Francesco Ferrante
email: francesco.ferrante@gipsa-lab.fr

Ricardo G Sanfelice
email: ricardo@ucsc.edu

Sophie Tarbouriech

Hybrid Regional Stabilization of Linear Systems with Actuator Saturation and Multi-Rate Samplers

Regional stability analysis of linear systems with multi-rate samplers and actuator saturation is studied. A hybrid controller is used to perform a fusion of measurements sampled at different times. In between sampling events, the controller behaves as a copy of the plant. When a new measurement is available, the controller state undergoes a jump. The resulting system is analyzed in a hybrid system framework. Sufficient conditions in the form of matrix inequalities are given to determine estimates of the basin of attraction of the closed-loop system. Finally, the effectiveness of the proposed methodology is shown in an example.

I. INTRODUCTION

Control of sampled-data systems has been attracting the attention of researchers for a long time and its interest is still vivid in the community as pointed out by the recent survey [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]. Capturing the effect of sampled-data information in control loops is indeed fundamental in addressing the challenges posed by modern engineered systems in which the use of embedded systems and communication networks is central. One of the most investigated aspects in the literature of sampled-data systems pertains to the study of aperiodic sampled-data systems, that is systems in which the exchange of information from the plant to the controller happens in an intermittent aperiodic fashion; see for example [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF], just to cite a few. In fact, aperiodic sampled-data systems may be used to capture different phenomena occurring in data networks as dropouts, sampling jitter, etc; see [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF].

A common assumption in sampled-data control consists of considering synchronous sampling of different variables. However, in certain applications due to technological constraints, some variables may happen to be measured with a different sampling-rate than others, making the assumption of synchronous sampling restrictive; see, e.g., [START_REF] Moarref | Multiloop controller design for a fly-by-wireless uav quadrotor based on a multirate sampled-data model[END_REF]. This need promoted the development of new tools devoted to the analysis of multi-rate control systems; see [START_REF] Lall | An LMI solution to the robust synthesis problem for multi-rate sampled-data systems[END_REF], [START_REF] Yen | Variable sampling rate controller design for brushless dc motor[END_REF].

Another limitation that is naturally present in actual control systems concerns to actuator saturation. The study of saturated control systems is by now a very mature research area as testified by the large amount of publications in this field [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF]. Many approaches have been proposed in the literature to analyze the effect of actuator saturation in (single-rate) sampled-data control systems; see [START_REF] Dai | Output feedback synthesis for sampled-data system with input saturation[END_REF], [START_REF] Seuret | Taking into account period variations and actuator saturation in sampled-data systems[END_REF], [START_REF] Gomes Da | Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation[END_REF]. However, to the best of our knowledge, no results are found in the literature to deal with actuator saturation in a multi-rate setting.

To fill this gap, in this paper, building on the modelbased controller architecture presented in [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF], originally proposed for single-rate non-saturated linear systems, we propose sufficient conditions for regional stability analysis of linear systems in the presence of actuator saturation and multi-rate samplers. The approach we propose is enabled by the hybrid systems framework in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. Sufficient conditions in the form of matrix inequalities are given to characterize estimates of the basin of attraction of the closed-loop system. Semidefinite programming is used to reduce the conservatism of such estimates.

The remainder of this paper is structured as follows. Section I-A presents some preliminaries on hybrid systems. Section II presents the problem we solve and the adopted modeling approach. Section III is dedicated to the main results. Section IV shows the effectiveness of the proposed approach in a numerical example. Due to space constraints proofs will be published elsewhere.

Notation: The set N is the set of positive integers including zero, the set N>0 is the set of strictly positive integers, R ≥0 represents the set of nonnegative real scalars, R n×m represents the set of the n × m real matrices, and S n + is the set of n × n symmetric positive definite matrices. The identity matrix is denoted by I, whereas the null matrix is denoted by 0. For a matrix A ∈ R n×m , A T denotes the transpose of A, A (i) the i-th row of A, A T (i) the i-th row of A T , and He(A) = A + A T . For two symmetric matrices, A and B, A > B, A ≥ B mean that A -B is, respectively, positive definite and positive semidefinite. In partitioned symmetric matrices, the symbol • stands for symmetric blocks. The matrix diag{A1, A2, . . . , An} := n i=1 An is the block-diagonal matrix having A1, A2, . . . , An as diagonal blocks. For a vector x ∈ R n , |x| denotes the Euclidean norm and for a matrix A ∈ R n×n , |A| denotes the Euclidean induced-norm. Given two vectors x, y, we denote (x, y) = [x T y T ] T . For two vectors x, y ∈ R n , x y means that the components of xy are nonnegative. Given a vector x ∈ R n and a closed set A, the distance of x to A is defined as |x|A = infy∈A |x -y|. For any function z : R → R n , we denote z(t + ) := lim s→t + z(s). Given P ∈ S n + and c > 0, then E (P, c) := {x ∈ R n : x T P x ≤ c}. Given a set S, we denote co S the convex-hull of S. The symbol × denotes the Cartesian products of two sets, while given sets S1, S2, . . . , Sn, we denote

× n i=1 Si = S1 × S2 × • • • × Sn. Given two sets X ⊂ R n+m and Y ⊂ R n , ΠY (X) := {x ∈ Y : ∃y ∈ R m s.t. (x, y) ∈ X}.

A. Preliminaries on Hybrid Systems

We consider hybrid systems with state x ∈ R nx of the form

H ẋ = f (x) x ∈ C x + ∈ G(x)
x ∈ D

In particular, we denote the function f : R nx → R nx as the flow map, C ⊂ R nx as the flow set, the set-valued map G : R nx ⇒ R nx as the jump map, and D ⊂ R nx as the jump set.

A set E ⊂ R ≥0 × N is a hybrid time domain if it is the union of a finite or infinite sequence of intervals [t j , t j+1 ] × {j}, with the last interval (if existent) of the form [t j , T ) with T finite or T = ∞. Given a hybrid time domain E, we denote sup j E = sup{j ∈ N : ∃t ∈ R ≥0 s.t. (t, j) ∈ E}.

A function φ : dom φ → R nx is a hybrid arc if: dom φ is a hybrid-time domain and φ(•, j) is locally absolutely continuous for each j. A hybrid arc φ is a solution to H if it satisfies the dynamics of H; see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] . A solution φ to H is maximal if it cannot be extended and is complete if dom φ is unbounded. In particular, given S ⊂ R nx , we denote S H (S) the set of maximal solutions φ to H with φ(0, 0) ∈ S, if S is omitted, φ ∈ S H means that φ is a maximal solution to H; see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] for more details.

II. PROBLEM STATEMENT

A. System description

We consider continuous-time linear time-invariant systems with saturated actuators of the form

P ẋp = Ax p + Bu u = sat(v) (1)
where x p ∈ R np is the plant state, u ∈ R nu is the plant input and v ∈ R nu is the actuator input. The function v → sat(v) is the symmetric decentralized saturation function with saturation levels v1 , v2 , . . . , vnu ∈ R >0 , whose components for each v ∈ R nu are defined as

sat(v) i = sat(v i ) := min(|v i |, vi ) sign(v i ) i = 1, 2, . . . , n u (2
) Moreover, A and B are constant matrices of appropriate dimensions.

As in [START_REF] Moarref | Sensor allocation with guaranteed exponential stability for linear multi-rate sampled-data systems[END_REF], assume that sporadic and asynchronous measurements of the components of the plant state x p are available. More precisely, assume that the components of the plant state x p are clustered into n s ≤ n p subvectors x

(1) p ∈ R np 1 , x (2) p ∈ R np 2 , . . . , x (ns) p ∈ R npn s , i.e., x p = (x (1) p , x (2)
p , . . . , x (ns) p

) and that the measure of each x (i) p is available only at some time instances t

(i) k , k ∈ N, not known a priori. Let, for each i ∈ N := {1, 2, . . . , n s }, M i ⊂ R ≥0 be the set of measurement times of x (i) p , i.e., M i = {t (i) k } ∞ k=0 , let M = ns i=1
M i the set of all measurement times, and let K ∈ R nu×np be a given control gain. Moreover, suppose that the arrival of a new measurement can be instantaneously detected and that the plant model is known 1 . Then, inspired by [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF], we want to study the stability of the closed-loop system obtained by interconnecting the following controller with state jumps to the plant (1)

K    η(t) = Aη(t) + B sat(Kη(t)) t / ∈ M η (i) (t + ) = x (i) p (t) t ∈ M i , i ∈ N ζ(t) = Kη t ∈ R ≥0 (3)
In particular, η ∈ R np is the controller state, ζ ∈ R nu is the controller output and the interconnection of the controller and the plant is realized by setting v = ζ.

