Y such that the sucient condition that solves the H ∞ -LPV-problem is given by the following LMIs 

    A(ω i )X + B 2 ĈK (ω i ) + ( * ) * ÂK (ω i ) + A T (ω i ) + C T 2 DT K (ω i )B T 2 Y A(ω i ) + BK (ω i )C 2 + ( * ) B T 1 (ω i ) + D T 21 DT K (ω i )B T 2 B T 1 (ω i )Y + D T 21 BT K (ω i ) C 1 (ω i )X + D 12 ĈK (ω i ) C 1 (ω i ) + D 12 DK (ω i )C 2 . . . . . . * * * * -γI nu * D 11 (ω i ) + D 12 DK (ω i )C 2 -γI ny     ≺ 0, (2.80) X I I Y 0, (2.81)
where ω i are the vertices of the polytope formed by all the extremities of each varying parameter ρ ∈ P ρ and terms denoted ( * ) are induced by symmetry, for example

M + N + ( * ) * K L = M + M T + N + N T K T K L The controller K(ω i ) at vertex i is then reconstructed as D K (ω i ) = DK (ω i ). C K (ω i ) = ( ĈK (ω i ) -D K (ω i )C 2 X)M -T . B K (ω i ) = N -1 ( BK (ω i ) -Y B 2 D K (ω i )). A K (ω i ) = N -1 ( ÂK (ω i ) -Y A(ω i )X -Y B 2 D K (ω i )C 2 X)M -T . -B K (ω i )C 2 XM -T -N -1 Y B 2 C K (ω i ).
(2.82)

where M and N are dened such that M N T = I n -XY and which can be solved through a singular value decomposition and a Cholesky factorisation. The nal controller K(ρ) dened in (2.79) is then the convex combination of local controllers (see also [START_REF] Sato | Gain-scheduled outputfeedback controllers using inexact scheduling parameters for continuous-time {LPV} systems[END_REF]).

Conclusion

In this chapter the H ∞ -control theory for LTI and LPV systems has been presented. The stability issue and the H ∞ -synthesis procedure regarding the two system classes have been discussed.

The investigation on the resulted H ∞ -controllers shows the Achilles heels of this advanced control technique which is the probable high order of the controllers. One evident solution to remedy this drawback is to set this order before the design step. The big exibility oered by the design method through the LMI tools make the order constraint very easy to incorporate in the basic H ∞ -problem. However, the problem become non-convex and by the way the controller is then harder to nd.

Many approaches are recently developed and summarised in this chapter. As shown in the last Section, convexication step is achieved after many approximation. In addition, the application of these algorithms has more chance to generate numerical diculties.

This fact limits the use and the reliability of this direct path to get a low order controller. This disadvantage come from the fact that the solutions are dependent on the developed numerical solvers. These last are still developing. In the next chapter, The order reduction way to get a low order controller is exterminated. Methods to reduce the order of controllers are presented and could give some alternative to given ones on this chapter. In fact numerical problems are less involved and optimisation problems are avoided.

methods combining model-order reduction and controller synthesis are suggested as more manageable way to get a low order controller.

Chapter 3

Introduction

As shown in Chapter 2, the obtained H ∞ -controllers have at least the same order as the plant. Since this latter is built from the physical model and the frequency weighting functions (for performance requirements), the order of plant may be very high. In this case, the full-order controllers may be of limited use in practical applications. In practice, low-order controllers are preferred to high-order controllers as they requires less complicated and more easy available hardware to understand, to maintain and to implement in the real world. In the early days, the development in this eld has made the order reduction for control a needful design step. In order to achieve these advantages, new methods which yield low order controllers are developing. These methods can be divided into three classes:

Full order plant Full order controller [START_REF] Anderson | Controller reduction: concepts and approaches[END_REF] 1. The rst approach is to design a reduced-order controller directly from the high order system [START_REF] Yeh | Fixed-order H ∞ compensator design[END_REF], Ly et al., 1985, Gangsaas et al., 1986, Bernstein, 1989, Hyland and Richter, 1990, Fischer and Gutman, 1991].

Reduced order controller Reduced order plant

The main lack of this kind of methods is that the order of the reduced-controller has to be chosen by an ad-hoc way [Goddard, 1995].

2. The second way is to reduce the model using one of the existing open-loop model order reduction methods [START_REF] Antoulas ; Antoulas | A new result on passivity preserving model reduction[END_REF], and then design a controller for the obtained reduced model. The disadvantage of this indirect method is that the resulting error from reduction process will not be taken into the controller design step. And as a result, the reduced order controller is not guaranteed to stabilise the full order model [Kong, 2012].

3. The third way is to rstly design a full-order controller and then, reduce it while preserving the properties of the closed-loop system.

Note that reducing the designed controller for a high order plant is more eective than designing a controller for the reduced plant for many reasons:

• Error might expand when the reduction step is achieved before controller design.

• The plant approximation needs knowledges of the controller while the later has not yet been designed.

Thus, in this dissertation the third way is used to design a low-order controller. In this Chapter, well known methods of model order reduction methods are rst reviewed in Section 3.1. Based on this, controller order reduction method for LTI-systems is shown in Section 3.2. Then, an extension to the LPV-systems is developed in Section 3.3.

Model Order Reduction

Model Order Reduction (MOR) is a very active research eld in system and control theory. Indeed, it allows the conception of compact systems from an initial complex high order models. Mathematically speaking, the model reduction problem can be formulated as follows: given Σ, the LTI system introduced in (1.20) with the transfer function G given by G(s) = D + C(sI -A) -1 B, nd a reduced order model with the transfer function G r (s)

G r (s) = D r + C r (sI -A r ) -1 B r (3.1)
with A r ∈ R r×r , B r ∈ R r×nu , C r ∈ R ny×r , D r ∈ R ny×nu and r < n, such that the following properties are satised

• Approximation error against the initial model is small.

• System stability is preserved.

• Reduction procedure is computationally ecient and reliable.

Various approaches are developed. They deal on the error expression by deleting the less important states or matching some parameters of the original and reduced order systems. These methods can be classied two main categories (Figure 3. Before introducing these methods, a balancing step is needed. It helps to split the states into two sets so-called dominant and non-dominant states:

Balancing Procedure

Let Σ, be the LTI-system of order n with n u inputs and n y outputs given in (1.20) with G(s) = C(sI -A) -1 B + D its transfer function. Then, let us associate to this system the following Lyapunov equations

AW r + W r A T + BB T = 0 A T W o + W o A + C T C = 0. (3.2)
If Σ is asymptotically stable (Re(λ i (A)) < 0 ∀i) and its state space realisation (1.20) is minimal (i.e. reachable and observable), then (3.2) has two unique denite positive symmetric solutions W r , W o ∈ S n 0 called: reachability Gramian and observability Gramian respectively. The square root of the eigenvalues of the product W r W o gives the so-called Hankel singular values, i.e.

σ i (G) = λ i (W r W o ). (3.3)
Theorem 15: [START_REF] Pernebo | Model reduction via balanced state space representations[END_REF] Let S be a minimal realisation of the asymptotically stable LTI-system (1.20). It can be expressed under its balanced realisation S such as: S :

  A 11 A 12 A 21 A 22 B 1 B 1 C 1 C 2 D   (3.7) with W r = W o = diag(Σ 1 , Σ 2 )
where

Σ 1 = diag(σ 1 I m 1 , . . . , σ k I m k ) and Σ 2 = diag(σ k+1 I m k+1 , . . . , σ q I mq ) with: σ 1 > σ 2 > • • • > σ q > 0, m i : i = 1, • • • , q are multiplicities of σ i and m 1 + • • • + m q = n.
Then, the reduced order realisation:

A r B r C r D r
, obtained by truncation is asymptotically stable, minimal and satisfy

G -G r ∞ ≤ 2 q k+1 σ i (3.8)
where G and G r are the full and the reduced order model transfer functions respectively. Note that r = k i=1 m i .

The presented approach is valid for all stable system with minimal realisation.

For unstable systems see [START_REF] Zhou | Balanced realization and model reduction for unstable systems[END_REF]. Remark

Singular Perturbation Approximation

The Singular Perturbation Approximation (SPA) method is an alternative to the truncation technique if the model is stable with minimal realisation and internally balanced.

These two methods constitute complementary model order reduction techniques for continuous-time systems, and both of them conserve the stability. Even though, the upper bound for both reduction methods is the same, the BT aims to reach a smaller error at high frequencies and tends to be larger at low frequencies, whereas the reduced order models through SPA method behave otherwise, i.e. the error goes to zero at low frequencies and tends to enlarge at high frequencies. In SPA method, all balanced states are sorted and divided into the slow and fast modes by representing the smaller Hankel singular values as the fast mode, and the rest as the slow mode. Thus, a reduced-order model can be obtained by setting the derivative of all states corresponding to the fast mode equal to zero.

Let S be the balanced realisation of the LTI-system given in (3.7), i.e.

S :

   ẋ1 (t) = A 11 x1 (t) +A 12 x2 (t) +B 1 u(t) ẋ2 (t) = A 21 x1 (t) +A 22 x2 (t) +B 2 u(t) y(t) = C 1 x1 (t) +C 2 x2 (t) +D r u(t)
∀ t ≥ 0.

Since the system is balanced, x2 represents states corresponding to smaller HSV, i.e. the fast dynamics of the system. Based on the concept of the SPA method [Liu and Anderson, 1989], the derivative of all states corresponding to x2

are set to zero. Then, the reduced order model is given by the transfer function

G r (s) = C r (sI -A r ) -1 B r + D r with A r = A 11 -A 12 A -1 22 A 21 , (3.9)
B r = B 1 -A 12 A -1 22 B 2 , (3.10) C r = C 1 -C 2 A -1 22 A 21 , (3.11)
D r = D -C 2 A -1 22 B 2 (3.12)
and assuming that A 22 is invertible.

Theorem 16: [Glover, 1984] Let S be the minimal balanced state space representation of the asymptotically stable system given in (3.7). Then, the reduced order model obtained by SPA is asymptotically stable and satisfy Example In order to showing the eectiveness of these approaches, a numerical application to a physical system is given. Then, a building (the Los Angeles University Hospital) with 8 oors each having 3 degrees of freedom is modelled by a state space form of order 48 [START_REF] Chahlaoui | A collection of benchmark examples for model reduction of linear time invariant dynamical systems[END_REF]. the two obtained reduced order models using BT and SPA are compared to the initial model. Note that BT-based method shows good performance in high frequencies but not for the static gain. As expected, the characteristics of both methods are contrary to each other. Indeed, the SPA-reduced order model ts the model in low frequency but not over 10Hz.

G -G r ∞ ≤ 2 n i=k+1 σ i . ( 3

Frequency Weighted Balanced Truncation

The BT and the SPA approaches are performed with the aim to limit the error between the full-order model and the reduced-order one in the whole frequency range.

However, for many applications, a reduction in specic frequency range is more important. To do this, a Frequency Weighted FW technique is given in order to t the reduction step to a limit frequency range. This idea has been introduced rst in [Enns, 1984] where the BT is extended to give birth to the so-called Frequency Weighted Balanced Truncation FWBT. Nevertheless, the stability of the reduced order model is not guaranteed. To tackle this problem, a second variant that guarantee the stability is proposed in [START_REF] Lin | Model reduction via frequency weighted balanced realization[END_REF]. However an additional assumption is needed which limit its use. Thereafter, the method is modied in [START_REF] Sreeram | Frequency weighted balanced reduction technique: a generalization and an error bound[END_REF] and [START_REF] Kim | Error bound for transfer function order reduction using frequency weighted balanced truncation[END_REF] to introduce an explicit a priori upper bound of the approximation er- ror. In [START_REF] Wang | A new frequencyweighted balanced truncation method and an error bound. Automatic Control[END_REF], then in [START_REF] Varga | Accuracy enhancing methods for the frequency-weighted balancing related model reduction[END_REF], a simpler upper bound is given. Moreover, another version that focus on the relative approximation error is proposed in [Zhou, 1995]. All these developed techniques require the determination of an input and output weighting functions which is not an easy task. Thus, the classical version of the FWBT is modied in [START_REF] Gawronski | Model reduction in limited time and frequency intervals[END_REF] where a reduced order model in a limited frequency range can be found without setting the input and output weighting functions. However, the stability in this approach is not guaranteed.

Recently [START_REF] Gugercin | A survey of model reduction by balanced truncation and some new results[END_REF] propose an improved version that guarantee the stability with an upper error bound.

Enns' Method

The FWBT technique can be achieved by introducing two weighting function V and W in input and the output of the system. Then, the desirable approximation error is

W (G -G r )V .
Let Σ be the square LTI-system and G(s

) = C(sI -A) -1 B + D its transfer function. Dene V (s) = C V (sI -A V ) -1 B V +D V and W (s) = C W (sI -A W ) -1 B W +D W a stable
input and output weighting functions respectively. Then, two augmented systems are given as

GV (s) = C i (sI -A -1 i )B i + D i , (3.14) W G(s) = C o (sI -A -1 o )B o + D o , (3.15)
where

A i B i C i D i =   A BC V 0 A V BD V B V C DC V DD V   , (3.16)
and

A o B o C o D o =   A W B W C 0 A B W D B C W D W C D W D   .
(3.17)

Then, let P i = P 11 P 12 P T 12 P 22

and Q o = Q 11 Q 12 Q T 12 Q 22
, be the solutions of following Lyapunov equations

A i P i + P i A T i + B i B T i = 0 A T o Q o + Q o A o + C T o C o = 0 (3.18)
Then, solving (3.18) is equivalent to solve

AP 11 + P 11 A T + X = 0 A T Q 22 + Q 22 A + Y = 0 (3.19) with X = BC V P T 12 + P 12 C T V B T + BD V D T V B T (3.20) and Y = C T B T W Q 12 + Q T 12 B W C + C T D T W D W C. (3.21)
The method is based on balancing P 11 and Q 22 , i.e

P 11 = Q 22 = diag(σ 1 I m 1 , . . . , σq I mq ) (3.22) with: m i : i = 1, • • • , q are multiplicities of σi and m 1 + • • • + m q = n.
σi are the weighted singular values of G(s).

