Didier Theilliol Mickaël Rodrigues

Dominique Sauter

Design of a Robust Polytopic Unknown Input Observer for FDI: Application for Systems described by a Multi-Model Representation

The main goal of this paper consists in fault diagnosis of systems represented by a multi-model approach. Partial knowledge of the system representation around operating points must allow to obtain its dynamic behaviour under modeling errors consideration. The goal of this paper is to decouple modeling errors through a wide operating range by a dedicated polytopic Unknown Input Observer (UIO). The robust polytopic observer allows an optimization of modeling errors matrix distribution and permits to develop a fault diagnosis strategy on a wide operating range. Polytopic observer stability is guaranteed by pole assignment established through Linear Matrix Inequality (LMI). The effectiveness and performances of the method are illustrated in simulation considering a CSTR.

I. INTRODUCTION

The increasing demands for good performance and higher standards of safety and reliability result in more emphasis on fault diagnosis. Fault detection and isolation (FDI) refers to the task of inferring the occurrence of faults in a process and finding the root causes of the faults. Fault diagnosis based on analytical models is developed for exact and uncertain linear mathematical description of the system, several books are dedicated to these topics: [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] and [START_REF] Gertler | Fault detection and diagnosis in engineering systems[END_REF]. FDI on nonlinear systems remains a challenge due to the problem of discriminating between disturbances and faults through a wide range of operating conditions. Different techniques based on an exact knowledge of the nonlinear system allow to generate residuals insensitive to fault by specific decoupling methods [START_REF] Alcorta-Garcia | Deterministic nonlinear observer based approaches to fault diagnosis: a survey[END_REF], [START_REF] Kinnaert | Robust fault detection based on observers for bilinear systems[END_REF] or geometric approach [START_REF] Persis | A geometric approach to nonlinear fault detection[END_REF], [START_REF] Hammouri | A geometric approach to fault detection and isolation for bilinear systems[END_REF]. An attractive alternative to nonlinear modeling problem is to use a multi-linear model approach. This approach is successfully used for some nonlinear systems in control field but rarely in FDI, and consists in partitioning the operating range of a system into separate regions in order to synthesize a global representation. The reader could refer to [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF] for a global review on multiple models strategy, and also for well developed identification method and modeling problems. A polytopic representation is sometimes used in multi-model for nonlinear systems modeling as for example in fault-free case in [START_REF] Angeli | Constrained predictive control of nonlinear plants via polytopic linear system embedding[END_REF], [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF], [START_REF] Wan | Efficient scheduled stabilizing model predictive control for constrained nonlinear systems[END_REF] and [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF], and in faulty case in [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF]. Multiple Model Adaptive Estimation (MMAE) [START_REF] Maybeck | Multiple model adaptive algorithms for detecting and compensating sensor and actuator/surface failures in aircraft flight control systems[END_REF] or Interacting Multiple Model (IMM) [START_REF] Blom | The interacting multiple model algorithm for systems with Markovian switching coefficients[END_REF] introduce a multi-model approach for FDI, but these techniques are developed around an unique operating point (OP). Indeed, This work was not supported by any organization M. Rodrigues, D. Theilliol and D. Sauter are with Centre de Recherche en Automatique de Nancy -CNRS UMR 7039, Université Henri Poincaré, BP 239 -54506 Vandoeuvre Cedex -France.

Phone: +33 383 684 465 -Fax: +33 383 684 462mickael.rodrigues@cran.uhp-nancy.fr these methods consider a particular multi-model approach where each model is dedicated to a specified fault. A fuzzy logic development is also mentioned in [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] for diagnosis but only with Kalman or Luenberger filters which do not permit to deal with modeling errors. Moreover, the activation functions for each model may be not robust to faults. This paper addresses a different method that could allow to detect both actuator or sensor fault in nonlinear systems which are known and exploited around few local models, i.e. we suppose only local identification as in some industrial systems around specific operating point. Compared to methods that could detect both operating conditions evolution and detect, isolate and estimate faults in nonlinear system as proposed by [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF] and [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF], this paper considers time varying modeling errors. Thus, the main contribution of this paper consists in the synthesis of a residual generator based on a Polytopic Unknown Input Observer (PUIO) which generates robust residual against error modeling under fault occurrence. Another kind of UIO was developed in [START_REF] Millerioux | Unknown input observers for messageembedded chaos synchronization of discrete-time systems[END_REF] for communication purposes but not for fault diagnosis and more over we would like to consider possible multiple different unknown inputs matrices which can be re-designed in a unique matrix in order to ensure fault detection during models transitions. The paper is organized as follows:

Section II deals with the nonlinear system representation under a convex set through a multi-model approach with taking into account modeling errors in the state space representation. Section III presents an optimal representation of modeling errors and the synthesis of the PUIO. In Section IV, stability of the PUIO is performed by use of Linear Matrix Inequalities (LMI) for observer gain synthesis. In the last Section, the performances of the PUIO are illustrated through a nonlinear CSTR simulation.

II. NONLINEAR REPRESENTATION AND FDI FOR POLYTOPIC SYSTEMS

A. Nonlinear Representation

Consider a discrete-time nonlinear dynamical system in fault-free case described by:

x k+1 = g x k , u k y k = h x k , u k (1)
where x k ∈ X ∈ R n represents the state vector, u k ∈ U ∈ R p is the input vector and y ∈ R m is the output vector. We assume that functions g x k , u k and h x k , u k are continuously differentiable.

It is assumed that dynamic behaviour of the system operating at different operating points can be approximated by a set of N linear time invariant models. Consider the following state space representation of a nonlinear system around i-th operating point, ∀i ∈ [1, . . . , N ], with additive actuator faults:

x k+1 -x i e =A i (x k -x i e ) + B i (u k -u i e ) + E i d k + F i f k y k -y i e = C i (x k -x i e ) + D i (u k -u i e ) (2
) Matrices (A i , B i , C i , D i ) are invariant matrices defined around the i th operating point (OP i ) generally obtained from a nonlinear system using a first-order Taylor expansion around (x i e , u i e ) with y i e = C i x i e + D i u i e or identification of a nonlinear system around predefined operating points as for example in chemical processes in [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF] and [START_REF] Wan | Efficient scheduled stabilizing model predictive control for constrained nonlinear systems[END_REF]. The fault distribution matrix is represented by F i ∈ R n×p . f k ∈ R p represents the actuator fault vector and in faultfree case it is obviously equal to zero. The modeling errors matrix are noted as E i ∈ R n×q and are supposed to be full column rank for each operating point and d k ∈ R q is the associated modeling error vector. These matrices E i can be obtained from a second-order Taylor expansion or through an identification procedure as treated in [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]. In the following, we consider that D i = 0. As proposed in fault-free case in [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF], [START_REF] Tayebi | Iterative learning control for nonlinear systems described by a blended multiple model representation[END_REF] and in faulty-case in [START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[END_REF](where E i was not considered), (2) can be specified by the set of system matrices:

M i = A i B i E i F i C i , ∀i ∈ [1 . . . N ] (3)
Let S k be a matrix sequence varying within a convex set, defined as:

M k := N i=1 ρ i k M i : ρ i k ≥ 0, N i=1 ρ i k = 1 (4)
So, M k characterizes at each sample the nonlinear system from an OP to another. Consequently, the dynamic behavior of nonlinear system can be defined by a convex set of a multiple LTI models, noted Υ Υ := [M 1 , . . . , M N ] . The representation (4) under a convex set can be considered as a conventional modeling approach for nonlinear smooth plant where ρ i k is an appropriate weighting function. The function ρ i k embodies the nonlinearity of the plant and

ρ i k ∀i ∈ [1 . . . N ] lie in a convex set Ω = {ρ i k ∈ R N , ρ k = [ρ 1 k ....ρ N k ] T , ρ i k ≥ 0 ∀i and N i=1 ρ i k =
1} and these functions are directly generated via works of [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] and [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF], which permit to generate insensitive residual to faults. Under the assumptions that these weighting functions ρ i k are considered as scheduling variables which are not affected by faults or modeling errors as proposed by [START_REF] Adam-Medina | Fault diagnosis in nonlinear systems through an adaptive filter under a convex set representation[END_REF] and [START_REF] Theilliol | Adaptive filter design for FDI in nonlinear systems based on multiple model approach[END_REF], the nonlinear system (1) can be described by the following state space representation:

x k+1 = N i=1 ρ i k A i x k + B i u k + E i d k + F i f k + ∆x i y k = N i=1 ρ i k C i x k (5
) with the term ∆x i , around an ith operating point, equals to:

∆x i = -A i x i e -B i u i e + x i e (6)
As referred in [START_REF] Alcorta-Garcia | Deterministic nonlinear observer based approaches to fault diagnosis: a survey[END_REF], a new set of matrices is designed with the additional term ∆x i such that:

S i = A i B i E i F i ∆x i C i , ∀i ∈ [1 . . . N ] (7)
Based on the representation (5), we focus our attention on the synthesis of the Polytopic Unknown Input Observer. In order to realize an efficient FDI, a decoupled residual against modeling errors should be generated and modeling errors matrices definition must be reconsidered.