The operating principle of the controller in ( 3) is as follows. The arrival of a new measurement x (i) p for some i ∈ N triggers an instantaneous jump in the controller state. Specifically, at each jump, the measured vector x (i) p is instantaneously stored in η (i) . If different components of x p are measured simultaneously, then all the corresponding components of η get updated accordingly. Then, in between consecutive measurements, η is continuously updated according to continuous-time dynamics so to mimic the plant dynamics and its value is continuously used in place of the plant state

x (i)
p in a static state-feedback controller scheme. It appears clear that, as long as the sampling events occur sufficiently often, the controller state η provides an accurate estimate of the plant state, which in turn suggests that closedloop stability should be preserved. In particular, we consider the following assumption on the event times: Assumption 1: For each i ∈ N , the sequence {t

(i)
k } ∞ k=1 is strictly increasing and unbounded and there exist two positive real scalars

T (i) 1 ≤ T (i) 2 such that 0 ≤ t (i) 1 ≤ T (i) 2 T (i) 1 ≤ t (i) k+1 -t (i) k ≤ T (i) 2 ∀k ∈ N (4)
△ Remark 1: The lower bound in condition (4) prevents the existence of accumulation points in the sequences {t

(i) k } ∞ k=1
and hence avoids the existence of Zeno behaviors, which are typically undesired in practice. In fact, for each i = 1, 2, . . . , n p , T (i) 1 defines a strictly positive minimum time in between consecutive measurements of x

(i) p . Furthermore, for each i = 1, 2, . . . , n p , T (i)
2 defines the maximum time in between consecutive measurements of x (i) p . Remark 2: Due to asynchronous measurements of the plant state components, as opposed to [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF], one cannot assume the controller state to converge in finite-time to the plant state and this renders the analysis of the closed-loop system more challenging. Nonetheless, it is worthwhile to observe that for each initial condition (x 0 , η 0 ) such that x 0 = η 0 , the resulting trajectories (φ p , φ c ) of the closedloop system are such that φ p = φ c . Thus, for such initial conditions the effect of the sampling is totally compensated by the controller. On the other hand, in real-world settings, the (unavoidable) presence of plant uncertainties renders the occurrence of such a behavior quite unlikely.

By following a similar approach as in [START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions[END_REF], our goal is to provide sufficient conditions for local asymptotic stability for the closed-loop system obtained by interconnecting controller (3) with plant (1). More precisely, the problem we solve is as follows:

Problem 1: Given the closed-loop system obtained by interconnecting controller (3) with the plant (1) via v = ζ, provide an estimate of the basin of attraction for the origin of the closed-loop system.

By interconnecting controller (3) with plant (1) via v = ζ, by defining η := x pη, and by denoting x cl = (x p , η), the closed-loop system is modeled by the following dynamical system with jumps:

   ẋcl (t) = A cl x cl (t) +B cl dz (K cl x cl (t)) t / ∈ M x cl (t + )= I 0 X M (t) 0 x cl (t) t ∈ M (5)
where

A cl := A + BK -BK 0 A B cl := B 0 , K cl := K -K M ∋ t → X M (t) = diag{X (1)
M1 (t), X (2)
M2 (t), . . . , X (ns)
Mn s (t)} (6) where for each t ∈ M, i ∈ N

X (i) Mi (t) = 0 np i ×np i if t ∈ M i I np i elsewhere
and for each s ∈ R nu , s → dz(s) := sat(s)s.

B. Hybrid modeling

The fact that the closed-loop system experiences jumps when a new measurement is available suggests that the dynamics of the closed-loop system (5) can be described via a hybrid system. We provide a hybrid model in the framework in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] that captures the behavior due to each possible evolution generated by any sequence {t

(i) k } ∞
k=1 , for i ∈ N , satisfying (4). This approach leads to a model with nonunique solutions allowing one to establish a strong result for all family of sequences {t

(i) k } ∞ k=1 , for i ∈ N , satisfying (4)
. The proposed approach requires to model the hidden time-driven mechanism triggering the jumps of the controller. To this end, similarly as in [START_REF] Ferrante | Observer-based control design for linear systems in the presence of limited measurement streams and intermittent input access[END_REF], we add n s timer variables τ 1 , τ 2 , . . . , τ ns to keep track of the duration of flows and to trigger jumps according to the mechanism in [START_REF] Lall | An LMI solution to the robust synthesis problem for multi-rate sampled-data systems[END_REF]. To accomplish that, we make τ 1 , τ 2 , . . . , τ ns decrease as ordinary time t increases and, whenever τ i = 0 for some i ∈ N , reset it to any point in [T

(i) 1 , T (i)
2 ], so as to enforce [START_REF] Moarref | Multiloop controller design for a fly-by-wireless uav quadrotor based on a multirate sampled-data model[END_REF]. In particular, define τ := (τ 1 , τ 2 , . . . , τ ns )

T := ns × i=1 [0, T (i) 2 ]
and for all i ∈ N , τ ∈ T

G τi (τ ) := χ ∈ R ns : χ i ∈ [T (i) 1 , T (i) 2 ], ∀k ∈ N \{i}, χ k = τ k (7)
Then, we consider the following dynamics for τ

τ = -1 ns τ ∈ T τ + ∈ i∈{k∈N : τ ∈D k } G τi (τ ) τ ∈ D τ (8)
where for each i ∈ N

D i := {τ ∈ T : τ i = 0} D τ := i∈N D i
Let x := (x cl , τ ) ∈ R 2np × T be the vector state. We define for each x ∈ C := R 2np × T , where C is the flow set, the flow map as

f (x) := A cl x cl + B cl dz(K cl x cl ) -1 ns
For each i ∈ N and η ∈ R np , define

J i (η) = diag{E (i) 1 , E (i)
2 , . . . , E (i) ns }η ∀i ∈ N where for each i, j ∈ N

E (i) j = 0 np j ×np j if i = j
I np j elsewhere and let for every

(η, τ ) ∈ R np × D τ J (η, τ ) := i∈{k∈N : τ ∈D k } (J i (η) × G τi (τ ))
We define for each x ∈ D := R 2np × D τ , where D is the jump set, the jump map as

G(x) = (x p , J (η, τ )) (9)
Therefore, the dynamics of the closed-loop system can be represented by following hybrid system

H cl ẋ = f (x) x ∈ C x + ∈ G(x) x ∈ D (10)
An important question that needs to be addressed to guarantee the robustness of the closed-loop system with respect to small perturbations, it is whether H cl is well-posed; see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] for more details and consequences of well-posedness.

A positive answer to this question is given by the following result.

Lemma 1: The hybrid system (5) is well-posed. Remark 3: Notice that when τ ∈ D i ∩ D j for some i = j ∈ N , then, only one component of the vector τ undergoes a change and multiple consecutive jumps occur. This phenomenon directly follows from having defined an outer semicontinuous jump map.

Concerning the existence of solutions to system H cl , it is straightforward to check that for every initial condition φ(0, 0) ∈ C ∪ D, the corresponding maximal solution φ to H cl exists and is complete, i.e., sup dom φ = ∞. Moreover, the definition of the data of H cl , similarly as in [START_REF] Ferrante | Observer-based control design for linear systems in the presence of limited measurement streams and intermittent input access[END_REF], ensures that for each solution, at most n s jumps can occur consecutively without flowing. Such a property, along with (4), ensures that for every maximal solutions φ to H cl and each (t, j) ∈ dom φ such that (t, s) ∈ dom φ for some s ∈ {j + 1, j + 2, . . . , j + n s }, one has [t, t + min{T } and offset N 0 = n s , which for all (t, j) ∈ dom φ implies τ D j ≤ t + n s τ D ; see, e.g., [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Example 2.15]. Such a property imposes a strictly positive uniform lower bound on the length of every flow interval, preventing from the existence of Zeno solutions.

To solve Problem 1, our approach is to determine sufficient conditions to ensure that the following set2

A = {0} × T ( 11
)
is asymptotically stable for H cl . In particular, in this work, we consider the following notion of asymptotic stability for a compact set:

Definition 1 (Local asymptotic stability [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]): Let A ⊂ R 2np+ns be compact. The set A is said to be Definition 2 (Basin of attraction [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]): Let A ⊂ R 2np+ns be locally asymptotically stable for H cl . The basin of attraction of A is the set of points B A ⊂ R 2np+ns such that each φ ∈ S H cl (B A ) is complete and converges to A.