In addition, in this basis, the system can be expressed under the following realisation:

S :

  A 11 A 12 A 21 A 22 B 1 B 2 C 1 C 2 D   . (3.23) Finally, A 11 B 1 C 1 D
is the reduced-order system obtained by truncation.

1. The reduced order model is not guaranteed stable but only when one weighting function is identity, i.e. V = I or W = I.

2. It is noted that the equations system (3.19) has semi denite pair of solutions P E and Q E if D is not singular (D = 0). When the system is singular, [Willems, 1972] proposes to replace D by (D + ε 2 I) for any ε > 0.

Remark Then-after, an upper error bound is expressed in [START_REF] Kim | Error bound for transfer function order reduction using frequency weighted balanced truncation[END_REF], but it is very hard to compute it: Theorem 17: [START_REF] Kim | Error bound for transfer function order reduction using frequency weighted balanced truncation[END_REF] Consider the minimal realisation of the asymptotically stable LTI-system given in (1.20). Then, the reduced order model obtained by the FWBT with Enns' method satisfy

W (G -G r )V ∞ ≤ 2 q i=k+1 σ2 k + (α k + β k )σ 3/2 k + α k β k σk (3.24)
where G r is the reduced order model transfer function and α k , β k are H ∞ -norms of certain functions dependant on W and V . In addition, if W = I or V = I, the reduced-order model is guaranteed stable.

When W = V = I, relation (3.24) become (G -G r ) ∞ ≤ 2 q i=k+1
σ k which is the absolute error given on the BT method. Remark 3.1.4.2 Lin and Chiu's Method

Enns' method has been modied in [START_REF] Lin | Model reduction via frequency weighted balanced realization[END_REF]] in order to conserve the system stability when reducing.

Later, this method is modied in [START_REF] Sreeram | Frequency weighted balanced reduction technique: a generalization and an error bound[END_REF] in order to give an upper error bound to the approximation error.

Let T LC i = I P 12 P -1

22 0 I and T LCo = I -Q 22 Q 12 0 I
, two transformations applied to the augmented systems (3.14) and (3.15) respectively. Then, two transformed systems are given as

Āi Bi Ci Di =   A X 12 0 A V B LC B V C CP 12 P 22 + DC V DD V   (3.25) and Āo Bo Co Do =   A W Y 12 0 A B B W D + Q -1 11 Q 12 B C LC D W D W D   (3.26) with X 12 = AP 12 P -1 22 + BC V -P 12 P -1 22 A V , (3.27)
Y 12 = Q -1 22 Q 12 A + B W C -A W Q -1 11 Q 12 (3.28)
and

B LC = BD V -P 12 P -1 22 B V , (3.29)
C LC = D W C -C W Q -1 22 Q 12 .
(3.30)

Then, transformed reachability and observability Gramians namely P LC and Q LC are satisfying the two Lyapunov equations

AP LC + P LC A T + B LC B T LC = 0 A T Q LC + Q LC A + C T LC C LC = 0. (3.31)
Simultaneous diagonalisation of P LC and Q LC gives

P LC = Q LC = diag(σ 1 I m 1 , .
. . , σq I mq ).

(3.32)

The reduced order model is obtained by transforming and partitioning the original system. Assuming no pole-zero cancellations between the weights and the original system, the realisation {A, B LC , C LC } is then minimal and Lin and Chiu's technique yields stable models in the case of double-sided weighting.

Theorem 18: [Lin andChiu, 1990, Sreeram and[START_REF] Sreeram | [END_REF] Consider the minimal realisation of the asymptotically stable LTI-system given in (1.20). Then, the reduced order model obtained by the Lin and Chiu's FWBT method is stable and satisfy

W (G -G r )V ∞ ≤ 2 q i=k+1 (σ 2 k + α k + λ k )(σ k + β k + ω k ) (3.33)
where α k , β k , λ k and ω k are the H ∞ -norm of certain functions that depend on W and V .

Note that the given upper bound is less easier to compute than the one introduced previously and given by [START_REF] Kim | Error bound for transfer function order reduction using frequency weighted balanced truncation[END_REF].

Varga and Anderson's Method

As shown previously, the assumption that no pole-zero cancellations occur when forming the augmented system limits the applicability of Lin and Chiu's method. A solution to overcome this drawback has been proposed in [START_REF] Varga | Accuracy enhancing methods for the frequency-weighted balancing related model reduction[END_REF]. The idea is to balancing P va and Q va instead of P 11 and Q 22 where P va = P 11 -α 2 P 12 P -1

22 P T 12 , Q va = Q 22 -β 2 Q T 12 Q -1 11 Q 12 , (3.34) with 0 ≤ α ≤ 1 and 0 ≤ β ≤ 1.
Note that α c = α o = 0 corresponds to Enns' method and α = β = 1 leads to Lin and Chiu's method. Remark 3.1.4.4 Wang's et al. Method The problem on stability in the case of two sided weighted method viewed in Enns' method, is resolved here. Then, a method has been proposed in [START_REF] Wang | A new frequencyweighted balanced truncation method and an error bound. Automatic Control[END_REF] in order to guarantee that X and Y of (3.20) and (3.21) respectively are semi-positive denite matrices. Then, the two Gramians of reachability and observability P wsl and Q wsl are obtained by solving the following two Lyapunov equations AP wsl + P T wsl + B wsl B T wsl = 0

A T Q wsl + Q wsl A + C T wsl C wsl = 0 (3.35)
where B wsl = U wsl |S wsl | and C wsl = |R wsl |V T wsl . The matrices U wsl , S wsl , V wsl and R wsl are obtained by an eigenvalue decomposition of X and Y given in (3.20) and (3.21), i.e. X = U wsl S wsl U T wsl (3.36)

Y = V wsl R wsl V T wsl (3.37) with S wsl = diag(s 1 , s 2 , . . . , s n ) (3.38) R wsl = diag(r 1 , r 2 , . . . , r n ) (3.39)
where

|s 1 | ≥ |s 2 | ≥ • • • ≥ |s n | ≥ 0 and |r 1 | ≥ |r 2 | ≥ • • • ≥ |r n | ≥ 0.
Theorem 19: [START_REF] Wang | A new frequencyweighted balanced truncation method and an error bound. Automatic Control[END_REF] Given G, let G r be obtained by Wang's et. al frequency weighted balanced trun-cation as above. Then G r is stable. In addition if rank([B, B wsl ]) = rank(B wsl )

(3.40) rank([C T , C T wsl ]) = rank(C T wsl ), (3.41)
then G r is asymptotically stable and satises

W (G -G r )V ∞ ≤ 2 W L ∞ KV ∞ n i=r+1 κ i (3.42) with L = CV wsl diag(|r 1 | -1/2 , |r 2 | -1/2 , . . . , |r ni | -1/2 , 0, . . . , 0) (3.43) K = diag(|s 1 | -1/2 , |s 2 | -1/2 , . . . , |r no | -1/2 , 0, . . . , 0)U T wsl B (3.44)
where ni = rank(X) and no = rank(Y ).

Gawronski's et al. Method

A scheme where the frequency weights are not predened and approximation is considered in a given frequency interval is introduced in [START_REF] Gawronski | Model reduction in limited time and frequency intervals[END_REF].

Let P and Q be the reachability and the observability Gramians introduced in (3.2). Using Parseval's relationship it follows that in the frequency domain, P and Q are given by

P = 1 2π +∞ -∞ H(w)BB T H * (w)dw, (3.45) Q = 1 2π +∞ -∞
H * (w)C T CH(w)dw, (3.46) where H(w) = (wI -A) -1 and H * (w) = (-wI -A * ) -1 . For a given frequency band Ω = [ω 1 ; ω 2 ], the frequency weighted Gramians can be dened as

P Ω = P (ω 2 ) -P (ω 1 ) (3.47) Q Ω = Q(ω 2 ) -Q(ω 1 ) (3.48)
where

P (ω) = 1 2π +ω -ω H(w)BB T H * (w)dw, (3.49) Q(ω) = 1 2π +ω -ω
H * (w)C T CH(w)dw.

(3.50)

Note that P (ω) and P (ω) are both positive denite. From the fact that BB T = -AP -P A T = (wI -A)P + P (wI -A) * , the weighted reachability Gramian can be expressed as

P (ω) = 1 2π +ω -ω [P H * (w) + H(w)P ] dw (3.51)
The nal equation yields P (w) = P S * (w) + S(w)P (3.52) where

S(ω) = 1 2π +ω -ω H(w)dw =  2π ln (ωI + A)(-ωI + A) -1 (3.53)
A similar argument leads to

Q(w) = S * (w)Q + QS(w).
AP Ω + P Ω A T + W c (Ω) = 0 (3.58) where W c (Ω) = W c (ω 2 ) -W c (ω 1 ). A similar argument yields A T Q Ω + Q Ω A T + W o (Ω) = 0 (3.59) where W o (Ω) = W o (ω 2 )-W o (ω 1 ) and W o (w) = S * (w)C T C+C T CS(w). Therefore S(w)
can be computed whenever a balanced realisation can be computed. Gawronski and Juang's frequency weighted method is obtained by balancing P Ω and Q Ω , i.e. nding a basis so that

P Ω = Q Ω = diag(σ n 1 I n 1 , . . . , σ nq In q ) (3.60)
where n i are the multiplicities of each singular value σ n 1 and n 1 + • • • + n q = n. Then the reduced order model is obtained by truncation. The advantage of this method is that the construction of input and output weights is avoided by dening the Gramians over the specied frequency range.

since W c (Ω) and W o (Ω) are not guaranteed to be positive denite, stability of the reduced model cannot be guaranteed. Remark 3.1.4.6 Gugercin's et al. Method In this method, a modication is introduced to Gawronski's method in order to obtain a frequency balancing method which guarantees stability and provides a simple error result.

B G = M diag(|λ 1 | 1 2 , . . . , |λ u | 1 2 , 0, . . . , 0), (3.63) C G = diag(|δ 1 | 1 2 , . . . , |δ v | 1 2 , 0, . . . , 0)N T .
(3.64)

H ∞ -LTI-Controller Order Reduction

Methods proposed in the last section deal with an open-loop order reduction, where only the stability of the model is considered. However, in a closed-loop conguration the stability of the closed-loop must be preserved when reducing. In addition, performance of the reduced order should be kept as close as possible to the original controller performance.

Full Order Controller

First, let us recall the standard interconnected closed loop system of the Figure 3.5 where the augmented plant P is represented as follow:

P K y u z w Figure 3.5: H ∞ -controlled system P = P zw P zu P yw P yu s =   A B w B u C z D zw D zu C y D yw D yu   . (3.65)
Then, nding the stabilising controller K is equivalent to reduce the H ∞ -norm of the transfer T zw such that:

T zw ∞ = F (P, K) ∞ = P zw + P zu K(I -P yu ) -1 P yw ∞ (3.66)
A general solution for this problem base in Youla parametrisation [START_REF] Youla | Modern wiener-hopf design of optimal controllerspart ii: The multivariable case[END_REF], Vidyasagar, 1985] is given in [START_REF] Zhou | Robust and optimal control[END_REF] (Fig. 3.6):

K = F (M, Q) = M 11 + M 12 (I -M 22 Q) -1 M 21 , (3.67)
where M is a xed transfer function matrix of the form:

M K Q y u
Q ∈ RH ∞ Q ∞ < γ.
(3.69)

The solution obtained by setting Q = 0 is called the central controller K c = M 11 . This solution is used in general as H ∞ -controller [Zhou et al., 1996, Nagado andUsui, 2009].

FWBT for Controller Order Reduction

The problem to be considered here is to nd a controller K r with a minimal possible order such that the H ∞ performance requirement F (G, K r ) ∞ ≤ γ is satised (Figure 3.7). This is clearly equivalent to nding a Q so that it satises the above constraint and the order of K r is reduced. However, directly nding such a Q has proven to be very dicult.

M K r Q r y u Figure 3.8: Block Diagram Representation of K r K r = M 11 + M 12 Q r (I r -M 22 Q r ) -1 M 21 (3.70) The reduction error ∆ K is dened by ∆ K K r -K c . Then, ∆ K = M 12 Q r (I -M 22 Q r ) -1 M 21 . (3.71)
Based on this:

Q r = (I r + M -1 12 ∆ K M -1 21 M 22 ) -1 M -1 12 ∆ K M -1 21 (3.72)
Then, nding a low-order-H ∞ -controller is equivalent to search the stable free param- eter Q r such that Q ∞ < γ. The expression (3.72) can be represented by the block diagram in Figure 3.9:

∆ K M -1 21 M -1 12 M -1 22 + - Figure 3.9: Block Diagram Representation of Q r
Based on the small gain theorem, the closed-loop system is stable if:

∆ K M -1 21 M 22 M -1 12 ∞ < 1 (3.73) or M -1 21 M 22 M -1 12 ∆ K ∞ < 1 (3.74)
Therefore, the above inequality can be interpreted as frequency-weighted model reduction problem given in the previous section w.r.t. the following settings:

V = I r and W = M -1 21 M 22 M -1 12 or V = M -1 21 M 22 M -1
12 and W = I r

Order Reduction Procedure

By considering the scheme in Figure 3.10, we summarise the contents of this approach in Algorithm 2.