III. MODELING ERRORS REPRESENTATION

Modeling errors E i are likely different for each operating point with different distribution matrices. One way to achieve a fault diagnosis robust to modeling errors is to make disturbance de-coupling conditions hold true (in an optimal sense) for all disturbance distribution matrices. This can be done by using a single optimal disturbance distribution matrix E * to approximate all of them as proposed by [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]. This optimal matrix E * with full column rank makes possible de-coupling residual from modeling errors between different OP. An exact de-coupling for each OP is robust against modeling errors around the OP but not during transitions between the different OP.

Let consider a set R which contains N modeling errors modeling matrices such that

[E 1 E 2 . . . E N ] = R (8)
An optimal matrix E * that can approximate the set R is equivalent to the following optimization problem:

min R * R -R * 2 F (9) subject to rank(R * ) ≤ m
where . 2 F denotes the Frobenius norm. The matrix R * is thus chosen so that the sum of the squared distances between the columns of R and R * is minimized subject to the constraint R * ≤ m. Moreover, the rank condition is due to the fact that FDI scheme can distinguish faults and modeling errors with all most m independent outputs. This optimization problem can be solved by using Singular Value Decomposition (SVD) on the set R * .

The second step is to obtain the required distribution matrix E * with appropriate dimension by decomposing the rank deficient matrix R * as:

R * = R 1 R 2 (10)
where R 1 ∈ R n×q is full column rank and R 2 ∈ R q×(N ×q) . An optimal approximation for the modeling errors distribution matrix is R 1 with rank q equivalent to matrix E * . Consequently, the following state space representation could be considered to describe the dynamic behaviour of the nonlinear system:

x k+1 ≃ N i=1 ρ i k A i x k + B i u k + E * d k + F i f k + ∆x i y k = Cx k (11)
Moreover, it is often possible to consider only one output matrix C if there is no nonlinearity on the outputs.

So, based on equation [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF], a robust polytopic unknown input observer against modeling errors is developed in the next section.

IV. POLYTOPIC UNKNOWN INPUT OBSERVER DESIGN

Definition 1: (Polytopic Unknown Input Observer) A polytopic observer is defined as a polytopic unknown input observer for the system described by [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF] without fault (f k = 0), if the estimation error tends asymptotically to zero despite unknown disturbances on the system. This polytopic unknown input observer is defined such that:

z k+1 = N i=1 ρ i k S i z k + T B i u k + K i y k + ∆z i x k+1 = z k+1 + H * y k+1
where x is the state space estimation of x. The estimation error is equivalent to

e k+1 = x k+1 -x k+1 = x k+1 -(z k+1 + H * y k+1 ) = x k+1 -H * (Cx k+1 ) -z k+1 = [I -H * C]x k+1 -(S(ρ)z k + T B(ρ)u k + K(ρ)y k + ∆z(ρ)) (12) with notation ( )(ρ) stands for N i=1 ρ i k ( ) i
. By taking into account gain decomposition K(ρ) such as K(ρ) = K 1 (ρ) + Π(ρ), the equation ( 12) leads to:

e k+1 = [I -H * C](A(ρ)x k + B(ρ)u k + E * d k + F (ρ)f k + ∆x(ρ)) -∆z(ρ) -T B(ρ)u k -S(ρ)(x k -e k -H * y k ) -K 1 (ρ)Cx k -Π(ρ)y k (13)
The estimation error is equivalent to:

e k+1 = S(ρ)e k -[S(ρ) -(I -H * C)A(ρ) -K 1 (ρ)C]x k + (I -H * C)E * d k -[Π(ρ) -S(ρ)H * ]y k + [I -H * C]∆x(ρ) + (I -H * C)F (ρ)f k -∆z(ρ) -[T -(I -H * C)]B(ρ)u k (14)
S(ρ), T, K(ρ) and H * are designed so as to ensure the stability and the convergence of the estimation error e k = x kx k without fault on the system (f k = 0). To obtain an exact decoupling, based on an optimal modeling errors distribution matrix E * , the following conditions should be satisfied:

S(ρ) = T A(ρ) -K 1 (ρ)C Π(ρ) = S(ρ)H * (15) (I -H * C)E * = 0 T = I -H * C T ∆x(ρ) = ∆z(ρ)
The robust polytopic UIO design is realized when equations (15) hold true and S(ρ) stable. So, estimation error without fault occurrence, denoted ēk , tends asymptotically to zero if all these conditions (15) are satisfied. The gain design K 1 (ρ) is addressed in the next section. The necessary and sufficient conditions for the existence of a polytopic UIO are:

(i) Rank(CE * ) = Rank(E * ) (ii) (T A i , C) are detectable pairs, ∀i ∈ [1, 2, . . . , N ].
These conditions are related in linear case [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF].

If condition (i) is true, the synthesis of matrix H * , which can permit to avoid unknown input effects, is performed by:

H * = E * (CE * ) + (16)
It should be noted that the matrix E * is a constant matrix ∀i ∈ [1, 2, . . . , N ]. If conditions (15) hold true, the estimation error e k and the residual r k are described as:

e k+1 = S(ρ)e k + (I -H * C)F (ρ)f k r k = Ce k (17)
Moreover, condition (ii) ensures that a gain K 1 (ρ) can be synthesized in order to obtain a Hurwitz matrix S(ρ) = T A(ρ) -K 1 (ρ)C and an estimation error which tends to zero without fault. E * is a fixed matrix over the operating range and for detection and isolation purposes, it is important to constraint q + f < m, i.e. the rank q of modeling errors matrix and the rank f of the faulty distribution matrix have to be smaller than the number m of the outputs. For an actuator fault isolation, a bank of p polytopic UIO has to be used where each polytopic UIO is computed with a different unknown input such as E new = [E * F j (ρ)](with rank of E new equals to q + f ), with F j (ρ) is the jth column of matrix F (ρ). If ( 15) and ( 16) are fulfilled, the estimation error e k and residual r k lead to:

e k+1 = S(ρ)e k + T F (ρ)f k r k = Ce k (18)
with an exact de-coupling on E * . However, it is essential to examine faulty distribution matrices F i with modeling error matrix E * which have to generate different spaces. Indeed, de-coupling can reject perturbations through the term T E * in equation ( 15), whereas fault are represented by the term T F (ρ). If it is very important in FDI scheme to cancel perturbation on a system, it is widely important to be able to detect and isolate fault through estimation error e k and r k . Thus, some relations have to be verified around all the operating points:

Rank[T F (ρ)] = rank[F (ρ)], ∀ρ ∈ Ω (19)
This condition allows to verify that de-coupling do not affect fault detection by the estimation error. This condition can be translated in a geometric way such as:

N i=1 Im(F i ) ⊆ Im(T T ) (20)
If the union image F i is included in the image of decoupling matrix T T , then fault detection is possible over all the operating range. It is equivalent to:

N i=1 Im(F i ) Ker(T ) = {0} = Im([F 1 . . . F i . . . F N ]) Ker(T ) = {0} (21)
If condition [START_REF] Gatzke | Multiple model approach for CSTR control[END_REF], which can be checked off-line, is fulfilled, de-coupling do not affect fault detection. It could be noticed that all F i are supposed to be full column rank in order to avoid fault compensation of non-independent columns. The estimation error is not corrupt by modeling errors based on an approximated matrix E * over the all operating range; but there should remain a little difference due to the minimization criterion (9).

V. GAIN SYNTHESIS BY LMI

A. Introduction on LMI Region

Pole assignment by LMI is dedicated for convex sets [START_REF] Oliveira | A New Discrete-time Robust Stability Condition[END_REF]. The main objective is to ensure, at first, polytopic observer stability and in a second time to constraint its poles in a specified region of the complex plane. The observer dynamic is performed by decomposing complex plane in LMI region.

Definition 2: LMI region [19] A subset D of the complex plane is called an LMI region if there exists a symmetric matrix α ∈ R m×m and a matrix

β ∈ R m×m , such that D = z ∈ C : f D (z) < 0 (22) with f D (z) = α + zβ + zβ T where f D is called the characteristic function.
Eigenvalues of a real matrix A lie in D [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF], if and only if there exists a symmetric positive definite matrix P > 0, such that

M D (A, P ) = α ⊗ P + β ⊗ (AP ) + β T ⊗ (AP ) T < 0 (23)
Then, it is both possible to constraint poles of the observer to a prescribed region and ensure it stability.

B. Poles Placement of the Robust Polytopic UIO under LMI Constraints

According to [START_REF] Oliveira | A New Discrete-time Robust Stability Condition[END_REF], the estimation error ēk , without fault, is expressed as

ēk+1 = S(ρ)ē k = (I -H * C)A(ρ) -K 1 (ρ)C ēk = Ā(ρ) -K 1 (ρ)C ēk (
24) Proposition 1: Let D(r) a LMI region defining a disk (included in the unit circle) with a center (0, 0) and a radius 0 < r < 1. The estimation error (24) is called quadratically D-stabilizable (all the complex poles lie in LMI region D) if there exists matrices R i for i = [1 . . . N ] and a positive symmetric matrix X > 0 with appropriate dimensions such that ∀i ∈ [1 . . . N ]:

-rX ĀT i X -C T R T i X Āi -R i C -rX < 0 (25) with R i = XK 1 i . Proof: If (25) is fulfilled ∀i ∈ [1 . . . N ], then by using notation R i = XK 1 i , it follows that -rX ( Āi -K 1 i C) T X X( Āi -K 1 i C) -rX < 0 (26)
The poles of the matrix ( Āi -K 1 i C) lie in a circle with a center (0, 0) and a radius 0 < r < 1 (Cf. Definitions of such elementary LMI regions in [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]). By weighting each LMI (26) by ρ i and summing all of them with N i=1 ρ i = 1, the inequality (26) leads to

-rX N i=1 ρ i ( Āi -K 1 i C) T X X N i=1 ρ i ( Āi -K 1 i C) -rX < 0 (27) where matrix N i=1 ρ i ( Āi -K 1 i C
) can be written as ( Ā(ρ)-K 1 (ρ)C) under the convex set Ω, ensuring that poles of the estimation error ēk in (24) lie in region D with gains

K 1 (ρ) = N i=1 ρ i X -1 R i , ∀i ∈ [1, . . . , N ].
The estimation is said quadratic D-stable.