In light of the above definitions, it turns out that Problem 1 can be solved by determining the basin of attraction of A defined in [START_REF] Gomes Da | Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation[END_REF]. On the other hand, the determination of such a set is a difficult problem and cannot be solved in general. To overcome this drawback, in the following section we provide sufficient conditions in the form of matrix inequalities to determine subsets of the basin of attraction of A. Building on such conditions, we show how semidefinite programming techniques can be used to reduce the conservatism of the proposed estimates.

III. MAIN RESULTS

A. Sufficient Conditions

In this section we provide sufficient conditions for the solution to Problem 1. To accomplish this task, we pursue a Lyapunov approach. In particular, we state sufficient conditions in the form of matrix inequalities guaranteeing local asymptotic stability of the set A in [START_REF] Gomes Da | Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation[END_REF] for H cl and providing an estimate of the basin of attraction of A. The derivation of such a result builds from [13, Corollary 8.4, Corollary 8.9] and on the generalized sector condition for the deadzone nonlinearity in [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF].

Theorem 1 (Local asymptotic stability):

Let W ∈ S np + , R 1 ∈ S np 1 + , R 2 ∈ S np 2 + , . . . , R ns ∈ S np ns + , S ∈ D nu + , σ 1 , σ 2 , . . . , σ ns ∈ R >0 , Z, J ∈ R nu×np . Assume that    W 0 Z T (i) 0 ns j=1 R j J T (i) • • v2 i μ    ≥ 0 i = 1, 2, . . . , n u (12)
and that for each 3

τ ∈ T   He((A + BK)W ) -BK BS -W K T -Z T • He(R(τ )A) -ΣR(τ ) K T -J T • • -2S   < 0 (13)
where4 Σ := ns j=1 σ j I np j and for all τ ∈ T , R(τ ) := ns j=1 e σj τj R j . Define5

T ∋ τ → P (τ ) := W -1 ⊕ R(τ )
Then, set A defined in [START_REF] Gomes Da | Regional stability analysis of discrete-time dynamic output feedback under aperiodic sampling and input saturation[END_REF] is locally asymptotically stable for H cl and the set

E μ := x ∈ C : x T cl P (τ )x cl ≤ μ
is included in the basin of attraction of A.

B. Numerical Issues

Theorem 1 turns the solution to Problem 1 into the feasibility problem of some matrix inequalities. On the other hand, the search for feasible solutions to ( 13) is quite challenging since such conditions need to be checked over the box T . To overcome this problem, next we show that (13) can be turned without any loss into a finite set of matrix inequalities.

Proposition 1:

Let W ∈ S np + , R 1 ∈ S np 1 + , R 2 ∈ S np 2 + , . . . , R ns ∈ S np ns + , S ∈ D nu + , σ 1 , σ 2 , .
. . , σ ns ∈ R >0 , and Z, J ∈ R nu×np be given. The following two items are equivalent 6(i) [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] holds

(ii) let Q := ns i=1 R i N (Ψ):=   He((A + BK)W ) -BK BS -W K T -Z T • He(QΨA) -ΣQΨ K T -J T • • -2S   < 0 ∀Ψ ∈ ns i=1 ψ i I ni : ψ i ∈ {1, e σiT (i) 2 } (14)
Another issue that needs to be tackled in Theorem 1 is related to the fact that the resulting estimate of the basin of attraction is a subset of C, hence intrinsically dependent on the "artificial" variable τ . Indeed, from a practical standpoint, one is interested in dealing with sets of initial conditions for the closed-loop system [START_REF] Lall | An LMI solution to the robust synthesis problem for multi-rate sampled-data systems[END_REF]. The result given next allows one to achieve this goal.

Corollary 1: Assume that there exist W ∈ S np

+ , R 1 ∈ S np 1 + , R 2 ∈ S np 2 + , . . . , R ns ∈ S np ns +
, S ∈ D nu + , σ 1 , σ 2 , . . . , σ ns ∈ R >0 , and Z, J ∈ R nu×np such that ( 12)-( 13) hold. Let

T ∋ τ → P (τ ) := W -1 ⊕ R(τ )
and 7 τ ⋆ := (T

(1) 2 , T (2)
2 , . . . , T

)

Then, the set E( P (τ ⋆ ), μ) × T is included in the basin of attraction of A for [START_REF] Seuret | Taking into account period variations and actuator saturation in sampled-data systems[END_REF].

Remark 4: Having an estimate of the basin of attraction that is "τ -independent" allows one to define convenient size criteria that can be conveniently exploited in an optimization setup to optimize the size of the resulting set. This aspect is shown in the subsequent section.

Remark 5: It is interesting to notice that the "size" of the estimate of the basin of attraction provided by Corollary 1 depends on the vector τ ⋆ . In particular, one can remark that as τ ⋆ goes to infinity, E( P (τ ⋆ ), μ) approaches the origin while as τ ⋆ goes to zero, E( P (τ ⋆ ), μ) approaches E( P (0), μ). Thus, on the one hand, E( P (0), μ) provides the largest estimate of the basin of attraction achievable via our methodology, on the other hand, increasing τ ⋆ tends to give rise to smaller estimates of the basin of attraction.

C. Optimization Issues

With the objective of reducing the conservatism of the proposed analysis, Problem 1 can be formalized as a suitable optimization problem. In particular, similarly as in [START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: Achieving regional, nonlinear performance[END_REF], [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF], [START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions[END_REF], the following optimization problem can be considered:

Problem 2 (Estimation of the basin of attraction): "Maximize" the set E( P (τ ⋆ ), μ).

To properly formulate the above optimization problem, one needs to define an adequate measure for the ellipsoidal set E( P (τ ⋆ ), μ). To this aim, different criteria are proposed in the literature; see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. In this work, we consider trace( 1 μ P (τ ⋆ )) = 1 μ trace(W -1 +R(τ ⋆ )) as a size criterion. In particular, larger values of trace( 1μ P (τ ⋆ )) are associated to a smaller ellipsoid. Therefore, the objective consists of 7 The symbol τ ⋆ is undefined in the conference version.

minimizing 1 μ trace(W -1 + R(τ ⋆ )). On other hand, to avoid dealing in the solution to the resulting optimization problems with W -1 as a decision variable, one can consider the following additional constraint

M W I • W ≥ 0 (15)
which is equivalent to M W -W -1 ≥ 0. That allows one to achieve the minimization of the considered objective by minimizing the objective 1 μ trace(M W + R(τ ⋆ )). By setting μ-1 := δ, this leads to the following formulation for Problem 2

                   minimize ω 1 δ + ω 2 trace(M W + R(τ ⋆ )) s.t. W > 0, Q > 0, (14), ( 15
), S ∈ D nu +   W 0 Z T (i) 0 ns j=1 R j J T (i) • • δv 2 i   ≥ 0 i = 1, 2, . . . , n u
where ω 1 , ω 2 ∈ R >0 are tuning parameters. Whenever the scalars σ 1 , σ 2 , . . . , σ ns are selected, the above optimization problem is a semidefinite program, i.e., an optimization problem with linear objective and linear matrix inequality constraints. Therefore, the solution to this problem can be efficiently obtained via numerical available software; see [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. On the other hand, the nonlinearities involving the real numbers σ i , for i = 1, 2, . . . , n s , are easily manageable (at least when n s is small enough) in a numerical scheme by treating these scalars as tuning parameters or by selecting those via an iterative search.