(K r -K)M -1 21 M 22 M -1 12 ∞ < 1 W o (K r -K)W i ∞ < 1
Frequency weighted balanced truncation

W i = M -1 21 M 22 M -1
12 and W o = I 5. Truncate the balanced realisation of K to obtain K r at the order r.

In this section, the balanced truncation method is used to reduce the order of H ∞ -LTI-Controller. In fact, the FWBT detailed in Section 3.1 is tted to the controller order reduction issue. The full order controller is rstly designed as shown in Section 2.1, then the FWBT with its various variants (Enns, Lin and Chiu, Wang, etc.) is used to obtain a stabilising reduced order one. Performance of the used techniques will be checked in Chapter 4

H ∞ -LPV-Controller Order Reduction

Thanks to their shown good properties, BT method has been proposed to reduce the order of controllers for LTI systems [START_REF] Liu | [END_REF]Anderson, 1989, Zhou et al., 1995]. The extension of these reduction techniques to LPV-systems is proposed in this section.

The idea is to substitute the use of LTI Gramians by using parameter/time varying equivalents [Wood et al., 1996, Sandberg and[START_REF] Sandberg | [END_REF]. A generalised method with unbounded rate parameter model is given in [START_REF] El-Zobaidi | Robust control and model and controller reduction of linear parameter varying systems[END_REF]].

Then-after, an eective BT method for H ∞ -LPV-controller order reduction is proposed in [START_REF] Widowati | Controller reduction of parameter dependent systems[END_REF].

The proposed method adapts the FWBT to reduce the order of an H ∞ -LPV controller.

For the development of this approach, the generalised Gramian framework is used [START_REF] Shaker | Generalized gramian framework for model/controller order reduction of switched systems[END_REF]. Then, the closed-loop system represented by T zw (ρ) P K y u z w

T zw (ρ) = A(ρ) B(ρ) C(ρ) 0 (3.75)
where

A(ρ) = A(ρ) + B 2 (ρ)D K (ρ)C 2 (ρ) B 2 (ρ)C K (ρ) B K (ρ)C 2 (ρ) A K (ρ) , B(ρ) = B 1 (ρ) + B 2 (ρ)D K (ρ)D K (ρ)D 21 (ρ) B K (ρ)D 21 (ρ)
and

C(ρ) = C 1 (ρ) + D 12 (ρ)D K (ρ)C 2 (ρ) D 12 (ρ)C K (ρ) .

Gramians and Generalised Gramians

The proposed method is based on the FWBT method where the key notions are the so-called reachability and observability Gramians.

Denition 3.3.1 (Gramians) Considering the quadratic stable and minimal realisation of the closed-loop LPV-system T zw (ρ) given in (3.75), Let us dene R(ρ) and O(ρ), the related reachability and observability Gramians respectively, they are dened as the solution of the two parameter dependent Lyapunov equations

- s i=1 v i ∂R(ρ) ∂ρ i + A(ρ)R(ρ) + R(ρ)A T (ρ) + B(ρ)B T (ρ) = 0, (3.76) s i=1 v i ∂O(ρ) ∂ρ i + A T (ρ)O(ρ) + O(ρ)A(ρ) + C T (ρ)C(ρ) = 0. (3.77)
For a given parameter trajectory ρ, let Φ ρ (t, 0) be the state-transition matrix. Then, the functional R(ρ) and O(ρ), respectively the reachability and observability Gramians of the closed-loop LPV-system (3.75), are expressed such that

R(ρ) = 0 -∞ Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) dt, (3.78) O(ρ) = ∞ 0 Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) dt.
(3.79) Denition 3.3.2 (Generalised Gramians) Let P(ρ) and Q(ρ) be respectively the generalised reachability and observability Gramians of the quadratic stable and minimal closed-loop LPV-system (3.75) given as

∀(ρ, v) ∈ ∆ ρ × ∆ v - s i=1 v i ∂P(ρ) ∂ρ i + A(ρ)P(ρ) + P(ρ)A T (ρ) + B(ρ)B T (ρ) ≺ 0 (3.80) s i=1 v i ∂Q(ρ) ∂ρ i + A T (ρ)Q(ρ) + Q(ρ)A(ρ) + C T (ρ)C(ρ) ≺ 0.
(3.81) Indeed, Lyapunov inequalities (rather than Lyapunov equations) are solved to compute generalised Gramians. This linear matrix inequality (LMI) approach to the model reduction problem is particularly useful when some structures need to be preserved in the process of model reduction. Controller reduction is a typical example of this type of problems [START_REF] Shaker | Generalized gramian framework for model/controller order reduction of switched systems[END_REF]. Note that the physical interpretations of generalised Gramians are similar to ordinary Gramians. Considering T zw (ρ), R(ρ) and O(ρ), the following lemma introduces useful results about the relation between Gramians and generalised Gramians.

Lemma 7

Let T zw (ρ) be a minimal state-space realisation of the quadratic stable and minimal closed-loop system dened by (3.75). Then, ∀ρ ∈ P ρ R(ρ) ≺ P(ρ(0)) O(ρ) ≺ Q(ρ(0)). Proof: By multiplying on the left by Φ ρ (0, t) and on the right by Φ T ρ (0, t), the relation

(3.80) yields -Φ ρ (0, t) s i=1 v i ∂P ∂ρ i Φ T ρ (0, t) + Φ ρ (0, t)A(ρ(t))P(ρ)Φ T ρ (0, t) + Φ ρ (0, t)P(ρ)A T (ρ(t))Φ T ρ (0, t) + Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) ≺ 0. (3.83) Using fact that ∂ ∂t 0 Φ ρ (t, t 0 ) = -Φ ρ (t, t 0 )A(ρ(t 0 )), then (3.83) yields - d dt Φ ρ (0, t)P(ρ)Φ T ρ (0, t) + Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) ≺ 0. (3.84)
which on integrating over the semi-innite time axis (-∞, 0] and considering that lim t→-∞ Φ ρ (0, t) = 0, gives

P(ρ(0)) 0 -∞ Φ ρ (0, t)B(ρ(t))B T (ρ(t))Φ T ρ (0, t) dt R(ρ) (3.85)
Similarly, let us multiply on the right by the closed-loop state-transition matrix Φ ρ (t, 0) and on the left by Φ T ρ (t, 0), the relation (3.81):

Φ T ρ (t, 0) s i=1 v i ∂Q ∂ρ i Φ ρ (t, 0) + Φ T ρ (t, 0)A T (ρ(t))Q(ρ)Φ ρ (t, 0) + Φ T ρ (t, 0)Q(ρ)A(ρ(t))Φ ρ (t, 0) + Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) ≺ 0 (3.86)
Using the fact that

∂ ∂t Φ ρ (t, t 0 ) = A(ρ(t))Φ ρ (t, t 0 ), then (3.86) yields d dt Φ T ρ (t, 0)Q(ρ)Φ ρ (t, 0) + Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) ≺ 0. (3.87)
Integrating the last expression from 0 to +∞ gives

Q(ρ(0)) +∞ 0 Φ T ρ (t, 0)C T (ρ(t))C(ρ(t))Φ ρ (t, 0) dt O(ρ) (3.88)

Frequency-Limited Order Reduction by FWBT

For a given parameter trajectory ρ, let consider Φ ρ (t, 0) the state-transition matrix of the closed-loop system (3.75). Then, let us dene

f ρ (t) = Φ ρ (0, t)B(ρ(t))H(-t) g ρ (t) = C(ρ(t))Φ ρ (t, 0)H(t) ∀t, (3.89)
where H is the heaviside step function.

By considering Ω = [ω 1 ; ω 2 ] the frequency range where order-reduction is desired to be better, the following denition is given Denition 3.3.3 (Frequency Limited Gramians) Let R Ω (ρ) and O Ω (ρ) two frequency dependent terms dened as

R Ω (ρ) = R ω 2 (ρ) -R ω 1 (ρ) and O Ω (ρ) = O ω 2 (ρ) -O ω 1 (ρ)
where

R ω (ρ) = +ω -ω F ρ (w)F * ρ (w)dw (3.90) O ω (ρ) = +ω -ω G * ρ (w)G ρ (w)dw (3.91) with F ρ (resp. G ρ ) is the Fourier transform of f ρ (resp. g ρ ).
Then, the functional RΩ (ρ) and ÔΩ (ρ), respectively the frequency limited reachability and observability Gramians of the closed-loop LPV-system (3.75), are dened as the solutions of

∀(ρ, v) ∈ ∆ ρ × ∆ v - s i=1 v i ∂ RΩ (ρ) ∂ρ i + A(ρ) RΩ (ρ) + RΩ (ρ)A T (ρ) + R Ω (ρ) = 0 (3.92) s i=1 v i ∂ ÔΩ (ρ) ∂ρ i + A T (ρ) ÔΩ (ρ) + ÔΩ (ρ)A(ρ) + O Ω (ρ) = 0. (3.93)
The quantities R Ω (ρ) and O Ω (ρ) have the following eigenvalues decomposition

R Ω (ρ) = U Ω (ρ) diag (λ 1 (ρ), . . . , λ n K (ρ)) U T Ω (ρ) (3.94) O Ω (ρ) = V Ω (ρ) diag (δ 1 (ρ), . . . , δ n K (ρ)) V T Ω (ρ) (3.95) with |λ 1 | ≥ . . . |λ n K | ≥ 0 and |δ 1 | ≥ . . . |δ n K | ≥ 0. Let u ρ ≤ n K and v ρ ≤ n K be respectively the ranks of R Ω (ρ) and O Ω (ρ).
Based on these denitions, let us dene the two quantities:

B Ω (ρ) = U Ω (ρ) diag(|λ 1 (ρ)| 1 2 , . . . , |λ uρ (ρ)| 1 2 , 0, . . . , 0), and (3.96)
C

Ω (ρ) = diag(|δ 1 (ρ)| 1 2 , . . . , |δ vρ (ρ)| 1 2 , 0, . . . , 0)V T Ω (ρ).
(3.97) Denition 3.3.4 (Modied Frequency Limited Generalised Gramians) Consider K(ρ), the full-order stabilising LPV-controller given in (2.79). Let

PΩ (ρ) = P1 (ρ) 0 0 P2 (ρ) 0 and QΩ (ρ) = Q1 (ρ) 0 0 Q2 (ρ)
0 be the modied frequency limited reachability and observability Gramians dened as the solutions of the following Lyapunov equations

∀(ρ, v) ∈ ∆ ρ × ∆ v - s i=1 v i ∂ PΩ (ρ) ∂ρ i + A(ρ) PΩ (ρ) + PΩ (ρ)A T (ρ) + B Ω (ρ)B T Ω (ρ) = 0 (3.98) s i=1 v i ∂ QΩ (ρ) ∂ρ i + A T (ρ) QΩ (ρ) + QΩ (ρ)A(ρ) + C T Ω (ρ)C Ω (ρ) = 0. (3.99)
For the generalisation, we have the following inequalities:

- s i=1 v i ∂P Ω (ρ) ∂ρ i + A(ρ)P Ω (ρ) + P Ω (ρ)A T (ρ) + B Ω (ρ)B T Ω (ρ) ≺ 0 (3.100) s i=1 v i ∂Q Ω (ρ) ∂ρ i + A T (ρ)Q Ω (ρ) + Q Ω (ρ)A(ρ) + C T Ω (ρ)C Ω (ρ) ≺ 0. (3.101) with P Ω (ρ) = P 1 (ρ) 0 0 P 2 (ρ) 0 and Q Ω (ρ) = Q 1 (ρ) 0 0 Q 2 (ρ) 0.
If the block diagonal solutions P Ω (ρ) and Q Ω (ρ) exist, then let T 1 (ρ) and T 2 (ρ) be two nonsingular matrices given such that

T -1 1 (ρ)P 1 (ρ)T -T 1 (ρ) = T T 1 (ρ)Q 1 (ρ)T 1 (ρ) = Σ 1 (ρ) = diag(ξ 1 (ρ), . . . , ξ n (ρ)), (3.102) with ξ 1 (ρ) ≥ ξ 2 (ρ) ≥ • • • ≥ ξ n (ρ), and T -1 2 (ρ)P 2 (ρ)T -T 2 (ρ) = T T 2 (ρ)Q 2 (ρ)T 2 (ρ) = Σ 2 (ρ) = diag(γ 1 (ρ), . . . , γ r (ρ) Σ 21 , γ r+1 (ρ), . . . , γ n K (ρ) Σ 22 ), (3.103) with γ 1 (ρ) ≥ γ 2 (ρ) ≥ • • • ≥ γ r (ρ) > γ r+1 (ρ) ≥ γ r+2 (ρ) ≥ • • • ≥ γ n K (ρ)
are the frequency limited generalised Hankel singular values of K(ρ) and r is the desired order for the reduced-order controller. The balanced realisation of K(ρ) can be written as

K(ρ) = T -1 2 (ρ)A K (ρ)T 2 (ρ) T -1 2 (ρ)B K (ρ) C K (ρ)T 2 (ρ) D K (ρ) = ÃK (ρ) BK (ρ) CK (ρ) DK (ρ) . (3.104)
Further, K(ρ) is partitioned as conformably with Σ 2 (ρ) as

K(ρ) =     ÂK (ρ) ÃK12 (ρ) ÃK21 (ρ) ÃK22 (ρ) BK (ρ) BK2 (ρ) ĈK (ρ) CK2 (ρ) DK (ρ)     . (3.105)
Finally, a truncation step is performed to obtain a reduced-order controller.