Remark: if parameter r = 1 then LMI in (25) expresses quadratic stability in discrete case with notation R i = XK 1 i and with a Schür Complement, we find after some computations:

( Āi -K 1 i C) T P ( Āi -K 1 i C
) -P < 0 and P > 0 that is the quadratic stability of estimation error with respect to

X -1 = P > 0 at each vertex ∀i ∈ [1, . . . , N ].
We should note that using LMI, observer gain synthesis is made through a polytopic form and estimation error (24) satisfies to the condition of quadratic D-stability with X > 0 for the Lyapunov matrix. This proposed solution allows to ensure a quadratic D-stability for the estimation error (without fault) and to fixe the dynamic of the polytopic UIO for the considered system. Poles constraints are specified with LMI defining a prescribed region of the complex plan.

VI. CSTR: CONTINUOUS STIRRED TANK REACTION

A. System Description

The performance of the robust polytopic UIO is illustrated through a Continuous Stirred Tank Reaction simulation (CSTR). A full description of the CSTR could be found in [START_REF] Chen | Nonlinear predictive control of a benchmark cstr[END_REF]. This system is a nonlinear chemical reactor that exhibits many interesting properties like input multiplicity, a gain sign change, asymmetric response and both minimum and nonminimum phase behavior [START_REF] Gatzke | Multiple model approach for CSTR control[END_REF]. The, so called van der Vusse reaction, is summarized by the following reaction scheme: A ⇛ B with B ⇛ C and 2A ⇛ D. The dynamic behaviour of the reactor is classically described by the following nonlinear differential equations:

                   ĊA = V Vr (C A0 -C A ) -k 1 (v)C A -k 3 (v)C 2 A ĊB = V Vr C B + k 1 (v)C A -k 2 (v)C B v = V Vr (v 0 -v) -1 ρCp k1(v)C A ∆H RAB + k2(v)C B × ∆H RBC k1(v) + k 3 (v)C 2 A ∆H RAD + kw Ar ρCpVr (v -v k ) vk = 1 m k C P K × (( Qk -v k ) + kwAr(v -v k )) (28)
with

k i (v) = k i0 e E i v+273
. 15 . The model has four states: concentration of A (C A ) with an initial concentration C A0 , concentration of B (C B ), temperature v with a fixed value for the feed temperature v 0 and cooling jacket temperature v k . It is assumed that states are directly measurable except for v k . The input variables are the flow rate normalized by the reactor volume V Vr and the heat removal Qk .

B. Operating Conditions

A specific condition, reported in the benchmark [START_REF] Chen | Nonlinear predictive control of a benchmark cstr[END_REF], underlines the fact that concentration of B should be taken into the range [0.8; 1.09] mol.l -1 for manipulated variables V Vr lying in [3h -1 ; 35h -1 ] and [-9000kJ.h -1 ; 0] for Qk . The feed temperature v 0 is considered as a fixed value at 105 o C as proposed in [START_REF] Chen | Nonlinear predictive control of a benchmark cstr[END_REF]. In the proposed study, the CSTR is controlled in an open-loop way with multiple desirable concentrations of product B, each of them defining an operating point. Thus, CSTR could produce different concentrations C B and the system is described around predefined operating points. By taking into account the static characteristic illustrated in figure [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] and in order to exploit the nonlinear system in a wide range, concentrations of B are C B1 , C B2 , C B3 with respectively 0.85, 1.00, 1.09mol.L -1 . 4) that permits to cover the wide operating range conditions. It could be noticed that there is non-minimum phase behaviour appearing on concentrations of A and B during a transition.

C. Experiments and Results

Based on the physical dynamic equation of the system, linear state space models were obtained around four OP using a first order linearization. These four operating points developed are: C B1 = 0.85mol.L -1 , C B3 = 1.09mol.L -1 and C B2 = 1.00mol.L -1 with V21 Vr = 7.8570h -1 and C B2 = 1.00mol.L -1 with V22 Vr = 27.90h -1 . The nonlinear model has been discretized through a Tustin method in order to provide discrete linear models represented in equation [START_REF] Ozkan | Control of a solution copolymerization reactor using multi-model predictive control[END_REF]. Disturbance matrices of modeling error E i are directly obtained by a second order linearization and an unique matrix E * is computed through an optimization technique based on Singular Value Decomposition as proposed in [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF]. An unique actuator fault on the manipulated variables V Vr is considered, fault matrices F i are related with the column of B i . According to the four linear models and under the assumptions that the interpolation functions ρ i are insensitive to faults , figure [START_REF] Alcorta-Garcia | Deterministic nonlinear observer based approaches to fault diagnosis: a survey[END_REF] shows the sequence of activation functions ρ i generated from [START_REF] Rodrigues | Fault Diagnosis on Industrial Systems based on a Multiple Model approach[END_REF]. Based on the unique disturbance matrice E * ∈ R 4×2 , the polytopic observer gain synthesis is performed by pole placement developed in the previous paragraph by a LMI region defined with r = 0.5. The different matrices are given in the Appendix.

In a first experiment, the polytopic UIO is evaluated without fault with the control input illustrated in figure [START_REF] Kinnaert | Robust fault detection based on observers for bilinear systems[END_REF]. This residual norm is presented in figure [START_REF] Persis | A geometric approach to nonlinear fault detection[END_REF]. It can be observed that this norm is different from zero around operating points because polytopic UIO is synthesized in order to obtain an exact decoupling against E * , but this matrix is just an optimal approximation of modeling errors. During transitions, uncertainties are minimized and the norm is very closed to zero. The magnitude of the residual norm is almost constant through all operating points and represents the modeling errors due to approximations of modeling error representation. We should underline that an adaptive threshold could be taken into account the specificities of the polytopic UIO synthesis in order to generate an efficient residual evaluation.

In a second experiment, an actuator fault with a loss of 65% from the nominal value, is introduced as in figure [START_REF] Kinnaert | Robust fault detection based on observers for bilinear systems[END_REF] during the second and the third operating point. The faulty control input is sent to the system and the computed control input is exploited by the polytopic UIO. This experiment simulates an actuator fault that would be stuck between OP 1 and OP 2 , then, a partial lost of an actuator fault is simulated during transitions of OP 2 and OP 3 . At the last change of operating point OP 3 -OP 4 , the actuator fault disappears. Due to the fact that we consider only one actuator fault which can occur into the system, the fault detection is equivalent to a fault isolation result. According to the synthesis of the robust polytopic UIO, the residual norm is sensitive to the fault occurrence. As illustrated in figure [START_REF] Persis | A geometric approach to nonlinear fault detection[END_REF], the residual norm of In this paper, a new residual generator robust against modeling errors was developed. This residual generation is dedicated to fault diagnosis of systems based on a multimodel approach which operate around specific and predefined OP. The modelling errors approximation over a set of OP allows an unique decoupling for all the system range even during transitions. Our contribution is underlined by a polytopic UIO that generates decoupled residuals from modeling errors. The observer gain synthesis is realized by LMI so as to ensure both stability and pole assignment of the polytopic UIO.

Fig. 1 .

 1 Fig. 1. Steady state gainsIt can be noted that concentration C B2 = 1.00mol.L -1 could be obtained for two different control values V Vr : two operating points are associated to this concentration.

Fig. 2 .

 2 Fig. 2. Dynamic evolution of the measurable states

4 Fig. 3 .

 43 Fig. 3. Weighting functions

Fig. 4 .

 4 Fig. 4. Faulty control input and nominal control input

Fig. 5 .

 5 Fig. 5. Residual norms evolution with and without fault

  Actuator fault diagnosis of nonlinear systems is realized under modeling errors. The illustration is made through a CSTR simulation around multi operating conditions.

				-0.8142	APPENDIX 1.3831   0.42	-0.49	-0.03	
	E	* =	  	-0.0032 11.7210	-1.6090 0.0956	   H	* =	  	-0.49 -0.03	0.57 -0.03	-0.03 0.99	  
					0	0						0			0	0
					0.43	0.51	0.03				0.38	0.47	0.02	
		K 1 = 1	  	0.54 0.04	0.24 0.03	0.02 -0.19	   K 2 = 1   	0.50 0.04	0.20 0.03	0.02 -0.19	  
					3.78	3.27	0.79					3.64	3.21	0.78
		K 3 = 1	   	0.41 0.52 0.04 3.81	0.50 0.23 0.03 3.25	0.03 0.02 -0.19 0.79	    K 4 =  1     	0.30 0.43 0.03 3.84	0.41 0.15 0.02 3.12	0.02 0.02 -0.19 0.75	     
	X = 10 3 ×	     		1.50 0.24 0.01 -0.06	0.24 1.42 0.01 -0.05	0.01 0.01 1.21 -0.01		-0.06 -0.05 -0.01 0.01	     	C =	 	0 0 1	0 1 0	1 0 0	0 0 0	



Matrices R 1 , R 2 , R 3 , R 4 are not reported. It could be done by the relation R i = XK 1 i .