IV. NUMERICAL EXAMPLE

Consider the example in [START_REF] Hu | Sampled-data control of networked linear control systems[END_REF], which is defined by the following data:

= 0.3, T To better assess the conservatism of the estimate of the basin of attraction (given by E( P (τ ⋆ ), μ) × T ), we pick initial conditions for H cl in U := (E( P (τ ⋆ ), μ) ∩ Y) × T , were Y := {(x p , η) ∈ R 2np : η = x p }, i.e., we enforce zero initial conditions for the controller (3). The projection of U onto R 2 (the plant state space) is shown in Fig. 1 8 Numerical solutions to SDP problems are obtained in Matlab R via Mosek [START_REF] Aps | The MOSEK optimization toolbox for MATLAB manual[END_REF] thanks to YALMIP [START_REF] Lofberg | Yalmip: A toolbox for modeling and optimization in matlab[END_REF]. Simulations of hybrid systems are performed in Matlab R via the Hybrid Equations (HyEQ) Toolbox [START_REF] Sanfelice | A toolbox for simulation of hybrid systems in matlab/simulink: Hybrid equations (HyEQ) toolbox[END_REF]. Code at https://github.com/HybridSystemsLab/IntermittentCtrlSatMrate along with some trajectories for the plant state. In particular, let φ = (φ xp , φ η , φ τ1 , φ τ2 ) be a solution to H cl , in this simulation we selected initial conditions of the form φ(0, 0) = (x 0 , x 0 , T

2 , T

2 ), x 0 ∈ R 2 , and for each (t j , j + 1) ∈ dom φ

  {s} ⊂ dom φ. Essentially, the domain of the solutions to H cl , manifests an average dwell-time property, with dwell-time τ D = min{T

1 TK

 1 = -1.2625 -1.2679 assume the control input is saturated with saturation level ū = 1, and set T

2 . 1

 21 By solving Problem 2 via a grid search for the scalars σ 1 , σ 2 , one gets 8 : = 0.041342, R 2 = 0.018985σ 1 = 1.8, σ 2 = 2.3, μ = 2.1380

2 Fig. 1 :Fig. 2 :

 212 Fig.1: The set U projected onto R 2 (dashed gray) and some trajectories of the plant state: x0 = (-4, 2) (dash-dotted black), x0 = (-2, 4) (black dotted), x0 = (2, 4) (solid black), x0 = (-2, -4) (gray solid), and x0 = (1.5, 4) (dashed black).

Although our construction relies on the model of the plant, closedloop asymptotic stability is preserved in the presence of small parametric uncertainties. Due to space constraints, robustness issues are not discussed in this paper and will be presented elsewhere.

By the definition of the system H cl and of the set A, for every x ∈ C ∪ D ∪ G(D), |x| A = |(xp, η)|.

There is a missing S in (13) and (14).

The symbol Σ is undefined in the published version.

In the published version, the first definition of the symbol P (τ ) appears in Corollary 1.

There is a missing I in the set defined in equation (14).

This work has been partially developed when the first author was with Department of Computer Engineering, University of California, Santa Cruz. This research has been partially supported by the National Science Foundation under CAREER Grant no. ECS-1450484, Grant no. ECS-1710621, and Grant no. CNS-1544396, by the Air Force Office of Scientific Research under Grant no. FA9550-16-1-0015, by the Air Force Research Laboratory under Grant no. FA9453-16-1-0053, and by CITRIS and the Banatao Institute at the University of California. This version fixes some typographical errors/undefined symbols in the published paper.

 [START_REF] Ferrante | Observer-based control design for linear systems in the presence of limited measurement streams and intermittent input access[END_REF]In particular, observe that

2 ) ∈ E( P (τ ⋆ ), μ) × T if and only if x 0 ∈ Π R 2 (U).

Fig. 1 points out that for some initial conditions outside the set E( P (τ ⋆ ), μ) × T the plant state diverges. On the other hand, the conservatism of our approach is revealed by the existence of converging trajectories from points outside E( P (τ ⋆ ), μ) × T (x 0 = (1.5, 4)). In Fig. 2, we reported the evolution of the control input when x 0 = (-2, 4). The figure emphasizes that, as foreseen through our methodology, despite the saturation of the control input, asymptotic stability is preserved.

V. CONCLUSION Regional stability analysis of linear systems with multirate samplers and actuator saturation was studied. The approach we proposed is able to provide explicit estimates for the basin of attraction by the use of semidefinite programming.

The results presented in this paper open the door to several research directions. In particular, the extension of these results to L 2 -external stability is currently an ongoing work. Other valuable extensions of this work pertain to the design of state estimators to address more realistic scenarios in which the vector state is not fully measurable and the analysis of the effect of plant uncertainties.


X Margueron

J-P Keradec

A Besri

Complete Analytical Calculation of Static Leakage Parameters. Application to HF Transformer Optimization

Keywords: leakage inductances, analytical calculation, transformer parameters, planar technology. I

The pre-determination of transformer leakage inductances is essential for component designers. Except for special winding disposal, usual analytical methods of evaluation are useless and fem simulation evaluation, even more widely applicable, requires a time unsuitable with any optimization process. Work presented here is based on an analytical method we published recently about leakage inductances calculation. In the present paper, an explicit expression of leakage inductances is given. This expression does not appear as series so it is requires short computer time and it opens a lot of future applications relying on optimization software. Moreover, the calculation is not restricted to leakage inductances: it has been extended to all the parameters characterizing the whole leakage behavior part of any transformer.

INTRODUCTION

Leakage inductances of a transformer have a strong impact on stresses sustained by switches so they are of major concern for power electronic designers. Contrary to magnetizing inductances, their values can't be easily predetermined by computation so they are often measured in a post-conception phase or evaluated before, using long fem simulations.

Because leakage inductances may cause over voltages on and extra losses in power electronic switches, designers must predict their values to be sure that converters will work properly. Some analytical methods are available [START_REF] Abraham Dauhajre | Modelling and estimation of leakage phenomena in magnetic circuits[END_REF][START_REF] Ferrieux | Alimentations à découpage, convertisseurs à résonance[END_REF][START_REF] Binns | The analytical and numérical solution of electric and magnetic fields[END_REF] but they are restricted to particular geometries. Some others, which rely on series expansion, are more widely applicable but they require a lot of terms that slow the calculation.

Three years ago we worked on a semi-analytical method based on 2D PEEC formulae [START_REF] Margueron | Analytical calculation of static leakage inductances of H.F. transformers Using PEEC Formulas[END_REF]. Potential vector was analytically expressed but, in order to reach energy values, a last integration was still carried out numerically. This last integration is now solved analytically so leakage inductances can be evaluated using only one explicit formula. The method can be applied to any kind of transformer winding, with quite no restriction on conductor's arrangement. Such a presentation suits very well to an optimization process [START_REF] Atienza | A methodology for the sizing and the optimization of an electromagnetic release[END_REF].

This energy calculation only assumes that total current is null in transformer window, so it can be used also to determine all the parameters which characterize the behavior of any nwinding transformer (which has, at least, one winding shortcircuited). These parameters are grouped in what we call "leakage transformer" [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF]. This approach allows these parameters to be evaluated at frequencies where eddy current effects are negligible. The method is very useful, particularly in the case of planar transformer, where windings arrangements may be more complicated than in usual round winding transformers. An industrial planar transformer supplies a realistic example. Measurements carried out on it will be compared to our analytical results.

II. LEAKAGE ENERGY CALCULATION

A. Definition of Leakage Inductance

Inductive equivalent circuit of a two winding transformer can be represented by two inductances, three resistances and one perfect coupler (Fig. 1):

• 1 r and 2 r : Winding resistances seen from primary.

• m L : Magnetizing inductance.

• f L : Leakage inductance.

• R : Magnetic losses resistance.

• η : Coupler ratio. If coupling is strong, transformation ratio η is very close to the turn number ratio and leakage inductance f L is far smaller than magnetizing one m L . f L influence appears on an impedance measurement when secondary is short circuited (Fig. 2). Its measurement can be realized only in a specific frequency range (zone 4). Others zones are respectively dominated by primary resistance (zone 1), magnetizing inductance (zone 2), resistances of both windings (zone 3) and capacitances in high frequency (zone 5).

If both windings (primary and secondary) are supplied by opposites currents whose ratio is equal to coupling ratio η , current in magnetizing inductance is null. Ampere turns are then compensated and, in frequency range 4, magnetic energy is stored in leakage inductance (1). In this paper we focus on static leakage inductances. They correspond to the inductances seen at low frequencies (i.e. such that current inside conductors is uniform). For higher frequencies, eddy currents reduce these values.