Denition 3.3.5 Given the balanced realisation K(ρ) dened in (3.116), let K(ρ) be the truncated realisation to the rth order and denoted as follows

K(ρ) = ÂK (ρ) BK (ρ) ĈK (ρ) DK (ρ) (3.106)
Furthermore, the reduced-order parameter dependent closed-loop system is given as

Tzw (ρ) = Â(ρ) B(ρ) Ĉ(ρ) D(ρ) =     A(ρ) + B 2 DK (ρ)C 2 B 2 ĈK (ρ) B 1 (ρ) + B 2 DK (ρ) DK (ρ)D 21 BK (ρ)C 2 ÂK (ρ) BK (ρ)D 21 C 1 (ρ) + D 12 DK (ρ)C 2 D 12 ĈK (ρ) D 11 (ρ) + D 12 DK (ρ)D 21     . (3.107)
Theorem 20

Suppose K(ρ) is the stabilising parameter dependent controller dened in (2.79) such that the closed-loop transfer T zw (ρ) dened in (3.75) is minimal, quadratic stable and there exist Lyapunov inequality solutions P Ω (ρ) and Q Ω (ρ) such that (3.100) and (3.101) are satised. Let K(ρ) be the reduced-order controller dened in (3.106) and obtained by truncation. Then, the closed-loop system with the reduced-order controller Tzw (ρ

) dened in (3.107) is stable. If in addition rank[B(ρ), B Ω (ρ)] = rank[B Ω (ρ)], (3.108)
and

rank[C T (ρ), C T Ω (ρ)] = rank[C T Ω (ρ)], (3.109)
then, Tzw (ρ) is quadratic stable and satises

T zw (ρ) -Tzw (ρ) i,2 ≤ 2 J B (ρ) ∞ J C (ρ) ∞ n K i=r+1 γ i,ρ (3.110)
where J B (ρ

) := diag(|λ 1 | -1 2 (ρ), . . . , |λ uρ | -1 2 (ρ), 0, . . . , 0)U Ω (ρ)B(ρ) and J C (ρ) := C(ρ)V Ω (ρ) diag(|δ 1 | -1 2 (ρ), . . . , |δ vρ | -1 2 (ρ), 0, . . . , 0),
Proof: The reachability and the observability Gramians given in (3.78),(3.79) can be expressed as

R(ρ) = +∞ -∞ f ρ (τ )f * ρ (τ ) dτ (3.111) O(ρ) = ∞ -∞ g * ρ (τ )g ρ (τ ) dτ (3.112)
where f ρ (τ ) and g ρ (τ ) are given in (3.89).

Then, using Parseval relationship, the reachability and the observability Gramians could be expressed as follows

R(ρ) = 1 2π +∞ -∞ F ρ (w)F * ρ (w)dw (3.113) O(ρ) = 1 2π +∞ -∞ G * ρ (w)G ρ (w)dw. (3.114)
By considering F ρ (resp.g ρ ) the Fourier transform of f ρ (resp.g ρ ), we can dene R ω (ρ) and O ω (ρ) as the limited reachability and observability Gramians given in (3.91) and (3.91). Then, by setting Ω = [ω 1 ; ω 2 ] the frequency range where approximation is desired to be better. Then, a new modied terms R Ω (ρ) and O Ω (ρ) are dened. Since R Ω (ρ) and O Ω (ρ) are not guaranteed to be positive denite, stability of the reduced-order controller is not guaranteed. Then, an idea based on eigenvalues decomposition presented in [START_REF] Gugercin | A survey of model reduction by balanced truncation and some new results[END_REF] is proposed here to guarantee stability by providing an upper error bound. Indeed, the solution of (3.100) and Let Tzw (ρ) be the frequency limited balanced realisation of the full-order closed-loop system. Then, Tzw (ρ) is dened by

Tzw (ρ) = T -1 (ρ)A(ρ)T (ρ) T -1 (ρ)B(ρ) C(ρ)T (ρ) D(ρ) =     Â(ρ) Ã12 (ρ) Ã21 (ρ) Ã22 (ρ) B(ρ) B2 (ρ) Ĉ(ρ) C2 (ρ) D(ρ)     , (3.115)
where

T (ρ) = diag(T 1 (ρ), T 2 (ρ)).
By considering the assumptions (3.108) and (3.109), there exist J B (ρ

) and J C (ρ) such that B(ρ) = B Ω (ρ)J B (ρ) and C(ρ) = J C (ρ)C Ω (ρ).
On the other hand, the reduction error is expressed as

T zw (ρ) -Tzw (ρ) i,2 = C(ρ)(sI -A(ρ)) -1 B(ρ) -Ĉ(ρ)(sI -Â(ρ)) -1 B(ρ) i,2 = J C (ρ) C Ω (ρ)(sI -A(ρ)) -1 B Ω (ρ) -Ĉ(ρ)(sI -Â(ρ)) -1 B(ρ) .J B (ρ) i,2 ≤ 2 J B (ρ) ∞ J C (ρ) ∞ n K i=r+1 γ i,ρ
The Assumptions (3.108) and (3.109) mean that there exist J B (ρ) and J C (ρ) such that B(ρ) = B Ω (ρ)J B (ρ) and C(ρ) = J C (ρ)C Ω (ρ). In addition, by following the steps in [START_REF] Anderson | Algebraic characterization of xed modes in decentralized control[END_REF], Wang et al., 1999, Imran et al., 2014] it was shown that assumptions (3.108) and (3.109) are almost always true. Hence we expect that our approach will apply in most of the cases. Indeed, during our simulations, the assumptions have always been satised.

Remark Algorithm 3 H ∞ -LPV-Controller Order Reduction

Considering the LPV-plant described in (2.78) the reduced-order controller can be computed as follows Inputs: (A(ρ), B(ρ), C(ρ), D(ρ)).

Outputs: ( Â(ρ), B(ρ), Ĉ(ρ), D(ρ)).

Assumptions: (A(ρ), B(ρ), C(ρ), D(ρ)) minimal. Algorithm:

1. Compute K(ρ) the full-order controller according Section 2.3.2.

2. Compute the closed-loop system T zw given in (3.75).

3. Compute the generalised Gramians P(ρ) and Q(ρ) solutions of (3.80) and (3.81) respectively.

4. Compute the balanced realisation K(ρ) of the full-order controller K(ρ) by: (a) Find T 2 , the basis change matrix according to (3.103).(For instance, Procedure in [START_REF] Moore | Controller reduction methods maintaining performance and robustness[END_REF]] can be used).

(b) Compute the balanced realisation K(ρ) as dened in (3.116).

5. Compute the reduced-order controller K(ρ) from K(ρ) by truncation.

Frequency-Limited Order Reduction by SPA

The balanced realisation of K(ρ) given in (3.116) can be partitioned as conformably with Σ 2 (ρ) as

K(ρ) =    ÃK11 (ρ) ÃK12 (ρ) ÃK21 (ρ) ÃK22 (ρ) BK1 (ρ) BK2 (ρ) CK1 (ρ) CK2 (ρ) DK (ρ)    (3.116)
Then, the equation ( 3.116) can be written as

ẋK1 (t) ẋK2 (t) = ÃK11 (ρ) ÃK12 (ρ) ÃK21 (ρ) ÃK22 (ρ) x K1 (t) x K2 (t) + BK1 (ρ) BK2 (ρ) y(t) u(t) = CK1 (ρ) CK2 (ρ) x K1 (t) x K2 (t) + DK (ρ) y(t) (3.117)
In this balanced realisation, states corresponding to the rst smallest singular values represent the fast dynamics of the system (i.e. states that have fast transient dynamics and decay rapidly to certain steady value). based on this concept of SPA method [START_REF] Oh | Controller order reduction using singular perturbation approximation[END_REF], the system (3.117) is approximated by setting x K2 = 0.

Then (3.117) becomes    ẋK1 (t) = ÃK11 (ρ)x K1 (t) + ÃK12 (ρ)x K2 (t) + BK1 (ρ)y(t) 0 = ÃK21 (ρ)x K1 (t) + ÃK22 (ρ)x K2 (t) + BK2 (ρ)y(t) u(t) = CK1 (ρ)x K1 (t) + CK2 (ρ)x K2 (t) + DK (ρ)y(t) (3.118)
Denition 3.3.6 Given the balanced realisation K(ρ) dened in (3.116), let K(ρ) be the Frequency Weighted Singular Perturbation Approximation FWSPA to the rth-order and denoted as follows

K(ρ) = ÂK (ρ) BK (ρ) ĈK (ρ) DK (ρ) (3.119)
where

ÂK (ρ) = ÃK11 (ρ) -ÃK12 (ρ) Ã-1 K22 (ρ) ÃK21 (ρ), (3.120) BK (ρ) = BK1 (ρ) -ÃK12 (ρ) Ã-1 K22 (ρ) BK2 (ρ), (3.121) ĈK (ρ) = CK1 (ρ) -CK2 (ρ) Ã-1 K22 (ρ) ÃK21 (ρ), (3.122) DK (ρ) = -CK2 (ρ) Ã-1 K22 (ρ) BK2 (ρ).
(3.123) Furthermore, the reduced-order parameter dependent closed-loop system is given as

Tzw (ρ) = Â(ρ) B(ρ) Ĉ(ρ) D(ρ) (3.124) =     A(ρ) + B 2 DK (ρ)C 2 B 2 ĈK (ρ) B 1 (ρ) + B 2 DK (ρ) DK (ρ)D 21 BK (ρ)C 2 ÂK (ρ) BK (ρ)D 21 C 1 (ρ) + D 12 DK (ρ)C 2 D 12 ĈK (ρ) D 11 (ρ) + D 12 DK (ρ)D 21    
Theorem 21

Suppose K(ρ) is the stabilising parameter dependent controller dened in (2.79) such that the closed-loop transfer T zw (ρ) dened in (3.75) is minimal, quadratic stable and there exist Lyapunov inequality solutions P Ω (ρ) and Q Ω (ρ) such that (3.100) and (3.101) are satised. Let K(ρ) be the reduced-order controller dened in (3.119). Then, the closed-loop system with the reduced-order controller Tzw (ρ)

T zw (ρ) -Tzw (ρ) i,2 ≤ 2 J B (ρ) ∞ J C (ρ) ∞ n K i=r+1 γ i,ρ (3.125)
where J B (ρ both highlighted. After a detailed review on these methods, it is shown that they are more ecient in quite high frequency or rather in low frequency only. Then, a third variant is introduced which performs in a limited specic frequency range. In fact, the Frequency Weighted Balanced Truncation FWBT allows better order-reduction in a specic frequency range. Thus, the several existing methods that deals with the FWBT are reviewed and discussed while showing their advantages and drawbacks. This study, leads to perform the controller order-reduction in closed-loop scheme in the rest of the chapter.

) := diag(|λ 1 | -1 2 (ρ), . . . , |λ uρ | -1 2 (ρ), 0, . . . , 0)U Ω (ρ)B(ρ) and J C (ρ) := C(ρ)V Ω (ρ) diag(|δ 1 | -1 2 (ρ), . . . , |δ vρ | -1 2 (ρ)
In Section 3.2, the LTI-case is considered. In fact, after analysing the FWBT methods.

It appeared that a link can be established between the H ∞ -control synthesis approach and the relative bounded H ∞ -relative error in the FWBT technique. Then, a method to reduce the order of an LTI-H ∞ -controller is derived. Regarding the various vari- ants of the FWBT approach, a comparison between these variants is achieved in the next chapter. Finally, in Section 3.3, a new method is proposed in order to extend the study to the LPV-case. Indeed, the limited frequency BT and SPA are managed to reduce the order of an H ∞ -LPV controller. To do this, the generalised Gramian framework is introduced and shown to be of big interest. The obtained reduced-order controller is proven stable and the degradation in the closed-loop performance is guaranteed bounded. An application of the proposed method to reduce the order of a semi-active suspension-controller is given in the next chapter.

Introduction

In this chapter, the numerical evaluations of the presented methods in Chapter 2 and Chapter 3 are given.

• In Section 4.1, the xed-order methods studied in Section 2.2 are evaluated on some benchmark examples of the COMPl e ib library. The evaluation is completed by two other available Matlab toolboxes namely HIFOO and HINFSTRUT.

Finally, a comparison between the xed-order way and the reduced order one is performed (Figure 3.1).

• In Section 4.2, an H ∞ -LTI-control problem is considered. It consists of an active suspension system. The mathematical model is rst given. Then, the full-order H ∞ -controller is designed. Based on this, the FWBT approach is used to reduce the designed controller order. Numerical simulations are performed and results are analysed in the frequency and the time domain. The evaluation is completed by a comparison with the xed-order approaches.

• In Section 4.3, the developed method to reduced the order of an H ∞ -LPVcontroller presented in Section 3.3 is evaluated. A problem of a semi-active suspension control is considered. Then, an LPV-model of a magneto-rheological suspension is rst set. Then, a reduced-order controller is obtained from the fullorder one. Finally, performance of the reduced-order controller are tested and approximation error evaluated.

Benchmark Library COMPl e ib

The synthesis of an H ∞ -xed order-controller has been reviewed in Section 2.2. Then, the studied methods are evaluated in this section. First, a description of the benchmark problems library COMPl e ib is given. Then, the two design methods are tested and compared on the COMPl e ib examples. In addition, comparison with two other existing toolbox namely hifoo and hinfstruct is also performed. This evaluation is performed by testing the performance parameter γ and the H ∞ -norm of the closedloop system. Finally, a comparison between the direct xed-order way and the indirect order-reduction one is performed.

Note that the simulations are performed on a intel core i7 CPU (3.07 GHz) running with Matlab 2015a.

COMPl e ib Problems

The evaluation of methods in Section 2.2 is achieved over the benchmark problem library COMPl e ib (COnstraint Matrix-optimization Problem library), which is presented in [START_REF] Leibfritz | Compleib: Constrained matrix-optimization problem library -a collection[END_REF]. This library contains systems collected from engineering literature and also pure academic problems. The set of problems includes for instance models of aircraft, helicopters, jet engines and reactors. Higher is the dimension, harder is to solve this problems. The library is easily available from a website 1 .