Nicolas Blin
email: nicolas.blin@univ-lorraine.fr

Pierre Riedinger
email: pierre.riedinger@univ-lorraine.fr

Jamal Daafouz
email: jamal.daafouz@univ-lorraine.fr

Louis Grimaud
email: louis.grimaud@safrangroup.com

Philippe Feyel
email: philippe.feyel@safrangroup.com

A comparison of harmonic modeling methods with application to the interconnection and the control of switched systems

Keywords: harmonic modeling, harmonic analysis, frequency couplings, switched systems, interconnected systems, harmonic-based control

come

A comparison of harmonic modeling methods with application to the interconnection and the control of switched systems

Introduction

Electricity has transformed the way we communicate, produce, travel and relate to the world. Electrical power is an essential element of modern society which provides for increasingly sophisticated needs. To meet the new needs in transmission and conversion of electrical power, the power systems are more and more complex with non-linear components and complicated frequential behaviors. That leads to several new challenges. The non-linear components introduce couplings between frequencies. In particular, high-frequency converters inherently generate numerous harmonics and can interact with each other. These interactions may cause unpredicted instability problems involving many different frequencies, like in the Swiss locomotives case [START_REF] Mollerstedt | Out of control because of harmonics-an analysis of the harmonic response of an inverter locomotive[END_REF]. Historically, in power electronics, the early models only represent the average signal behavior through the State-Space Averaging (SSA) method which involves averaging on a sliding window [START_REF] Middlebrook | A general unified approach to modelling switching-converter power stages[END_REF]. These models have been widely used to study small signals. However, they only allow the harmonic analysis of the steady-state system and they have revealed limitations in modeling large signals [START_REF] Erickson | Large-signal modelling and analysis of switching regulators[END_REF]. This has motivated the researches for an analysis approach capable of characterizing the harmonic dynamics and the frequency couplings.

A review of the literature in this area shows that various frequency domain based approaches have been introduced. Generalized State-Space Averaging (GSSA) [START_REF] Sanders | Generalized averaging method for power conversion circuits[END_REF] and Dynamic Phasors (DP) [START_REF] Mattavelli | Phasor dynamics of thyristorcontrolled series capacitor systems[END_REF] are generalized averaging approaches to model the harmonic content of general linear systems. They employ Fourier series of time-varying coefficients and have been widely used in power system modeling and control, e.g. [START_REF] Javaid | Arbitrary order generalized state space average modeling of switching converters[END_REF], [START_REF] Mattavelli | Phasor dynamics of thyristorcontrolled series capacitor systems[END_REF], [START_REF] Stankovic | Modeling and analysis of FACTS devices with dynamic phasors[END_REF]. Extended Harmonic Domain (EHD) [START_REF] Madrigal | Modelling of power electronics controllers for harmonic analysis in power systems[END_REF] and Dynamic Harmonic Domain (DHD) [START_REF] Chavez | Dynamic harmonic domain modeling of transients in three-phase transmission lines[END_REF] have been introduced to analyze the frequency behavior of Linear Time-Periodic (LTP) systems trough Linear Time-Invariant (LTI) models in the harmonic domain. These approaches have also been widely used, e.g. [START_REF] Karami | Single-Phase Modeling Approach in Dynamic Harmonic Domain[END_REF], [START_REF] Rico | Dynamic harmonic evolution using the extended harmonic domain[END_REF]. Meanwhile, Harmonic State Space (HSS) has been introduced for the harmonic analysis and the control analysis of LTP systems [START_REF] Wereley | Analysis and control of linear periodically time varying systems[END_REF]. The HSS approach makes it possible to represent a LTP system as a LTI state space of infinite dimension and to define a harmonic transfer function for these LTP systems.

The main contribution of this paper is to clearly establish the links between the various harmonic modeling techniques and to develop a general harmonic methodology in a rigorous theoretical framework that unifies them. As a result, the literature approaches have different scopes of application and specificities but some strong links are stated in Section 2 and a new harmonic model is proposed from which all the literature models can easily be derived. This has been addressed in [START_REF] Blin | A comparison of harmonic modeling methods with application to control of switched systems with active filtering[END_REF], of which this paper is an extended and updated version. Also, the decomposition of non-periodic signals in generalized Fourier series with time-varying coefficients is specified with strong proof and slightly different from the literature. One more contribution of the paper is to show the new possibilities that are given by harmonic modeling techniques for the interconnection and the control of systems, in particular switched systems. These are illustrated through two practical examples. In Section 3 we take advantage of the harmonic modeling to analyze the interconnection of a Boost converter and a Flyback converter. In Section 4 we present a control of a Boost converter that uses the harmonic techniques to act directly on the signals harmonics and to achieve active filtering by rejecting periodic perturbations.

Harmonic modeling

In this section, we discuss the various harmonic modeling approaches that have been introduced in the literature. Their links are clearly identified and a general harmonic methodology in a rigorous theoretical framework that unifies them is developed. It results in a harmonic model presented in Proposition 4 from which they can all be derived. We also propose a strong proof for the use of time-varying Fourier coefficients through general Fourier decomposition which is slightly different from the literature. In this context, let us consider a linear time-varying system

ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) (1)
with x(t) ∈ R nx , y(t) ∈ R ny , u(t) ∈ R nu , and where A(t), B(t), C(t), D(t) are some real-valued matrices of appropriate dimension. If the matrices A(t), B(t), C(t), D(t) are periodic of same period, the system is a LTP system. If they are constant, the system is a LTI system.

Generalized averaging

In this section, the various tools of generalized averaging techniques, namely GSSA, DP, EHD and DHD are presented. We describe their scopes of application and their specificities and we identify strong links between the methods. Based on the generality of each approach, a unifying rigorous theoretical framework for harmonic modeling is proposed. It allows to deal with linear time-varying systems and to take into account an arbitrary number of harmonics, possibly infinite. This section also highlights the general Fourier decomposition which is slightly different from the literature by providing a strong proof.

The conventional SSA models only consider the averaging behavior of signals but it is known that higher order harmonics are needed to describe the harmonic behavior accurately [START_REF] Erickson | Large-signal modelling and analysis of switching regulators[END_REF]. By extension, the SSA approach has resulted in the GSSA. The GSSA approach is a general Fourier series development with time-varying coefficients, first introduced by Sanders [START_REF] Sanders | Generalized averaging method for power conversion circuits[END_REF]. The DP approach has been introduced by Mattavelli, Verghese, and Stankovic [START_REF] Mattavelli | Phasor dynamics of thyristorcontrolled series capacitor systems[END_REF] to emphasize the time-varying nature of the Fourier coefficients, but it essentially shares the same formulation. Proposition 1. Any piecewise C 0 signal x(•) with bounded variations can be represented using a general Fourier series of arbitrary period T by

x(t) = 2 ∞ k=-∞ X k (t)e jkωt -x(t -T ) a.e. ( 2
)
and for a real number δ ∈]0, T [, by the non-causal relation

x(t -δ) = ∞ k=-∞ X k (t)e jkω(t-δ) a.e. ( 3
)
with time-varying coefficients

X k (t) = 1 T t t-T x(p)e -jkωp dp ( 4
)
where ω = 2π T and "a.e." means almost everywhere. Proof. Let x(•) be a piecewise continuous signal with bounded variations and let us introduce the signal

xT (t, τ ) = ∞ k=-∞ X k (t)e jkωτ ( 5
)
with time-varying coefficients (4). This signal depends on two time scales expressed by the two variables t and τ and is T -periodic with respect to τ . Actually, it is a general Fourier series with respect to the variable τ , plotted for two different values of t in Figure 1. By application of Dirichlet's theorem on Fourier series convergence: for any real number t and for any τ ∈]t -T, t[,

xT (t, τ ) = x(τ-)+x(τ+) 2 (6)
where x(τ -) and x(τ + ) stands for the left and right limits of x(•) in τ . For a real number δ ∈]0, T [, by writing ( 6) for τ = t -δ, xT (t, t -δ) =

x((t-δ) -)+x((t-δ) + ) 2 so that (3) holds almost everywhere, except at the discontinuities of x(•). Otherwise, for τ = t, xT (t, t) =

x(t-)+x((t-T ) + ) 2 so that (2) holds almost everywhere, except at the discontinuities of x(•). The coefficients X k (t) are called generalized Fourier coefficients in GSSA but also referred to as phasors in DP. They are functions of time since the interval under consideration slides as a function of time. X k (t) is the k th coefficient (or k-phasor) at time t that represents the k th harmonic of the signal at frequency k T . If the signal x(t) is T -periodic after a certain time t r , then its general Fourier coefficients are constant for t ≥ t r + T , so the steady state is defined by constant harmonics, as it can be seen in Figure 2. In other words, in periodic steady state, (2) becomes the well-known

Fourier formula x(t) = ∞ k=-∞
X k e jkωt . This is why a good suggestion to analyze the harmonic content of DC-DC switching converters is to choose the period T equal to the switching period, so that the harmonics of the signals are constant for a fixed duty cycle. However, it is important to highlight that the proposition holds for any period. The Proposition 1 is different from the general Fourier formulas of literature, e.g. [START_REF] Sanders | Generalized averaging method for power conversion circuits[END_REF], [START_REF] Mattavelli | Phasor dynamics of thyristorcontrolled series capacitor systems[END_REF], at time t (for δ = 0) because the discontinuity of the periodic function extrapolated from one signal period is considered. It results that the Fourier series can not exactly match this signal in general but only if the function extrapolated is continuous, that is to say if the original signal is periodic. Proposition 2. The harmonic dynamics of a C 0 , piecewise C 1 signal x(•) of derivative of bounded variations can be described as

Ẋk (t) = < ẋ > k (t) -jkωX k (t) a.e. ( 7
)
where Ẋk (•) stands for the derivative of the k th general Fourier coefficient of x(•), < ẋ > k (•) stands for the k th general Fourier coefficient of the derivative of x(•) and with the term 'jkω' which is the derivative of the k th complex exponential function e jkωt .