B. Leakage Energy Calculation Method

Leakage inductance calculation methods are based on magnetic energy stored when ampere-turns are compensated [START_REF] Abraham Dauhajre | Modelling and estimation of leakage phenomena in magnetic circuits[END_REF][START_REF] Ferrieux | Alimentations à découpage, convertisseurs à résonance[END_REF][START_REF] Binns | The analytical and numérical solution of electric and magnetic fields[END_REF]. This energy can be computed with magnetic excitation H (2) or potential vector A [START_REF] Binns | The analytical and numérical solution of electric and magnetic fields[END_REF]. Integral of (2) must be extended to infinity while integral (3) is only limited to conductors, where J is not null.

dV H µ W ⋅ ⋅ = ∫∫∫ 2 2 1
(2)

dV J A W ⋅ ⋅ = ∫∫∫ 2 1 (3)
with µ : permeability J : current density In order to simplify calculations, transformer windings are considered as 2D plan systems with a length equal to the mean length turn. Energy per unit of length is noted Wl (4) [START_REF] Atienza | A methodology for the sizing and the optimization of an electromagnetic release[END_REF].

dS H µ Wl ∫∫ ⋅ ⋅ = 2 2 1 (4) dS J A Wl ∫∫ ⋅ ⋅ = 2 1 (5)
According to conductor shape and arrangement in planar transformer (Fig. 3), the method introduced in [START_REF] Margueron | Analytical calculation of static leakage inductances of H.F. transformers Using PEEC Formulas[END_REF] is based on exact formulation of potential vector created by a straight rectangular cross section conductor (Fig. 4). All materials are considered as linear, homogenous and isotropic. Then, three more hypotheses have to be made:

• Studied device is supposed to be 2D plan. Currents are perpendicular to conductor sections and they are also uniform inside each wire (static case). • Transformer is in leakage inductance "measurement situation": Total current in each window is null. • Windings are made of either rectangular wires either round conductors grouped into rectangular regions.

Our elementary field source is an infinite straight wire, having a rectangular cross-section (Fig. 4), in which a homogenous current flows. Vector potential [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF] is obtained by integrating that of a thin wire, from x = -a to +a and from y =b to +b. In a Oxyz base, potential vector is directed along Oz .

In our previous paper [START_REF] Margueron | Analytical calculation of static leakage inductances of H.F. transformers Using PEEC Formulas[END_REF], the double integration which appears in (5) still had to be made numerically. Recently, we find the analytical expression [START_REF] Stratton | Electromagnetic Theory[END_REF] of this integral which is that of the ( )

Y X F ,
function written in [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF].

C. Example for Two Rectangular Conductors

To illustrate the computing method and then, obtain only a unique formula for leakage inductance calculation, let's take the example of a transformer window occupied by only one pair of rectangular wires (Fig. 5) indexed 1 and 2.

( ) ( ) [ ] [ ] b y Y b y Y a x X a x X Y X F J y x A - = + = - = + = ⋅ - = , 4 , 0 π µ with ( ) ( )       +       + + = Y X Y X Y X Y X XY Y X F arctan arctan ln , 2 2 2 2 and b a I J 2 2 ⋅ = (6) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 4 2 2 4 24 7 3 1 6 24 1 Y X Y X Y X Y X XY Y X Y Y X X dXdY Y X F Y X G -               +       + + ⋅ + - - = ⋅ = ∫∫ arctan arctan ln , , (7)
Each wire is supplied with a total current equal respectively to 1 I and 2 I , that, at the end, are supposed to be opposite.

Vector potential due to these two wires ( 8) is the sum of two contributions similar to [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF]. Combining ( 5) and ( 8) leads to the expression of total energy per unit of length Wl [START_REF]Agilent 4294A Precision Impedance Analyser-Operation Manual[END_REF].

( ) ( ) ∑ = - - = 2 1 , , k k k k yr y r x x Az y x A (8) ( ) ( ) [ ] ( ) ( ) dxdy A A J dxdy A A J dxdy yr y r x x A yr y r x x A J Wl Wcs Wcs Wcs Wcs ⋅ + + ⋅ + = ⋅ - - + - - ⋅ = ∫∫ ∫∫ ∫∫ 2 2 1 2 1 2 1 1 2 2 2 2 1 1 1 2 1 2 1 , , 2 1 U (9)
with 1 Wcs and 2 Wcs : Wire cross section areas Value of ( 9) is obtained by varying the ( ) Y X G , function [START_REF] Stratton | Electromagnetic Theory[END_REF] over wire cross section Wcs [START_REF] Margueron | Identifying the magnetic part of the equivalent circuit of n-winding transformer[END_REF].

Vector potential created by the conductor centred in ( ) 0 , 0 is deduced from the double variation [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF]. This variation must be included in the ( ) Y X G , expression. Other wires are located by ( )

k k yr xr ,
and they all must be taken into account.

Formula (10) can then be rewritten as [START_REF] Skutt | Leakage Inductance and Termination Effects in High-Power Planar Magnetic Structure[END_REF] to obtain the exact formulation of magnetic energy per length unit for two rectangular conductors. With

1 2 J J - =
, This energy is proportional to 2 1 J . This leads to the inductance per length unit. Neglecting the influence of the magnetic core, the leakage inductance is obtained by multiplying the previous value by the mean turn length.

D. Generalization for Transformer Winding

Formula ( 11) is easily extended to N conductors (12). Always assuming that total current in the window is null, this expression gives the exact value of energy per length unit. This leakage energy depends on

1 - N independent currents and it is characterized by ( ) 2 1 - N N independent coefficients.
The formula is completely analytical and explicit so it is quickly computed and well suited to be used it an optimization software.

III. ACCOUNTING FOR MAGNETIC MATERIAL

Although magnetic core has a small impact [START_REF] Margueron | Analytical calculation of static leakage inductances of H.F. transformers Using PEEC Formulas[END_REF] on values regarding short-circuit conditions (often less than 5% on leakage inductances), it must be taken into account, even roughly, to obtain corresponding parameters with a suitable accuracy.

Magnetic image technique [START_REF] Stratton | Electromagnetic Theory[END_REF] supplies a simple way to approximately account for magnetic material influence.

( ) ( ) [ ] [ ] [ ] [ ] 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 1 2 2 1 1 2 2 1 2 1 2 1 1
G G J G G J dxdy A A J dxdy A A J Wl + - + - + - + - + + + = ⋅ + + ⋅ + = ∫∫ ∫∫ (10) ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 0 1 1 1 1 0 2 2 2 2 2 0 1 1 1 1 0 1 , 4 , 4 2 1 , 4 , 4 2 1 b yr y b yr y a xr x a xr x b y Y b y Y a x X a x X b y Y b y Y a x X a x X b yr y b yr y a xr x a xr x b y Y b y Y a x X a x X b y Y b y Y a x X a x X yr Y xr X G J yr Y xr X G J J yr Y xr X G J yr Y xr X G J J Wl + = - = + = - = - = + = - = + = - = + = - = + = + = - = + = - = - = + = - = + = - = + = - = + =               - - - - - - +               - - - - - - = π µ π µ π µ π µ (11) ( ) [ ] [ ] n n n n n n n n k k k k b yr y b yr y a xr x a xr x N k b y Y b y Y a x X a x X k k k N n n yr Y xr X G J J Wl + = - = + = - = = - = + = - = + = =                       - - - = ∑ ∑ 1 1 0 , 4 2 1 π µ (12)

A. Magnetic Images for Infinite Material

The goal of this technique is to replace the magnetic material by currents located in its volume in order to obtain the same magnetic field in the air.

To begin, let's recall a well known problem. A thin straight wire, parallel to the plan which limits two semi-infinite materials of respective relative permeability 1 and r µ (Fig. 6) carries a current I . One can show that the field in the air is the sum of the one ( Hi ) created by the wire alone and another one ( Hr ) created by another wire, called "image wire". This wire is symmetrically located on the other side of this interface, and a current ' I flows through it (13).

B. Influence of Ferrite Thickness on Images

Before applying the image technique to the magnetic core, it is indispensable to answer this question: is the same technique still applicable if magnetic material has a finite thickness? The answer cannot be always positive, especially when the thickness tends to zero or the relative permeability reaches 1.

With this in mind, the influence of magnetic core thickness has been investigated. Semi-infinite material of Fig. 6 has been replaced by a magnetic layer of thickness ep (Fig. 7). In this case, reflected field r H , is expressed as (14).

This expression (14) is divided in two terms:

• The first one is identical to the field reflected by an infinite media. This field is created by an image wire located in ( )

h - , 0
and supplied by a current given by the expression given above (13).

C. Comparison Between Finite and Infinite Thickness

Both analytical expressions have been presented in previous paragraph. In order to know what result (infinite or finite) must be used, results of both techniques must be compared. First term of ( 14) is identical to the field of infinite media. The second one decreases as n increases because

( ) ( ) 1 1 1 < + - r r µ µ
. Thus, as n increases, image wire is less and less influent. For some situations, infinite model may be sufficient.

To find an acceptable approximation, we consider a wire placed at mm h 50 = and mm x 0 = of a ferrite media (Fig. 7). Tangent magnetic field on the surface at mm x 50 = is used to check both formulations (finite and the infinite) for two different values of permeability 40 = r µ and 2000 ) (Fig. 8).