Fixed-Order Controller Evaluation

In this section, the comparison between the studied methods and their performance is

Numerical Considerations for Augmented Lagrangian Method

The studied methods in Section 2.2 are based on the augmented Lagrangian approach.

Algorithms used to implement these methods are given in Appendix A. For their implementation, some numerical considerations about parameter initialisation are given.

The initial penalty parameter has been set to µ 0 = 10 -3 and the multiplier is set to λ 0 = µ 0 (X 0 a Y 0 a -I) for the Apkarian's method and λ 0 = µ 0 q(X 0 , Y 0 ) for the Ankelhed's method, where X 0 a , Y 0 a and X 0 , Y 0 are solutions of the convex set X, X a respectively. The parameters ρ and ρ 0 determine the updating rules for the penalty and multiplier in general tests. They are chosen as ρ = 9 and ρ 0 = 0.9. However, in other few tests these parameters have to be changed in order to improve the results. Then, the values of these parameters are changed and low values in the case where γ T zw ∞ are given (remember that our objective is to nd T zw ∞ < γ). Then, the idea is to slow down the optimisation process to achieve the objective by obtaining T zw ∞ < γ. This technique is valid for the greater part of tests when γ T zw ∞ .

Results and Discussion

The studied methods are rst tested between them on some chosen COMPl e ib Problems.

Then, an extended evaluation against another existing toolboxes is performed. Finally, the general scheme in Figure 3.1 is considered and a comparison between the direct xed-order control and the controller order reduction is achieved.

Ankelhed and Apkarian's Methods Comparison After implementing the two methods, the evaluation results are shown in Table 4.1. To evaluate and to compare methods, we focus our reviews only on results shown in bold in order to simplify explanation and avoid repeating the same remarks. The rst column presents the system name, the system order, the number of the control inputs and outputs, and the order of the designed controller. The second and the third columns show the γ and H ∞ -norm obtained by each method. N.P. means Numerical Problems. The rst evaluation is done using the systems HE1 and HE3. Note that similar results are obtained for both methods and for dierent orders. The evaluation using the system ROC1 shows that the Apkarian's method give slightly better result than the Ankelhed's method. For this test, the Ankelhed's method give worse results in the case n K = 2 where the objective of the H ∞ -synthesis is not achieved (i.e. γ > T zw ∞ ). However, the evaluation from the system JE3 and for n K = 0 (static gain) shows that the Ankelhed's method has better results compared to the Apkarian's method which fails with this system.

As conclusion, both methods almost give the same results when synthesising H ∞controllers for systems of order n ≤ 20. However, in few cases one of them can gives better results than the other. When 20 ≤ n ≤ 24, the Ankelhed's method give better results compared to Apkarian's method. Note that, in some cases, methods do not show any results. This can be explained by the high number of decision variables.

Once the order of the system begins to be higher n ≥ 24, several problems encountered in the algorithm initialisation. In fact, it is even dicult to nd a starting point of decision vector x 0 and x 0 a for both methods. This problem can be explained also by the fact that decision vectors x 0 , x 0 a contain many decision variables, and it is very dicult to solve this problem by one of the existing solvers. Even if a starting point of decision vectors is found, but in optimisation step more numerical problems are encountered.

In few words, we can say that studied methods achieve good results for medium order systems. (i.e. n ≤ 24)

Comparison with Other Existing Methods In addition of the studied techniques, two numerical toolboxes namely hinfstruct and hifoo are evaluated hinfstruct is included in the Robust Control Toolbox of Matlab and based on [START_REF] Apkarian | Nonsmooth H ∞ synthesis. Automatic Control[END_REF] work. The method uses subgradient calculus to solve the problem [Clarke, 1990]. First, it minimises the spectral abscissa of the closed-loop system in order to nd parameters for a stable controller. Then, these parameters are used as a starting point to minimise the H ∞ -norm. hinfstruct is a deterministic technique which does not involve any random elements. However, extra starting points can be randomised upon request.

hifoo (H-Innity Fixed-Order Optimisation) is a software package that can be run in Matlab. hifoo is described in [START_REF] Burke | [END_REF], [START_REF] Gumussoy | Timings for numerical experiments on benchmark examples for xed order H-innity controller design[END_REF] and [START_REF] Gumussoy | Multiobjective robust control with HIFOO 2.0[END_REF]. The later optimisation problem can be written as minimise

A K ,B K ,C K ,D K T zw ∞ .
When hifoo has nished the optimisation, the controller with the best H ∞ -norm of the three candidates is the output. Due to the randomisation of the initial points and the randomisation in gradient sampling phase, the experiments are not repeatable with the same result each time. In [START_REF] Gumussoy | Timings for numerical experiments on benchmark examples for xed order H-innity controller design[END_REF], the authors suggested that hifoo is evaluated by running it ten times on each problem and choosing the best result. The hifoo is also shown very well compared to several others methods, and it has also been used several times in dierent applications.

It is noted that the hinfstruct should be initialised with two extra starting points when comparing its performance with hifoo, since this last one uses three randomised starting points.

Results and Discussion

The comparison between studied methods and available toolboxes is giving according to Table 4.2. From the system AC2 and EB4, Note that all methods achieve good results but without forgetting that systems order is between 4 ≤ n ≤ 20. In general, the examined methods achieve a comparable results in most cases.

hifoo performs well but the best results are obtained by hinfstruct. From the system IH where the orders of the controller are n K = 5 and n K = 7, hinfstruct achieves better results than the other methods. Then, we can say that this method keeps good properties when augmenting the order of the controller. For the controllers obtained from the system JE3, hinfstruct and hifoo show good performance. However, as said previously, the studied methods encounter numerical problems due to the high number of decision variables when n ≤ 24.

Fixed-Order vs. Reduced-Order Controllers

As shown in Section 3.2, there are two dierent ways to obtain a low order controller even if the model has a high order. These two approaches are evaluated here on some

COMPl e ib problems. Then, results are presented in Table 4.3. Since all methods in this evaluation attempts to minimise the closed-loop H ∞ -norm T zw ∞ , this latter is compared at several orders for dierent examples. The comparison is done with the best results given by each way. Note that, for very low orders, the xed-order approach deals better and gives a smaller H ∞ -norm. However, for medium or higher orders, the order reduction methods gives better results than the xed-order one. This can be explained

Quarter Vehicle LTI-Model

After having introduced the LTI model, an H ∞ -control law is designed with respect to the performance specications. Once the full-order controller is obtained, the controller order reduction method based on FWBT (Section 3.2) is used to reduce the controller order. The obtained reduced-order controller is rst compared with a similar reducedorder one based on the unweighted BT. Then, a comparison between the several variants of the FWBT methods is given according to Section 3.2. Results are discussed in both frequency and time domain. Finally the robustness of the reduced-order controller is analysed by considering uncertainties on the system physical parameters.

System Modelling

A quarter-vehicle model (Figure 4.1) can be used to represent the vertical physical behaviour for the suspension system.

m us zus = k s (z s -z us ) + c s ( żs -żus ) + k us (z r -z us ) -f s (4.2)
The parameters values, chosen accordingly to [Sammier, 2001] are presented in Ta-

Performance Specication

The main objective of an active suspension system is to improve the comfort in the vehicle simultaneously to the performance on road handling. The passenger comfort can be improved by isolating the vibrations transmitted from the road surface. It is evaluated considering the response of the vehicle chassis with excitation coming from the vertical disturbances. According to industrial performance specications [Sammier, 2001], the main objective is to mitigate the resonance peak in low frequencies without degradation at the high frequencies:

• The maximal gain of the transfer Z sacc /Z r should be limited to 2 in the frequency range [1 ; 5] Hz, where Z sacc = Z s .s 2 is the Laplace transform of zs .

• The road holding is evaluated from the unsprung mass (wheel) oscillations with respect to the road prole. The maximal gain of the transfer Z us /Z r should be 2 in the frequency range [1 ; 15] Hz.

H ∞ -Controller Design

The standard H ∞ -control problem is formulated according to the Figure 4.2 where the vector of output variables to be regulated is chosen as Hedrick andButsuen, 1990, Sammier, 2001]) for the transfer Z s /Z r and Z us /Z r in specic frequencies. The weighting function W u penalises the control signal for frequencies higher than 20 Hz. The transfer Z s /Z r and Z us /Z r are templated by the inverse of W zs and W zus respectively, in order to reduce their gains in the low frequencies ([1 ; 10] Hz and [5 ; 15] Hz respectively). The weighting function W zs is chosen constant. Hence, W z = diag(W zs , W zus , W zs ).

: z = [z s z us zs ] T . Then, W = z r , U = u = f s , Z = [u T W T u z T W T z ] T , Y = (z s -z us ).
Note that W zs and W zus are second order lters (for more details see [Sammier, 2001]) which increase the order of the plant. Then, the Figure 4.3 gives the weighting functions frequency representation.

Considering the above specications, a 14-order H ∞ -controller is obtained. It has a high order considering such systems. Then, a reduction step will be of great interest.

Simulation Results and Discussion

The simulations are given in this section to evaluate the performance of the reducedorder H ∞ -controller. Experiments are performed on a intel core i7 CPU (3.07 GHz)

running with Matlab 2012b.

Then, the following systems are compared:

• Passive suspension system (open-loop).

• Active suspension system based on 14 order H ∞ -controller.

• Active suspension system based on reduced 5 order H ∞ -controller.

Note that the designed full-order controller is singular. According to Remark 2 (page 65), a small singular perturbation is applied to D with ε = 0.08.

As shown in Section 3.2, method based on the FWBT is used to reduce the order of the LTI-controller. Then, a rst evaluation is represented in Table 4.5. where the applied method is compared to the classical BT method that reduces the H ∞ -norm of the absolute error ∆ K .

Note that, the closed-loop system is unstable below the tenth order for the BT reduced order controller. However, the FWBT method maintains the stability of the closed loop

Frequency Domain Analysis

In the Figure 4.4, the frequency response magnitude of Z s to the road disturbance Z r is plotted. Note that the template γ/W z designed in Section 4.2.2 limits the amplication of this transfer in the frequency range [1; 10] Hz where the human sensitivity to vertical vibrations is important. The other techniques are less eective. Moreover, the reduced-order H ∞ -controller by Enns' method is very closed to the full-order one. However, the Lin and Chiu's method amplies the response in [1; 5] Hz.

Time Domain Analysis

For the time domain analysis, the road prole Z r is set with a bump of 0.01 m x 2 m and the vehicle is travelling with a constant velocity of 30 km/h. The response of the several designed controllers is shown in response conrms the contribution of the H ∞ -control design. In fact, the wheel is stabilised rapidly without overtake on the suspension deection, The most important observation is that reduced-controller by Enns gives similar performance as the fullorder controller on the diverse controlled outputs. However, with the Lin and Chiu's method, the controller fails to steer the system correctly, which is an expected result according to the increased weighted error.

Robustness Analysis

Variations on model parameters are considered as follows: Those parametric uncertainties can be expressed in the block ∆ r = diag(δ s , δ us , δ k , δ kp ) (Figure 4.9) and are used to analyse the RS . For NP analysis, the input and the output weights P i and P o respectively, are taken the same as the given ones for performance specication, i.e. W z = diag(W zs , W zus , W zs ) for the outputs, but not W u . In addition, a normalisation is performed to obtain:

k = k(1 + p k δ k ) k p = k(1 + p kp δ kp ) m s = k(1 + p s δ s ) m us = k(1 + p us δ us ) where {p k , p kp , p s , p us } ∈ [-1; 1], {δ k , δ kp , δ s , δ us } ∈ [ 0; 1].
||N ew || ∞ = ||P o HP i || ∞ = 1
Then, the upper µ measurement of the transfer matrix N (Figure 4.10) determines the admissible interval for the RP. For the presented example, the uncertain model parameters are initially chosen as follow: m s = 415 ± 15% kg, m us = 15 ± 15% kg, k s = 22000 ± 5% N/m, and k us = 270000 ± 5% N/m. A rst test is shown in Figure 4.11 where a set of closed-loop systems is generated. It consists of the nominal system with a frequency-dependent amount of uncertainty. The stability margin analysis indicates that the reduced order closed-loop remains stable and tolerates variability in k, m and c. More consistently, the µ-analysis tool is used to get a reliable test on the controller robustness. Then, the NP, RS and RP problems of the reduced-order closed-loop system are evaluated: 4.12 shows that the maximal measurement µ is lower than 1 which means that nominal performance are reached. This was expected because the nominal performance are the same as the closed-loop performance. given here. Then, comparison is done with the best results given by each way. From the fact that, this example gives details about performance specication, then a frequency and time domain analysis can be performed. This allows a better evaluation compared

∆

N G K P i P o ∆ r ∆ f z ∆ v ∆ w e

LPV-Model

In this section, the given method to reduce the LPV-controller is applied on a semiactive automotive suspension presented in [START_REF] Do | Multiobjective optimization by genetic algorithms in H ∞ /LPV control of semi-active suspension[END_REF]. Actually, when suspension modelling and control are considered, the vertical quarter car model is often used. This model allows to study the vertical behaviour of a vehicle according to the suspension characteristic (passive or controlled). Figure 4.18 shows the so-called vertical quarter car. Then, the dynamical equations of the system are given by

m s zs = -k s z def -F mr m us zus = k s z def + F mr -k t (z us -z r ) (4.4)
where F mr is the magneto-rheological force generated by the semi active suspension. According to the non-linear model of Guo [START_REF] Guo | Dynamic modeling of magnetorheological damper behaviors[END_REF], F mr can be expressed as follows

F mr = a 2 żdef + v 0 x 0 z def + a 1 tanh a 3 żdef + v 0 x 0 z def (4.5)
with z def = z sz us is the damper deection (must be measured or estimated) and żdef = żsżus is the damper velocity. Parameters a 2 , a 3 , v 0 and x 0 are constant, and a 1 is the controllable force such that a 1 ∈ [a 1min ; a 1max ].