Proof. This proposition can be easily verified by differentiating (4).

Proposition 3. The harmonics of a product of two piecewise C 0 signals y(•) and x(•) can be expressed using the following convolution-based procedure

< yx > k (t) = ∞ i=-∞ Y k-i (t)X i (t) a.e. ( 8
)
where Y k (t), X k (t) are the k th general Fourier coefficients of y(t), x(t).

Proof. This proposition can be easily verified using (3) for both signals.

In the literature, GSSA has been applied by taking few harmonics into account, although it was acknowledged that higher order harmonics must be incorporated for a more accurate model of a large-signal system. This is because GSSA equations do not allow to add harmonics easily. That has led to EHD approach that systematically describes a LTP system by its state space in the harmonic domain with an arbitrary number of harmonics, possibly infinite [START_REF] Madrigal | Modelling of power electronics controllers for harmonic analysis in power systems[END_REF], [START_REF] Rico | Dynamic harmonic evolution using the extended harmonic domain[END_REF]. The term DHD has also been introduced in [START_REF] Chavez | Dynamic harmonic domain modeling of transients in three-phase transmission lines[END_REF] but it shares an identical formulation. EHD/DHD approach is more general than GSSA/DP since it allows to take into account an arbitrarily large number of harmonics. Nevertheless, GSSA/DP is more general since it allows the study of non-periodic systems. It is proposed on the following to take into account an arbitrarily large number of harmonics and not to be limited to LTP systems. Basically, GSSA equations are used to generate the state-space of a general linear system in a chosen harmonic basis. This requires the use of Toeplitz form [START_REF] Wereley | Analysis and control of linear periodically time varying systems[END_REF] and generalized here to time-varying Fourier coefficients. Proposition 4. A linear time-varying system [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF] with piecewise C 0 input signal and matrices can be represented in the harmonic basis of arbitrary period T as a state space model of the form where the matrix of differentiation N contains the vector of all the derivative terms of the complex exponential functions e jkωt of the harmonic basis replicated as many times as there are state variables in [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF]

Ẋ(t) = (A(t) -N )X(t) + B(t)U (t) Y (t) = C(t)X(t) + D(t)U (t) (9) with X(t) = [... X -k (t) ... X 0 (t) ... X k (t) ...] T Y (t) = [... Y -k (t) ... Y 0 (t) ... Y k (t) ...] T U (t) = [... U -k (t
(•) are < ẋ > k (t) = < Ax > k (t)+ < Bu > k (t) Y k (t) = < Cx > k (t)+ < Du > k (t) (11)
By the convolution-based procedure [START_REF] Hill | On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon[END_REF],

< ẋ > k (t) = ∞ i=-∞ A k-i (t)X i (t) + ∞ i=-∞ B k-i (t)U i (t) Y k (t) = ∞ i=-∞ C k-i (t)X i (t) + ∞ i=-∞ D k-i (t)U i (t) (12)
The infinite vectors of the harmonics of ẋ(t) and y(t) can be written with the Toeplitz form of matrices as

< ẋ > (t) = A(t)X(t) + B(t)U (t) Y (t) = C(t)X(t) + D(t)U (t) (13)
By [START_REF] Ghita | Harmonic state space feedback control for AC/DC power converters[END_REF], we also write

< ẋ > (t) = Ẋ(t) -N X(t) (14)
so that the derivative of the infinite vector X(t) of the harmonics of x(t) can be expressed as

Ẋ(t) = (A(t) -N )X(t) + B(t)U (t).
The state-space formulation ( 9) is an extension of GSSA approach to an infinite number of harmonics, and an extension of EHD approach to the study of non-periodic systems. It provides the state space of a general linear system in a harmonic basis of chosen period. If the investigated system is LTP of period T , then by choosing T or a multiple of T as the harmonic basis period, the Fourier coefficients of the matrices appear to be time-invariant, so are their Toeplitz form. In other words, a LTP model of period T is converted into a LTI model of the form [START_REF] Hwang | Harmonic state-space modelling of an HVdc converter with closed-loop control[END_REF] with

A(t) = A, B(t) = B, C(t) = C, D(t) = D.
Similar results are presented in EHD and DHD approaches for which the harmonic basis period is imposed equal to the system matrices period, but the proposed modeling is more general since it can also be applied to non-periodic systems, opening new possibilities as to model AC-DC and DC-AC converters.

Harmonic state-space

In this section, the tools of HSS approach are presented and we identify the strong links with the generalized averaging methods and with the theoretical framework developed in section 2.1. The fundamental notion behind the development of the traditional analysis tools like Bode diagram for LTI systems is that a sinusoidal input at a given frequency is mapped by the LTI transfer function into a sinusoidal output of the same frequency, but with possibly different amplitude and phase. In contrast, if a sinusoid is input to a LTP system, several sinusoids may appear in the output signal. The corresponding output signal may oscillates at the frequency of the input signal plus integer multiples of the system fundamental frequency [START_REF] Wereley | Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties[END_REF]. As a result, the LTI analysis Laplace domain cannot be used as it stands in LTP context. This has motivated the researches for a solution to LTP systems analysis. The Floquet solution is a geometrically periodic signal whose transient state has a complex exponential behavior [START_REF] Floquet | Sur les équations linéaires a coefficients périodiques[END_REF]. EMP signals have been introduced by Hill [START_REF] Hill | On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon[END_REF] through periodic extension to the kernel function of Laplace to make the connection with the assumed Floquet solution. More details about Floquet theory resulting in EMP signals can be found in [START_REF] Bittanti | Periodic systems: filtering and control[END_REF]. Definition 1. An exponentially modulated periodic (EMP) signal x(•) can be expressed as the complex Fourier series of a periodic signal of fundamental period T = 2π ω modulated by a complex exponential

x(t) = e st ∞ k=-∞ X k e jkωt ( 15
)
where s ∈ C. It can be written on compact form

x(t) = e st Γ(t)X (16)
with

Γ(t) = [... e -jkωt ... 1 ... e jkωt ...] X = [... X -k ... X 0 ... X k ...] T ( 17
)
Proposition 5. EMP signal spaces defined for a period T are fundamental signal spaces for LTP systems of period T .

Proof. A LTP system of period T maps an EMP signal of period T to an EMP signal of same fundamental frequency but with different amplitudes and phases, see [START_REF] Wereley | Analysis and control of linear periodically time varying systems[END_REF].

This frequency separation property reveals the strong analogy that EMP signals are to LTP systems as complex exponentials are to LTI systems, as illustrated in Figure 3.

Proposition 6. A LTP system (1) of matrices A(t), B(t), C(t), D(t) of period T can be written in the harmonic basis of period T as an infinite LTI system of the form

sX = (A -N )X + BU Y = CX + DU (18) with X = [... X -k ... X 0 ... X k ...] T Y = [... Y -k ... Y 0 ... Y k ...] T U = [... U -k ... U 0 ... U k ...] T N = diag(... -jkω ... 0 ... jkω ... -jkω ... 0 ... jkω ...) ( 19
)
where A, B, C, D are the Toeplitz matrices of A(t), B(t), C(t), D(t) and s ∈ C.

Proof. The proof is based on the harmonic balance approach using EMP fundamental signals [START_REF] Wereley | Analysis and control of linear periodically time varying systems[END_REF].

The state-space form [START_REF] Rico | Dynamic harmonic evolution using the extended harmonic domain[END_REF] with the frequency separation property of EMP signals through LTP systems allows to define a transfer function for LTP systems in the harmonic domain. The harmonic transfer function of a LTP system [START_REF] Rico | Dynamic harmonic evolution using the extended harmonic domain[END_REF] is given by the infinite matrix G(s) = C((sU I -(A -N )) -1 B + D where U I stands for the identity matrix. This time-invariant matrix contains the transfer functions between each input harmonic and each output harmonic. It is a linear operator of the harmonic domain that allows to analyze harmonic dynamics and couplings using traditional LTI analysis techniques, as pole-zero map [START_REF] Ormrod | Harmonic state space modelling of voltage source converters[END_REF] or Nyquist plot [START_REF] Mollerstedt | Out of control because of harmonics-an analysis of the harmonic response of an inverter locomotive[END_REF]. Another strong analogy is given in this paper: HSS domain is to LTP systems what Laplace domain is for LTI systems. As opposed to the Laplace domain of unique reference frame e st , HSS domain contains an infinite range of reference frames (e (s+jkω)t ) k∈Z . The complex exponential e st is related to the harmonic transient evolution. Its derivative is se st so that the multiplication by s denotes differentiation over time on direction s. The product sX stands for the dynamic of the state harmonics.