                        + - + + - -         + - + = ∑ ∞ =1 2 1 2 1 1 1 1 , 2 , 1 4 1 1 , , 2
1 n n r r i r r r r i r I nep h y x H µ µ I h y x H H µ µ µ µ π (14)
For both permeabilities, field obtained with finite thickness reaches the one obtained with an infinite magnetic material when thickness reaches a sufficient value. As thickness decreases test field reaches a limit value independent of the permeability: that of the incident field. However, this value is reached for a lower thickness when permeability is high. To sum up, as thickness grows, test field increases from the limit related to no magnetic material to another limit given by an infinite media. This growth reaches 90% of its maximum at . For thickness higher than these values, reflected field is very close to that of a semi-infinite material.

In practical cases, thickness is not an important parameter for material whose permeability is upper than 2000. For permeability around 40, computation must be done using multiple reflections ( 14). The right resulting field is obtained by adding at least 20 to 25 images.

D. Application to Leakage Energy Calculation

In order to compute leakage energy, magnetic material can be taken into account with the use of magnetic images. Formulation depends on permeability and ferrite thickness. Application of such method is not very different with infinite or finite material. Only current magnitude in images, wire positions and necessary numbers of images are different.

In Fig. 9, 8 magnetic images are added to initial transformer windows. In most cases, with high permeability ferrite, theses images are sufficient. Indeed, because total current in the windows is null, radiated field decreases so fast that images located further has negligible influence on that existing inside the window. Formula (12) is then modified into (15) in order to account for the effect of magnetic material.

Finally, (15) is the complete expression of leakage energy per unit of length inside transformer windows. This formula is totally analytical and can be used with optimization software. Not only leakage inductances can be computed with this formula (15) but, in case of a transformer owning more than two windings, if leakage inductances are coupled, parameters depicting these properties, can be too.

The leakage transformer (which is defined for any transformer) describes the magnetic behavior that remains when one winding is short-circuited [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF]. It is composed by leakage inductances and couplers that describe coupling between these inductances. For a 3 winding transformer (Fig. 10), leakage energy is computed when current in magnetizing inductance is null (16) (i.e. assuming primary is shortcircuited). In this case, Ampere-turns are compensated.

In order to determine all the parameters of the leakage transformer, computation is made as in section II and III. Stored energy appears as a quadratic function of two independent currents so this function is fully described by three coefficients. Firstly, to evaluate inductance a Lf (17), secondary current must be null ( 0 = s I

). Secondly, for inductance s Lf defined by (18), auxiliary current must be null ( 0 = a I

).

Finally, a third combination of winding currents, the sum of which is always null ( It is useful to evaluate the coupling coefficient of leakage inductances (20). This coefficient is indispensable to decide if coupling of leakage inductances can be neglected or not and, as a consequence, if the related coupler can be removed. Finally, all the magnetic static parameters depicting leakage transformer can be evaluated analytically. This process is general and can be applied to an n -winding transformer (having an ( )

i i b y Y b y Y a x X a x X k k k N n n MI yr Y xr X G J J Wl + = - = + = - = = - = + = - = + = =                 +       - - - = ∑ ∑ ∑ 1 1 0 , 4 2 1 π µ (15) With ( ) [ ] [ ] ∑ = - = + = - = + =         - - - - + - = N k b y Y b y Y a x X a x X i k i k r r k i k k k k y yr Y x xr X G J MI
1 - n
winding leakage transformer) [START_REF] Margueron | Design of equivalent circuits and characterization strategy for n-input coupled inductors[END_REF]. Only the number of current combinations (having a null sum) needed to obtain all the parameters varies.

V. APPLICATION TO PLANAR TRANSFORMER An automatic tool for computing leakage inductances (and transformers) has been developed under Mathcad ® software [START_REF][END_REF]. This tool has been use to identify static leakage transformer of an industrial planar transformer (Fig. 11). This flat device contains two identical 3-winding transformers located side by side. We now focus on one of them to compare experimental results to those given by our static leakage computing tool.

A. Experimental Caracterization

Measurements has been carried out using an Agilent 4294A impedance analyzer [START_REF]Agilent 4294A Precision Impedance Analyser-Operation Manual[END_REF], following the method developed in [START_REF] Margueron | Identifying the magnetic part of the equivalent circuit of n-winding transformer[END_REF]. Lossless equivalent circuit of such a transformer is presented in Fig. 10.

Leakage inductances are plotted in Fig. 12 and Fig. 13 versus frequency. As it has been told in section II.A, in low and high frequency, inductances are difficult to evaluate because they are masked by other components. Moreover, leakage inductances are often modified by eddy currents so, to model them in a wide frequency range, we place some Lp-Rp cells [START_REF] Margueron | Identifying the magnetic part of the equivalent circuit of n-winding transformer[END_REF] in series with a pure inductance. In this case, static leakage inductance corresponds to the sum of all these serial inductances. Searched values appear on the low frequency side of model related curves shown in Fig. 12 . All these values are summed up in Table I. Finally, agreement of our computations and our experimental results is quite good: discrepancies are always less than % 20 . These discrepancies can be attributed to both measurements and analytical calculation. In [START_REF] Margueron | Identifying the magnetic part of the equivalent circuit of n-winding transformer[END_REF] we have shown that, even with meticulous care, we are unable to measure inductances with incertitude lower than about nH 8

. On the other hand, our calculation tool assumes the transformer to be a 2D system, that is not exact, and it neglects extra wiring which links turns to external terminals [START_REF] Skutt | Leakage Inductance and Termination Effects in High-Power Planar Magnetic Structure[END_REF]. For transformers having higher values of leakage inductances, agreement is generally far better.

Accuracy is not impressive for presented example. However, for wire arrangement shown in Fig. 14, usual methods of calculation are not applicable or they require a lot of term to converge toward a result accurate enough. Method presented here is also able to account for partial surrounding of windings by magnetic core [START_REF] Margueron | Analytical calculation of static leakage inductances of H.F. transformers Using PEEC Formulas[END_REF]. Evaluated leakage inductances are static ones. Their values generally decrease when frequency grows because of eddy currents effects. We now work on the extension of the presented technique to high frequency.

VI. CONCLUSION

A complete analytical formula for computing static leakage parameters of a transformer has been introduced. This paper is completing a previous one by adding a fully analytical formulation. Proposed method applies as soon as wires, or winding cross sections, are rectangular.

Leakage energy computation assumes device geometry can be considered as a 2D plane. It is based on expression of vector potential created by a rectangular conductor carrying a uniform current. Magnetic material is taken into account using magnetic images, even if it doesn't completely surround the wires. Influence of magnetic material thickness has been investigated too.

A tool has been developed using Mathcad ® software and it has been applied to industrial planar transformers. This tool proved to be effective for transformer designers. They are now able to quickly and accurately predict all static parameters, including inductances, regarding leakage.

Two extensions can now be accessed. The first one consists in coupling our developing tool with optimization software to improve device behaviour. The second one aims to account for eddy currents inside rectangular wires. This will permit these effects to be taken into account in transformer equivalent circuit.

Figure 1 .

 1 Figure 1. Two winding transformer inductive equivalent circuit

Figure 2 .

 2 Figure 2. Impedance measurement-seondary short circuited

  Figure 3. Planar transformer section Figure 4. Rectangular conductor

2 Figure 5 .

 25 Figure 5. Rectangular conductors

Figure 6 .

 6 Figure 6. Wire and infinite magnetic material

Figure 7 .

 7 Figure 7. Influence of finite magnetic material (thickness ep) • The second one corresponds to the field of an infinite set of thin wires. The ieth n is placed at ( ) ep n h × --2 , 0 and it carries a current ( ) ( ) [ ] n r r I

Figure 8 .

 8 Figure 8. Comparison of finite and infinite formulation (a) : µr=40 | (b) : µr=2000

Figure 9 . 8

 98 Figure 9. 8 magnetic images

  the determination of the coupling ratio sa η between secondary and auxiliary (19).

  Figure 10. Leakage transformer from 3-winding transformer

5 cmFigure 11 . 3 -

 5113 Figure 11. 3-winding planar transformer

Figure 14 .Figure 15 .