ρ 1 = tanh a 3 żdef + v 0 x 0 z def , c mr = a 2 , MR damping coecient, k mr = a 2 v 0 x 0 , MR stiness coecient.
Then, a state-space representation can be given by considering the state vector x s = [z s żs z us żus ] T and the exogenous input w = z r , as follows

     ẋs = A s x s + B s ρ 1 a 1 + B s1 w z = C s1 x s + D s1 ρ 1 a 1 y = C s x s (4.6) with A s =      0 1 0 0 -ks+kmr ms -cmr ms ks+kmr ms cmr ms 0 0 0 1 ks+kmr mus cmr mus -ks+kmr+kt mus -cmr mus      , B s =      0 -1 ms 0 1 mus      , B s1 =      0 0 0 kt mus      .
The measurement output is y = z sz us , and the controlled outputs are chosen as z = [z s z s ] T , respectively the acceleration and the displacement of the sprung mass. However, two constraints must be satised 1. The control signal a 1 must be positive (dissipative constraint)

2. The input matrices B s ρ 1 and D s1 ρ 1 must be constant to satisfy the LPV-H ∞ synthesis assumption.

The passivity problem is solved by dening a new control signal u = a 1 -F 0 where F 0 is the mean value of a 1 (F 0 = (a 1max -a 1min )/2). Then, the problem of the passivity on a 1 is recast to a simple saturation problem on u

(u ∈ [-F 0 ; F 0 ]). With these modications, (4.7) yields        ẋs = (A s + B s2 ρ 1 C s2 x s C s2 )x s + B s ρ 1 u + B s1 w z = C s1 x s + D s1 ρ 1 u y = C s x s (4.7)
where

B s2 = 0 -F 0 ms 0 F 0 mus T and C s2 = -a 3 v 0 x 0 a 3 -a 3 v 0 x 0 -a 3 T .
To overcome the second problem, [START_REF] Apkarian | A convex characterization of gain-scheduled H ∞ controllers. Automatic Control[END_REF] proposes to add a strictly proper lter F to make the controlled output matrices independent of the scheduling parameters

F : ẋf u = A f B f C f 0 x f u c (4.8)
Then, by dening ρ 2 = ρ 1 C s2 xs and x = (x s x f ) T the system (4.7) can be represented as

     ẋ = A(ρ 1 , ρ 2 )x + Bu c + B 1 w z = C 1 (ρ 1 , ρ 2 )x y = Cx (4.9)
where

A(ρ 1 , ρ 2 ) = A s + ρ 2 B s2 C s2 ρ 1 B s C f 0 A f , B = 0 B f , B 1 = B s1 0 , C 1 (ρ 1 , ρ 2 ) = C s1 ρ 1 D s1 C f and C = C s 0 .

H ∞ -Controller Synthesis

By considering the LPV-model (4.9), an H ∞ -controller is designed to guarantee the internal closed-loop stability and to satisfy some required performance. In fact, the main objective and challenge of a controlled suspension system is to improve the comfort for car passengers simultaneously to the performance on road holding. The passenger comfort can be improved by isolating the vibrations transmitted from the road surface.

Then, the frequency response from the road prole z r to the vehicle chassis acceleration zs must be kept small in the low frequency range. Furthermore, the road holding is evaluated from the unsprung mass (wheel) oscillations with respect to the road prole. This transfer should be kept small at high frequencies.

Then, W zus is designed as

Numerical Issue

The proposed method requests the solution of two Lyapunov inequalities with an innite number of constraints. These sets of innite LMIs can be solved by gridding techniques.

Then, some approximations must be made by gridding the set ∆ ρ with nite number L of points {ρ i } L i=1 [Lee, 1997]. Moreover, the innite variables P Ω (ρ) and Q Ω (ρ) in LMIs (3.100), (3.101) are approximated by combinations of scalar basis functions such as P Ω (ρ) = N P j=1 φ j (ρ)P j 0

Q Ω (ρ) = N Q j=1 ϕ j (ρ)Q j 0
where P j = P T j , Q j = Q T j . There is a large freedom in the choice of basis functions [Wood, 1995]. For this example, the following choice is made:

{φ j } 13 j=1 = {ϕ j } 13 j=1 = {1, ρ 1 , ρ 2 , ρ 2 1 , ρ 2 2 , ρ 1 ρ 2 , ρ 1 ρ 2 2 , ρ 2 1 ρ 2 , ρ 2 1 ρ 2 2 , ρ 3 1 , ρ 3 2 , ρ 1 ρ 3 2 , ρ 3 1 ρ 2 }.
The main consequence of this approximation is that the number of LMIs to be solved is nite and is 2L(2 s+1 + 1) where s is the number of parameters, i.e. s = 2 and L = 13.

The full-order controller is designed using the procedure developed in [START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF].

Then, an 8-order controller K full is obtained. The proposed method is used to reduce K full . Then, K FWBT is produced.

To test its eectiveness, a comparison with the method developed in [START_REF] Widowati | Controller reduction of parameter dependent systems[END_REF] (LPV balanced truncation) namely K BT , is performed.

The feasibility problems (3.100)(3.101) are convex. Using (Matlab LMI Control Toolbox), controllers are reduced to the 5th order obtained heuristically by trial-and-error approach. Therefore, a frequency and time analysis are performed.

Results and Discussion

The rst evaluation is represented in Table 4.9 where rst the assumption (3.108) and

(3.109) are checked. Note that for 25 frozen values of (ρ 1 , ρ 2 ), the rank assumption is satised for all these points. This fact conrms Remark 3.3.2 and allows us compute an upper bound. Precisely, Table 4.10 express the upper bounds and H ∞ -norm of the error values. These results evince that the upper bound is correctly positioned (the gap is positive). However, we note that this upper bound is not tight to error H ∞ -norm. Frequency analysis:

The Bode diagrams at several frozen values of ρ 1 and ρ 2 (25 points) of the three transfer function T zs , T zs and T zus are shown respectively in Figure 4.20,4.21 and 4.22. In fact, the frequency behaviours of the chassis position z s and acceleration zr are chosen to be analysed in order to observe the comfort performance regarding the road prole input z r . The wheel position signal z us is also analysed to test the road holding. Then, the weighting functions W zs and W zs designed in Section 4.3.2, limit the amplication of the previously cited transfers in low frequency range (around [1 ; 10] Hz). In fact, the human sensitivity to vertical vibrations is important in this frequency range [START_REF] Do | An LPV control approach for comfort and suspension travel improvements of semi-active suspension systems[END_REF].

For this reason, the frequency interval of the proposed frequency limited FWBT method is chosen as [1 ; 8] Hz.

In Figure 4.20, note the reduced-order closed-loop system produced by FWBT approximate well the full-order closed-loop system in the chosen frequency range [1 ; 10] Hz. In this interval, the reduced-order closed-loop system produced by the unweighted BT fails. In fact, an important gap appears a 2 Hz and 3 Hz which is exacted as th BT is known to guarantee good approximation at high frequency. The same comment is given in Figure 4.21 where FWBT ts the full order closed-loop system in all the shown range unlike BT method that miss the peak around 2 (a resonance frequency).

These results are more explicit when observing Figure 4.22. Indeed, FWBT gives a good approximation when BT fails (2 Hz and 8 Hz). The other important fact stated by these results, is the sensitivity against the parameters variation. Actually, a dispersion of ρ 1 and ρ 2 values is induced by the reduction step. This degradation is expected since the given application is a qLPV system where ρ 1 and ρ 2 are depending on the However, this loss when reducing is under control for two reasons: the rst one, the stability of reduced-order closed-loop is preserved and the error is guaranteed limited.

The second one is that this dispersion is weak in the required frequency range. Thus, performance are not aected.

As Bode diagram for non-linear systems is not possible, a pseudo-Bode plot is then proposed in [Poussot-Vassal, 2008]. Remark Time analysis:

In the time domain, the several controlled suspensions are travelling a bump of 0.01 m x 2 m for a vehicle speed 8.3 m/s (i.e. 30 km/h). It is observed that the time response conrms the contribution of the FWBT reduction method. In fact, in Figure 4.23, the chassis is stabilised rapidly (1 sec. after the perturbation) without overtake on the suspension unlike the suspension with K BT . This observation, preserves the required `comfort' performance. Moreover, for z us , the wheel equipped with a K FWBTsuspension, keeps almost the same prole of the road although its variations which which respects the `road handling' performance. The K BT -suspension generate an innite perturbation just after the bump (after 1 sec.). Note also, that temporal test draw two output signals (chassis and wheel positions) regarding the input (the road prole) and by the way there will be just one plot of each transfer besides the several plots in the frequency responses. The dierent methods to obtain a xed-order controller have been rst tested: the Apkarian and the Ankelhed's algorithms are implemented and tested on some COMPl e ib benchmark problems. From the fact that the main objective is to minimise T zw ∞ , this H ∞ -norm of the closed-loop system is evaluated for dierent orders. First analysis, shows similar good results for the two techniques. On the contrary, they failed for some models with quit high order which was expected regarding the big number of decision variables. This evaluation has been completed by an additional comparison with two available Matlab toolboxes. These latters show a bit better result in most cases. Evaluation on COMPl e ib benchmark has been nished by a macro test where best results in the direct xed order way have been compared the ones of the indirect way. As expected, the xed order way show some numerical problems in many cases unlike the order reduction approach which give a stable results even if the order is big.

It has been noted also that the xed order way give better results than the reduced order approach for very low orders (0 or 1) . However for higher orders (> 2) reduction the best results are obtained by the reduced order controller.

Thereafter, The indirect way in turn is evaluated. An automotive-suspension with a controlled damper is consider. The system is rst represented with an LTI-model.

Then, performance specications have been achieved with a full-orderH ∞ -controller.

This latter has been the subject of an order reduction step where the FWBT technique is used. As shown in Section 3.2, various versions have been used to perform the order reduction. Then, these variants are implemented and tested through this system.

The shown results have conrmed their eectiveness and comparable good results have been obtained for each method. In fact, a remarkable gain in time and memory has been observed. Finally, a time and frequency domain analysis have been performed.

This evaluation has conformed the eectiveness of the implemented methods, Indeed, the performance specication have been respected even the controller is reduced to one third of its order.

Finally, the most important contribution concerning the H ∞ -LPV-controller order reduction, has been validated in the last section. The control of a car magneto-rheological damper is considered. This semi-active suspension system has been written under an LPV-model. Then, an H ∞ -LPV-controller has been designed. Thereafter, the developed method has been implemented and a reduced order controller is derived. Similar examination to the previous example is done by performing a time and frequency domain analysis. Subject to bump road prole, the controlled semi-active suspension shows good performance even the controller order is reduced. Unlike the xed order way, the developed method guarantee an upper bound of the approximation error. This latter has been computed and shown to be always bounded by the introduced upper bound.

Conclusion and Perspectives

The work presented in this dissertation is related to the H ∞ -LPV-controller order reduction. This latter consists of the design of a robust reduced-order LPV-controller for LPV-systems. The order reduction issue has been very fairly investigated, However, the case of LPV-control design is slightly discussed. This thesis focuses primarily on two topics : How to obtain an LPV-reduced-order controller even the high order generated by the classical synthesis and how this reduced order controller can deal with a practical engineering problem (semi-active suspension control).

In view of this, The order-reduction topic and the H ∞ -synthesis theory have been widely

studied in this thesis. This study, has allowed the development of a new method for H ∞ -LPV-controller order reduction.

In Chapter 1, Denitions and basic notions on system and control theory have introduced. Mathematical framework has been well detailed and required notions have been detailed. Essential notions in the linear algebra have been rst recalled. Then, the interested systems classes namely the LTI and LPV systems, have been described.

In Chapter 2, H ∞ -control design has been investigated. Firstly, the LTI-case has been considered. The stability and the performance requirements have been shown

guaranteed by the H ∞ -controller. The examination of this design methodology has shown its drawback regarding the high order of the resulting synthesised controller.

One way to tackle this problem was to set the order when synthesising the controller.

This approach has been examined in this chapter. Finally, the LPV-case is considered where the stability and the performance issues have been also treated. The same observation has been stated i.e. the high order of the LPV-controller. Then, reducing this latter has conducted to dene the main objective of this thesis.

In Chapter 3, the primary contribution of this thesis has been given. To achieve this latter, The model order-reduction matter has been rst introduced and discussed.

Methods based on singular value decomposition have been then studied and compared.

Their analysis has shown their usefulness for the control order reduction (instead of the model order reduction). Hence, they have been used to obtain H ∞ -LTI-reduced-order controllers. This study has completed the main summery scheme on how to get a low-order controllers given in Figure 1. Indeed, comparison work has been planned in the last chapter to evaluate every edge of the scheme for the LTI-case. Inspired by these stated developments of H ∞ -LTI-controller order reduction, a new method has been derived to achieve this objective for H ∞ -LPV controllers. A big advantage of this approach was to perform the order reduction in oriented limited frequency range.

tested in all the possible cases, e.g. chose a very low frequency and test the BT or chose a very high frequency range and test the SPA.

• It has shown in Section 2.1 that the weighting functions derived from the performance specications are the primary reason to increase the controller order. Then, the link between the several proposed methods and these weighting functions should be investigated with the aim to limit the order before the H ∞ -controller design.

• Almost all the proposed work in this thesis concerns the H ∞ -control. An extension to the H 2 -control seems to be feasible and could give an interesting contribution. Moreover, mixed H ∞ /H 2 control design problem could be treated and reduced-order controller procedure for this type of control design would be derived.