Discussion

In this section, we synthesyze the links between the literature approaches and we define the bonds with the unified model of Proposition [START_REF] Chavez | Dynamic harmonic domain modeling of transients in three-phase transmission lines[END_REF]. It is easy to derive all the literature harmonic models from the one proposed in this paper so that it is possible to combine their advantages. We also discuss the choice of the harmonic basis period. Many publications have focused on harmonic content, resulting in several methods. The literature has not described precisely the links between these approaches because of the differences in their writing and goals [START_REF] Hwang | Harmonic state-space modelling of an HVdc converter with closed-loop control[END_REF]. DP and GSSA are essentially identical, as they are based on the same generalized Fourier representation (3) with time-varying coefficients (4) and generate the same equations ( 7) and [START_REF] Hill | On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon[END_REF]. Equation ( 2) has been introduced in this paper to manage some issues that may occur when moving back to the temporal domain because the relation (3) is not causal. In the literature, these approaches have only been developed for few harmonics. Mostly, only the first harmonic at fundamental frequency is incorporated [START_REF] Sanders | Generalized averaging method for power conversion circuits[END_REF], [START_REF] Stankovic | Modeling and analysis of FACTS devices with dynamic phasors[END_REF], or some known harmonics of the system are selected to design a controller [START_REF] Ghita | Harmonic state space feedback control for AC/DC power converters[END_REF]. What is not clearly stated in the literature is that EHD and DHD approaches are a systematical writing of the LTI state space model that one can get by applying GSSA equations to a LTP system with an arbitrary number of harmonics. Moreover, HSS is a frequential analysis domain for LTP systems that is analogous to the Laplace domain for LTI system analysis. It is a purely frequential release of EHD that can be derived from EHD state space by Laplace transform. The links between the different literature harmonic modeling approaches which have been clearly identified in this paper are summarized in Figure 4.

The model developed in this article in Proposition 4 is a generalization of all the harmonic modeling models as we can not only choose the number of harmonics to take into account but also consider time-varying systems. From this model can be derived all the averaging models by truncation or selection of harmonics and HSS model by Laplace transform. It can be seen as a bridge model whose benefit is to combine easily the advantages of all the harmonic techniques. The bonds with the bridge model are summarized in Figure 5.

In practice, the Fourier series have to be truncated so that they can be computed. Then, the accuracy of the truncated model highly relies on the choice of the harmonic basis period and the number of harmonics. The impact of the basis period has been weakly developed so far. Actually, it is a compromise between model complexity and accuracy. A rigorous theoretical basis for determining the truncation order is difficult to develop as it depends on numerous parameters. For example, high-order harmonics can be excited with respect to the transient dynamics. It is still an open issue but we provide on the following some serious directions to assess the model accuracy with respect to the choice of the harmonic basis period and the maximal order of harmonics. The main idea is that the harmonic basis period T specifies the frequency spectrum precision as the frequencies represented in the harmonic basis are separated from 1 T and the maximal order of harmonics h specifies the frequency bandwidth as the frequencies represented are between -h

T and h T . The model order is then (2h + 1)n x , with n x the number of physical variables and (2h + 1) the number of harmonic variables for each physical variable (the dc component, h positive harmonics and h negative harmonics).

Practically speaking, we should choose the period large enough to capture the low frequencies and include enough harmonics to capture the high frequencies. Since the general harmonic model has the great advantage to allow to choose an arbitrary number of harmonics and to change this number easily, it is possible to add more and more harmonics until the accuracy suits our goal, or on the contrary to remove more and more harmonics until the model order is low enough for our needs. This compromise depends on the application and specifications. Moreover, by choosing wisely the harmonic basis period (if possible as a multiple of all relevant signal periods), we can both limit the model order and get to a LTI model with very little accuracy loss. As an example, let's consider the waveform x(t) = a 1 sin(ω 0 t) + a 3 sin(3ω 0 t). By choosing T = 2π ω0 , only three harmonics at ω 0 , 2ω 0 , and 3ω 0 would be needed to model perfectly the signal. More generally, by choosing T = 2πN ω0 , N ∈ N * , we need 3N harmonics. However, by choosing T arbitrarily, we may need much more harmonics and they have no reason to be constant. Similarly, the better choice to model a DC-DC switched system seems to be a harmonic basis of period equal to the switching period, so that the state-space model is LTI in steady state and few harmonics are needed for accuracy.

Harmonic modeling has proved to be an interesting tool to analyze the harmonic content of a system. It can also be used to make good choices when associating components by characterizing their harmonic couplings. This point has not been much explored and a striking example is developed in Section 3, where the models of two different converters of type Boost and Flyback are simply interconnected in a common harmonic basis. The choice of the harmonic basis period also allows to target some specific harmonics for example to reject oscillations, which is illustrated with the active filtering of a Boost converter output voltage in Section 4.

Harmonic analysis of associated switched systems

The harmonic modeling can be employed to improve the analysis of interconnected switched systems. Not only does it allow to represent switched systems by continuous models with good precision, but it also contributes to the analysis of their interconnection by capturing their harmonic couplings. We develop on the following a harmonic model of a switched affine system and show the resulting models for a Boost converter and a Flyback converter. Afterthat, we build the harmonic model of the interconnection of these two converters, from which we can analyze the harmonic couplings between them and the resulting global harmonic content. We neglect in this section the discontinuous mode of the converters. A particular attention will be paid to the choice of a common harmonic basis to these converters made possible by the discussion on the harmonic basis period.

Harmonic modeling of switched affine systems

A switched affine system can be described by a state-space model of the form

ẋ(t) = A(d(t))x(t) + B(d(t))u(t) (20)
where x(t) ∈ R nx is the state vector, u(t) ∈ R nu is the input vector and d(t) is the switching signal.

The time-varying matrices can be decomposed as

A(d(t)) = A ind + d(t)A dep B(d(t)) = B ind + d(t)B dep ( 21
)
where d(t) is scalar and A dep , B dep are some constant matrices. It is possible to control the behavior of a switched affine system by the switching signal d(t). It is a discontinuous signal that contains many harmonics at multiples of the switching frequency f sw . For a fixed duty cycle, the switching signal is periodic of period T sw . Thus, an interesting idea to describe a switched affine system by an almost LTI state-space model is to choose this period to build the harmonic basis. That is to say, we choose the basis of complex exponential functions (e jkωt ) k∈Z , with ω = 2πf sw . By Proposition 4, the corresponding state space is

Ẋ(t) = A(D(t))X(t) + B(D(t))U (t) (22)
with

X(t) = [X {1} (t) ... X {nx} (t)] T U (t) = [U {1} (t) ... U {nu} (t)] T A(D(t)) = A ind ⊗ U I -Ñ + A dep ⊗ D(t) B(D(t)) = B ind ⊗ U I + B dep ⊗ D(t) Ñ = diag(N , ..., N ) (23)
where X {j} (t) is the vector of harmonics of the j th state variable, U {l} (t) is the vector of harmonics of the l th input variable, D(t) is the Toeplitz matrix of the switching signal d(t), N = diag(... -jkω ... 0 ... jkω ...) is the matrix of differentiation, U I is the identity matrix, and where ⊗ stands for the Kronecker product. n x is the number of states and n u is the number of inputs. This state-space model is LTV, but a great advantage of it is that the Toeplitz matrices reveal the couplings between input, control and output frequencies. Moreover, each physical variable is described by many harmonic variables that vary much less over time, which reduces the simulation step time. Typically, for a constant duty cycle, the switching signal is periodic of the same period than the harmonic basis, so the Toeplitz matrix D is constant, and for periodic inputs of period T sw , the vector U is constant and the model is LTI.

Choice of a common harmonic basis

In this part, we raise concern about the choice of the harmonic basis, in particular its period. It has been presented in the last part that an interesting choice to model a switched system is to choose the period of the harmonic basis equal to the switching period, so that the model is almost LTI (LTI in steady state) and precisely describes the system behavior with a limited number of harmonics. However, by choosing any multiple of the switching period, similar results can be obtained. The drawback is that we have to increase the number of harmonics taken into account for the same precision. That is the reason why there has been little mention of it in the literature. However, it could be the key point to find a common harmonic basis for two switched systems in order to associate them.

We are interested in a common harmonic basis for two switched systems of different switching period. A harmonic basis can be said to be wisely chosen with respect to a system if all the significant frequencies of the system are contained in the basis. For a switched system, any complex exponential basis of period equal to a multiple of the switching period is a good choice, even more the switching period itself. Furthermore, for two switched systems with different switching period, a wise choice is a complex exponential basis of period equal to a common multiple of both switching periods. This choice must lead to an almost LTI state-space representation of the harmonic behavior of the interconnection. Unfortunately, the case where the frequencies do not have any common multiple is an issue. An idea could be to approximate a common multiple, but the sensitivity has to be studied since the harmonics would not be constant and the accuracy would be degraded. Another issue is that the number of harmonics has to be increased. Greater the common multiple is (with respect to the switching frequencies), more the number of harmonics need to be increased for accuracy. Taking the smallest common multiple is a wise choice to limit the number of harmonics.