 1415 Figure 12. Auxiliary leakage inductance variation vs frequency

TABLE I

 I

	.				LEAKAGE TRANSFORMER PARAMETERS
	Parameters				Measurement	Computation	Error
	P/S inductance leakage	Lf	s	. 69	8	nH	. 59	4	nH	15	%
	P/A leakage inductance	Lf	a	. 72	9	nH	. 61	4	nH	16	%
	A/S coupling ratio	η	sa	0	.	64	0	.	77	17	%
	A/S
	coefficient coupling	k	sa	0	.	65	0	.	78	17	%


Frederic Alexandre
email: frederic.alexandre@inria.fr

Creativity explained by Computational Cognitive Neuroscience

Recently, models in Computational Cognitive Neuroscience (CCN) have gained a renewed interest because they could help analyze current limitations in Artificial Intelligence (AI) and propose operational ways to address them. These limitations are related to difficulties in giving a semantic grounding to manipulated concepts, in coping with high dimensionality and in managing uncertainty. In this paper, we describe the main principles and mechanisms of these models and explain that they can be directly transferred to Computational Creativity (CC), to propose operational mechanisms but also a better understanding of what creativity is.

Introduction

Artificial Intelligence (AI) has developed approaches based on knowledge manipulation and others based on data processing. The latter ones have made tremendous progresses recently, mostly due to technological improvements. Nevertheless, the development of AI remains constrained by a series of limitations, which are present from the birth of this domain and still unsolved. This is also the case for CC, seen as a subfield of AI [START_REF] Colton | Computational Creativity: The Final Frontier?[END_REF].

The three limitations of Artificial Intelligence

These limitations are mostly methodological, which means that they do not prevent the design of projects of AI or CC but they render difficult their development up to a realistic size and they miss a realistic description of certain characteristics, particularly when targeting their autonomy.

Giving a semantic grounding Probably, one of the most fundamental limitations of AI is the difficulty to make sense and exploit the meaning of the objects they are manipulating. This limitation is also important in CC where a novel point of view can come from such a semantic analysis. This is often related to the incapacity to ground these objects in the real world [START_REF] Harnard | The symbol grounding problem[END_REF]) and to propose solutions related to the intrinsic meaning of the objects in the task. This problem is generally addressed by developping ontologies of the task, but these ontologies are often built explicitly from knowledge, whereas in humans, intrinsic meanings are generally acquired implicitly and difficult to express explicitly.

Coping with high dimensionality Recently, the increase of computational resources allowed AI to explore tasks in high dimensional spaces but this progress remains negligible as compared to the virtually infinite size of state spaces in realistic tasks. Similarly, [START_REF] Colton | Computational Creativity: The Final Frontier?[END_REF] underline the growing tendency in CC to exploit web and other large resources. A strategy is to use heuristics and to explore a limited part of the state space. A side effect of long lasting exploration in high dimensional state spaces is that learning requires many examples and long training times, which is not realistic, as compared to the ability of animals and especially of humans to learn quickly with few examples.

Managing uncertainty Tasks in the real world can be stochastic, marring observations with noise; they can be volatile, implying that a relation learned one day can suddenly be no longer valid. A related important problem, when an intelligent system receives an error signal, is to decide if this is due to stochasticity (in the case, the best strategy is to insist) or to volatility (requiring to look for a new response). This is related to the balance between exploration and exploitation, questioned in many models in AI. Exploration can be purely random, but this is not consistent with observation of a certain degree of flexibility in changing environments, where the appropriate behavior is immediately reinstated each time the same context is revisited. Choosing between exploration and exploitation and going beyond random processes are also important issues in CC. All these problems are generally tackled in various probabilistic frameworks (like bayesian techniques). Some of them propose solid and mathematically grounded methods but they are computationally prohibitive, thus adding to the problem of high dimensionality defined in the previous item.

The CCN approach

CCN models are primarily developed as a way to operationalize proposed cognitive concepts, together with their brain implementation. They can sometimes give a biological basis to classical models in AI as it is the case in Reinforcement Learning. They can also help study how the brain solves some problems that remain difficult with classical mathematical approaches. This is specifically the case with the three limitations mentioned above, as we describe now in more details.

Giving a semantic grounding It is generally considered that access to the semantic meaning of objects is due to the fact that we have a body that we can perceive externally (sense of exteroception, including proprioception) and internally (sense of interoception [START_REF] Craig | How do you feel -now? the anterior insula and human awareness[END_REF]). Considering the consequences of taking these sensations into account onto cognition is studied in what is called embodied AI [START_REF] Pfeifer | How the body shapes the way we think: a new view of intelligence[END_REF].

Internal representation of the body is built in the insular cortex. On the one hand, rich signals of pain and pleasure allow to elaborate reinforcement signals much more complex than the unique scalar generally used in reinforcement learning. Learning to anticipate these signals (pavlovian conditioning, see a model in [START_REF] Carrere | A pavlovian model of the amygdala and its influence within the medial temporal lobe[END_REF]) elicits what is called emotions and plays a pivotal role in decision making by allowing to compare different signals under a common currency, as summarized by the principle of the somatic markers [START_REF] Damasio | The Somatic Marker Hypothesis and the Possible Functions of the Prefrontal Cortex[END_REF].

On the other hand, other important internal signals form motivations and can be related to physiological needs (extrinsic motivations) as well as needs for certain kinds of information (intrinsic motivations). Such signals are particularly important to generate internal goals and to prevent from obeying only stimuli in the environment. This is an important dimension for goal-directed behaviors, see computational frameworks in [START_REF] Pezzulo | Thinking as the control of imagination: a conceptual framework for goaldirected systems[END_REF].

Coping with high dimensionality One important contribution has arisen from CCN about the way humans can learn in high dimensionality with the Complementary Learning System (CLS) framework [START_REF] Mcclelland | Why there are complementary learning systems in the hippocampus and neocortex: insights from the successes and failures of connectionist models of learning and memory[END_REF]. In this framework, it is proposed that humans can, at the same time, learn slowly concepts in semantic memory, in the cortex, and store quickly episodes in the hippocampus. Later on, by a phenomenon called consolidation, the hippocampus will send episodes back to the cortex to train it off-line and ensure good properties to the learning process, for example to avoid mix-up, also called catastrophic forgetting. This kind of principles has been used in Machine Learning to propose ways of training large architectures (for example deep networks) with "not so big data", cf a model in [START_REF] Drumond | Bioinspired Analysis of Deep Learning on Not-So-Big Data Using Data-Prototypes[END_REF].

Recently, progresses in neuroscience about the hippocampus and in the corresponding models led to more precise explanation about information encoding in the hippocampus, and particularly about place cells and grid cells (model in [START_REF] Stachenfeld | The hippocampus as a predictive map[END_REF]) and their proposed contribution in the goal-oriented exploration of high-dimensional spaces. It has particularly been proposed that replays of episodes for consolidation are not random but obey subtle strategies (implemented in [START_REF] Mattar | Prioritized memory access explains planning and hippocampal replay[END_REF]) and that they can be organized in time to participate to the elaboration of knowledge-based strategies in the frontal cortex (cf computational framework in [START_REF] Daw | Are we of two minds?[END_REF]). It has also been experimentally shown (Derdikman and Moser 2010) that the hippocampus not only replays stored episodes but is also able to create new episodes combining pieces of old episodes, thus promoting an imaginative training with virtual episodes.

These concepts and models are today transfered to the domain of AI to elaborate what is called Episodic Reinforcement Learning [START_REF] Gershman | Reinforcement Learning and Episodic Memory in Humans and Animals: An Integrative Framework[END_REF]) and participate to the design of more realistic and faster learning in high dimensional data spaces.

Managing uncertainty Defining the stochastic or volatile nature of the environment has been related to the role of neuromodulation and exploited in models [START_REF] Alexandre | Modeling Neuromodulation as a Framework to Integrate Uncertainty in General Cognitive Architectures[END_REF], adapting a general framework of behavior selection to the estimated kind and level of uncertainty.

As for the flexible adaptation of behavior to a volatile environment, the role of the prefrontal cortex has been mentioned for a long time in neuroscience, observing that patients with a frontal lesion demonstrate perseveration and are unable to adapt to changes [START_REF] Nauta | The problem of the frontal lobe: A reinterpretation[END_REF]). Accordingly, it has been proposed that the prefrontal cortex is the place where nondominant behaviors are defined [START_REF] Wise | Forward Frontal Fields: Phylogeny and Fundamental Function[END_REF] or, to tell it differently, the place where top-down modulations are sent to other regions of the brain to insert internally generated priorities and to help resist to immediate responses driven by stimuli [START_REF] Mesulam | Representation, inference, and transcendent encoding in neurocognitive networks of the human brain[END_REF], thus defining two important mechanisms in the prefrontal cortex: the inhibition of dominant and occasionally non-adapted behaviors and the triggering and maintenance by working memory of attentional focus towards characteristics supposedly important to generate the presently appropriate behavior.