• The proposed method in Section 3.3 concerns the order-reduction way for LPVcontrollers. Even if research for a such complicated problem, the xed-order way for H ∞ -LPV-controller should be investigated and evaluated.

• Passivity notion is essential in dynamical systems. Unlike the proprieties like stability and performance, this passivity criterion has not been discussed. It would be useful if a study could be conducted in this direction. Some recent contributions are treating this issue [Li et al., 2015, Li and[START_REF] Li | [END_REF].

• Even thought the design of an H ∞ -LPV controller is considered robust against the parameters variation, it is more consistent to consider uncertainties for LPVsystems. This uncertain LPV-controllers would necessary have a higher order, Then, approximating such controllers would be very useful.

  values, i.e. ρ ∈ ∆ ρ . V ρ : Set of vertices of ∆ ρ , i.e. V ρ = vert {∆}. ∆ v : Set of the parameter derivative values, i.e. ρ ∈ ∆ v . ω i : The i th vertex of a polytope. S n 0 : cone of n × n symmetric positive denite matrices. ≺ ( ), ( ) : Negative (semi-negative), positive (semi-positive) denite. S * : Complex conjugate transpose of S. A -T :(A -1 ) T = (A T ) -1 , inverse transpose of A.

: 2 :

 2 The ith frequency weighted generalised Hankel singular value. L Time domain Lebesgue space = the space of square integrable signals in R.

  scalars a and b. arg min f : Minimising argument of f . diag(A) : Diagonal elements of A. H(f (x)) : Hessian of f (x). H δ (f (x)) : Approximated Hessian of f (x).

  Figure 1: Low Order Controller Design

Figure 2 .

 2 Figure 2.1: Closed-loop for Stability Analysis

  2.2), and let N R and N S denote orthonormal bases of the null spaces of B T 2 , D T 12 and

Figure 2 . 4 :

 24 Figure 2.4: LFT of the uncertain model

  Then we construct the modication matrix D m in order to ensure the positiveness of L(D + D m )L T by following these steps: 1. Compute the eigenvalue decomposition D = Q DQ T . (2.41) 2. Then, compute the modication matrix D m = QEQ T .

3 ( 2

 32 .46) which after multiplication by R Y from the left and by R T Y from the right becomes

  included in Robust Control Toolbox of Matlab. Remark Gradient and Hessian Computation Before starting to compute the derivatives, rst we compute the Jacobian of the function e(x a ) = X a Y a -I, J(x a ) = (Y a ⊗ I)D (I ⊗ X a )D 0 (2.63) where D denotes the transformation mappping from vech to its vec representation as shown in Denition 1.2.5. The gradient of the Lagrangian function is given by

Figure 2 .

 2 Figure 2.5: H ∞ -Controller Synthesis Methods Comparison

  Figure 2.6(b)) in which the controller is time-invariant and without adaptation even the uncertainties values are changing. However in the LPV-framework the LPV-controller is updating its values whenever the model changes.Even if guarantees for overall stability of the designed LPV-controllers have not been

Figure 2 .

 2 Figure 2.6: Robust Controller

Denition 2 .

 2 3.1 (Quadratic Stability) System (1.35) is said to be quadratically stable if the positive denite quadratic form V : x → x T P x, P ∈ S n 0 (2.67) is a Lyapunov function for (1.35). A such Lyapunov function is often referred to a common Lyapunov function or a parameter-independent Lyapunov function.

Figure 2 .

 2 Figure 2.7: LPV-controllers types

Figure 3

 3 Figure 3.1: Controller Order Reduction Scheme[START_REF] Anderson | Controller reduction: concepts and approaches[END_REF]

2 ) 1 .Figure 3 . 2 :

 2132 Figure 3.2: Flowchart of Model Order Reduction Methods [Antoulas, 2005a]

  .13) where G and G r are the full and the reduced order model transfer functions re-

Figure 3 .Figure 3 . 3 :

 333 Figure 3.3 shows the model Hankel singular values HSV that/whose can give a rst idea about the order of the reduced order model by eliminating the smallest HSV. Remark that the rst ten HSV are quite bigger than the rest of the values. Then, when truncating, the rst ten states are kept. Figure 3.4 shows the model frequency response, where

Figure 3 . 4 :

 34 Figure 3.4: Frequency Response of the Building Model.

  denitions of S(w) and P in(3.53) and(3.45), and from the fact that H(ω 1 )H(ω 2 ) = H(ω 2 )H(ω 1 ) for any ω 1 , ω 2 ∈ R, one obtains S(ω)BB T H * (φ)dφ.

  )W c (w)H * (φ)dφ.(3.56) where W c (w) = S(w)BB T + BB T S * (w). Since A is asymptotically stable, P (w) is the solution to the Lyapunov equationAP (w) + P (w)A T + W c (w) = 0 (3.57)Therefore, the frequency weighted Gramian P Ω in (3.47) is obtained by solving

  Given the Lyapunov equation(3.58) and(3.59), let W c (Ω) and W o (Ω) have the following EVD:W c (Ω) = M ΛM T = M diag (λ 1 , . . . , λ n K ) M T , (3.61) W o (Ω) = N ∆N T = N diag (δ 1 , . . . , δ n K ) N T , (3.62) where M M T = N N T = I n with |λ 1 | ≥ . . . |λ n | ≥ 0 and |δ 1 | ≥ . . . |δ n | ≥ 0.From the fact that W c (Ω) and W o (Ω) are both symmetric, such decompositions exist. Let u ≤ n and v ≤ n be respectively the ranks of W c (Ω) and W o (Ω). Based on these denitions, let us dene the two quantities:

Figure 3

 3 Figure 3.6: LFT of the Controller K

( 3

 3 .68) should be stable and should satisfy the following inequalities:

Figure 3 . 7 :

 37 Figure 3.7: From Full to Reduced Order H ∞ -controller

Figure 3 .

 3 Figure 3.10: FWBT for controller order reduction

Figure 3 .

 3 Figure 3.11: Closed-loop Scheme

  ρ), O(ρ), P(ρ) and Q(ρ) are the solutions of (3.78), (3.79), (3.80) and (3.81) respectively.

( 3 .

 3 101) is performed instead of(3.92) and(3.93). In addition, the modied frequency limited Gramians in(3.98) and(3.99) are used instead of the ordinary ones dened in(3.98) and(3.99). In fact, Lemma 7 shows that the generalised Gramians could be an approximation of the ordinary Gramians. The transition to generalised Gramian framework might induce less accurate approximation but the order-reduction error still bounded. Finally the expression of the upper error bound given in(3.110) is found according to this

  , 0, . . . , 0), similar to the previous proof Proof Conclusion In this chapter, order reduction issue is discussed. The model order-reduction problem is rst stated in Section 3.1 where an open-loop conguration is considered. Approaches based on the Singular Value Decomposition SVD are introduced. Specically, the Balanced Truncation BT and the Singular Perturbation Approximation SPA are

  performed on 11 systems. The problems included in COMPl e ib library are of dierent order from 4 to 24 states. The systems chosen for the evaluation are: Aircraft models (AC2 and AC5 and AC18), Helicopter models (HE1 and HE3), Jet Engines model (JE3), Euler Bernoulli Beams (EB4), Reduced Order Control (ROC1), Academic test problems (NN11) and mathematical model of position and velocity control for a string of high speed vehicles (IH).

  , the disturbance, the input and the measured output signals are respectively : x = [z s z us żs żus ] T , w = z r , u = f s and y = [z s z us zs ] T respectively.

Figure 4 . 2 :

 42 Figure 4.2: H ∞ -control Scheme for the Active Suspension

  Figure 4.3: Weighting Functions

Figure 4 .

 4 Figure 4.5 shows the performance of the full order and the reduced-order controllers with respect to the road holding specications.The full-order controller and the reduced-order controller by Enns mitigate the frequency response in [8; 14] Hz which is amplied in the passive case. In this frequency range, the maximal gain is reduced to 1.5 which responds to the performance specications in Section 4.2.2 as shown in Figure 4.7 and Figure 4.6.

  Figure 4.8. It is observed that the time

Figure 4 . 4 :

 44 Figure 4.4: Bode-diagram of the Transfer Function Z s /Z r

Figure 4 .

 4 Figure 4.5: Bode-diagram of the Transfer Function Z us /Z r

Figure 4 .

 4 Figure 4.6: Frequency Response of Z r to Z s

  K 14 Reduced-order K 5

Figure 4 .

 4 Figure 4.7: Frequency Response of Z r to Z us

Figure 4 .Figure 4

 44 Figure 4.8: Time Responses on a bump of 0.01x2 m

  Figure

Figure 4 .

 4 Figure 4.10: Real and ctive uncertainties of ∆

Figure 4 .

 4 Figure 4.11: K 5 -time and Frequency Analysis with Uncertainties

µ

  of RS = 1.2), which means that these initial uncertainty margins do not guarantee the stability performance. Then, a new uncertainty range are redened as: m s = 415 ± 10% kg, m us = 52 ± 10% kg, k s = 22000 ± 4% N/m, and k us = 270000 ± 4% N/m. The new upper µ measurement value show that performance robustness are satised for these margins (Figure 4.15).

Figure 4 .

 4 Figure 4.12: Upper and Lower µ for Nominal Performance

Figure 4 .

 4 Figure 4.13: Upper and Lower µ for Robust Stability

Figure 4 .

 4 Figure 4.14: Upper and Lower µ for Robust Performance

Figure 4 .

 4 Figure 4.15: Upper and Lower µ for Robust Performance

Figure 4 .

 4 Figure 4.16: Bode-diagram Magnitude of the Transfers T zszr and T zuszr

Figure 4 .

 4 Figure 4.17: Time-response on a Bump of 0.01x2 m

Then

W zus = w zus s 2 + 2 21s 2 + 2 22W

 2222 ξ 21 w 21 s + w 2ξ 22 w 22 s + w zr = 5 × 10 -3 is the road prole gain. Finally, the lter introduced in (4.8) is givenas: F = w f s+w f. It is designed with a large bandwidth to decouple the input and the varying parameters, where w zs = 1, ξ 11 = 0.1, ξ 12 = 1, w 11 = 2π × 1 rad.s -1 , w 12 = 2π × 3 rad.s -1 ,

Figure 4 .

 4 Figure 4.20: Bode-diagram Magnitude of the Transfer T zszr

Figure 4 .

 4 Figure 4.21: Bode-diagram Magnitude of the Transfer T zszr

Figure 4 .

 4 Figure 4.22: Bode-diagram Magnitude of the Transfer T zuszr

Figure 4 .

 4 Figure 4.23: Time-response on a Bump of 0.01x2 m

  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Signal and System Descriptions . . . . . . . . . . . . . . . . . . .

	Chapter 1
	Preliminaries and Background

Contents 2 1.1.1 Signal Spaces and Norms . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 System Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2.1 State-Space Realisation . . . . . . . . . . . . . . . . . . . 4 1.2 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Linear Fractional Transformation . . . . . . . . . . . . . . . . . . . 5 1.2.2 Hamiltonian Matrix and Riccati Equation . . . . . . . . . . . . . . 6 1.2.3 Polynomial Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4 Matrices Vectorisation Operators . . . . . . . . . . . . . . . . . . . 8 1.3 LTI-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 LTI-representaion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Controllability and Observability . . . . . . . . . . . . . . . . . . . 10 1.3.3 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Schur Complement Formula . . . . . . . . . . . . . . . . . . . . . . 12 1.3.5 Bounded Real Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 LPV-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

  , D 21 ) respectively In this formulation, LMIs (2.31a),(2.31b) and (2.31c) replace assumptions A1 to A4 in Section 2.1.2.2 for H ∞ -controller design. Finally the condition (2.31d) seems to allow the design of H ∞ -controllers with lower order. Either the condition(2.31d) is easy to satisfy, however research of xed-order controller is very dicult to resolve. Indeed, this constrain is not convex. and the problem is bilinear.

	T 12 and
	(C 2

  H ∞ -Controller Order Reduction Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . 57 3.1.1 Balancing Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.2 Balanced Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1.3 Singular Perturbation Approximation . . . . . . . . . . . . . . . . 60 3.1.4 Frequency Weighted Balanced Truncation . . . . . . . . . . . . . . 62

3.1.4.1 Enns' Method . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.4.2 Lin and Chiu's Method . . . . . . . . . . . . . . . . . . . 65 3.1.4.3 Varga and Anderson's Method . . . . . . . . . . . . . . . 67 3.1.4.4 Wang's et al. Method . . . . . . . . . . . . . . . . . . . . 67 3.1.4.5 Gawronski's et al. Method . . . . . . . . . . . . . . . . . 68 3.1.4.6 Gugercin's et al. Method . . . . . . . . . . . . . . . . . . 70 3.2 H ∞ -LTI-Controller Order Reduction . . . . . . . . . . . . . . . . 70 3.2.1 Full Order Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2.2 FWBT for Controller Order Reduction . . . . . . . . . . . . . . . . 72 3.2.3 Order Reduction Procedure . . . . . . . . . . . . . . . . . . . . . . 73 3.3 H ∞ -LPV-Controller Order Reduction . . . . . . . . . . . . . . . 75 3.3.1 Gramians and Generalised Gramians . . . . . . . . . . . . . . . . . 75 3.3.2 Frequency-Limited Order Reduction by FWBT . . . . . . . . . . . 78 3.3.3 Frequency-Limited Order Reduction by SPA . . . . . . . . . . . . 84

Table 4 .

 4 1: Evaluation and Comparison between Apkarian's and Ankelhed's methods Sys(n, n u , n y , n k ) rst chooses three random starting points to initialise the algorithm. Then, it tries to nd stabilising controllers before optimising it locally with respect to the closed-loop H ∞ -norm. Both optimisation problems are non-smooth and non-convex.