Application to the interconnection of a Boost converter and a Flyback converter

The harmonic techniques can be applied to a Boost converter and a Flyback converter. The resulting harmonic models are shown on the following. Then, a common harmonic basis is chosen to model their interconnection and capture the couplings between their frequencies.

The Boost converter is a high-frequency switched system shown in Figure 6. Its instantaneous temporal model is expressed as a switched affine representation [START_REF] Stankovic | Modeling and analysis of FACTS devices with dynamic phasors[END_REF] with

x(t) = i L (t) v out (t) u(t) = v in (t) i out (t) A(d(t)) = -R L -1-d(t) L 1-d(t) C 0 B(d(t)) = 1 L 0 0 -1 C (24)
where i L (t) is the current through inductance, v out (t) is the output voltage, v in (t) is the input voltage, i out is the output current that is required from the load (i out = vout R load in case of constant resistive load), and d(t) is the discontinuous switching signal, usually coming from pulse width modulation (pwm). For a constant duty cycle, this last signal is periodic at switching frequency f boost .

The harmonic model of a Boost converter in a complex exponential basis (e jkωt ) k∈Z of given frequency f is expressed as in [START_REF] Wereley | Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties[END_REF] with

X(t) = I L (t) V out (t) U (t) = V in (t) I out (t) A(D(t)) = -R L U I -N -1 L (U I -D(t)) 1 C (U I -D(t)) -N B(D(t)) = 1 L U I Z Z -1 C U I (25)
with ω = 2πf , where I L (t), V out (t), V in (t), I out (t) are the harmonic vectors of the inductance current, the output voltage, the input voltage and the output current, D(t) is the Toeplitz form of the switching law d(t), U I is the identity matrix and Z is the zero matrix.

This model of infinite dimension is relevant for any given frequency of the harmonic basis. However, in practice, the model has to be truncated, which can affect its accuracy. The harmonic model of the Boost converter gives very accurate results by choosing the complex exponential basis of frequency f boost and by taking into account three harmonics at least to approximate well the switching effects. Same results can be obtained by choosing the complex exponential basis of frequency f = f boost N and by taking into account h = 3N harmonics at least. The Flyback converter is a high-frequency switched system shown in Figure 7. Let us define the transformer rate n = n1 n2 . Its instantaneous temporal model is expressed as a switched affine representation [START_REF] Stankovic | Modeling and analysis of FACTS devices with dynamic phasors[END_REF] with

x(t) = i L (t) v out (t) u(t) = v in (t) i out (t) A(d(t)) = -R L -n(1-d(t)) L n(1-d(t)) C 0 B(d(t)) = d(t) L 0 0 -1 C ( 26
)
where i L (t) is the current through inductance, v out (t) is the output voltage, v in (t) is the input voltage, i out is the output current that is required from the load, and d(t) is the switching signal. For a constant duty cycle, this last signal is periodic at switching frequency f f lyback .

The harmonic model of a Flyback converter (with n = n1 n2 ) in the complex exponential basis (e jkωt ) k∈Z of given frequency f is expressed as in [START_REF] Wereley | Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties[END_REF] with

X(t) = I L (t) V out (t) U (t) = V in (t) I out (t) A(D(t)) = -R L U I -N -n L (U I -D(t)) n C (U I -D(t)) -N B(D(t)) = 1 L D(t) Z Z -1 C U I (27)
with ω = 2πf , where I L (t), V out (t), V in (t), I out (t) are the harmonic vectors of the inductance current, the output voltage, the input voltage and the output current, D(t) is the Toeplitz form of the switching law d(t), U I is the identity matrix and Z is the zero matrix.

The harmonic model of the Flyback converter gives very accurate results by choosing the complex exponential basis of frequency f f lyback and by taking into account three harmonics at least. Same results can be obtained by choosing the complex exponential basis of frequency f =

f f lyback N
and by taking into account h = 3N harmonics at least.

V 1-2 = V boost out = V f lyback in and I boost out = D f lyback I f lyback L .
The harmonic model of the interconnection of a Boost converter and a Flyback converter with a constant resistive load in the complex exponential basis (e jkωt ) k∈Z of given frequency f is expressed as in [START_REF] Wereley | Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties[END_REF] with

X(t) =     I L1 (t) V 1-2 (t) I L2 (t) V out (t)     U (t) = V in (t) A(D(t)) =     -R 1 L 1 U I -N -1 L 1 (U I -D1) Z Z 1 C 1 (U I -D1) -N -1 C 1 D2 Z Z 1 L 2 D2 -R 2 L 2 U I -N -n L 2 (U I -D2) Z Z n C 2 (U I -D2) -1 R load C 2 U I -N     B(D(t)) =     1 L1 U I Z Z Z     (28)
with ω = 2πf , where index 1 refers to the Boost converter and index 2 refers to the Flyback converter, where V in is the harmonic vector of the input voltage of the Boost converter (viewed as a perturbation), V out is the harmonic vector of the output voltage of the Flyback converter, V 1-2 is the harmonic vector of the intermediate voltage between them, D 1 (t), D 2 (t) are the Toeplitz forms of the switching signals of the Boost and the Flyback (time indices have been left for clarity), U I is the identity matrix and Z is the zero matrix.

A wise choice of common harmonic basis for this association is the complex exponential basis of frequency f = GCD(f boost , f f lyback ) equal to the greatest common divisor of both switching frequency. Here, f = 25kHz. As f = f boost 4 , we need h = 12 harmonics at least for the Boost commutation modeling accuracy. As f = f f lyback 3

, we need h = 9 harmonics at least for the Flyback commutation modeling accuracy. Thus, the complex exponential basis of frequency f = 25kHz may be a wise choice and requires only h = 12 harmonics for accuracy. The output voltage of the interconnection of a Boost converter and a Flyback converter is plotted in Figure 9 by open-loop simulation of the continuous harmonic model (28) with f = 25kHz and truncated to 12 harmonics. It is compared to the output voltage from the interconnected discontinuous temporal models to demonstrate its accuracy. Constant duty cycles are applied to both converters. The converters parameters can be found in Figure 10. We have considered a constant resistive load R load = 10Ω. The harmonic model (25) has several advantages. First, it is a continuous model that reveals the high-frequency behavior of both converters. It is possible, by simulation of this model, to analyze the influences of commutations with larger time steps than for temporal discontinuous models but still very accurate results. In addition, it highlights the couplings between the harmonics generated by the commutations in both converters. Finally, it becomes a LTI model in a steady state given by constant duty cycles, so it allows to use traditional control laws. It has to be reminded that these advantages may disappear if the common harmonic basis is not wisely chosen, for example if we do not know the switching periods of the converters or if they do not have any common multiple.

In this section, we raised concern about the choice of the harmonic basis, in particular its period. We have taken advantage of this new parameter to interconnect two switched systems in a common harmonic basis, so that we can analyze their interconnection with an almost LTI continuous model. Actually, by making another choice for the period of the harmonic basis, other possibilities can be opened. In disturbed environment, by choosing a harmonic basis at a perturbation frequency, it could permit to directly control the harmonics and reject this frequency. This point has not been much explored yet and it is considered in the next part.

Control with active filtering of switched systems

The harmonic modeling can be employed to improve the control of industrial systems. Not only does it allow the analysis of frequency behavior, but it also extends the notion of equilibrium points. Indeed, a T -periodic signal is constant in the harmonic basis of period T , so that a periodic steady state can be viewed as an equilibrium point. It also enables to act separately on harmonics and to analyze their interactions. These advantages have been used to design a LQ controller for a full-bridge rectifier in [START_REF] Ghita | Harmonic state space feedback control for AC/DC power converters[END_REF]. The harmonic model has been linearized for a few harmonics and a feedback control synthesis in harmonic domain has given harmonic rejection criteria. We develop on the following a globally asymptotically stable control law for switched affine systems with periodic perturbation rejection. This control takes advantage of harmonic modeling without linearization to eliminate a periodic input perturbation.

Equilibrium points in the harmonic domain

Any periodic steady state of period T can be viewed as an equilibrium point in the harmonic basis of same period T . Then, the harmonic equilibrium point is defined by a vector of constant harmonics for each original periodic state variables. In instance, as we are dealing with switched systems, an equilibrium is defined both by the state variables and the switching signal (the control signal). The equilibrium points defined in the harmonic domain are called harmonic equilibrium points on the following. Definition 2. The set of harmonic equilibrium points of a switched system of the form ( 22) is defined for a periodic input signal of period equal to the period of the harmonic basis by

E = {(X e , D e ) | X e = -A(D e ) -1 B(D e )U } (29)
where X e contains the harmonics of all state variables in steady state and D e contains the harmonics of the switching signal in steady state. For control purposes, we are interested in accessible harmonic equilibrium points E a = {(X e , D e ) | X e = -A(D e ) -1 B(D e )U, A(D e ) Hurwitz}

Now that equilibrium points have been defined in the harmonic basis, a control law can be built to reach one of them.