Corresponding models describe the prefrontal cortex as a region where tasks in specific contexts are represented, thus defining the notion of Task Sets [START_REF] Domenech | Executive control and decision-making in the prefrontal cortex[END_REF], learning in certain contexts to inhibit the default behavior and to suggest new behaviors by an attentional process (cf model in [START_REF] O'reilly | Prefrontal cortex and dynamic categorization tasks: representational organization and neuromodulatory control[END_REF])) biasing the current perception. In order to select the pertinent nondominant behavior, some interesting mathematical frameworks have been proposed [START_REF] Collins | Reasoning, Learning, and Creativity: Frontal Lobe Function and Human Decision-Making[END_REF] to tame the combinatorial exploration and generate a limited number of hypotheses. Other interesting and bio-inspired computational mechanisms have been proposed to control and monitor the execution of such behaviors, particularly in the case of hierarchical planning [START_REF] Pezzulo | Thinking as the control of imagination: a conceptual framework for goaldirected systems[END_REF].

These models and concepts are also presently transferred to AI in so-called Meta Reinforcement Learning [START_REF] Wang | Prefrontal cortex as a meta-reinforcement learning system[END_REF], considering that, to adapt to the changing world, a meta learner must quickly learn to select a specialized (and slowly learning) learner, depending on the context.

Another important limitation in AI: Creativity

It is often mentioned though disputable [START_REF] Boden | Computer Models of Creativity[END_REF]) that creativity remains one of the rare cognitive phenomena that AI cannot replicate (with other phenomena like sense of humor, not to say consciouness). Remarkably, studying solutions proposed by CCN to remedy limitations in AI (and consequently in CC), we argue (and will establish below) that they massively rely on cognitive characteristics related to creativity. We consequently propose that CCN could be pivotal to fuel CC with fresh ideas and particularly add a global view often missing in AI.

Defining creativity

Whether dealing with creativity in specific domains (like musical improvisation or scientific creativity) or from a general point of view [START_REF] Beaty | Creative Cognition and Brain Network Dynamics[END_REF], authors often mention that creativity has two main steps: creating a novel idea and verifying that this idea is useful or appropriate to the task [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF]; else the idea is adapted or rejected and another novel idea is elaborated. These two steps are respectively termed divergent and convergent thinking in human creativity research [START_REF] Jung | The structure of creative cognition in the human brain[END_REF]. Corresponding mechanisms in CC are called "generate and evaluate". The first step can be associated to insight [START_REF] Kounios | The cognitive neuroscience of insight[END_REF] and originate from an emotional or a more cognitive process [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF]. As for the second step, it is generally proposed that the assessment of the value of what has been proposed is carried out by brain circuits associated to executive functions. The brain circuits responsible for these two steps have been studied in different ways as it will be discussed in more details below.

The two steps of creativity

Concerning the divergent thinking step, it is reported that self generated thoughts can be spontaneous or goal directed (to meet specific task demands) [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF]. This can correspond to the reinterpretation of a situation to produce a nondominant interpretation [START_REF] Kounios | The cognitive neuroscience of insight[END_REF] or to using generative models to try to explicitly build a novel solution, which has been qualified as a more restricted but more efficient solution [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF]. [START_REF] Boden | Computer Models of Creativity[END_REF] proposes that novel ideas can be produced by combination, exploration or by transformation. [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF] proposes four types of creativity: In the first step, novelty can come from emotional or cognitive structures and the processing can be deliberate or spontaneous.

Among neuronal mechanisms evoked above, it can be remarked that the hippocampus is a good candidate to generate spontaneous novel ideas and that the mechanisms in the prefrontal cortex, responsible for selecting a new Task Set, might participate to the explicit elaboration of new thoughts. Let us also underline that the model proposed in [START_REF] Collins | Reasoning, Learning, and Creativity: Frontal Lobe Function and Human Decision-Making[END_REF] explicitly mentions that if no existing Task Set is appropriate to the task, a new Task Set might be created by the combination between two existing Task Sets or by the random generation of a new one.

The convergent thinking step is often related to cognitive control, the role of which is also to ensure appropriateness in the execution of a behavior. Appropriateness can be syntactic (checking that procedural constraints are respected, which is easy to do with a generative model) or semantic (checking that manipulated objects have consistent values, which is probably less easy [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF][START_REF] Boden | Computer Models of Creativity[END_REF]). In all cases, it is proposed that these verifications are made by the PFC, with the corresponding information set in working memory for their conscious evaluation.

The two steps of creativity in brain circuitry

The functional description of brain circuits is often made through networks of brain regions [START_REF] Mesulam | Representation, inference, and transcendent encoding in neurocognitive networks of the human brain[END_REF], highlighting some regions as critical hubs for the spreading and processing of information. The attentional, semantic, default, cognitive control and salience networks are the major ones and play a central role in creativity [START_REF] Jung | The structure of creative cognition in the human brain[END_REF].

The default network (activated by default, for spontaneous thinking, including parietal cortex, medial prefrontal cortex and hippocampal regions) is much involved in creativity and is known for its links to episodic memory in the hippocampus, spontaneous retrieval, replay activity and simulations based on personal past experiences [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF].

To evaluate the semantic and syntactic appropriateness of candidate ideas, a reference is made toward an emotional system for biological significance of events (involving the semantic network, associated to the insula and other limbic regions) and an information processing system to perform detailed feature analysis involving attentional control networks [START_REF] Beaty | Creative Cognition and Brain Network Dynamics[END_REF]). As a synthesis, pertinence, adaptation or rejection of candidate ideas is made by the cognitive control network, involving the lateral prefrontal cortex and the anterior cingulate cortex, performing top-down modulation of self generated information for efficacy evaluation, selection and adaptation to the task [START_REF] Beaty | Creative Cognition and Brain Network Dynamics[END_REF].

Dynamics of inhibition and excitation in hubs and networks are observed with the analysis of EEG activity in behavioral and psychological tasks [START_REF] Kounios | The cognitive neuroscience of insight[END_REF]. It must be remarked that corresponding tests of creativity are in fact relying on measures of fluency, flexibility, originality and elaboration, all measures also related to problem solving. This is also a strong argument to propose that the same brain circuits and cognitive mechanisms are used for higher brain functions and for creativity [START_REF] Dietrich | The cognitive neuroscience of creativity[END_REF]).

Discussion

CCN models presented in this paper insist on the need to build, inspired from neuroscience, different kinds of representations in different regions of the brain [START_REF] Alexandre | Feature, Configuration, History : a bio-inspired framework for information representation in neural networks[END_REF] and to organize the selection of behavior through the interactions between different kinds of memories [START_REF] Alexandre | Biological Inspiration for Multiple Memories Implementation and Cooperation[END_REF].

Understanding these mechanisms and corresponding brain circuitry is particularly important to address current limitations in AI. As a central claim for this paper, we propose that these mechanisms and cognitive processes are also central to understand creativity as it is carried out in the brain. Perhaps this is not so surprising if we consider that among the mechanisms the brain has developed to circumvent the tedious and systematic analysis of some situtations, creativity might be a choice mechanism to efficiently explore novel approaches.

The present paper mentions operational models that could be directly exploited to implement CC. It also proposes paths of research to implement specific mechanisms. Spontaneous processing could be related to the hippocampus and its role in the default network, whereas deliberate processing would be more frontal and related to the combination of Task Sets. The more difficult aspect of combinational creativity could be linked to the limitation of semantic grounding, thus pleading for a closer look to the semantic network. Using CCN models to implement CC could be also an easier way to choose the type of creativity to generate or even to study fundamental differences between human and nonhuman creativity [START_REF] Colton | Computational Creativity: The Final Frontier?[END_REF].

The very nature of creativity itself can be questioned from the elements reported here. Creativity is often associated to the generation of completly novel ideas. What is reported here about convergent but also divergent thinking is that most of the time, old memories are used to create new ideas. Here also, it is the old that makes the new. What is also reported here (in line with [START_REF] Boden | Computer Models of Creativity[END_REF] claiming that creativity is not magic) is that CC, indeed considered as a scientific questioning, suffers mainly from fragmentation in AI research [START_REF] Colton | Computational Creativity: Coming of Age[END_REF] and could highly benefit from the global framework proposed by cognitive neuroscience.