		Apkarian	Ankelhed
		γ	T zw ∞	γ	T zw ∞
	HE1(4,2,1,0)	0.19	0.16	0.20	0.18
	HE1(4,2,1,1)	0.20	0.17	0.20	0.12
	HE1(4,2,1,2)	0.19 0.16 0.22 0.15
	HE1(4,2,1,3)	0.19 0.16 0.23 0.16
	HE3(8,4,6,0)	0.95 0.92 0.97 0.92
	HE3(8,4,6,1)	0.94 0.93 0.91 0.87
	HE3(8,4,6,2)	0.95	0.93	0.97	0.89
	HE3(8,4,6,3)	0.95	0.94	1.01	0.91
	HE3(8,4,6,4)	0.94	0.93	0.97	0.90
	ROC1(9,2,2,2)	2.63 2.48 1.34 5.24
	ROC1(9,2,2,3)	2.11	1.96	1.28	2.09
	ROC1(9,2,2,4)	1.82	1.71	3.46	3.29
	ROC1(9,2,2,5)	1.78 1.60 3.45 3.28
	ROC1(9,2,2,7)	1.95	1.80	3.42	4.21
	NN11(16,3,5,0)	0.10	0.10	0.10	0.10
	NN11(16,3,5,1)	0.10	0.10	0.11	0.11
	NN11(16,3,5,2)	0.10	0.10	0.11	0.11
	NN11(16,3,5,3)	0.10	0.10	0.11	0.10
	NN11(16,3,5,4)	0.10	0.10	0.12	0.11
	NN11(16,3,5,5)	0.10	0.10	0.41	0.17
	JE3(24,3,6,0)	9.10 8.60 5.10 5.10
	JE3(24,3,6,3)	N.P	N.P 2.90 2.89

hifoo

Table 4 .

 4 2: Comparison between hinfstruct, hifoo, Apkarian's and Ankelhed's Meth-

	ods.
	Sys(n, n u , n y , n k )	Apkarian	T zw ∞ Ankelhed hinfstruct	hifoo
	AC2(5,3,3,0)	0.11	0.11	0.11	0.11
	AC5(4,2,2,0)	658	660	665	669
	AC5(4,2,2,3)	665	662	658	643
	EB4(20,1,1,0)	N.P	2.46	2.06	2.06
	EB4(20,1,1,3)	1.81	1.80	1.82	1.82
	IH(21,11,10,0)	4.45	2.34	1.59	1.90
	IH(21,11,10,1)	4.12	1.96	1.80	1.80
	IH(21,11,10,3)	3.69	1.97	1.57	1.74
	IH(21,11,10,5)	4.01	1.86	1.15	1.69
	IH(21,11,10,7)	3.88	1.75	0.79	1.72
	JE3(24,3,6,0)	8.6	5.10	5.10	5.10
	JE3(24,3,6,3)	N.P	2.89	2.90	2.89
	by the fact that, the number of decision variables is increasing when n K increases for
	xed-order case.

Table 4 .

 4 3: T zw ∞ in Direct and Indirect Low-order Approaches SYS(n, n u , n y , n K ) Fixed-order controller Reduced-order controller s zs = k s (z usz s ) + c s ( żusżs ) + f s

	AC2(5,3,3,0)	1.01	2.16
	AC2(5,3,3,1)	0.12	0.43
	AC2(5,3,3,2)	0.24	0.21
	AC2(5,3,3,3)	0.23	0.22
	AC2(5,3,3,4)	0.14	0.14
	AC2(5,3,3,5)	N.P	0.14
	HE1(4,2,1,0)	0.16	0.55
	HE1(4,2,1,1)	0.12	0.15
	HE1(4,2,1,2)	0.12	0.08
	HE1(4,2,1,3)	0.15	0.08
	HE1(4,2,1,4)	N.P	0.07
	AC18(10,2,2,0)	N.P	140.33
	AC18(10,2,2,1)	N.P	48.28
	AC18(10,2,2,2)	N.P	17.80
	AC18(10,2,2,3)	2.09	19.74
	AC18(10,2,2,4)	2.34	4.51
	AC18(10,2,2,5)	2.48	4.99
	AC18(10,2,2,6)	3.28	4.98
	AC18(10,2,2,7)	13.4	10.30
	AC18(10,2,2,8)	13.3	4.31
	AC18(10,2,2,9)	13.56	2.01
	AC18(10,2,2,10)	N.P.	2.02

m

Table 4 .

 4

	4: Parameter Values

Table 4 .

 4

	Controller order	12	8	5	3
	Enns	9.5	6.9	1.4 x 10 2 2.6 x 10 2
		stable	unstable	stable	unstable
	Lin & Chiu	9 x 10 7	6.9 x 10 10 5 x 10 10	6.8 x 10 10
		stable	stable	stable	stable
	Varga & Anderson	3 x 10 6	1.4 x 10 9	2 x 10 10	3.5 x 10 10
		unstable	stable	unstable	unstable
	Wang	1.5 x 10 3 3.6 x 10 5	9 x 10 8	7.6 x 10 9
		stable	stable	unstable	unstable

7: Closed-loop Stability of Reduced-order Controllers and Error Approximation and Lin and Chiu's methods are elected to be compared in frequency and time domain for the order 5.

Table 4 .

 4 Several Frozen Values of (ρ 1 , ρ 2 )

	H H ρ1	H	H H ρ2	-1.0	-0.5	0.0	0.5	1.0
		0		(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)
	0.25	(14/14 ; 14/14)	(13/13 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)
	0.5		(11/11 ; 14/14)	(13/13 ; 14/14)	(11/11 ; 14/14)	(12/12 ; 14/14)	(11/11 ; 14/14)
	0.75	(12/12 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)
	1.0		(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)	(14/14 ; 14/14)

9: Values of rank[B(ρ), B Ω (ρ)]/rank[B Ω (ρ)] ; rank[C T (ρ), C T Ω (ρ)]/rank[C T Ω (ρ)] for

Table 4 .

 4 10: Values of the T zw (ρ) -Tzw (ρ) i,2 and the Upper bound for Several Frozen Values of (ρ 1 , ρ 2 ) ≤ 5.07 1.35 ≤ 3.32 1.33 ≤ 4.08 1.31 ≤ 3.37 1.30 ≤ 3.72 0.25 0.32 ≤ 9.88 0.32 ≤ 9.93 0.32 ≤ 10.38 0.31 ≤10.39 0.31 ≤ 9.80 0.5 0.00 ≤ 1.79 0.00 ≤ 0.13 0.00 ≤ 2.22 0.00 ≤11.81 0.00 ≤ 2.03 0.75 0.31 ≤ 9.97 0.31 ≤ 9.92 0.30 ≤10.58 0.30 ≤9.86 0.29 ≤ 9.89 1.5 1.22 ≤ 3.77 1.20 ≤ 3.04 1.19 ≤ 4.46 1.18 ≤ 2.88 1.17 ≤ 2.90

	H H ρ 1	H	H H H ρ 2	-1.0	-0.5	0.0	0.5	1.0
		0.0	1.36

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Available from: http://www.compleib.de/

 Denition 3.1.1 ([Moore, 1981]) The reachable, observable and stable LTIsystem Σ is called balanced if W r = W o = diag(σ 1 I m 1 , ....., σ q I mq ). (3.4) with σ 1 > σ 2 > • • • > σ q > 0, m i : i = 1, • • • , q are the multiplicities of σ i and

To balance the system, a basis change matrix T can be dened so that ẋ = Ãx + Bu y = C x + Du In this new basis, Gramians are expressed as

Note that Moore & Laub procedure [Moore, 1981] is the historical well-known one to compute the matrix T . It is given as follows 1. Compute Gramians W r and W o of a minimal realisation (A, B, C, D). Note that another algorithm based on the use of the Cholesky factors of the two Gramians is given in [Li, 2000].

Balanced Truncation

Balanced Truncation BT concept was introduced rst by [START_REF] Mullis | Synthesis of minimum roundo noise xed point digital lters[END_REF],

Then, generalisation for system theory is given in [Moore, 1981]. The main idea is to nd another equivalent state space representation where the system states are as reachable as observable. In this new basis, the reduced order model is obtained by states truncating at the desired order. Applied to stable systems, this approach preserves stability with a guaranteed error bound [START_REF] Pernebo | Model reduction via balanced state space representations[END_REF]. For medium order models (dozens of states), this method is very ecient however for very high order model ( 100), the implementation of the BT seems to be expensive. Indeed, dense matrices factorisation is required, which needs a higher memory capacity. at lower order. This rst result conrms the choice of this approach.

In Table 4.6. The following indicators are analysed:

• Rising time (step response): The time it takes for the output to rst reach 90%, from 10% of its nal value, which is usually required to be small.

• Settling time (step response): the time after which the output remains within ± 2% of its nal value, which is usually required to be small.

• Storage memory: the memory allocated to the controller.

Simulation results from a step response applied on the full-order (K 14 ) and the reduced- order one (K 5 ) are shown. Note that the time-rise for K 5 is 61% rise-time of the one obtained with K 14 . The settling-time of K 5 remains close to the K 14 . It is shown also that the required memory for the designed reduced-order controller is reduced to less than the half of the memory for the full order which is very interesting when packaging the controller in a simple calculator. The third evaluation is represented in Table 4.7 where the approximation errors obtained using the several methods are compared. The test on stability of the closed-loop system is also given.

Rise time

Note that although the FWBT by Enns' method can produce unstable reduced-order controller (closed-loop stability), it yields the lower approximation error. The Lin and Chiu's method already gives a stable reduced-order controller. However, the increased weighted error can aects the frequency behaviour of the stabilising controller. Enns

Then, an interconnection between the LPV-model and these weighting functions are presented in

For more details and explanation on H ∞ -LPV-control synthesis, see [Apkarian andGahinet, 1995, Scherer et al., 1997].

The design method for LPVsystems is used like in [START_REF] Apkarian | Self-scheduled H ∞ linear parameter-varying systems[END_REF].

The obtained reduced-order controller is guaranteed stabilising and the approximation error has been proven bounded. performance with comparable results for each method. However, considerable numerical resources and some technical troubles have been encountered when applying them.

Thereafter, the control of an automotive suspension has been concerned. According the actuator technology, two control laws are derived: an H ∞ -LTI controller for an active-suspension and H ∞ -LPV-controller for a magneto-rheological semi-active one.

For the rst one, methods reviewed in Section 3.2 has been evaluated with this system.

Performance regarding the passenger comfort and the road handling are tested and results have shown their eectiveness. The frequency and time analysis have shown also their ecacy comparing to the xed-order methods. Ultimately, the developed H ∞ -LPV-controller method with its two variants developed in the Section 3.

Perspectives

Among the presented work, several improvements are proposed

• The H ∞ -synthesis for LTI-systems is known to ensure the controller robustness.

This latter is investigated in this thesis and a Robustness analysis is performed through the µ-analysis tool in Section 2.1 and Section 4.2. Then, the stability and the performance robustness of the reduced-order controllers have been tested.

However, no guarantee is ensured by such control approach. Recently, various techniques are developing that guarantee a robustness degree while synthesising the controller. The most known one to achieve this objective is the so-called DK-iteration procedure [START_REF] Doyle | Design Examples Using µ-Synthesis: Space Shuttle Lateral Axis FCS During Reentry[END_REF]. It would be interesting if the orderreduction procedure and this controller synthesis would be mixed in order to ensure the robustness even the order-reduction.

• The frequency limited method for LPV-case proposed in Section 3.3 is developed with two variants: the BT way that approximate better in high frequency and the SPA one that approximate error in low frequency. The two variant have tested for medium frequency range. It will be of great interest if the method would be Numerical Considerations

A.1 Gradient and Hessian

A.1.1 Gradient computation, [Helmersson, 2009] Let C i ∈ R n×n and dene

These matrices can be computed easily using the coecients c k . The rst-order derivatives of a coecient c k (Z) in (1.12) are given by

with the matrix E ij dened as

where e i , e j ∈ R n are the ith and jth unit vectors respectively. The expressions in (A.1) can be written in another form using the invariance under cyclic (trace(ABC ) = trace(CAB) = trace(BCA)). For an example where i = j, we get the following relations

which is equivalent to extract two elements from X, Y and E ij and summing them.

A.1.2 Hessian computation, [Helmersson, 2009]

A.2 X, Y and X a , Y a Searching Algorithms Algorithm 4 X and Y searching 1. Initial phase. (b) Dene the variable k and set k = 0. Choose starting value for α, λ 0 , µ 0 , ρ, ρ 0 and the tolerance ε (ρ > 1, 0 < ρ 0 < 1, α = 0.98).

Optimisation phase:

or equivalently:

x k = x k-1 + α dx.

3. Update penalty and multiplier.

Terminating phase: If q(x) > ε, go to the phase 2, else check the following

• If γ k < 0.99 γ k-1 for three consequent iterates, it can be that we are near a local optimum. Finish the program.

• Otherwise, the objective function value is still decreasing, hence we continue the optimisation, i.e. go back to phase 2.

Algorithm 5 X a and Y a searching 1. Initial phase: (b) Dene the variable k and set k = 0. Choose starting value for λ 0 , µ 0 , ρ, ρ 0 and the tolerance ε (ρ > 1, 0 < ρ 0 < 1). 3. Update penalty and multiplier

4. Stopping test: In the case where X a Y a -I n+n K F < ε and the necessary optimality conditions are satised then the algorithm stops progress. Otherwise increase the counter k and go to phase 2.

5. Terminating phase: If X a Y a -I n+n K F < ε, we give the solution x a and try to construct the controller as explained in the next subsection, but if this fails, reduce ε, increase k and go to phase 2.