Embedded control of switched systems in the harmonic domain

The embedded control for switched systems has been proposed in [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF]. It has the advantage of been an adaptive switched control design methodology with a global asymptotic stability property. Please note that other more traditional control laws like Proportional Integrator (PI) or Linear Quadratic (LQ) can also be used after linearization in the harmonic domain. The main asset of the harmonic domain is that we can use the same control techniques than in temporal domain meanwhile we control the harmonic behavior. The drawback to keep in mind is the number of variables that increases. Proposition 7. Considering the system [START_REF] Wereley | Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties[END_REF], any accessible equilibrium point (X e , D e ) ∈ E a is globally asymptotically stable by the control law

D(t) = D e -KB T (X(t), U )P(X(t) -X e ) ( 30
)
with K > 0, P > 0 solution of the Lyapunov equation A(D e ) T P + PA(D e ) + Q = 0, and

B(X(t), U ) = nx j=1 (A dep {C j } ⊗ X {j} T (t)) + nu l=1 (B dep {C l } ⊗ U {l} T ) ( 31
)
where A dep {C j } , B dep {C l } are respectively the j th column of A dep and the l th column of B dep , and with X {j} T (t), U {l} T the Toeplitz matrices of the harmonics of the j th state variable and the l th input variable.

Proof. This proposition can be verified by a Lyapunov function approach similar to the one in [START_REF] Beneux | Adaptive stabilization of switched affine systems with unknown equilibrium points: Application to power converters[END_REF] applied in the harmonic domain.

Q is the Lyapunov gain to spread the control effort and K is the gain to adjust the control significance. This control law can be exploited either to solve a tracking problem or to eliminate some unwanted harmonics, simply by reaching a harmonic equilibrium of chosen harmonics amplitude and phase.

Application to the control of a Boost converter

In this part, we present some results about a controller using harmonic modeling to eliminate unwanted harmonics. The chosen system to demonstrate is a Boost converter shown in Figure [START_REF] Floquet | Sur les équations linéaires a coefficients périodiques[END_REF], of switching frequency f sw = 100kHz and connected to a constant resistive load. Its temporal model is expressed as in [START_REF] Stankovic | Modeling and analysis of FACTS devices with dynamic phasors[END_REF] with

x(t) = i L (t) v out (t) u(t) = v in (t) A(d(t)) = -R L -1-d(t) L 1-d(t) C -1 R load C B(d(t)) = 1 L 0 ( 32
)
where i L (t) is the current through inductance, v out (t) is the output voltage, v in (t) is the input voltage (viewed as a perturbation), R load is the constant resistive load, and d(t) is the switching signal coming from pulse width modulation (pwm). High-frequency input perturbations are not relevant as they are filtered by the system and low-frequency ones can be overcome by adaptive control techniques, but a specific frequency range remains critical, as shown in Figure 11. We can see in our case that the critical range of frequency is between 200Hz and 10kHz with the worst case around 5kHz, quite far from the high switching frequency of 100kHz. That leads to the assumption that high frequencies due to commutations can be neglected for rejection of such frequencies. This assumption may be checked by using same techniques than in the last section to find a common harmonic basis for switching harmonics and perturbation harmonics. Another assumption is that we know the period T in of the input perturbation, so that by choosing the harmonic basis of period T = T in , the periodic perturbation is viewed as some constant harmonics perturbations. This second assumption is a strong one, but in practice the perturbation would come from another associated system whose frequency may be known, measured or estimated with good precision.

The harmonic model of a Boost converter with a periodic input perturbation of period T in in the complex exponential basis (e jkωint ) k∈Z is expressed as in [START_REF] Wereley | Linear time periodic systems: transfer function, poles, transmission zeroes and directional properties[END_REF] with

X(t) = I L (t) V out (t) U (t) = V in (t) A(D(t)) = -R L U I -N -1 L (U I -D(t)) 1 C (U I -D(t)) -1 R load C U I -N B(D(t)) = 1 L U I Z (33)
where I L (t), V out (t), V in (t), I out (t) are the harmonic vectors of the inductance current, the output voltage, the input voltage and the output current, D(t) is the Toeplitz form of the switching law d(t), U I is the identity matrix and Z is the zero matrix. The control objective is to regulate the output voltage and avoid its oscillations. The corresponding reference output voltage harmonic vector contains zeros everywhere except at its dc component. Some integrators are used to reject static errors on each harmonics. Figure 12 shows the better rejection of a periodic perturbation v in (t) = 30 + 3sin(2πf t) of critical frequency f = 5kHz by harmonic-based control. What is remarkable about this result is that the control laws are of the same type and complexity. What is new in the harmonic-based control law is that we control directly the harmonics of the output voltage so we have more parameters and we can adjust them to eliminate some unwanted harmonics and still regulate the dc component (the output voltage average). This is illustrated by Figure 13 in which the harmonics of the regulated output voltage are plotted.

To reject the unwanted harmonics, the controller takes into account the couplings between input, control and output harmonics, so that it generates a control law with some harmonics to compensate the oscillations on the output voltage. It can be understood as some harmonic injections to compensate oscillations. The control law to reject the input oscillation is shown in Figure 14 and its corresponding harmonics are shown in Figure 15. Obviously, the relevance of this control method depends on the assumption that we know the frequency of the perturbation with sufficient accuracy. The sensitivity of the approach to the perturbation frequency knowledge is an issue where several parameters are involved and should be studied in further works.

Conclusion

The harmonic techniques can capture the harmonic content of a system and provides a better understanding of frequency phenomena. Many different harmonic approaches have been developed but this article shows that they are based on the same equations and ideas. The differences in their writing are only due to their different scopes of application, for control design or harmonic analysis, for LTP systems or non-periodic ones. In this paper, a more general state-space model is developed to reveal the links between the different approaches. From this unified modeling methodology can be easily derived all the different methods presented in the literature. It has been revealed that the number of harmonics taken into account is of relevance. This article shows that one key point of harmonic modeling techniques is the choice of the harmonic basis period. The period has to be large enough to capture the significant low frequencies and the number of harmonics has to be large enough to capture the significant high frequencies. Moreover, a wise choice of the period as a common multiple of all the significant periods of the signals oscillations may lead to a LTI state-space harmonic model, which allows the use of traditional analysis and control tools with extensive results as for harmonic dynamics modeling, frequency couplings analysis and control with harmonic rejection. The influence of the period on the type of the resulting harmonic model is illustrated in the table in Figure 16.

One advantage of harmonic modeling is the possibility to capture the high-frequency behavior of a switched system through an almost LTI continuous model. However, this result requires the knowledge of the switching frequency and the model has no reason to be LTI if it is not precisely known. It also reveals the couplings between the input, control and output harmonics and, under the assumption that we find a common harmonic basis, it may reveal the frequency couplings within interconnected systems, even associated switched systems. Once more, the results would be simpler to analyze with a wise choice of harmonic basis, which means not only to know precisely their switching frequencies but also to find a common divisor of both frequencies.

Figure 1 :

 1 Figure 1: The signal x(τ ) is plotted in blue and the signal xT (t, τ ) is plotted for t = t 1 in red and for t = t 2 in green, versus its second time scale τ .

Figure 2 :

 2 Figure 2: The generalized Fourier coefficients X k (τ ) of signal x(τ ) (blue signal in Figure 1) are plotted against time. The periodic steady state is clearly defined by constant harmonics.

Figure 3 :

 3 Figure 3: EMP signals have frequency separation property through LTP systems (down) as complex exponential signals have through LTI systems (up).

Figure 4 :

 4 Figure 4: Links between the different harmonic modeling approaches.

Figure 5 :

 5 Figure 5: Links between the bridge model unifying the different harmonic modeling approaches. All harmonic models can be derived from the state space of the bridge model in Proposition 4.

Figure 6 :

 6 Figure 6: Circuit diagram of a Boost converter.

Figure 7 :

 7 Figure 7: Circuit diagram of a Flyback converter.

Figure 8 :

 8 Figure 8: Circuit diagram of the interconnection of a Boost and a Flyback converter.

Figure 9 :Figure 10 :

 910 Figure 9: Comparison of the output voltage from temporal model and harmonic model in the exponential basis of frequency f = 25kHz and with 12 harmonics.

Figure 11 :

 11 Figure 11: Influence of a periodic input perturbation v in (t) = v inc + v invar sin(2πf t). A traditional control law is used, improved by adaptive techniques. The maximal amplitude error on the output voltage is plotted against the frequency of the perturbation .

Figure 12 :

 12 Figure 12: Output regulation with a periodic input perturbation of critical frequency f = 5kHz. The resulting output voltages are plotted for traditional (temporal) control in green and harmonic-based control in red.

Figure 13 :

 13 Figure 13: Output regulation with a periodic input perturbation rejection by harmonic-based control. The resulting output voltage harmonics are plotted.

Figure 14 :

 14 Figure 14: Output regulation with a periodic input perturbation of critical frequency f = 5kHz. The resulting duty-cycle commands are plotted for traditional (temporal) control in green and harmonic-based control in red.

Figure 15 :

 15 Figure 15: Output regulation with a periodic input perturbation rejection by harmonic-based control. The resulting command law harmonics are plotted.

$ This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Meanwhile, harmonic modeling techniques can be exploited to control industrial systems. One can define and verify some harmonic criteria that are difficult to express in temporal domain for control design, for example to reject some oscillations. Furthermore, it is possible to control directly the harmonics. By such an harmonic-based control approach, either periodic tracking problems can be solved as several simple regulation problems on harmonic variables or some harmonics can be rejected. This implies to deal with many additional variables and control parameters.